High-Frequency Circuit Design and Measurements
High-Frequency Circuit Design and Measurements Peter C.L. Yip Department of Electronic Engineering City Polytechnic of Hong Kong Hong Kong
mll
CHAPMAN &. HALL London· Glasgow· Weinheim . New York· Tokyo· Melbourne· Madras
Published by Chapman & Hall, 2-6 Boundary Row, London SEt 8HN, UK Chapman & Hall, 2-6 Boundary Row, London SEI 8HN, UK Blackie Academic & Professional, Wester Cleddens Road, Bishopbriggs, Glasgow G64 2NZ, UK Chapman & Hall GmbH, Pappelallee 3,69469 Weinheim, Germany Chapman & Hall USA, One Penn Plaza, 41st Floor, New York, NYlO119, USA Chapman & Hall Japan, ITP - Japan, Kyowa Building, 3F, 2-2-1 Hirakawacho, Chiyoda-ku, Tokyo 102, Japan Chapman & Hall Australia, Thomas Nelson Australia, 102 Dodds Street, South Melbourne, Victoria 3205, Australia Chapman & Hall India, R. Seshadri, 32 Second Main Road, CIT East, Matlras 600 035, India First edition 1990 Reprinted 1991, 1995
© 1990 P. Yip Typeset in 10/12pt Times by Best-set Typesetter Ltd, Hong Kong ISBN-13: 978-0-412-34160-1 e-ISBN-13: 978-94-011-6950-9 DOl: 10.1007/978-94-011-6950-9 Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the UK Copyright Designs and Patents Act, 1988, this publication may not be reproduced, stored, or transmitted, in any form or by any means, without the prior permission in writing of the publishers, or in the case of reprographic reproduction only in accordance with the terms of the licences issued by the Copyright Licensing Agency in the UK, or in accordance with the terms of licences issued by the appropriate Reproduction Rights Organization outside the UK. Enquiries concerning reproduction outside the terms stated here should be sent to the publishers at the London address printed on this page. The publisher makes no representation, express or implied, with regard to the accuracy of the information contained in this book and cannot accept any legal responsibility or liability for any errors or omissions that may be made. A Catalogue record for this book is available from the British Library Library· of Congress Cataloging-in-Publication Data available
Contents Acknowledgements Preface 1 Introduction 1.1 Trends in electronic circuits and systems 1.2 High-frequency circuits 1.3 Examples of high-frequency systems Further reading
ix x I 1 2 3 6
2 Transmission-line Theory and Microstrips 2.1 Transmission lines in high-frequency circuits 2.2 Transmission-line parameters 2.3 Terminated transmission line 2.4 Terminated lossy line 2.5 Smith chart 2.6 Microstrip as a transmission line 2.7 An example of the application of microstrip 2.8 Static TEM parameters 2.9 Formulae for the synthesis and analysis of micros trips 2.10 Frequency dependence of Eeff 2.11 Effect of finite strip thickness and metallic enclosure 2.12 Fabrication of microstrips Problems Further reading
7 7 8 10 14 15 18 20 20 24 25 26 26 27 29
3 s-pararneters 3.1 Network characterization 3.2 Scattering parameters 3.3 Measurement of s-parameters 3.4 s-parametets and signal flow graphs Problems Further reading
31 31 31 33 34 37 41
4 Impedance Matching 4.1 Introduction 4.2 Concept of operating Q-factor
42 42 42
GJC_-
-I
CONTENTS .
-~~.-~---------~
4.3 Two-element L network 4.4 Three-element matching 4.5 Designing with the Smith chart 4.6 Transmission-line matching network Problems Further reading
5 Transistors at High Frequencies 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9
Introduction Transistor equivalent circuit Input impedance Output impedance Gain Feedback Small-signal two-port parameters Understanding high-frequency transistor data sheets Biasing of high-frequency transistors Problems Further reading
43 46 51 59 68 70 71 71 71 74 74 75 75 76 77 85 86 91
6
Small-signal Amplifier Design 6.1 Characterization of high-frequency amplifiers 6.2 Power gain 6.3 Unilateral amplifier design 6.4 Non-unilateral amplifier design 6.5 Stability criteria 6.6 Load and source stability circles 6.7 Constant power gain circles 6.8 Low-noise amplifier design 6.9 Broadband considerations 6.10 Summary of design procedures Problems Further reading
92 92 93 96 98 100 102 104 109 113 114 115 117
7
Power Amplifiers 7.1 Introduction 7.2 Biasing of power transistors 7.3 Power transistor design data 7.4 Power amplifier design Problems Further reading
119 119 121 122 128 137 138
L-~~~~~~~~~~~_C_O_N_T_E_N_T_S~~~~~~~~~~~~I ~ 8 Oscillators 8.1 General overview of oscillator design 8.2 Conversion of the s-matrix 8.3 Theory of oscillation 8.4 Oscillator design 8.5 Summary of design procedures Problems Further reading
139 139 140 142 146 150 157 161
9 The Spectrum Analyser and its Applications 9.1 Introduction 9.2 Operating principle of a spectrum analyser 9.3 Characteristics of a spectrum analyser 9.4 Tracking generator 9.5 Applications of spectrum analysers Problems Further reading
162 162
164 168 173 174 182 184
10 Microwave Frequency Counting 10.1 Basics of digital frequency counters 10.2 Microwave frequency counting 10.3 Performance of down-converted frequency counters Problems Further reading
185 185 185 192 193 195
11
196 196 198 200 204 205 206 206 207 208 208 210
Noise Measurement 11.1 Noise and noise figure 11.2 Effective input noise temperature 11.3 Measurement of noise 11.4 Noise source 11.5 Noise-figure measurement (single frequency) 11.6 Wide-band noise-figure measurement 11. 7 Noise-figure measurement at microwave frequencies 11.8 Single-sideband and double-sideband measurements 11.9 Summary Problems Further reading
12 Swept Measurements and Network Analysers 12.1 Network analysis 12.2 Signal source for swept measurements 12.3 Vector or scalar measurement? 12.4 Scalar network analysis
211 211 211 212
213
Iviii II
CONTENTS
L -_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ __ _
12.5 12.6 12.7 12.8
Index
Other scalar network-analyser systems Vector network analyser Source synchronization Power-splitter circuit Problems Further reading
217 217 219 220 220 223 224
Acknowledgements I would first like to acknowledge Dr B. Jefferies, Head of Electronic Engineering, City Polytechnic of Hong Kong, for his encouragement and help in establishing an elective stream in high-frequency circuit design, which has resulted in the preparation of this book, in the BEng(EE) programme of the Polytechnic. I would like to express my most profound gratitude to two persons in the City Polytechnic, without whose help the writing of this book could not have been possible. Thanks are due to Dr T. Lund for patiently reading the manuscript and making many invaluable suggestions, and to Mr M.W. Luk for typing out the final manuscript as well as preparing all the computergenerated diagrams. I would also like to thank Dr J.S. Dahele of the Royal Military College of Science and Dr S.H. Tan of NTI-Singapore, from whom I learnt about the subject at different stages of my career. Finally, I would like to thank Dr B. Lago of Stafford for the happy and profitable years I spent working under his supervision.
Preface An elective course in the final-year BEng progamme in electronic engineering in the City Polytechnic of Hong Kong was generated in response to the growing need of local industry for graduate engineers capable of designing circuits and performing measurements at high frequencies up to a few gigahertz. This book has grown out from the lecture and tutorial materials written specifically for this course. This course should, in the opinion of the author, best be conducted if students can take a final-year design project in the same area. Examples of projects in areas related to the subject matter of this book which have been completed successfully in the last two years that the course has been run include: low-noise amplifiers, dielectric resonator-loaded oscillators and down converters in the 12 GHz as well as the 1 GHz bands; mixers; varactor-tuned and non-varactor-tuned VCOs; low-noise and power amplifiers; and filters and duplexers in the 1 GHz, 800 MHz and 500 MHz bands. The book is intended for use in a course of forty lecture hours plus twenty tutorial hours and the prerequisite expected of the readers is a general knowledge of analogue electronic circuits and basic field theory. Readers with no prior knowledge in high-frequency circuits are recommended to read the book in the order that it is arranged.
~______In_t_ro_d_u_c_tl_·o_n______~1 ~ 1.1 TRENDS IN ELECTRONIC CIRCUITS AND SYSTEMS Before the mid 1960s electronics to most people meant AM (or FM) radio receivers and audio amplifiers. 'Electronics', in those days, was all implemented with vacuum tubes; to those who used to play with electronics in that era, valves like 12AT7, 6BQ5 and 7189 should bring about some sweet memories. The late 1960s and early 1970s saw the replacement of vacuum tubes by transistors in AM/FM receivers and audio amplifiers. Television receivers also began to get into the lives of ordinary people. To 'electronics' people televisions signified the beginning of problems related to high frequencies. High-frequency circuits, such as the VHF and UHF circuits in television tuners, require a lot of tuning in order to achieve frequency selectivity and impedance matching. In the early days, and even nowadays with low-end AMlFM receivers, tuning was achieved by employing coils (transformers). The major disadvantage of coil-tuned high-frequency circuit design is the upper frequency limit of coils; the inductances required are getting too small to be implemented by traditional core-tuned coilsl transformers as the frequency increases. With the introduction of microprocessors in the early 1970s, 'electronics' has been 'digitized' and has almost become synonymous with the words 'digital' and 'computer'. For a long time since the mid 1970s, 'electronics' people have been indulging themselves in microprocessors in the era best described as the (microprocessor) technology-led era. A senior undergraduate in electronic engineering once (around 1983/84) came to seek the opinion of his professor of a certain electronic system which he had designed to his own satisfaction. He had wanted to design a 'system' which was capable of magnifying a time-varying signal. The way he had planned to implement it, as he told his professor, was first to convert the input signal into digital form by using an AID converter, then write a certain multiplication algorithm into a microprocessor and finally convert the output signal back to analogue form by using a D/A converter. In fact, all this student wanted was an amplifier which could easily be implemented by one or two transistors or an operational amplifier. Although this story
~ I~
__________________I_N_T_RO_D_U_C_T_I_O_N__________________~
may seem a little exaggerated, it somehow sounds familiar to those who are teaching electronics in this technology-led era of microprocessors. Microprocessors are often used independently of whether they were actually needed or not. In the last few years 'electronics' has entered into yet another era which can best be described as the (microprocessor) application-led era. People began to realise that with the advances in microprocessors a lot of applications which had not been realistic became possible. The last few years has seen rapid advancement, with the aid of microprocessors, in all modes of communications such as those in computer networking, FAX, video phones and high-definition television transmissions, etc. With the additional features of telecommunications made available by the application of microprocessors, demand on the radio-frequency (Lf.) end of communication systems becomes increasingly heavy, and as a consequence, Lf. systems are being pushed to higher carrier frequencies and higher transmission power for larger signal-transmitting capacities. And all of a sudden there is a severe shortage of electronic engineers and technicians capable of designing and testing circuits to operate at UHF frequencies and above. To compound this world-wide manpower shortage problem, cellular mobile telephony, direct broadcast satellite television and high-definition TV transmission standards, etc., have already come or are around the corner, which make the training of high-frequency (as opposed to digital) engineers something of a necessity.
1.2 HIGH-FREQUENCY CIRCUITS 'High frequencies' in the context of this book means any frequency above a few tens of megahertz and below, say, 18 gigahertz. Most high-frequency systems, whether they be a communication system or an item of test equipment, can be divided into small circuit blocks. High-frequency systems are generally made up of some or all of the following circuit blocks: 1. Small-signal amplifiers - narrow or wideband; 2. Low-noise amplifiers; 3. Small-signal oscillators - fixed tuned, varactor tuned or dielectric resonator tuned (DRO), voltage controlled (VeO); 4. Power oscillators; 5. Power amplifiers; 6. Non-linear circuits - mixers (up or down convertors), phase detectors, frequency multipliers, modulators and demodulators, switches; 7. Filters.
In this book we are only dealing with the design principles of small-signal amplifiers, low-noise amplifiers, small-signal oscillators and power ampli-
c===_______E_XA__M_P_L_E_S_O_F_H_IG__H-_F_R_EQ_U_E_N_C_y__Sy_S_T_E_M_S________~I ~ fiers. It is hoped that in a future edition or in a separate volume filters and some non-linear circuits can be included. Theories for designing power oscillators employing non-class-A bias of BJTs or FETs are still not well developed, and it will be some time before they can appear in a text book. This book was originally written as the lecture notes for the final-year undergraduate course in high frequency circuit design at the City Polytechnic of Hong Kong, and the lecture hours available in this course tend to set a limit on the coverage of this book.
1.3 EXAMPLES OF HIGH-FREQUENCY SYSTEMS Readers may have already seen quite a few communication systems in block schematic form, where high-frequency circuits are used in the implementation. In this section we are going to look at the block schematics of two applications of high-frequency circuits, namely: (i) the direct-broadcast satellite (TV) receiver (DBS-TV), and (ii) the cellular mobile radio 'phone (MRP). Both DBS and MRP are consumer products, the demand for which was expected to take off towards the end of the 80s and the saturation market potentials of which are expected to be too large to be ignored by any traditional electronic appliance manufacturing country. Figure 1.1 shows the block diagram of a typical DBS-TV receiver and shows how the circuit blocks listed in Section 1.2 are connected to form a DBS receiver. There are basically four frequency bands in this receiver, namely the 12 GHz band for the low-noise front-end amplifier, the 1 GHz band for the first IF, the 70 MHz band for the second IF and the baseband for video and audio outputs. While the circuits in different bands can be designed by using similar methods, their implementation and the types of components used may be very different. For example, both the 12 GHz LNA and the first IF amplifier (1 GHz band) are small-signal amplifiers, and they can be designed using the techniques discussed in Chapter 6. However, it may require a GaAs FET as the active element for the LNA while a much cheaper silicon bipolar transistor or MOSFET can do the job nicely for the first IFA. While the 1 GHz circuits can be built on a printed circuit board using micros trip technology, the same technology may result in circuits too large for implementation when applied to the 70 MHz band. Figure 1.2 shows the block schematics of the r.f. section of a cellular mobile radio phone (MRP). The duplexer in Fig. 1.2 is just a parallel combination of two bandpass filters, one in the transmitting and one in the receiving band. Here again we see how circuit blocks such as those listed in Section 1.2 are connected to form a high-frequency system, an MRP in this case. Design and implementation techniques for MRPs, especially the hand-held type where circuit size is an important factor amongst other criteria, are not mature yet. For example, the size of the duplexer and the size as well as spectral purity of the VCOs still have plenty of room for
ITJI
INTRODUCTION
L -_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
~
~-------------------------l
I
Dish antenna
I
11.7-12.2GHz
I I
I
-----\ss!
I
1st IF amplifier
1st local OUTDOOR oscillator UNIT (1O.7GHz) L__________________________
I I
I I I I I
I I
~
II INDOOR -------------------------l UNIT I
I
I
~~
~I
><
I
filter
amplifier
I I
I I I I I
r-------.J I
+----+--I~
Limiter
I
FM demodulator
f
L ________________ I ~
Fig. 1.1
Block diagram of a typical DBS-TV receiver.
To video monitor or TV
RX
VCO
FrO
L----.,I PA 1"""1411-----...., MODI4
Duplexer
TX
IVCO
Frequency synthesizer
Logic unit Voice
0 Digital data
1--..1----<0 Digital data
..
Voice
Fig.1.2 r.f. Section of a cellular mobile radiophone. MIXI is the first mixer, which down converts the received signal to an IF of, say, 70 MHz. MIX2 is the second mixer which down converts to a second IF of, say, 10.7 MHz. Fl and F2 are bandpass (IF) filters. VCORX is a voltage-controlled oscillator for receive-channel selection. MOD is a modulator. PA is the power amplifier for transmit.
~~
Carriers in 8()()-900 MHz band
~ ~I
I_N_T_RO __ D_UC_T_I_O_N__________________~
___________________
improvement. It is exactly the immaturity of this field which makes the subject of high-frequency circuit design an interesting one. In the last four chapters of this book, we will be dealing with the principles and applications of commonly used high-frequency test equipment. There again, we will see how circuit blocks in Section 1.2 are utilized to form various types of high-frequency test equipment.
FURTHER READING Douville, R.J. (1977) A 12GHz Low-cost Earth Terminal for Direct TV Reception from Broadcast Satellites, IEEE Transaction on Microwave Theory and Techniques, Vol-MIT 25, No. 12, December. Gibson, S. W. (1987) Cellular Mobile Radiotelephone, Prentice-Hall. Gould, R.G. (1984) Transmission Standards for for Direct Broadcast Satellites. IEEE Communications Magazine, 22, No.3, March. lTV (1982) Provisions for ALL Services and Associated Plan for the Broadcasting Satellite Service in Frequency Bands 11.7-12.2 GHz (in Regions 2 and 3) and 11.7-12.5 GHz (in Region /), Appendix 30, lTV Radio Regulations, Geneva. Johnstone, B. (1988) Programming better quality TV. Far Eastern Economic Review, 11 August. Johnstone, B. (1988) Getting a clearer picture. Far Eastern Economic Review, 11 August. Rainger, P., Gregory, D., et at. (1985) Satellite Broadcasting, Wiley.
Transmission-line Theory and Microstrips
2.1 TRANSMISSION LINES IN HIGH-FREQUENCY CIRCUITS Resistance, capacitance and inductance are the three main passive elements commonly used in circuits operating up to about 300MHz, i.e. up to the lower end of the UHF band for various functions. High-frequency circuits from, say, a few tens of megahertz upwards, normally consist of a.c. and d.c. (bias circuit) paths. Bias circuits in high-frequency circuits, like those at low frequencies, employ resistive networks to set the biasing voltages, capacitances to decouple and to act as d.c. blocks and inductances to act as a.c. blocks to prevent a.c. signals from getting into the d.c. paths. One of the most important considerations in high-frequency circuit design is the use of reactive elements to achieve impedance matching between consecutive stages. Capacitances and inductances are normally used to implement impedance-matching networks up to a few hundred megahertz. However, as frequency increases the value of the capacitance and inductance required will eventually become too small to be realized, in the sub-picofarad and sub-nanohenry region. When frequencies are too high for discrete capacitors and inductors to be practical, transmission-line or distributed circuits have to be employed. Transmission-line circuits are not normally designed to replace directly discrete capacitors and inductors in a design based on discrete reactance; but rather, they require distinctly different methods and they provide more circuit varieties than circuits merely implemented by discrete capacitors and inductances. Examples of transmission-line circuits without discrete 'LC' equivalence are stub matching networks and interdigital filters. A transmission line can be a pair of twisted wires, a coaxial cable, a strip line or a microstrip. In the next few sections, we will briefly go through the transmission-line theory relevant to high-frequency circuit design and measurements. This theory does not only apply to the two-conductor type of transmission lines such as coaxial cables and microstrips, but it is also valid for one-conductor transmission lines such as rectangular and circular waveguides. A study of the general transmission-line theory does not only show how signals are propagated in a line but also leads to the definition of
2
[TIC
TRANSMISSION-LINE THEORY AND MICROSTRIPS
~--------------------------------
some terminologies such as reflection coefficients and VSWR, which are important quantities in describing the terminal behaviour of high-frequency circuits and systems. In the later sections of this chapter, we will describe how microstrips are designed and fabricated. The microstrip is singled out for discussion in this book because it is the most commonly used structure in the implementation of transmission-line circuits in applications such as amplifiers and oscillators, at frequencies ranging from a few hundred megahertz to above 10 GHz.
2.2 TRANSMISSION-LINE PARAMETERS Physical dimensions such as the inner and outer radii of a coaxial cable or the height-to-width ratio of a microstrip, and the dielectric constant of the material separating the two conductors of a transmission line can be related analytically, in the case of coaxial cable, or empirically, in the case of a microstrip, to the circuit parameters of the transmission model representing the coaxial cable or microstrip line. These transmission-line circuit parameters are normally represented by R, G, Land C, which respectively denote the series resistance, shunt conductance, series inductance and shunt capacitance, per unit length, of the transmission line. In terms of these primary circuit parameters we can express other secondary parameters of the transmission line such as the characteristic impedance (Zo), propagation coefficient (y), phase constant (J3), attenuation constant (a) and phase velocity (v). A brief revision of transmissionline theory is given below. The voltage and current propagating along a transmission line are generally functions of both time and distance (z) in the direction of propagation, and are expressed by the transmission-line equation d 2 V(z) ~ - (RG - w2LC)V(z) - jw(RC
and
d /(z) dT 2
(RG - w2LC)/(z) - jw(RC
+ LG)V(z) =
+
°
LG)/(z) = 0,
(2.1) (2.2)
where the time variation is assumed to be sinusoidal of angular frequency w, i.e. i(z, t) = f(z )e jwf and v(z, t) = V(z )e jwf . The solution of (2.1) may be written as (2.3)
where V+ and V- are amplitude constants for waves propagating in the +z and -z directions, respectively, and to be determined by boundary conditions, and y, known as the propagation coefficient, is given by 1
Y = [-w 2LC + RG + jw(RC + LG)l' = [(R
1
+ jwL)(G + jwC)]'.
(2.4)
~_____________TRA __N_S_M_I_S_SI_O_N_-L_I_N_E_P_A_R_A_M_E_T_E_R_s_____________·~l ~ Similarly, the solution for 1(z) may be written as
re-
I(z) =
re
Yz -
(2.5)
Yz ,
where rand r are current amplitude constants for waves propagating in the +z and -z directions, respectively. A negative sign before r is inserted because the reflected current wave will flow in the opposite direction to The total voltage V and the total current 1 at any point z are related by either
r.
dV ( dz = - R
~~
or
=
. ) + JwL 1
(2.6)
-(G + jwC)V.
(2.7)
Substitution of (2.3) and (2.5) into (2.7) gives
Y + jwC
V+
r
and the ratio V+ I is equal to
r
G
is called the characteristic impedence of the line, which
(RG ++ jWL)' jwC . I
Z _ o- G
Y _ + jwC -
(2.8)
From the definition of Zo, it follows that
v+
Zo = [+
V-
=-r'
(2.9)
For a line without loss, R = G = 0, then y = jp = jw V(LC) (loss-free line)
and
Zo =
(2.10)
~ ( ~) (loss-free line),
(2.11)
where f3 is called the phase constant and both f3 and Zo are real. Since y is totally imaginary, the voltage and current wave propagate in the + z and -z directions without attenuation. For most microwave transmission lines the losses are very small, i.e. R <.g wL and G <.g we, and the term RG may be neglected in the following expression for y. I
Y = [-w 2 LC + RG + jw(RC + LG)l' I
"'" [-w 2 LC
+ jw(RC + LG)l'
"'" jwV(LC)
+
V(~C) (~
+
~)
= a
where use has been made of the binomial theorem
+ jp,
(2.12)
o
L I_
_
_
T~RA_N~S~M~IS~S~IO~N~-~L~IN~E~T~H~E~O~R~Y~A~N~D_M~IC~R~O~S~T~R~IP~S_ _ _
----1
_
1
= 1 + (1I2)x for Ixl
(1 + x)'
~ 1.
The phase constant fJ is the same as that for loss free lines, and the attenuation constant a is given by V(LC)
a = -2-
where Yo
=
fJ = wV(LC),
(RL + G) 1 C 2(RYo + GZo),
(2.13)
=
1
1/Zo = (c/L)' is the characteristic admittance of the line.
2.3 TERMINATED TRANSMISSION LINE
Consider a transmission line (the discussion in this section also applies to waveguides) with characteristic impedance Zo terminated by an arbitrary load ZL as shown in Fig. 2.1. The line is assumed to be loss-less with propagation coefficient y = jfJ. The expressions for the voltage and current are functions of position z only, i.e.
+
V(z) = V+e-i/l z
I(z) = re-i/l z
At the load end, i.e. at z
=
-
1= IL =
zor
= V+ and
zor
(2.14) (2.15)
0, V = VL = V+
but
V-ei/lz rei/l z .
+
V-
(2.16) (2.17)
r -r
= V-, hence (2.17) may be written as h
=
~o (V+
(2.18)
- V-).
Generator
v+ + v-
Z=Z
1= -z .....II---+--...... ~ z
z=O Fig. 2.1
Terminated transmission line.
TERMINATED TRANSMISSION L_IN_E_ _ _ _ _ _-----"I
Combining (2.16) and (2.18), we have VL
Z V+ + V°V+ - V
_
h -
but
Vdh is the load impedance
(2.19)
ZL, hence
ZL _ 1 + TL _ Zo - 1 - TL -
Z
L,
(2.20)
where ZL (= ZL/Zo) is the normalized (by Zo) load impedance and r L is the voltage reflection coefficient at the load, defined as
V-
(2.21)
TL = V+·
Hence
ZL - 1
=--. ZL + 1
TL =
(2.22)
r L = 0 the load is said to be matched to the line. r L = 0 indicates that there is no voltage being reflected by the load. Under matched condition, all the incident power is transmitted to the load without being reflected back to the line (hence to the generator). The power P delivered to the load is thus given by
If ZL = Zo then
P
or
1 2
P
= -Re(VI*) at z = 0
=
IRe(v+r*)
2
= lyo 2
1V+1 2 = IYL 1V+1 2
2
(2.23)
where ,*, denotes the complex conjugate of the quantity preceding it. If ZL Zo, the load is said to be mismatched to the line and a reflected wave is produced. The power P delivered to the load is given by
"*
P
= ~Re(Vdt) = ~Re[(v+ + =
~YoIV+12 (1
-
V-)(l+ - r)*]
ITLI2).
(2.24)
In the absence of reflection, the magnitude of the voltage (at a specific point z) along the line is a constant equal to IV+ I . When a reflected wave also exists, the incident and reflected waves interfere to produce a standing-wave pattern along the line. The voltage at any point on the line, i.e. for any z ::; 0, is given by
+ TLV+eij3z 11 + TLe2ij3zl 11 + TLe-2ij3ll,
V(z) = V+e- ij3z IVI = IV+ I = Iv+1
(2.25)
0
o
T_RA_N_S_M_I_SS_IO_N_-_L_IN_E_T_H_E_O_R_y_A_N_D_M_I_C_RO_S_T_R_IP_S_ _ _ _--.J
L I_ _ _ _
where l = -z is the distance measured from load (z = 0) towards the generator. Since r L is in general complex, it may be expressed in polar form as (2.26)
where Q is the magnitude of the load-reflection coefficient and OL is the phase difference between V- and V+. Equation (2.25) may then be written as
Similarly for the current function we have
III
=
~ Zo [(1
+ (If + 4(1sin 2 (f31
1
- Od2)]'.
(2.28)
Equation (2.27) shows that the magnitude of the voltage oscillates back and forth between maxima and minima in z. For maxima f31and for minima
f31 -
OL
2
2OL = =
nJr
(2.29)
mr Jr
+ "2'
(2.30)
where n = 0,1,2,3 .... This simply means that voltage maxima occur when the incident and reflected waves are in phase and that minima occur when they are 1800 out of phase. Successive maxima (and minima) are spaced at a distance d = nl{3 = n)../2n = ),,12, where).. is the wavelength for TEM waves in the medium surrounding the conductors. The distance between a maximum and a minimum is ),,14. The plot of the voltage and current waves for ZL = 3Zo is shown in Fig 2.2. From (2.27), the maximum and minimum values of Ivi are given by
iVlmax = 1v+1 (1 + (I) iVlmin = iV+1 (1 - (I).
(2.31) (2.32)
The ratio of these two values is termed the voltage standing wave ratio, or VSWR for short, VSWR = .!.....±..g. 1-(1
(2.33)
VSWR is an important parameter in a transmission system. At high frequencies, e.g. at microwave frequencies, direct measurement of absolute
~____________T~E~R~M~IN~A~T~E~D~T~R~A~N~S~M~IS~S~IO~N__L~IN~E____________~I ~ 1=0
).
1.--
Fig. 2.2
Voltage and current distribution for ZL
2
=
1=0
3Zo.
voltage and current is very difficult if not impossible. On the other hand, it is possible to construct devices to measure voltage ratio. Hence VSWR is a readily measurable quantity. From the VSWR the magnitude of r L , i.e. (J, can be found. A complete knowledge of the load impedance demands the knowledge of the angle of r L , i.e. (JL, which can be measured by noting the distance of the first minimum from the load. At a point z = ~l, i.e. at a distance I from load, the reflection coefficient r( I) is
or
(2.34)
where rL is the reflection coefficient of the load. The normalized impedance, looking towards the load, at z denoted by (I), is given by
z:
~l,
[U 14UJ l__u__~_
TRANSMISSIONuLINE THEORY AND MICROSTRIPS
Z m
(I) = Zin(l) = V(l) Zo /(/)Zo V+e iiJl + V-e- iiJl = V+e iiJl - V-e- iiJl
1+ Zin = 1 -
Replacing have
rL
by (ZL - Zo)/(ZL
Z
+
=
rL e- i 2/31 rLe -i2iJl'
~
1 + r(l) 1 - r(l) (2.35)
Zo) and e±if3[ by cos (31 ± j sin(3l, we
+ jZo tanf31 Zo + jZL tanf3I'
= ZL
m
......
(2.36)
similarly, Yin = YinlYo = 11 Zin is given by
y = m
YL + jYo tanf31 Yo + jYL tanf3l"
(2.37)
Of particular interest are two special cases, namely the immittances at (31 = n (or 1 = ,1./2) and at (31 = nl2 (or 1 = ,1./4), Zin(l = ,1/2) = ZL Z~ Zin(l = ,1/4) = ZL'
(2.38) (2.39)
The first of these is an ideal one-to-one impedance (or admittance) transformer whereas the second one, commonly known as the quarter-wave transformer, inverts the immittance with respect to the square of the characteristic immittance. It can be readily shown that the maximum and minimum of Zin are given by Zin(max) = VSWR X Zo Zin(min) = Z(/VSWR.
(2.40) (2.41)
2.4 TERMINATED LOSSY LINE The propagation coefficient of a lossy line will be y = a + j(3 instead of y = j(3 as in the case of loss-free lines. Assuming that the characteristic impedance Zo is real, which is valid for low-loss lines at microwave frequencies, the reflection coefficient as a function of 1 is given by (2.42)
r(l) decreases expontially with I. Thus when a load ZL is viewed through a long section of a lossy line, it appears to be matched to the line since r(l) is negligibly small as 1 ~ 00. Zin is given by
SMITH CHART
L -_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
or
Z
= ZL
III
Zo
+ Zotanhyl + ZLtanhyl
which shows that Zin ~ Zo as 1~
10
~
1 + rLe-j2fJle-2al Zin = Z01 _ rLe j2fJle 2al
(2.43)
00.
2.5 SMITH CHART The Smith chart is the plot of the families of the real and imaginary part of immittances on the complex plane of the voltage-reflection coefficient or the complex r-plane, where r = QeiO. It facilitates the solution of almost all problems arising from transmission lines in areas such as the design of a matching network. The most outstanding feature of the Smith chart is that, within a finite area of the r-plane, complete information relating all possible values of normalized immittances, reflection coefficients and standingwave patterns for all transmission-line and waveguide circuits involving only passive elements may be obtained. Smith charts can also be used in the design of active circuits. A modified form of the Smith chart, known as Linvill's chart (Linvill and Schimf, 1956), is the foundation of modern solid-state microwave amplifier design. The derivation of the Smith chart is based on the following equation r(l) = Zin(l) - Zo, Zin(l) + Zo
(2.44)
where r( I) is the reflection coefficient at a distance 1from the termination and Zin(l) is the transformed impedance of ZL when viewed through a length 1 of the transmission line with characteristic impedance Zoo Detailed derivation of the Smith chart can be found in many textbooks, e.g. Chipman (1968), so it is not necessary to go into this here. A Smith chart consists of a family of normalized resistance circles and a family of normalized reactance circles; these families of circles are respectively the real and imaginary parts of Zin(l) normalized by Zoo Figure 2.3 shows a family of resistance circles and Fig. 2.4 shows a family of reactance circles like those normally appearing on a standard Smith chart. On a standard Smith Chart there are normally eight radial scales in addition to the resistance and reactance circles. Some of these radial scales are more often used than the others in circuit design, but they are all briefly explained below (and shown in Fig. 2.5) for the sake of completeness. Scales A and B are both reflection coefficients. Distance from the centre on scale A represents Q = Irl, the absolute value of the voltage- (or electric field) reflection coefficient. Distance from the centre on scale B is the square of Q, i.e. Q2, representing the ratio of the reflected power to the incident power.
o
T_RA_N_S_M_I_SS_IO_N_-_L_IN_E_T_H_E_O_R_y_A_N_D_M_IC_RO_S_T_R_IP_S_ _ _----'
L I_ _ _ _
Imr
r-plane
Rer
'n = 0
Fig. 2.3
Normalized resistance (rn) circles on F-plane.
Imr
r-plane
Rer
-1
Fig. 2.4
Normalized reactance (xn) circles on F-plane.
~
0
>
tt
OJ
0::
~
~
Fig. 2.5
II I s
SO
;?,
SCALE H
SCALE G
SCALE D
~-
SO
~
I
I
';:
Iii r!
N
TOWARD LOAD
I
I (
:':
I
I
I'"
Sio
olE
-'10
~
~
0::
~
_ .0:=:
~
r:
~
P-t-Y+
'I
I
c;
N
iii
I
- -w:r:-
.:::
d E '~~ I " S ::; I Iii
:
I,:::
CENTRE
·I·OW· AR' D'LO' A'I)' ".
"';:
Radial scales on the Smith chart.
""
~
I" i (
SCALE C
.,i;,
,,",
TOWARD GENERATOR
H"
a o!- H ~ > ~ t;;
<0 o?~ -' _
?;
"cri-vi v 7. a: Q., ~ 0 ~ " _ rr~
'">
-
II·OWARD GENF·RA·rOR . _
~
"i3 B" i", 6 ~ I' ~" " ~ r
. ....: 0
§-'
:..w
- t;;
aQ:.
"5::
~ ":) w ...: 0 H
,
,
_
1-'
7':=:
E
I
IX
::;
-.I
<=:
SCALE F
_
C
SCALE E
SCALE B
tv
SCALE A
'
i I I I
"1'1
'!5 ~
-
;:!::!
E
os:; ~ ;:
;:;
iii
x-
~
RADIALLY SCALED PARAMETERS
c
0-
~
i
I
_
I I'
~
"c I
~
S
I
3
.j..,.
c
::;
"J>
b
" ~
c
2l g
~
,
-J ::;
I
:lC :=
:g
'
J' <3
~
C
6
'I I
.8
~
~
r::c
~ ~
L
~ I~
~ ~ ?
~
~< 0m"
0
c
8
,...
t:g
::0
0
"
~,...:2~ - i
'"
'" "0 c:E
I
0
I H
!
eo
:...
- ,... .
~ I I,I" ~ I oc
I i i I i~
15
I
~:;;;
::;. . .
2!
z
c
C)
..;
'"
o
L I_
_
_
T_RA_N_S_M_IS_S_IO_N_-_L_IN_E_T_H_E_O_R_y_A_N_D_M_IC_R_O_S_T_R_IP_S_ _ _-----'
_
Scales G and H are standing-wave ratios. Distance from the centre on scale H represents the voltage standing wave ratio s, also denoted by VSWR, defined by 1 + (2 VSWR=s=--. 1 - (2
(2.45)
Distance from the centre on scale G represents s in dB, i.e. SdB
= 20log lO s.
(2.46)
Scale E is the return loss, Ln in dB defined by Lr = 20log JO
1
-.
(2
(2.47)
Lr is the square of the reciprocal of the reflection coefficient in dB. Scale F is the reflection loss, L" in dB defined by 1
L J = 10 IOglO -1- - 2 -(2
(2.48)
which is the ratio in dB of the power of the incident wave to the power absorbed by the load. The expression of L1 is arrived at by considering P ab = Pi - P refl
hence
P ab = 1 _ P refl = 1 _ (Erefl )2 = 1 _ (22 Pi Pi Ei 1 L) = 1OIoglO Pa~ = 1OIoglO 1 _ (22'
( P.)
Scale D is the line attenuation. Each step stands for the effect of 1 dB line attenuation loss on the radial scale parameters Q or s. Whether measured towards generator or towards load depends on the direction in which the point of interest in the line is moving. Note that the value of Q or s is always the distance from the centre to the particular step.
2.6 MICROSTRIP AS A TRANSMISSION LINE The types of transmission lines to be used in active high-frequency circuits must provide suitable geometries so as to enable discrete active and passive components to be solidly connected to form hybrid circuits or to enable total integration of the transmission lines and other active/passive components on the same substrate. There are a few such geometrical structures suitable for high-frequency transmission, which are commonly known as striplines. Examples of striplines are microstrip, slot line and coplanar waveguide, as shown in Fig. 2.6. Of these three, microstrip line is the most commonly used for high freqencies from a few hundred megahertz to over 10 GHz, because of its simplicity in geometry and hence ease of
~___________M_IC_R_O_S_T_R_IP_A_S_A__T_RA__N_SM__IS_SI_O_N_L_I_N_E__________~I ~ Conducting strip
(a)
(b)
Conducting plane
(c)
Fig. 2.6 Types of striplines: (a) microstrip, (b) slotline and (c) coplanar waveguide.
~
L I_
_
_
T_RA_N_S_M_IS_S_IO_N_-_L_IN_E_T_H_E_O_R_y_A_N_D_M_IC_R_O_S_T_R_IP_S_ _ _-----"
_
fabrication. Microstrips consist of metallic strips and a ground metallic sheet separated by a layer of dielectric. The metallic strips and the ground metallic sheet could be made by depositing (by evaporation or sputtering) the metal on the dielectric substrate or by etching the unwanted metallic parts from a double-sided printed circuit board made with a dielectric such as Teflon (PTFE). In the next few sections, microstrips will be discussed qualitatively and some design equations for the physical dimensions in terms of the desired values of the electrical parameters such as the characteristic impedance and the permittivity, will be stated. No derivation of these design equations is given as it is more important for a circuit designer to be able to use rather than to derive these equations. 2.7 AN EXAMPLE OF THE APPLICATION OF MICROSTRIP In order to illustrate the problems involved in designing a micros trip circuit, we consider the input circuit of an amplifier as shown in Fig. 2.7. The input circuit consists of a 50-ohm microstrip section connected to the transistor base through a 31.3-ohm quarter-wave transformer (also made of microstrip). The base is shunted by a 66.7-ohm, AgiS section, that provides a purely reactive impedance to neutralize the reactance of the transistor input. If the optimum admittance looking into the transistor input is given by Yin =
(0.051
+ jO.015)s.
Then, by using the configuration shown, it can easily be verified that the characteristic impedance of the AgiS section is equal to 66.7 Q. The calculation is left as an exercise to the readers. It is noted that the Ag/S shunt section is there to neutralize the imaginary part of Yin, thus leaving a purely resistive input admittance to be matched to the 50-ohm line via the 31.3-ohm quarter wave transformer. After the circuit shown in Fig. 2.7 has been designed, with all the electrical parameters obtained, the main design problem is to evaluate the physical width (w) and lengths for a given substrate of thickness (h) and dielectric constant (fr) at a given frequency, as shown in Fig. 2.S.
2.8 STATIC TEM PARAMETERS Unlike coaxial lines and hollow waveguides which support one mode of propagation, at least within a range of frequencies, microstrips support more than one mode of propagation at all frequencies. However, the bulk of the energy transmitted along a micros trip is through a field distribution which resembles that of a TEM mode. The deviation from the pure TEM mode of propagation on a microstrip increases with frequency. Analysis
ST_A_T_I_C_T_E_M_P_A_RA __M_E_T_E_R_S______________~I
L -_ _ _ _ _ _ _ _ _ _ _ _ _ _ _
I+-- 19/4 - - - + I 50Q
Fig. 2.7
31.3Q
Amplifier input circuit using microstrip.
based on the TEM mode, usually referred to as the quasi-TEM approach, is generally fairly accurate up to 1 or 2 GHz and if a frequency-correction function is incorporated it can be extended beyond 10 GHz within 1% error. A commonly used technique to relate the strip width wand strip length I to its corresponding characteristic impedance Zo and electrical length (), with the thickness h and permittivity COC r as parameters, is to consider the microstrip as a statically charged capacitor where all fields, electric and magnetic, are in the transverse plane only. Hence, the technique is known
Dielectric f
= fOfr
Top conducting strip Air
Ground plane (conducting)
Fig. 2.8 Microstrip geometry.
~
o
1"-----____T_RA_N_S_M_IS_S_IO_N_-_L_IN_E_T_H_E_O_R_y_A_N_D_M_IC_R_O_S_T_R_IP_S_ _ _------1 as the static-TEM method. Parameters so derived are quite accurate up to a few gigahertz. At higher frequencies the method can still be valid if a frequency-correction function is incorporated.
2.8.1 Characteristic impedance For any TEM-type transmission line, the characteristic impedance Zo is given by Z
o
=
~(L) C
=
v
p
L= _1 vpC'
(2.49)
where Land C are the inductance and capacitance per unit length, respectively, and vp is the phase velocity given by v
p
1 V(LC)'
=--
(2.50)
If the substrate of permittivity
EoET is (effectively) removed, the microstrip becomes an air-filled line and the velocity of propagation is equal to c, the velocity of light in free space (air). The characteristic impedance of this airfilled line, Z01 is thus Z(ll =
~( CL
I
) = eL = _1 , eC I
(2.51)
where C 1 is the new capacitance per unit length and L remains the same, as the change of dielectric does not affect inductance. Combining the last three equations, we have Z 0-
1 eV(CC,),
(2.52)
Equation (2.52) shows that the required characteristic impedance is known if we can calculate the capacitances per unit length, with and without the presence of the dielectric substrate.
2.8.2 The effective microstrip permittivity For an air-filled microstrip the propagation velocity is given by e = -1- -
V(LC,) .
(2.53)
Dividing (2.53) by (2.50), we have (2.54)
~_______________ST_A_T_I_C_T_E_M_P_A_RA __M_E_T_E_R_S______________~I ~ The capacitance ratio ClC I is termed the effective microstrip (relative) permittivity, feff' or (2.55)
With
feff'
Zo and Z{)1 can be related by Z - ZOI o - ve;:-;:;.
(2.56)
For very wide lines such as that shown in Fig. 2.9(a), where w ~ h, nearly all the electric field is confined in the substrate dielectric, just like in a parallel-plate capacitor, and we have for w
feff ~ fr
jl>
(2.57)
h.
For very narrow line such as that shown in Fig. 2.9(b), where w «:: h, the electric flux lines are almost equally distributed in the air and the dielectric region, hence it may be approximated that feff
The range of
feff
=
1
+ 1) for
2( fr
w ~ h.
(2.58)
is therefore 1
2(fr
+
1)
(2.59)
::0; feff::O; fro
It can be convenient to express the effective microstrip permittivity as feff
= 1
+ q(fr
(2.60)
- 1),
where the new quantity, the filling factor q, has the bounds 1
2::0;
q
::0;
1.
E,
E,
;7 ~~l t.-WL2:7. w
h
Wide (a) (w
h
T T
(a)
Fig.2.9
1
jl>
(b)
h) and narrow (b) (w ~ h) microstrip.
OD ,- -I____T_RA_N_S_M_IS_S_IO_N_-_L_IN_E_T_H_E_O_R_y_A_N_D_M_IC_R_O_S_T_R_IP_S_ _ _~ 2.8.3 Wavelength Ag and physical length 1 For any propagating wave the velocity is given by the product of the operating frequency and the wavelength. In free space c = fAo and in the microstrip vp = fAg, where Ag is the guided wavelength and vp is the velocity of propagation usually known as the phase velocity. Equation (2.56) can be written in terms of Ag as Eeff
Ag
or
(TAO)2
=
g
=~ ~.
(2.61)
In designing a microstrip circuit, the length calculated is normally expressed as so many Ag, hence with Ag obtained from (2.61), the physical length I can easily be found.
2.9 FORMULAE FOR THE SYNTHESIS AND ANALYSIS OF MICROSTRIPS
There are many methods to calculate the two static capacitances C and CI . In this section, the results of one such method are given. The results given can easily be programmed.
2.9.1 Synthesis formulae Synthesis means finding wlh and Eeff with the knowledge of the values of Zo and Er . For narrow strips (Zo > (44 - 2Er) ohms)
(e
w
1
H
--,;= 8- 4e where
and
H
=
Eeff =
ZOV2(Er
+ 1) +!
119.9
Er + 1 [ -2- 1 -
2
H
)-1
1 (In~ 2
Er Er
+1
1 Er 2H Er +
1
(2.62)
'
+ ~In~) Er
lr
(lr + -zln;. 1 4) J
-2
1 In"2
(2.63)
.
(2.64)
Note that (2.64) was derived under a slightly different changeover value of Zo> (63 - Er) ohms. For wide strips (Zo < (44 - 2Er) ohms)
~= ~[(dt h
lr
where
- 1) - In(2d, - 1)]
+
Er - 1 lrE r
d = t
[In(d f
59.95~
ZoVc; ,
-
1)
+ 0.293 - 0.517J, Er
(2.65) (2.66)
and under a slightly different changeover value of Zo > (63 - 2Er) ohms
~~~~~~~_F_RE_Q_U_E_N_C_Y~D_E_P_EN_D~EN_C_E~O_F_E_e_ff~~~~~__~t ~ Eeff
Er
+1
= -2-
+
Er -
1(
-2~
h ) -0.555 1 + 10-;:;;-
(2.67)
2.9.2 Analysis formulae Analysis means finding Zo with the knowledge of wlh and strips, i.e. wlh > 3.3,
fro
For wide
Zo = 119.9JT {~ + ~ + In(eJT2/16) (Er ~ 1) 2VEr 2h JT 2JT Er
(w
Er + 1 [JTe + 2JTEr In 2: + In 2h + 0.94
)]}-l
For narrow strips, i.e. wlh < 3.3, Zo =
119.9 {In V2(Er + 1)
[4hw + -y116(~)2 + 2] _ .!(Er - 1)(ln ~ +.! In i)}. w 2 Er + 1 2 Er JT
(2.68)
2.10 FREQUENCY DEPENDENCE OF Eeff The formulae given in the last section and derived from the quasi-TEM assumption are accurate up to 1 or 2 GHz. For higher frequencies, the effect of the frequency dependence of feff has to be taken into account. In the limiting case E (f) eff
~
{Eeff Er
as f ~ 0 as f ~ 00,
as shown in Fig. 2.10
Effective microstrip permittivity feff (j)
O~f
Frequency f-'>
Fig. 2.10
Plot of
Eeff(f)
against frequency.
~I
T_RA_N_S_M_IS_S_IO_N_-_L_IN_E_T_H_E_O_R_y_A_N_D_M_IC_RO_S_T_R_IP_S_ _ _ _---'
L ____
Many attempts have been made to calculate the actual frequency dependence of feff(f) and to express it in closed form. One such result is due to Edwards and Owens (1976) for the range of 2 GHz :::; f:::; 18 GHz, er - eeff ( ) eeff f = er - 1 + (hIZo)1.33(0,43f2 - 0.009f3) '
(2.69)
where h is in millimetres, f is in gigahertz and feff is the value calculated by (2.64) or (2.67), whichev'er is appropriate. With the value of feff(!) calculated, the more accurate formula for guided wavelength and hence the value for the physical length of a line should be
A=~ g
Veeff(f) .
(2.70)
It is noted that the actual physical length of a microstrip is always shorter
than the value calculated (through the value of feff) from Section (2.9).
2.11 EFFECT OF FINITE STRIP THICKNESS AND METALLIC ENCLOSURE In all our previous discussions, the thickness of the strips were considered to be infinitely small and the dielectric was considered to extend infinitely. Microstrip circuits, however, are found in most cases to be enclosed by metallic boxes. Both the metallic enclosure and the finiteness of strip will alter the values of Zo and feft and hence the vp and Ag that we have calculated. There are correction formulae available to take these two factors into account (Edwards and Owens (1976)). However, they are beyond our present scope.
2.12 FABRICATION OF MICROSTRIPS Microstrips are normally made from a structure where a metallic layer is deposited on a low-loss substrate (also known as laminate). The micros trip circuits are then constructed by etching away the unwanted metal, leaving metallic strips sitting on top of the substrate. The parameters for a high-frequency substrate are dielectric constant, thickness of substrate and dissipation factor. The most commonly used substrates are alumina (a kind of ceramic) and PTFE fibreglass. PTFE stands for polytetrafluorethylene; sometimes it is also known as Teflon. Ceramic materials normally have a higher dielectric constant (fr for alumina:::::: 10) compared with PTFE fibreglasses (fr :::::: 2 to 3) and are about one order of magnitude less lossy than PTFE fibre glass. On the other hand, PTFE fibreglass materials are easy to drill and machine; they are normally supplied in the form of single or double copper-cladded printed circuit boards, similar in form to ordinary PCBs.
10
P~R~O~B~LE~M_S_ _ _ _ _ _ _ _ _ _
L - -_ _ _ _ _ _ _ _ _ _
The substrates commonly used in high-frequency work and their respective dielectric constant and loss tangents are listed in Table 2.1. Table 2.1
Dielectric constants and loss tangents of substrates
Substrate
Dielectric constant,
fr
Woven PTFE/glass
2.55,2.45,2.33,2.17
Microfibre Teflon or Non-woven PTFE/glass
2.33,2.20
-----------------------
Ceramic-filled PTFE/glass
6.0,10.2
------~--------
9.0 -10.0
Alumina
Dissipation factor approx. tan 0 at 10 GHz
= 0.0008 to 0.0021 ------~
= 0.003
= 0.0001
The term 'dissipation factor' or 'loss tangent (tan 6)' is the ratio of the energy dissipated to the energy stored in the material when excited. PTFEI glass substrates are normally supplied with a thickness of 118", 1116", 1132" or 1164". The thickness of copper cladding is usually described as 10z eu (or 1I20z or 2 oz). 1 oz of copper is cladded on to 1 square foot on ONE side of the substrate and is equivalent to a thickness of 0.0014" or 0.03556 mm. In general, non-woven PTFE/glass substrates are more uniform and have lower losses than woven PTFE/glass substrates.
PROBLEMS 1. A transmission line is 2.00 wavelength long at its frequency of operation. It is terminated in a normalized input impedance 0.25 -
j1.80. What is the normalized input impedance of the section if its total attenuation is (a) zero, (b) 1.0dB, (c) 3.0dB and (d) lOdB? Ans: (a) 0.25 - j1.8, (b) 0.68 - j1.62, (c) 1.11 - j1.06, (d) 1.08 - jO.17 2. A loss-free transmission line of characteristic impedance Zo = 50 Q is terminated by a load impedance ZL. Measurements on the line show a voltage standing wave ratio of 3.0. The distance between successive voltage minima was measured to be 15 cm. If the load is replaced by a short circuit, the position of the minima are seen to have moved a distance of 5 cm towards the generator. Determine the value of ZL by calculation and by Smith-chart method. Ans: 50 + j57.7 Q 3. For a distortionless transmission line, LG = RC, prove: 1. 2. 3. 4.
The The The The
attenuation coefficient a = V( GR); phase constant f3 = wV(LC); phase velocity vp = lIV(LC); and characteristic impedance Zo = V(R/G).
[~[
TRANSMISSION-LINE THEORY AND MICROS TRIPS
~----------------------------------------------------~
4. An isolator is inserted between a signal generator and a device of input VSWR 1.6. The isolator has an isolation of 18 dB. Neglecting the insertion loss of the isolator and its input and output reflection (i.e. input and output VSWR of isolator = 1), calculate the equivalent VSWR of the isolator plus the device as seen by the generator. Ans: 1.06 5. If the isolator in Problem 4 is a 'real' device with the following specifications (Radiall R462-205): 1. Isolation: min. 18 dB; 2. Loss (insertion): 0.5 dB max.; and 3. VSWR (liP and O/P): 1.3 max. what is the maximum VSWR of the isolator plus device as seen by the generator? Ans: 1.3059 6. For finite (non-zero) thickness (t) of a microstrip, the width (w) calculated from the formulae given in the text ((2.62) and (2.65)) can be modified for better accuracy to We
=
+ Llw,
4JTW) In -t-
where
t (1 Ll W = :;
+
or
Llw = -t ( 1
+ In--
JT
W
for wlh
2h) t
::0;
1/(2JT)
for wlh 2: 1/(2JT).
Devise a program capable of listing wlh against Zo and Eeff for various values of Er • Provide an option for the inclusion of the finite strip thickness and frequency (2.69) as input parameters. 7. The attenuation coefficient a of a microstrip is given by where a c is the loss due to finite conductivity of the strip and ad is the loss due to the dielectric. They may be calculated from _ 2 73 cr (Ceff
ad-·
-
1)
-"
l)~nu
2.73
CrCCeff -
Ag
Ceff(E r -
1) -" tan u 1)
~
dBm-1
or
ad = -
and
a = 2000 c
(
Ccff Cr -
In 10 wZo
where Rs is the surface resistance of the metal strip. Find the total loss coefficient for a 50 Q microstrip fabricated from a 1132", 2.55 PTFE/glass substrate with tan a= 0.0018. Repeat the calculation for alumina of the same thickness with Er = 10 and tan a = 0.0001.
F_U_R_T_H_ER__R_EA __ D_IN_G________________~I
L -_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
Fig. P.2.1
8. A length of micros trip is resonating at a frequency of second. The Q factor of the resonator is defined as Q -_
Wo
Wo
radians per
energy stored average power loss
Derive an expression relating Q to the total attenuation coefficient a. Ans: Q = n/( aAg) 9. A transmission line section of length / can be represented by an equivalent T network as shown in Fig. P.2.1. Show that the circuit parameters of the T network are given by ZI = 2Zo tanh
~
Z=~ 2
sinh y['
where y is the propagation coefficient.
FURTHER READING Cheung, W.S. and Levien, F.H. (1985) Microwave Made Simple, Artech House. Chipman, R.A. (1968) Transmission Lines, Schaum's Outline Series, McGraw-Hill. Collin, R.E. (1986) Foundation for Microwave Engineering, McGraw-HilI. Edwards, T.e. and Owens, R.P. (1976) 2-18GHz Dispersion measurement on 10-100 ohm microstrip lines on sapphire, IEEE Trans. MTT-24, No.8, August. Edwards, T.C. (1981) Foundation for Microstrip Cricuit Design, Wiley. Getsinger, W.J. (1973) Microstrip dispersion model, IEEE trans. MTT -21, No.1, January, 34-9.
~
01
TRANSMISSION-LINE THEORY AND MICROSTRIPS
~--------------------------------------------------~
Laverghetta, T.S. (1984) Practical Microwaves, Howard W. Sams. Laverghetta, T.S. (1984) Microwave Materials and Fabrication Techniques, Artech House. Linvill, J.G. and Schimf, L.G. (1956) The design of tetrode transistor ampifier. B.S. T.l., 35, 813-40. Product Bulletin for GT, GX, LX and £-10 Copper Clad Dielectric Laminates, Electronic Products Division, 3M. Sazonov, D.M., Gridin, A.N. and Mishustin, B.A. (1982) Microwave Circuits, Mir, Moscow. The Non- Woven Glass Microfiber- Reinforced PTF£ Structure, RT/duroid, Rogers Corporation.
~_____s_-_p_a_ra_m__e_te_r_s______~1 ~ 3.1 NETWORK CHARACTERIZATION In the theory of network analysis, it is well known that a network, or to be more specific a two-port network, can be completely specified by a set of four parameters. This set of four parameters could be anyone of the set of y-, z-, h- or ABeD parameters. All these are network characterizations based on the total voltages and currents appearing on the terminals of the two-ports. The term 'total' may at first sound strange, but it will become obvious if one relates it to the voltage or current on a transmission line because of the fact that any voltage or current on a transmission line can be considered as a combination of a forward and a backward travelling component. In high-frequency applications these parameters are seldomly used, although for historical reasons y-parameters may still be used up to a few hundred megahertz in transistor circuit design, because of the following problems at high frequencies.
1. Total voltages and currents are difficult to measure, and even the definition of these quantities may be questionable in some cases. 2. In the measurement of these two-port parameters, short and open circuits are required. However, they are difficult to realise over a broad band of frequencies. 3. Furthermore, most active devices or circuits are not open- or shortcircuit stable. In general, if these two-port (or, for that matter n-port) parameters cannot be measured, or readily measured, at high frequencies, they are not normally employed.
3.2 SCATTERING PARAMETERS The two-port network shown in Fig. 3.1 can be completely described, as far as its external electrical characteristics are concerned, by two equations via any two-port parameter set such as the y-parameters, i.e.
_ _ _S-~P~A~R~A~M~E~T~E__R~S_ _ _~~~-~~~--~-~_-----
~~]
/,
~
-
-
"
"
Two-port network
r 1'
Fig. 3.1
-
2
r
-
"
~
2'
Two-port network.
II
= YII VI + YI2 V 2
h
= Y2I V I
+ Y22 V2,
(3.1)
where the Vs and Is are total terminal quantities. The quantities VI, 110 V 2 and 12 can be expressed in terms of travelling voltage and current waves as VI = Vi + VI I_vi-vI 1Zo V 2 = vi
+ Vi
I_ Vi - Vi 2 Zo
(3.2)
where the' +' and '-' superscripts refer to whether the travelling wave is going into or coming out from the two-port network and Zo is an arbitrary impedance constant normally taken as the characteristic impedance of the system to which the two-port is intended to be connected or the system characteristic impedance of the equipment with which the two-port is to be measured. On substituting (3.2) to (3.1), we obtain
+ fI2(Y, Zo)Vi hl(Y, Zo)Vi + f2b, Zo)Vi
VI
= f11 (y, Zo)Vi
Vi
=
(3.3)
It is important to note that the fijs are functions of the y-parameters (or z or ABeD or h) and the impedance level Zo chosen. Equation (3.3) will not change if we divide throughout by and define
no
---~0
MEASUREMENT OF s- PARAMETERS b - Vi
ffo
2 -
and Su = tu(Y, Zo), such that
S12
= tI2(Y, Zo), S2] = fz](Y, Zo) and S22 = fz2(Y, Zo), bl = b2 =
slla] S21 a ]
+ Sl2a2
+ S22a 2,
(3.4)
where the SijS are known as the scattering parameters, or simply the sparameters, of the two-port network and they are only uniquely defined if the system impedance level Zo is fixed. Note that a], b], a2 and b 2 are the square roots of the incident and reflected (or scattered) powers at port 1 and port 2, respectively. These quantities can be related to the total terminal voltages and currents as al
=
vt = square root ate f h power mCI " d ent at port 1 rZo V] + IIZo
,(7
2ffo
a2
= ,Vi ( 7 = square rZo
V2
+
f h power mc] ··dent at port 2 root ate
12 Z o
2ffo
= ,~ = square root of power emitted at port 1
bl
rZo
VI - I]Zo
2ffo
b2
= ,~ = square root of power emitted at port 2
rZo _ V2 - 12 Z o
-
(3.5)
2VZ;;
The s-parameter description of a two-port is shown schematically in Fig. 3.2. 2
T
Two-port network (Sll
S12 )
S21
S22
..., , Fig. 3.2
~b2
I
I
1
T
Two-port s-representation.
~a2 I
1
2'
~
IL-__~~~~~~~~_S_-P_A_R_A_M_E_T_E_R_S~~~~~~~__~~ 3.3 MEASUREMENT OF s-PARAMETERS
In measuring the Z-, y-, h- or ABeD parameters it is required to have at least one port open or short circuited. However, from (3.4), it is seen that, in the measurement of Sll and S2b only a2 needs to be made zero in order that Sll be measured in one transmission measurement, i.e.
(3.6)
In these measurements, 'power' a1 is made to enter the two-port at port 1-1' and a2 is made zero by connecting a transmission line of characteristic impedance equal to Zo terminated in a load Zo to port 2-2' of the network. It helps to note that a2 is defined as the 'power' coming from outside into the network at port 2-2'. Even when a2 = 0, the output of the two-port is still not necessarily matched to the system impedance Zo, hence some of the power output will still be reflected at port 2-2' back into the network. However, this reflection only occurs within the 'walls' of the network between 1-1' and 2-2' and hence it is totally within the two-port network. It is important to note that the Sll and S21 so measured (or defined) will be different if the same network is measured by (or defined with respect to) a system of a different characteristic impedance. S22 and S12 can be similarly measured by interchanging the positions of port 1-1' and port 2-2' in the measuring system, according to
(3.7)
The measurement technique discussed so far can be extended to n-port networks by terminating all but one port at a time in the system characteristic impedance. 3.4 s-PARAMETERS AND SIGNAL-FLOW GRAPHS
A signal-flow graph is a pictorial representation of a system normally described by a set of simultaneous equations. In microwave circuit analysis, circuits are described in terms of travelling 'power' waves, as and bs, related to each other by s-parameters in the form of linear simultaneous equations. Hence, the signal-flow graph technique can easily be adopted to represent linear microwave circuits pictorially via s-parameters, and furthermore, it can also be used to simplify circuits for analysis. Let us first represent a two-port network with parameters S, i.e.
~_ _ _ _ _s_-P_A_R_A_M_E_T_E_R_S_A_N_D_S_IG_N_A_L_F_L_O_W_G_RA_P_H_S_ _ _ _------11 bi = b2 =
SUal
+
S2I a l
+ S22 a2·
S12 a 2
(3.8)
By considering the variables, both dependent (bs) and independent (as), as nodes and the SijS as branches, (3.8) can be represented pictorially by a signal-flow graph as shown in Fig. 3.3. Consider now a signal generator of source voltage Vs and source impedance Zs as shown in Fig. 3.4(a). The power-delivering capacity ofthe
Fig. 3.3
Signal-flow graph for a general two-port.
bs
Zs
b
I~b I I
Ts
I~a
I I
a
(a)
Fig. 3.4
Signal generator.
(b)
0
~ ~I___________________S_-_PA_R_A_M_E_T_E_R_S__________________~ generator can be described by P avso , defined as the power available from the source (generator) to be delivered to a load equal to a certain system impedance Zo, i.e. P avso =
(z V,+ Z ) 2Zo. s
(3.9)
0
By defining a 'power'-wave variable b s equal to VPavso , i.e. b s is defined as the square root of the power available from the source to a load equal to the system impedance (chosen) Zo, the generator can be represented by a signal-flow diagram as shown in Fig. 3.4(b). In Fig. 3.4(b), rs is the reflection coefficient of the source, defined with respect to the system impedance Zo as
r. S
= Z5 - Zo
(3.10)
Z5+ Z 0'
The signal-flow diagram for a load ZL can be similarly deduced by defining r L = (ZL - ZO)/(ZL + Zo) as shown in Fig. 3.5. Next, we consider the application of a signal generator to a two-port network loaded by ZL' The signal-flow graph for such a system can be arrived at by combining the last three figures, as shown in Fig. 3.6. It is noted that out ofthe five 'power' variables shown in Fig. 3.6(b), (b., ab bb a2 and b 2 ) only b s is independent. Signal-flow graphs such as that shown in Fig. 3.6 can be used to help evaluate the transfer functions and drivingpoint immittance functions. As an example, let us evaluate the power ratio b 2/b s of the circuit shown in Fig. 3.6. This quantity may be interpreted as the ratio of the power incident to the load and the power available from the source. Figure 3.6 can be redrawn such that b s is the input and b 2 is the output of the system, as shown in Fig. 3.7(a). The signal-flow diagram is reduced successively using the basic canonical form in the feedback system as indicated in Fig. 3.7. a
b (a)
Fig. 3.5 Load.
(b)
~____________________P_R_O_B_LE_M_S____________________~I ~ Zs
Two-port network
v.
(a)
Ts
(b)
Fig. 3.6 Loaded two-port network with source.
On simplifying the expression in Fig. 3.7(e) b2 = bs 1-
S21 s"rs -
S22rL -
S12S21rsrL
+ S"S22rsrL'
(3.11)
Other network functions of a loaded two-port network excited by a source of reflection coefficient rs can be evaluated in a similar way, the results of which will be made use of in deriving various power gains and impedance functions for an amplifier in Chapter 6. PROBLEMS 1. Determine the s-parameters of the networks shown in Fig. P.3.1
--
-
-
--
Ans: (a)Z/(Z + 2), 2/(Z + 2), 2/(Z + 2), Z/(Z + 2); (b) Dual of (a) 2. For a section of transmission line of characteristic impedance Zo and of electrical length (), determine its S matrix. Ans: 0, exp( -j()), exp (-j()), 0 3. For the loaded two-port network shown in Fig. P.3.2, find:
~I~____________________s_-_PA_R_A_M__ET_E_R_S___________
bs
(a)
b2
b2
S2!
a!
Us cU r~'l;
•• 1
bs
S21
•
bs
~
1
L 1-
bs
•• 1
•
L
Fig.3.7
•
~
•
S2!
1-
SIlTs
•
1 1 - sIlTs
..
SI2
s12 s 21
(1 -
Evaluation of b 2 /b s '
•
1
S22TL
T s TI
SlITs)(l -
1
•b
2
(b)
(c) 2
~••
1 1-
~
1
S22TL
T Ls 12 T s
1_
bs
lC:r,
•b
~
snTd
(d)
b2
•
(e)
I~
PROBLEMS 0
0
0
(a)
Fig. P.3.1
50Q
0
Z
: ~ : (b)
j.-- A/8 ---i~~1 - . . . alex)
Two-port network
Zo = 50Q
lOLO° volts
x=O Zl = 50
Zo = 50Q
50Q
x =A/8
+ j50Q
Fig. P.3.2
1. Z(O); 2. al (0), b1(0), al (/lIB), b1(/lIB) and a2(0); 3. The average input power at x = 0 and at x 4. S11 (0) and S11 (/lIB).
= /lIB;
4. A transistor must be imbedded in a transmission-line structure known as a test jig in order for its s-parameters to be measured as shown in Fig. P.3.3. The transmission-line sections in front of and behind the imbedded device are of length 11 and 12 , respectively. Assuming that the propagation velocity along these sections is v, derive an expression relating the measured s-parameters, Sm, to the true s-parameters, S, of the device. S. The common-emitter s-parameters of the npn transistor Motorola MRF914 at 1 GHz measured at V CE = 5 V and Ic = 5 rnA are given by Sll
= 0.16L - 150°
S12
= 0.19L55°
S21
= 2.17L55°
S22
= 0.55L - 45°.
If a capacitor of 1 pF is connected across collector and base, find the
modified s-parameters of the common-emitter transistor-capacitor combination at 1 GHz under the same bias conditions.
~
~_P_A_RA_M__ET_E_R_S__________________~
L I_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
THROUGH SECfION
TEST JIG
Fig. P.3.3
_ - - I (in
Fig. P.3.4
A.g)-----II.~I~I __-
DR is the dielectric resonator; Ag is the guided wavelength.
~_________________F_U_R_T_H_ER__R_EA_D_I_N_G________________~I ~ 6. For the signal-flow graph shown in Fig. 3.6 (in the text), find the ratio bl/b s •
7. For the partially completed circuit shown in Fig. P.3.4, derive an expression for the source loading, r s , presented to the gate of the FET by the dielectric-resonator loaded microstrips. Given: Zo, the loaded Q of the dielectric resonator and the coupling coefficient {3 between the dielectric resonator and microstrip.
FURTHER READING Gonzalez, G. (1984) Microwave Transistor Amplifier Analysis and Design, Prentice-Hall. Hejhall, R., RF small signal design using two-port parameters, Application Note AN-215A, Motorola Products Inc. Hewlett Packard, S-parameter Technique for faster, more accurate network design, Application Note 95-l. Hewlett Packard, S-parameter design, Application Note 154. Partha, R. and Sharma, M.L. (1986) design of dielectrically stabilized oscillators using Feedback Techniques, Proc. of RF Technology Expo 86, pp. 291-6, California. Roddy, D. (1986) Microwave Technology, Prentice-Hall. Sander, K.F. (1987) Microwave Components and Systems, Addison Wesley.
GI'- -__
Im_p_e_d_an_c_e_M_a_tc_h_in_g_------'
4.1 INTRODUCTION
One of the most important aspects of high-frequency circuit design and microwave engineering is the problem of impedance matching. Impedance matching is the design of a circuit to be inserted between a source and a load (both used in the general sense) so as to provide maximum power transfer between them. For example, the source could be a 50-ohm r.f. signal generator and the load could be the optimal input impedance of a transistor under certain bias conditions. Alternatively, the output voltage of a transistor and its output impedance could be considered as the source, and a 50-ohm termination could be considered as the load. In either case, a matching network inserted between the source and the load is necessary in achieving maximum power transfer. One of the most fundamental criteria is that a matching network must at least theoretically be lossless. The reason is obvious. As a consequence. matching networks in high-frequency circuit design always take the form of an LC (never R) circuit in the discrete case, or in the form of transmissionline sections and stubs in the distributed case. In this chapter, the following matching techniques will briefly be introduced with the aim of utilizing the design equations rather than delving into their derivation: 1. Two-element L networks; 2. Three- (or more) element networks; and 3. Transmission-line networks.
Smith-chart and computer-aided methods will be introduced in aiding the above designs where appropriate. 4.2 CONCEPT OF OPERATING Q-FACTOR
Since matching networks are basically constructed from inductive and capacitive elements, they invariably are frequency discriminative. In fact,
~_______________T_W_O_-_E_LE_M__EN_T__L_N_E_T_W_O_R_K______________~I one of the functions of a matching network is to provide harmonic attenuation within certain bandpass specifications. Matching networks are normally designed for a single frequency, known as the operating frequency or centre frequency, and the bandwidth requirement of the network is fulfilled by choosing a proper quality factor for the network. The quality factor Q of a circuit is roughly equal to the reciprocal of the percentage bandwidth, i.e.
-.l _ ,1f
(4.1)
Q - fa'
where LJf is the 3-dB bandwidth of the matching network. In two-element matching, the Q-factor cannot be assigned and is governed by the ratio of the real part of the impedances to be matched. This type of matching network normally has a low Q-factor. The value of the Q-factor can however be assigned by the designer in the case of three-element or transmission-line networks. It was noted that matching is normally calculated at a single frequency. Unfortunately, the exact operating Q of a circuit cannot be determined by calculations made at one frequency only. The operating Q-factor or loaded Q of a circuit is defined by calculating the Q values of each node in the circuit at the centre frequency and by taking the highest value as the operating Q of the circuit. The operating Q so defined is a good approximation of the actual Q for circuit-design purposes, and is a parameter to be assigned by the designer.
4.3 TWO-ELEMENT L NETWORK A two-element L-matching network inserted between two resistances Rl and R2 to be matched is shown in Fig. 4.1. The idea of L matching is to put the larger of the two resistances to be matched, R2 in this case, in parallel L-matching
r--------------l I
IL Fig. 4.1
I
______________ I
L-matching network.
~
[33]
01
IMPEDANCE MATCHING
L -_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
~
with a reactive element jX2 , so that the series resistance of the (R 2 ,jX2 ) combination has a smaller value than R2 , and X 2 is so chosen that this series resistance is equal to R I . In so choosing X 2 , X 2 could be positive or negative, depending on the designer's assignment. Xl is then chosen to be equal to and opposite in sign to the series reactance of the (R 2 , jX2) combination, so that the reactive part of the (RbjXI ) combination and that of the (R 2 ,jX2 ) combination neutralize each other (or resonate) at the operating frequency. In determining Xl and X 2 , Q has no part to play and is only a consequence of the circuit designed. The derivation of the design equations for the L-matching network is left as an exercise in Problem 4.3 and these equations are given below Ql
~ (~: - 1)
= Q2 =
=I~:I Q2 = I~~I· QJ
(4.2)
In general, an L-matching network can assume either the low-pass or the high-pass format as shown in Fig. 4.2. L
c
(a)
c
L
(b)
Fig.4.2
(a) Low-pass and (b) high-pass L-networks.
L-~~~~~~~T~W~O~-~E~LE~M~EN~T~L~N~E~T~W~O~R~K~~~~~~~~I ~ Example 4.1 Design a matching network to match a 100 Q source to a 1000 Q load at 200 MHz. It is also required that d.c. could be transferred from source to load through the matching network.
Solution First of all, it is noted that it does not matter which is source and which is load. What is important is the impedance level, 10011000, to be matched. Because of the d.c. requirement, the matching circuit should look like Fig. 4.3. L
r
o~------~~~----~o
r 100
Fig. 4.3
i X2
n
C
I
0-------------------------0
Circuit for Example 4.1.
From (4.2) Ql
=
Q2
=
~(\O~~ - 1) = 3
Xl = QIRI = 3 x 100Q = 300Q.
Note that Xl is chosen to be positive (inductive).
IX2 1 =
R2 = 1000
Q2
3
=
333Q
X 2 = -333 Q (capacitive).
Hence, at 200 MHz, L=
300 H 2.n(200 x 106 ) = 238.5nH
c= 333 X
1 F 2.n x 200 x 106 = 2.4 pF.
1000
n
~
IM_P_E_D_A_N_C_E_MA __T_C_H_I_N_G________________~
L t_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
Before leaving the topic of two-element matching, a certain question should be raised. What happens if one of or both the impedances to be matched are complex? Such a case would frequently occur if the impedances correspond to the input or output port of a transistor. The answer to this question is that it can be handled in two ways. 1. Method of absorption - neglect the reactive part of the load and source impedance and perform the calculation as before. Then try to absorb the reactive part of the source and load impedance into their corresponding adjacent reactive component of the matching network. This may not always be possible as the reactance to be absorbed may be bigger than the matching reactance or they may be of opposite polarity (e.g. trying to absorb capacitance into an inductor). 2. Method of resonance - when absorption is not possible or practical (resulting element value too small), the load and source reactance can be made to resonate (cancel) by adding an equal and opposite reactance at the frequency of interest. It is noted that the techniques stated in handling complex loads and sources apply not only to two-element matching, but to all other matching networks.
4.4 THREE-ELEMENT MATCHING The main disadvantage of the two-element matching is in not having the choice of the operating Q. The Q for two-element circuits is generally very low and is, in fact, the lowest possible value for a matching network matching R[ to R 2 . In narrow-band or high-Q applications, three-element circuits have to be employed. Three-element circuits provide the designer the freedom to specify an operating Q restricted only by the practicability of the resulting component values. A three-element matching network is usually arranged in either the II form or the T form as shown in Fig. 4.4. The mechanism by which a threeelement network provides the matching function and the freedom to choose the operating Q can be best understood by considering a II-network as two L-networks connected back to back as shown in Fig. 4.5. Looking from AA' towards R2 one should see a resistance, say R, if Lnetwork 2 is designed according to the procedures laid down in the last section. The operating Q of L-network 2 is given by
By the same token, if L-network 1 is also designed in the same manner matching Rl to R, the operating Q will be
T_H_R_EE_-_E_LE_M_E_N_T__MA __T_C_H_IN_G______________~I
L -_ _ _ _ _ _ _ _ _ _ _ _ _ _
(a)
(b)
Fig.4.4 Three-element matching networks: (a) II-form and (b) T-form.
Since the operating Q of a circuit is dominated by the branch of the circuit having the highest Q value, therefore the overall (4.3)
where RH is the bigger of RI and R 2 . The overall Q factor, Qn, can be assigned by fixing the value of R, which must be smaller than both RI and R 2 . Once R is fixed QLl and QL2 are determined. The values for Xl, X 21 , X 22 and X3 can then be calculated based on the two-element design method; L-network 1
L-network 2 -rI
I
rI I Virtual J,
R A'
I
I
L,J I
I
Resistance B'
~~J-~__---------<}-~--.,,--------~e---o-~
Fig. 4.5
II-network as two back-to-back L-networks.
~
~
IM_P_E_D_A_N_C_E_MA __T_C_H_I_N_G________________~
L I_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
If the value of the virtual resistance R is desired to be higher than R 1 and R z, II-networks cannot be used, instead the two shunt reactances should be
placed at the centre of the network to form a single shunt reactance, resulting in a T-network. The operating Q of a T-network is then given by
where RL is the lower of Rl and Rz . Instead of deriving the design equations for the three-element T- and 11networks, the design equations for five possible variations of three-element networks (which could result in more than three elements) with one termination being capacitively complex are given below. These circuits represent the most commonly used matching networks in active circuit design.
4.4.1 Network A Q Q ... T ...
Device to be matched to R2
r---------l
I
I I
1 XLI 1 1--4:)-rv-V--Y
I I
COUI
1..........- - 1
1----0----,
I I 1 RJ I I 1 1 1 1 1L _.::_______ ..JI Fig. 4.6 Network A. All X-values are positive.
Design equation for network A
Select the operating Q XLI = QR I X C2 =
-
X cout
(Xcout is -ve)
AR2
B
XCI
(4.4)
= Q _ A'
where A =
~[RI(1R: Q2) -
B = R I (1
+ Q2).
1] (4.5)
~_____________T_H_R_E_E_-E_L_EM __E_NT__MA __T_C_H_IN_G______________~I ~ 4.4.2 Network B Device to be matched to R2
.,. ·T· I I Q
Q
Q'
Q'
XL
I
Fig. 4.7 Network B.
Design equation for network B
Select the operating Q
(4.6)
Note that for this network we may have either RI > R2 or R2 > RIo However, when R2 > Rb the overall circuit Q is not equal to the Q value chosen, but is equal to Q' =
~[ :: (Q2 + 1) - 1]
0
4.4.3 Network C Device to be matched to Rz
Q
r-------~
.~.
Q
IR'~ fc~. t>--]-x-Ic~3JR' IL _______ --'I Fig. 4.8
Network C.
~ ~I_________________IM_P_E_D_A_N_C_E_M_A_T_C_H_I_N_G______________~ Design equation for network C
Select the operating Q
(4.7)
Note that when V[(R2 - Rd/Rd > Q, the actual circuit Q is V[(R2 - RJ)/Rd.
4.4.4 Network D Device to be matched to R2
r--------,
I I
Q
Co~ I----+-
! I I
Q
RJ
I
Ilrvv!....e>-----<>---,
I I
I I I
I I L _______ ...l
I
Fig. 4.9
Network D.
Design equation for network D
Select the operating Q
(4.8)
Note that the Q so defined is an approximated value for the circuit Q where C 1 <2i Cout is assumed. When V[(R 2 - RJ)/Rd > Q, the actual circuit Q is V[(R2 - RJ)/Rd·
D_E_SI_G_N_IN_G_W __IT_H_T_H_E_S_M_I_T_H_C_H_A_R_T__________~I
L -_ _ _ _ _ _ _ _ _ _ _
4.4.5 Network E Device to be matched to R2
Q
,--------, I
I
Cout
I
I
I
~~I~JV~n
I I
Q
R]
I I
IL _______ ..JI
Fig. 4.10 Network E.
Design equation for network E
Select the operating Q XLI = RJQ
+ X cout
X L2 = R2B XCI
= Q
A
+ B'
(4.9)
where A = R I (1 B =
+ Q2)
~(:2 - 1).
The five 'three-element' networks considered above can best be utilized by listing out the reactance values using a computer with Q as a parameter, and R J and R2 as variables. This will help the designer decide on which circuits to be chosen as this will give a quick reference as to whether the component values required by a certain circuit configuration are reasonable or not.
4.5 DESIGNING WITH THE SMITH CHART The two- and three-element network design discussed in the previous sections can also be calculated using an impedance Smith chart overlaying an admittance Smith chart. However, it should be noted that the Smithchart method is not only limited to the two- and three-element networks previously discussed.
~
~
I_M_P_E_D_A_N_C_E_MA __T_C_H_IN_G ________________~
L l_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
Fig. 4.11
Impedance Smith chart.
To understand the overlaid Smith chart, let us consider an impedance Z normalized by an arbitrary Zo (usually but not necessarily equal to 50 Q), i.e.
-
z
Z = Zoo
For example, let Z = 1 + jl. The reflection coefficient, r, created by Z = 1 + j1 on a system of normalized system impedance 2;) = Z(/Zo is given by
Z - 1 +1
r= Z
1 + jl - 1
= 1 + jl + 1 = 0.447L63.4°.
If admittance Y = Y/Yo were used, where Y = then
1/Z, Y =
1/ Z and Yo = 1/Zo,
1 Y = 1 + jl = 0.5 - jO.5 and
r=1- ~=11
+Y
1
0.5
+ jO.5 = 0 447
+ 0.5 - jO.5
.
L
63 40 .
.
Now, let us interpret these numerical calculation on a Smith chart as shown in Fig. 4.11. Z is entered on an impedance chart as point A. A circle with radius from the centre of the chart to A is drawn. Produce AO to meet the circle at point B. The value of the 'impedance' at B is read as 0.5 - jO.S,
D_E_S_IG_N_I_N_G_W __IT_H_T_H_E__ SM_I_T_H_C_H_A_R_T__________~I
L -_ _ _ _ _ _ _ _ _ _ _
t
+ve reactance (X) Series inductance
IMPEDANCE CHART
!
-ve reactance (X) Series capacitance
ADMITTANCE CHART
Fig. 4.12
f
-ve susceptance (B) Parallel inductance
~
+ve susceptance (B) Parallel capacitance
Impedance and admittance chart.
which is actually equal, numerically, to the admittance Y of the original impedanceZ. However, to enter an admittance Y = 0.5 - jO.5 on an impedance chart as point B shown is not correct since the phase of r at B is 180 + 63.4 instead of 63.4 although the magnitude Irl is the same at points A and B. Therefore, in order to enter properly an admittance on a Smith chart, the impedance chart has to be rotated about centre 0 by 180 resulting in an admittance Smith chart as shown in Fig. 4.12. Consider matching an impedance Z = 10 + j 10 ohms to a 50-ohm transmission line using the Smith-chart method. For convenience, Z is normalized by 50 Q, i.e. Z = 0.2 + jO.2 and Z is to be matched to 1 + jO. Z = 0.2 + jO.2 is entered on the impedance chart as point A in Fig. 4.13. Point A is then moved to point B via a constant R circle, i.e. A is transformed into B by adding a reactance to Z. The required reactance X AB is equal to 0
0
,
0
X AB
= XB
-
XA
= +0.4 - (+0.2) = 0.2.
,
0
~
[81
IMPEDANCE MATCHING
L -_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
Fig.4.13
~
An example of two-element matching using the immittance Smith chart.
Point B is then moved to the matched point C (corresponding toZ = 1 + jO or Y = 1 + jO) via the G = 1 circle. In other words, B is transformed to C by 'adding' a susceptance BB~ in parallel with the admittance value at point B. BBe is given by BBe
=
Be - BB
=
(0) - (-2.0)
=
+2.0
Hence the matching network is as shown in Fig. 4.14. It is noted that the network shown in Fig. 4.14 is not the only solution. The alternative approach is shown in Fig. 4.15. Point A is moved to point B, also via a constant R circle to meet the G = 1 circle at B, this time in the 'toward' load direction. The rest of the chart construction is self explanatory. The results are X AB BBe
or
= XB = Be
-
XA
- BB
= (-0.4) - (+0.2) = -0.6 = (0) - (+2.0) = -2.0
]·-X Be =
1 --:-2 -]
=]'05 . .
The alternative network is shown in Fig. 4.16. The two alternatives shown in Fig. 4.14 and Fig. 4.16 can also be obtained by calculation using (4.2). The choice between these two configurations depends on whether high-pass or low-pass characteristics are preferred, and whether d.c. is to be blocked.
D_E_S_IG_N_I_N_G_W __IT_H_T_H_E__ SM __ IT_H_C_H_A_R_T----------~I
L -_ _ _ _ _ _ _ _ _ _ _
jlO 0(j0.2)
100 (0.2) 500 (1 )
-j250 ( -jO.5) +j100 (j0.2)
Fig. 4.14 Two-element Smith-chart matching.
In designing two-element matching networks using Smith chart, the operating Q is not up to the designer's choice, but is rather predetermined by the terminating resistances R 1 and R 2 . In the example shown in Fig. 4.14 and Fig. 4.16, the Q-value can easily be calculated by setting (4.2) to be equal to 2. A question arises: 'How is the operating Q read from the Smith chart?' Each node in a circuit corresponds to a point on the Smith chart (either regarded as impedance or admittance), and each point on the Smith chart
Fig.4.15
An alternative solution to the example in Fig. 4.13.
~
~ ~I
I_M_P_E_D_A_N_C_E_M_A_T_C_H_IN_G ________________~
________________
-j30
n
(- jO.6) 100 (0.2)
+j2S n (+ jO.S)
50 n ( 1)
+jlO 0 (j0.2)
Fig. 4.16
An alternative solution to Fig. 4.14.
is associated with a Q-value equal to the ratio of the reactive part (or susceptance part) to the resistive part (or conductance part) of the immittance of that point. And the overall operating Q of the circuit is equal to the largest Q-value of the nodes. Consider the Q-values of nodes A, B and C of Fig. 4.13 and Fig. 4.15, they are listed below From Fig. 4.13 QA
= 1°·21 0.2 = 1
QB
=
Qc
=
1°.41 0.2 = 2
Iii = °
From Fig. 4.15 QA
= 1°·21 0.2 =
1
1- .41 2
0 = QB = 0.2 Qc
=
Iii
=
°
Hence, it is seen that each circuit has its operating Q (=2) determined by the Q-value at its corresponding point B. As was seen in the section on three-element matching, the operating Q needs to be specified, otherwise there will be too many solutions. So, in using the Smith-chart method for three- (or more) element matching, it is important that at least one of the nodes (points on the Smith chart) should have a Q-value equal to the specified Q, whereas the Q-value for all other nodes should be lower than the specified Q. In order to facilitate graphic determination of a three- (or more) element matching network involving the designer's choice of Q, a device known as the constant-Q curve is plotted on to the Smith chart first. A Q = 5 curve is plotted as shown in Fig. 4.17. In Fig. 4.17 it is noted that points of constant Q lie on the arc of two circles, one on the top half and the other on the lower half of the chart. It is also noted that these constant-Q arcs are the same for both an admittance and an impedance chart.
D_E_S_IG_N_I_N_G_W __ IT_H_T_H_E_S_M_I_T_H_C_H_A_R_T__________~I
L -_ _ _ _ _ _ _ _ _ _ _
Fig. 4.17 Constant-Q curves.
The use of the Smith-chart technique in designing a three-element matching network can best be illustrated by an example. Example 4.2
Design a T-network to match a ZI = 15 + j15 Q impedance to a Z2 225 Q impedance at 30 MHz with a loaded (operating) Q of 5.
=
Solution
First draw the arcs for Q = 5 as shown in Fig. 4.18. Choose a convenient normalizing value of Zo = 75 Q, then ZI = 0.2 + jO.2 Z2 = 3. ZI and Z2 are entered on an impedance chart as points D and A, respectively. It is required that point A, after being transformed by the T-
network will come to point C which is the complex conjugate of D shown in Fig. 4.18. Figure 4.19 together with Fig. 4.18 will help us understand the procedures. Z2, after being transformed by L J and C}, will come to a point, point I, having an impedance ZI (or YI). The real part of ZI should be equal to the real part of Zc = Zo, since further action by L2 would only change the imaginary part of ZI. To locate point I, the real part of Zc is extended to meet the Q = 5 arc at point I.
~
~
I_M_P_E_D_A_N_C_E_M_A_T_C_H_IN_G ________________~
L I_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
Fig. 4.18
Example 4.2.
Z2 at point A is moved through constant R to point B which has a conductance equal to that at point I. Point B is then moved to point I through constant G to point I, and that completes the procedures. The flow of transforming Z2(A) to Zl(C) is indicated by the direction of the arrows shown in Fig. 4.18. To read the reactance and susceptance from Fig. 4.18 XLI = X B
so
LJ
=
X A = +2.5 - (0) = +2.5 2.5(75) 2n(30 x 106) = 995 nH. -
BCI = B, - BB = (+0.97) - (-0.16) = +1.13
21 15+j15 f! (O.2+jO.2)
Fig. 4.19
Circuit for Example 4.2.
22
225 f! (3)
T_RA_N_S_M_I_SS_IO_N_-_LI_N_E_MA __T_C_H_IN_G __N_E_TW __O_R_K________~I
L -_ _ _ _ _ _ _ _ _
so
so
C1
_ -
1.13(1/75) _ X 106) - 80pF.
2Jl(30
X L2 = Xc - XI = (-0.2) - (-0.96) = +0.76 0.76(75) L2 = 2Jl(30 x 106) = 302 nH.
It is noted that the flow of matching actions from point A to point C as indicated by the direction of the arrows in Fig. 4.18 is in the direction of 'towards generator' on the Smith chart. Alternative solutions can be derived by matching point A to point C 'towards load', or by matching point D to point A 'towards generator' or 'towards load'. Apparently, there are four different solutions. However, it would be a good exercise for the readers to show that these are actually two distinct solutions. It should be further noted that the terms 'towards generator' and 'towards load' are only transmission-line terminologies and they bear no physical meaning when the Smith chart is used for discrete component design.
4.6 TRANSMISSION-LINE MATCHING NETWORK The techniques for matching-network design described in the previous section are based on lumped capacitors and inductors for their realization. However, as the operating frequency increases beyond a few hundred megahertz, and the values of Cs and Ls so calculated may become impractically small, then distributed elements have to be employed instead. The realization of distributed elements is based on the properties of a transmission line. The most readily available transmission-line structure in the high-frequency domain, especially for circuits involving active devices, is the open microstrip. In this section, two methods of matching using transmission lines are considered. 1. Stub matching: (i) Single stub, (ii) Double stub. 2. Quarter-wave transformer. In matching an immittance to a transmission line (or waveguide in general) or to another immittance, the main objective is to provide maximum power transfer between these two immittances. In using Smith charts to aid the calculations in transmission-line matching problems, there is a danger in misinterpreting the meaning of the terms 'towards load' and 'towards generator' which appear on the Smith chart. For example, if we are to match the input impedance of a FET or BJT, Zin, to a signal generator of internal impedance of 50 Q, it is natural to enter Zin as 'load' on the Smith chart and move it 'towards generator' until
~
ODI
IMPEDANCE MATCHING
L -_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
it reaches 50 Q. This is correct. However, if we are to match the output impedance of a FET or BJT, Zout, to a load of 50 Q, one is tempted to enter Zout as 'generator' on the Smith chart and move it 'towards load' until it reaches 50 Q. And, this time, it is incorrect. The correct way to interpret 'load' and 'generator' on a Smith chart, in the context of matching, is that we are always matching a 'load' to the 'generator' by moving the 'load' 'towards generator', independent of whether the signal is physically coming from the 'load' end or the 'generator' end.
4.6.1 Stub matching (a) Single-stub matching In matching a load to a generator (both used in the Smith-chart sense), either impedance or admittance can be used. However, a matching scheme using an impedance chart will end up with series stubs while that using an admittance chart will end up with shunt stubs. For circuits using microstrip implementation series stubs are not possible, hence the admittance chart, which results in shunt stubs, is always perferred. A stub is a short transmission line which can either be of the same characteristic impedance as the main line concerned, or otherwise. For the purpose of stub-matching calculations, a stub is considered to be loss-less. The immittance of a transmission line is given by
Z = ~ + j tan {31 1 + jZL tan {31 y = ~ + j tan {31 1 + jYL tan {31" In
and
In
For an open-circuited line, ZL =
00
Zino =
and YL = 0, hence -
j cot {31
Yino = j tan {31.
For a shorted line, ZL = 0 and Y L =
00,
(4.10) ( 4.11)
hence
Zins =
j tan {31
( 4.12)
Y ins =
-j cot{3l.
( 4.13)
In actual matching problems, the Smith chart is widely used. The choice of series or shunt stubs and shorted or opened stubs is arbitrary and the Smith-chart method is independent of these choices. The only difference in these choices is in the last step when the stub length I is read from the chart. Figure 4.20 shows a single-stub matching scheme which matches a microwave antenna ZL (Yd to a waveguide Zo (Yo), and its transmissionline equivalent.
T_RA_N_S_M_I_SS_IO_N_-_L_IN_E_MA __T_C_H_IN_G__N_E_~ __O_R_K________~I
L -_ _ _ _ _ _ _ _ _
Stub waveguide
Microwave antenna
Yo
Fig. 4.20 Single-stub matching.
To find d and 1 for perfect match, Y L = YdYo is first entered into the chart as shown by point A in Fig. 4.21. The immittance viewed from just right of the stub towards the load is effectively equal to moving point A towards the generator a distance d (in A), which transforms ZL (Yd to a new immittance, shown as point B, ZL (YL) such that Re(ZL)
= Re(YL) = 1.
The transformation just described takes place under constant-VSWR or constant-Q condition, hence the path from point A to point B is an arc of a circle centred at G = R = 1 and B = X = O. Viewed from just left of the stub towards the load, the new load Z'L = 1 (or Y'i. = 1), i.e. it is matched. To achieve the match condition, point B is moved to point C under a constant-resistance (conductance) condition. The reactance (or susceptance) change from point B to point C is provided by the stub length I. d is given (in A) by the length of the arc of the circle sustained by A and B with C as shown.
~
~
I_M_P_E_D_A_N_C_E_MA __T_C_H_IN_G ________________~
L I_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
E
.----f---+_--~~---+_-+'-t---::::ii.'1
Fig. 4.21
F
Single-stub matching.
To match B to C, an amount of reactance (or susceptance) equal and opposite to that of B is needed. Point D has this property. To determine I, we consider the following cases. (a) When impedances are used and the stub is open circuited then I = 110 is the arc length (in A) from F (corresponds to open circuit) to D' in the clockwise direction, hence
--
110 = arc FD'. lIs
(b) When impedances are used and the stub is short circuited, then I is the arc length from E to D' ~
lIs
=
arc ED'.
(c) When admittances are used and the stub is open circuited, then lAO =
~
arc ED'.
(d) When admittances are used and the stub is short circuited, then ~
lAs =
arc FD'.
Figure 4.22 shows the schematics of these four cases.
=
"'l
;':<1
~.
"'"
N N
'Tj
0
NI
:: .... '"S"
;
II
(JQ
!f '"2"
cr"
3~
~
~
~
8" S"
;:r
i !
(JQ (")
0 ::l
::n
(JQ
:: ....
1
"-
~
0"
::l
:"
~I
0
! "-
NI r
':<1
;'
NI
II
;'
II
~
:s
~
t
f
"-
t
1
"-
~I
NI
r
...-
~I
IMPEDANCE MATCHING
~--------------------------------------------~
(b) Double-stub matching
The main disadvantage of the 'single-stub method' is that the position of the stub, d, is a function of frequency and that every new load requires a new stub position. The 'double-stub method' which utilizes two stubs spaced at a fixed distance and located at fixed positions is devised to overcome the more frequency-sensitive single-stub method. However, the doublestub method cannot match all possible load immittance values. A doublestub matching scheme is set up as shown in Fig. 4.23. In double-stub matching the spacing between the stubs, d{, is arbitrarily fixed, but the best value for d{ is ,1,/8 or 3,1,/8 for minimum forbidden region. The procedure for double-stub matching using the Smith chart is listed below with reference to Fig. 4.24. 1. The immittance of load (ZL or Y d is entered as point A. 2. The G = 1 or R = 1 circle is rotated in the anticlockwise direction, i.e. towards the load, through an angular length of d{ in A. For minimum forbidden region, the rotation is 90° or 270°. 3. Looking from 'just right' of stub 1, Zin (or Yin) is equal to the transformed value of ZL (or Yd through a constant-VSWR section, hence it corresponds to moving point A towards the generator by a line length of d to point B. 4. Stub 1 transforms point B to either point C or point C' (depending on the stub length) through constant R or G circles until it meets the rotated unit circle at point C or point C'. 5. Point C or C' is then rotated in the clockwise direction by a length d{ to points 0 or 0' where both of these lie on the original unit circle. 6. The effect of stub 2 is to match either point 0 or 0' to the centre of the chart where matched condition is achieved.
The stub length I] is found by first letting X = Xc - X B (or I; is found by X' = Xc - X B ), where Xc and X B are the reactance or susceptance of points C and B, respectively. The imaginary part of the immittance, X, of stub 2 is then entered on the circumference of the chart. The length, I], of stub 2 can be found as the arc length from either the open-circuit or shortcircuit point of the chart 'towards generator' to X, depending on whether an open-circuit or short-circuit stub is chosen for stub 1. The length, 12 , of stub 2 is found by using the same procedures as were employed in determining the stub length for single-stub matching. The stub lengths I] and 12 depend on whether impedances or admittances are used (throughout the process) and whether short-circuit or open-circuit stubs are used. Straightforward stub matching leaves no room for choosing desirable Qvalues. However, with additional reactive elements placed properly, the operating Q could be controlled.
T_RA __N_SM __IS_S_IO_N_-_LI_N_E_MA __T_C_H_IN_G __N_E_TW __O_R_K________~I
L -_ _ _ _ _ _ _ _ _
Ld' Fig. 4.23 Double-stub matching.
Fig. 4.24 Double-stub matching.
--~4----
d - - - - - I••
~
o
I"-----________I_M_P_E_D_A_N_C_E_M_A_T_C_H_IN_G_ _ _ _ _ _ _ _ _-----" Example 4.3 is chosen to illustrate how single-stub matching can be applied in matching the input of a transistor amplifier to a 50 Q source. The problem is stated in a way that could easily mislead readers who come across the topic for the first time. Example 4.3 Design a single-stub network for matching the input of a common-emitter power amplifier to a 50 Q source. The desired source reflection coefficient, r s , as seen by the transistor is 0.614L160° on a 50 Q system.
Solution The problem is better understood with the aid of Fig. 4.25. The required source admittance is Y s = 2.8 - j1.9 and the required input admittance of the transistor is thus equal to Yin
= Y; = 2.8 + j1.9.
Y, and Yin are entered on to an admittance Smith chart as points A and B, respectively, as shown in Fig. 4.26. Point B (not point A) is required to be transformed by the matching network to point D which corresponds to 50Q or r = O. Point B is moved towards the generator through constant-VSWR (or constant-I rl) to meet the G = 1 circle at point C. Moving on a constantVSWR circle is equivalent to providing a (series) transmission path between two nodes. The length of the path is read from the chart as 0.097A.. Point C is then moved to point D (the matched point) by adding a shunt susceptance Bs of value equal to Bs = Bo - Be = 0 - (-l.6) =
+ 1.6.
In Smith-chart terms, point C is moved to point D through the G = 1 circle. The required additional susceptance Bs = + 1.6 can be provided by
50
n Matching network
Transistor amplifier
fs=0.614L160° or Ys=2.8-j1.9
Fig. 4.25
Formulation of Example 4.3.
T_RA_N_SM_IS_SI_O_N_-L_I_N_E_MA_T_C_H_IN_G_N_E_T_W_O_R_K_ _ _ _
L -_ _ _ _ _
- - - -'I OD
Short circuit
-~ B,=+0.16 Fig. 4.26
Smith chart for Example 4.3.
either an open-circuit stub or a short-circuit stub. The value of Bs = + 1.6 is entered on the rim of the chart as point X. Realization of Bs = + 1.6 by an open-circuit stub requires a stub length of 0.165,1, whereas the length required of a short-circuit stub is 0.165,1, + 0.25,1,. Suppose the shorter stub, the open-circuit stub, is chosen. The circuit is shown in Fig. 4.27. From Fig. 4.26 it can be read that the highest Q-value of the whole matching path is about 1.6, occurring at point C, hence the operating Q is 1.6. By merely looking at this low Q-value of 1.6, it is apparent that the matching network is good for broadband application. However, it should be noted that both the transmission path and stub length so calculated
50
Fig. 4.27
n
d.c. block
0.097.1e
II
Microstrip realization of Example 4.3.
~ ~I
I_M_P_E_D_A_N_C_E_MA __T_C_H_IN_G ________________~
________________
are only calculated for a single frequency and, because of the transmissionline property, electrical lengths tend to be extremely frequency sensitive. Hence, the calculated operating Q based on a single frequency is not a valid reference for saying that the network is wideband. To achieve broadband matching, the double- or triple-stub method can be used. To achieve narrower bandwidth, lumped reactances can be used replacing some of the stubs in the appropriate manner. (b) Quarter-wave transformer
The input impedance Zin of a quarter-wavelength transmission-line section of characteristic impedance Zo loaded with a load impedance ZL is given by (4.14)
If ZL is resistive, and ZL is to be matched to a real Zin, this can be done by designing Zo to satisfy (4.14). However, if either ZL or Zin or both are
complex, a quarter-wave transformation is not enough. Under such circumstances, additional series or shunt reactances can be placed at either end of the quarter-wave transformer to resonate off the reactive part of the impedance at that particular node, leaving the quarter-wave transformer to match the real part of the impedances. In general, a quarter-wave matching system depends mainly on the Q-value of Zin and ZL·
PROBLEMS 1. A load of ZL
=
1600
+
j800 Q is connected to a transmission line of = 400 Q as shown in Fig. P.4.1.
characteristic impedance Zo
1. Determine the location (d in Ag) and the length (I in Ag) of a shortcircuited series stub of the same characteristic impedance (400 Q)
such that there is no reflection beyond the plane of the stub. 2. Repeat the same problem if the short-circuited shunt stub is used instead. 3. What will the length (I) of stub be in parts 1 and 2 if the characteristic impedance of the stub is 600 Q? Ans.: 1. 0.086Ag, 0.168A g, 2. 0.198A g, 0.082A g, 3. 0.137,1,g, 0.058Ag. 2. Determine the lengths (in Ag) 11 and 12 for perfect match for the circuit shown in Fig. P.4.2. Ans.: 0.3056A g, 0.442A g. 3. Show that the design equations for the two-element L-matching network shown in Fig. P.4.3 are given by
~________________P_RO_B_L_EM~S====~__________~I~ QI = Q2 = ~(~~ - 1) QI = I~:I Q2 =
I~~I·
4. Derive the design equations for network B in Fig. 4.7. 5. Most test equipment has a system impedance of 50 Q. When a device under test is of a system impedance other than 50 Q, an impedance
1 14r--d Fig. P.4.1
Note: The line and stub are lossless.
Y L =O.24+jO.63
Fig. P.4.2 Note: The lines and stubs are of the same characteristic impedance and are lossless.
~ I~
_______________I_M_P_E_D_A_N_C_E_MA__T_C_H_I_N_G______________~ r-----------, L-Matching
I I
I
R]
I
jX]
L __________ -1
R2
R2>R]
Fig. P.4.3
converter may be required. Design aT-section 50 Q to 75 Q impedance converter for use in testing VHF TV tuners for channels 7-13 (175.25 to 211.25 MHz). Use the Smith-chart method or otherwise. 6. The output impedance of a Motorola power transistor MRF340 at 150 MHz operated in common-emitter mode with Vee = 27 V and output power 8.0W is (38.3 - j17.0) Q. Design a lumped circuit to match the power transistor output to a load of 50 Q, with a bandwidth of 30 MHz, using a Smith chart or otherwise. 7. The input reflection coefficient of the Motorola transistor MRF901 under certain bias conditions and optimal loading (for maximum gain) at 1 GHz is 0.5LI70° (system impedance = 50 Q). Devise a single-stub matching network to match the transistor input to a 50 Q source with an operating Q approximately equal to 2. 8. Show how the stub-matching input network would be implemented with microstrips using 3M LX-0625-50 PTFE PCB (substrate thickness 1132 inch = 0.625 inch, Er = 2.50).
FURTHER READING Becciolini, B., Impedance Matching Networks Applied to RF Power transistors, Motorola Application Note AN-721. Bowick, C. (1982) RF Circuit Design, Howard W. Sams. Davies, F., Matching Network Designs with Computer Solutions, Motorola Application Note AN-267. RCA Technical Series RFM-430 RF Power Transistor Manual, RCA Corp., 1971. Roddy, D. (1986) Microwave Technology, Prentice-Hall.
Transistors at High Frequencies
5.1 INTRODUCTION The gallium arsenide field-effect transistor (GaAsFET) and the bipolarjunction transistor (BJT) are the two most commonly used devices in the design of amplifiers, oscillators and mixers at high frequencies. BJTs used in UHF and microwaves are usually of planar npn silicon type. The advantages of silicon planar BJTs over other types of transistors at high frequencies are that they represent mature technology both in the understanding of the physics and the device design, low cost and proven reliability. Compared with its microwave BJT counterpart, the GaAsFET has high gain and lower noise figure and can operate at a higher maximum frequency. The difference in frequency-handling capacity between BJTs and GaAsFET is due to the slower minority carriers in the base region of the BJT whereas conduction in a GaAsFET depends mainly on majority carriers. GaAsFETs can be used from below 1 GHz to beyond 18 GHz whereas the BJTs can operate up to about 10 GHz. Research on high electron mobility transistors (HEMT) and GaAs heterojunction bipolar transistors (HBT) in recent years has promised the potential application of 'transistors' at millimetre-wave frequencies. In this chapter, we shall focus our attention on the BJT, which will be abbreviated from now on as transistor, with the objective that readers will know what to look for from a typical transistor data sheet. As for other types of 'transistors', their a.c. parameters, i.e. the s-parameters and the noise parameters, can be interpreted and ultilized in the circuit design in the same way.
5.2 TRANSISTOR EQUIVALENT CIRCUIT Figure 5.1 shows a hybrid-II model of a transistor in common-emitter mode at high frequencies. The various quantities used in Fig. 5.1 are as follow. 1. L B , L E , Lc represent base, emitter and collector series lead inductances
5
u
III U
....
r.J..l
'Su
....
'u ....t: Q)
c;; >'S
0'"
Q)
....
....0
'Vi '"t:
....'"
E-
....
r.J..l
II'i I:lil ~
T_RA_N_S_I_ST_O_R__ EQ_U_I_V_A_L_EN_T__C_IR_C_U_IT____________~I
L -_ _ _ _ _ _ _ _ _ _ _ _
2.
3. 4. 5. 6. 7. 8.
counted from the semiconductor to the external connection points. They are each of the order of 10 nH. 'bb' is the base spreading resistance. This is the junction resistance between the base contact (or terminal) and the base region of the semiconductor material. Smaller transistors such as those used at higher frequencies tend to have larger values for 'bb" 'bb' is usually of the order of tens of ohms. 'b'e is the input resistance. This is the junction resistance of the forwardbiased B-E junction of the order of 1 kQ. 'ee is the output resistance. This is the resistance as seen from port CE into the transistor. As this resistor consists of that of a reverse-biased CB junction, its value is fairly high, of the order of 100kQ. rb'e is the feedback resistance. This is a very large resistance of the order of 5MQ. Ce is the feedback capacitance. This is the junction capacitance across the reverse-biased CB junction. This is an important parameter at high frequencies, of the order of one picofarad. Ce is the emitter capacitance. This is the sum of the emitter-diffusion capacitance and the BE junction capacitance, where the former dominates. Ce is usually of the order of 100pF. {3 is the small-signal current gain. This is the ratio between Ie and IB of the order of few tens to about 200.
In order to simplify the complicated model shown in Fig. 5.1 so that useful circuit parameters of the transistor such as the input/output impedances and the feedback characteristic can be deduced, we may simply ignore the large feedback resistance 'b'e and transpose Ce from its B-C series connection to a position in parallel with Ceo The parallel combination of Ce and the transposed value of Ce is denoted by CT' The simplified model is shown in Fig. 5.2.
B
Eo---------------------------~------------------oE
Fig. 5.2 Simplified transistor model.
~
~
TRA __N_S_IS_T_O_R_S_A_T_H_IG_H__ FR_E_Q_U_E_N_C_IE_S__________~
L I_ _ _ _ _ _ _ _ _ _ _ _
5.3 INPUT IMPEDANCE
The transposed value of Cc can be obtained by the use of the Miller effect as Ce, = Ce (1 - j3Rd
where RL is the load resistance connected across CEo Hence CT = Ce + Ce (1 - j3Rd·
(5.1)
Since rce is much larger than the series impedance of the collector-emitter loop as shown in Fig. 5.2 and rce is in parallel with LE as far as the input impedance looking into port BE is concerned, therefore, the input equivalent circuit of the transistor can be approximated by one as shown in Fig. 5.3. B
r Eo-------------------------------~
Fig. 5.3 Input equivalent circuit of a transistor.
The input impedance
Zin
can easily be calculated from Fig. 5.3 as
Zin = jWLB + jwL E (l + j3) + rbb' + 1
:b'e
+ Jwrb'e
C. T
(5.2)
At d.c. and low frequencies, Zin = rbb' + rb'e, dominated by rb'e' As frequency increases, Zin becomes complex and negatively imaginary (capacitive) due to the normally small value of (LE + LB)' As frequency increases further, Zin eventually becomes complex and inductive with a real part roughly equal to rw.
5.4 OUTPUT IMPEDANCE The output impedance is a decreasing function of frequency and it can be deduced from the equivalent circuit shown in Fig. 5.1. It can be shown from simplifying assumptions that Zout at high frequencies approximates lI(w T Cc ), where WT is the unity operating gain frequency.
~_________________F_E_ED_B_A_C_K________________~I~ At low frequencies near d.c., the output impedance is ree' As frequency increases, Ce and Ce will soon determine the output impedance, together with rb'e at lower frequencies but independent of rb'e at higher frequencies. The Ce and Ce combination is the main contributor to the decreasing function (with frequency) nature of the output impedance. Another contribution is the feedback current through Ce which tends to increase I B , and hence in turn increase Ie, thus decreasing the output impedance. 5.5 GAIN
The gain of a transistor is usually expressed in terms of the power gain rather than of the voltage or current gain. Under proper operating conditions, the input and output impedance level of a transistor may be very different, hence voltage and current gain are not meaningful, and only power gain can truly represent the gain characteristic of a transistor. The gain of a typical transistor is shown in Fig. 5.4. The power gain trends to decrease at a rate of 6 dB/octave at high frequencies. This can be understood by ignoring the lead inductances of the simplified model shown in Fig. 5.1. Without the lead inductances, the transistor (at the input) functions as an RC low-pass filter, and this accounts for the 6 dB/octave roll-off. 5.6 FEEDBACK
From Fig. 5.1 it is seen that the main feedback elements are rb'e and Ce . Since rb'e is usually very large and can generally be ignored, feedback is mainly due to Ce • At low frequencies, instability is minimal because the high reactance of Ce reduces feedback to a minimum. However, at high frequencies, the impedance due to Ce decreases and Ce provides both a Power gain (dB)
o Fig. 5.4 Power gain of a transistor.
Frequency
~
T_RA __N_SI_S_TO __ RS__ A_T_H_IG_H__F_RE_Q_U_E_N_C_IE_S____________~
L I_ _ _ _ _ _ _ _ _ _ _ _
low-impedance path and a phase shift to the input circuit. This phase shift, together with the phase shift due to other stray reactances, may add up to the 180 required to render unstable a CE operated transistor. 0
5.7 SMALL-SIGNAL TWO-PORT PARAMETERS
When a high-frequency transistor is intended for small-signal (i.e. linear, class-A) operations such as small-signal amplification (as opposed to power amplification or oscillation) the manufacturer usually expresses its a.c. (high-frequency) behaviour such as input and output impedances, level of feedback and gain, in terms of a set of linear two-port parameters, usually in the form of s- or y-parameters. The definition of s-parameters and some of their manipulations have been discussed in Chapter 3. The use of s-parameters in circuit design is the main subject matter in the following chapters. For transistors intended for slightly lower frequencies, from VHF upwards, y-parameter specifications are normally supplied by manufacturers. The y-parameters are defined by II = YiV, 12 = YfVI
+ YrV2 + Yo V2,
(5.3)
where the subscripts 1 and 2 denote the input and output ports, respectively, and the subscripts i, r, f, and 0 specify whether the admittance is one of the input driving point, reverse transfer from port 2 to port 1, forward transfer from port 1 to port 2 or output driving point, respectively. Notice that the currents and voltages used in (5.3) are total terminal quantities, which means that they are the sum of the incident and reflected currents and voltages, respectively. Although we will not be using y-parameters for our design in later chapters, a conversion between the y-parameters and s-parameters will be most useful. s-parameter design methods can be applied to transistors whose y-parameters are given in the databook, through conversion formulas. It should be noted that the definition of s-parameters depends on the value of a system impedance Zo, whereas y-parameters are defined without any reference to Zoo The conversion equations are listed below. S = -(Yol
or
Sll =
+ Y)-I(y - Yo/)
(1 - Yi)(l + Yo) + M (1 + Yi )(1 + Yo) - YrYf
-2Yr
512
= (1 + Yi )(1 + Yo) - YrYf
521
-~2Y~f~_____ = _______ (1 + Yi )(1 + Yo) - YrYf
TRA __N_S_IS_T_O_R_D_A_T_A__ SH_E_E_T_S______________~I
L -_ _ _ _ _ _ _ _ _ _ _ _ _ _ _
_ (1 + Yi)(1 - Yo) + M (1 + Yi)(1 + Yo) - YrYr'
(5.4)
S22 -
where
y = yZo and I
is a 2 x 2 identity matrix, and
+ S)-l Sll) + SI2S21 x 1-
Y = Yo(J - S)(J
or
y. = (1 + I
Y r -
Yr =
Yo =
(1
S22)(1 -
+ sll)(1 + S22)
-
S12S21
-2s 12
(1
+ sll)(1 + S22)
ZO
1
-
-2s 21
SI2S21
xZO
1
x-
+ sll)(1 + szz) - SI2S21 ZO + sll)(l - szz) + S12S21 1 x(1 + sll)(l + S22) - S12S21 Zo· (1 (1
(5.5)
The dynamic (i.e. a.c. high-frequency) characteristics of power transistors are normally specified by manufacturers in the form of the input/output impedances as functions of frequency under specific bias conditions and mode of operation. 5.8 UNDERSTANDING HIGH-FREQUENCY TRANSISTOR DATA SHEETS Different manufacturers present data of their h.f. transistors (BJTs, FETs, etc.) in different formats. Many parameters are included by some manufacturers solely for historical reasons. In order to understand the data sheet of a high-frequency transistor, the data sheet of the Hewlett Packard lownoise transistor, HXTR 6102 (also known as 2N6742), is shown in Fig. 5.5. Let us go through the data sheet item by item. (a) Features
Under this heading are two important 'noise' terms, namely the noise figure and the associated gain. The noise figure F of a transistor is defined as the amount of noise added by the transistor, i.e. F = (S/N)in (S/N)out·
If the transistor is modelled as a two-port network, the noise figure could
be expressed as F = Fmin
Irs - r opt l2 Zo I1 + r opt 12 (1 - Irs 12)'
+ 4 Rn
(5.6)
where rs is the reflection coefficient of the source as seen by the input of the transistor, F min is the minimum (optimal) noise figure when rs is set at an optimal level, r opt . Zo is the system impedance used to define the rs
~
F/i#l HEWLETT
LOW NOISE TRANSISTOR
a:~ PACKARD
2N6742 (HXTR·6102)
Features LOW NOISE FIGURE 2.5 dB Typical FMIN AT 4 GHz HIGH ASSOCIATED GAIN 9.0 dB Typical G a HERMETIC PACKAGE
Description The 2N6742 (HXTR-6102) is an NPN bipolar transistor designed for minimum noise figure. The device utilizes ion implantation techniques in its manufacture and the chip is also provided with scratch protection over its active area. The device is supplied in the HPAC-70GT, a rugged metal/ceramic hermetic package, and is capable of meeting the environmental requirements of MIL -S-19500 and the test requirements of MIL-STD-750/883.
1.00 (0,039)
MAX
j i=di5L=:::J
0.838 (0033)
Absolute Maximum Ratings* (TCASE ~ 25°C) , Symbol VCBO VeEo VEBO Ie PT TJ TSTG
-
Parameter Collector to Base Voltage Collector to Emitter Voltage Emitter to Base Voltage DC Collector Current Total Device Dissipation Junction Temperature Storage Temperature Lead Temperature (Soldering 10 seconds each lead)
Limit 35V 20V 1.5V 20mA 300 mW 300°C -65°C to 200°C +250°C
0.102 (0.004)
0.533 (0.021)
TVP
DIMENSIONS IN MILLIMETERS AND INCHES)
Outline HPAC-70GT
* Operation in excess of anyone of these conditions may result in permanent damage to this device.
Notes: 1. A(-)JC maximum of 245°CfW should be used for derating and junction temperature calculations (TJ = PD x (-)JC + T CASE). 2. A MTTF of 1.0 x 10 7 hours will be met or exceeded when the junction temperature is maintained under TJ = 200°C (based on an activation energy of 1.1 eV). For operation above this condition, refer to page 10S. "Reliability Performance of Bipolar Transistors"
Fig. S.S(a) Hewlett Packard low-noise transistor. Copyright of Hewlett Packard. Used by permission,
Electrical Specifications at T CASE = 25°C Test MIL-STD-750 3001.1-
Units
Min.
V
30
Collector-Emitter Leakage Current at VCE=10V
3041.1
nA
Collector Cutoff Current at VCB= 1OV
3036.1
nA
hFE
Forward Current Transfer Ratio at VCE= 1OV, Ic=4mA
3076.1-
-
FMIN
Minimum Noise Figure f= 4GHz 2GHz
Symbol
Parameters And Test Conditions
BVCES
Collector-Emitter Breakdown Voltage at Ic=100,.A
ICEO ICBO
Max.
Typ.
500 100 50
dB
150
250
2.8 1.6
3.0
3246.1 Associated Gain f= 4GHz 2GHz
G.
dB dB
8.0
9.0 13.5
.
Bias Conditions for Above: VCE = 10V, Ic = 4mA 300 !AS Wide pulse measurement at :52 Yo duty cycle.
"""'" -,..... I
.....
I 1"-..
.....
r-...
ASSOCIATED
-GAIN (Gal
... r-...
I I
I'
-
1.5
2.0
3.0
12
""
9
~
'"
-~ 8 ~ ~ ~
~z
I I I
"'
6
4.0 5.06.0
:;
u.
11J. ASSOCIATED
GAlN- r (Gal
I I
I I
3
z
ioo-NOISE FIGURE (FMIN) 1.0
~
z
MAG
~
IOI~;:'~URi-
0
012345678
COLLECTOR CURRENT (mAl
FREQUENCY (GHz)
Figure 1. Typical MAG. FMIN and Associated Gain vs. Frequency at VCE = 10V, Ic = 4mA.
COLLECTOR CURRENT (mA)
Figure 2. Typical FMIN and Associated Gain vs. Ic at 4 GHz forVCE = 10V (Tuned for FMIN).
Figure 3. TypicallS2'El 2 vs. Bias at4GHz.
Typical S-Parameters VCE = 10V, Ic = 4mA S22
S'2
S2'
S" Freq. (MHz)
Mag.
Ang.
Mag.
Ang.
Mag.
Ang.
Mag.
Ang.
100 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 5500 6000 7000
0.917 0.782 0.635 0.598 0.589 0.570 0.575 0.560 0.548 0.530 0.518 0.500 0.489 0.491
-11 -54 -98 -127 -149 -163 -173 180 173 167 160 152 146 132
7.149 6.277 5.037 3.881 3.148 2.646 2.209 1.948 1.665 1.450 1.346 1.210 1.076 0.897
168 135 113 87 71 59 48 37 29 20 11 1 -7 -23
0.007 0.026 0.037 0.039 0.042 0.042 0.043 0.046 0.049 0.053 0.058 0.060 0.063 0.069
79 54 33 28 26 25 25 25 24 24 23 22 20 15
0.991 0.901 0.787 0.763 0.754 0.760 0.773 0.795 0.816 0.850 0.860 0.880 0.877 0.872
-4 -18 -30 -35 -43 -50 -58 -64 -71 -76 -84 -92 -99 -108
Fig_ 5.5(b) Hewlett Packard low-noise transistor.
Typical Noise Parameters VeE = 10V, Ie = 4mA
r.
Freq, (MHz)
(MagJAng,) .480/23' .450/61' .410/88' .425/121' .475/166' .530/-164' .520/-131'
1000 1500 2000 3000 4000 5000 6000
RN (Ohms) 23.31 15.57 15.73 10.72 3.50 2.81 7.23
FMIN (dB) 1.45 1.58 1.72 2.18 2.75 3.67 4.78
Low Power Bias Performance Bias VeE V
Ic rnA
FMIN dB
G. dB
RN Ohms
3 3 3
0.25 0.50 1.00
2.25 1.87 1.55
8.5 12.7 15.7
60.5 25.5 13.9
I'o
fL
.805/31' .713/38' .571/39'
.788/25' .779129' .774129'
Mag.lAng. Mag.lAng.
Figure 4. Noise Parameters at 1 GHz.
Frequency 1000 MHz
BIAS VeE V
Ie rnA
FMIN dB
2000 MHz
1500 MHz
G. dB
0.25 2.25 8.5 3 1.87 0.50 12.7 3 1.0 1.55 15.7 3 F,gure 5. NOIse Performance VS. Frequency and
FMIN dB
G. dB
FMIN
2.67 2.06 1.73 BIas.
5.0 9.9 11.7
3000 MHz
dB
G. dB
FMIN dB
G. dB
2.83 2.23 1.79
4.7 7.9 10.2
3.88 2.93 2.38
4.1 6.4 8.1
Typical S-Parameters VeE = 3V, Ie = 0.25 rnA Freq. (MHz)
Mag.
500 1000 1500 2000 3000
.988 .956 .929 .910 .888
VeE
5 21
5 22
5 '2
Ang.
(dB)
Mag.
Ang.
(dB)
Mag.
Ang.
Mag.
Ang.
K
-22 -42 -65 -81 -112
-6.9 -7.2 -7.5 -7.7 -8.1
.451 .438 .423 .412 .394
152 127 106 89 56
-28.2 -23.1 -20.6 -19.7 -19.3
.039 .070 .093 .104 .108
72 55 38 27 6
.993 .975 .956 .945 .938
-12 -22 -33 -42 -59
.220 .464 .586 .679 .821
Ang.
(dB)
Mag.
Ang.
(dB)
Mag.
Ang.
Mag.
Ang.
K
-24 -47 -72 -89 -121
-0.8 -1.3 -2.0 -2.5 -3.3
.991 .863 .792 .747 .688
152 128 107 91 60
-28.4 -23.6 -21.4 -20.6 -20.1
.038 .066 .085 .093 .099
70 52 35 24 7
.986 .955 .920 .906 .889
-13 -24 -34 -43 -60
.220 .423 .583 .682 .816
Ang.
(dB)
Mag.
Ang.
(dB)
Mag.
Ang.
Mag.
Ang.
K
-25 -54 -82 -102 -133
4.4 3.7 2.7 1.9 .77
1.67 1.54 1.36 1.25 1.09
149 125 104 88 59
-28.6 -24.3 -23.1 -22.6 -22.1
.037 .061 .070 .074 .079
66 47 31 23 10
.972 .919 .873 .854 .842
-14 -25 -36 -43 -59
.328 .492 .664 .793 .908
= 3 V, Ie = 0.50 rnA
Freq. (MHz)
Mag.
500 1000 1500 2000 3000
.976 .929 .887 .856 .818
VeE
5"
=
3 V. Ie
=
5 21
5' 2
522
1.0 rnA
Freq. (MHz)
Mag.
500 1000 1500 2000 3000
.952 .884 .821 .775 .738
Fig. S.S(c)
5"
5"
5 21
Hewlett Packard low-noise transistor.
5' 2
5 22
T_RA __ N_SI_ST_O_R_D_A_T_A__ SH_E_E_T_S______________~I
L -_ _ _ _ _ _ _ _ _ _ _ _ _ _
and Rn is an important parameter known as the equivalent noise resistance of the transistor. Noise measurements in general will be discussed in a later chapter. In order to measure these noise parameters, the output of the transistor under test is connected to a noise-figure meter through a stub tuner and the input of the transistor is connected to a noise generator through another stub tuner. The stubs are tuned until minimum noise is read and the noise figure is Fmin . The input stub tuner is then taken out, with the generator end loaded by a 50-ohm termination, and the Sl1 of the other end is measured by a vector network analyser. The Sl1 value measured is ropt. The input stub is then connected back to the original test arrangement and is detuned. The output stub is again tuned until the noise figure read is a minimum. This new noise figure, F, corresponds to the noise figure of the transistor when its input is connected to a termination of rs. rs can be measured by repeating the procedures for measuring ropt. The Sl1 measured is rs. With Fmin , F, ropt and rs measured, the equivalent noise resistance Rn can then be found (Zo = SOQ). In any noise measurement, the measuring equipment itself also contributes to the overall noise figure (Ftotal) measured. Assuming the noise figure of the measuring equipment to be F2 , then the noise figure F of a transistor is given by Ftotal = F
F2 - 1 +-a'
where G is the transducer gain of the transistor with its output port conjugately matched. When F = Fmin (i.e. rs matched at r opt), G = Go is known as the associated gain. (b) Description The manufacturer normally includes here a brief description of the fabrication process or other such information as will help the designer to choose a transistor to suit his purpose. (c) Absolute maximum ratings These are the maximum values of the static parameters, which are not to be exceeded. The term MTTF appearing in Note 2, denotes the 'mean time to failure', and is defined as the reciprocal of the rate of failure. (d) Electrical specifications at T CASE
= 25°C
TCASE refers to the case or ambient temperature. The first three items are static parameters which are not normally used in the design. hFE is the d.c. current gain used in the bias calculations. Note that the a.c. current gain h fe
~
01
TRANSISTORS AT HIGH FREQUENCIES
~--------------------------------------------~
is not given. This is because of the fact that h fe is normally given at 1 kHz and is thus not meaningful in h.f. operations. In the three graphs under this heading, there are two more gain terms which need to be defined. 1. MAG - maximum available gain - is the theoretical maximum of the transducer power gain, which occurs when the transistor is conjugately matched at both its input and output ports. It is a function of frequency and is only defined when the transistor is unconditionally stable. 2. 1S21E 12 is a transmission transducer power gain of the transistor in common-emitter mode with a 50 Q source and load. This is the power gain offered by the transistor even without matching. (e) Typical s-parameters There are four tables of typical values of s-parameters as a function of frequency under different bias conditions. In the last three tables, the stability factor k is given. k is Rollet's stability factor. A transistor is unconditionally stable when k > 1 and potentially unstable otherwise. k will be discussed further in Chapter 6. (f) Typical noise parameters/Low-power bias performance
In the table for typical noise parameters, ro is the optimal source reflection (ropt ) coefficient needed to achieve F = Fmin . The r L given is the output reflection coefficient as seen by the transistor when its input is loaded by
rs = ro (ropt ).
Instead of providing the s-parameters, some manufacturers supply with a transistor its y-parameters. A typical y-parameter representation of transistors is shown in Fig. 5.6. Figure 5.7 shows an alternative (but less precise and less convenient) way of presenting the noise figure of a transistor. Figure 3 of Fig. 5.7 shows the noise figure F (or NF) as a function of frequency measured at V CE = 6.0Vct .c . and Ic = LOrnA d.c. The source resistance (as seen by the transistor input) employed in these measurements is near to the optimal value. Very rarely would a designer be using a transistor under the conditions stated in obtaining Figure 3 (in Fig. 5.7), hence Figure 4 and Figure 5 (in Fig. 5.7 for two different frequencies) are more useful. To relate the quantities in these two diagrams with those of (5.6), we note that, from Figure 4 (Fig. 5.7): at Ic = 1.5mA d.c., Fmin = 3dB (or 2) when Rs = 220Q rs
Rs -
50
= Rs + 50 = 0.6296 = r opt (for Ie = 1.5 rnA d.c.);
TRA __N_S_IS_T_O_R_D_A_T_A__ SH_E_E_T_S______________~I
L -_ _ _ _ _ _ _ _ _ _ _ _ _ _ _
~
FIGURE 6 - CURRENT-GAIN-BANDWIDTH PRODUCT 2.0 B
6 4
2
0 O.
_V
l---
B ___
- --
~
-r----..
--
~
O. 6
i3
o.4
1.0
2.0
4.0
3.0
5.0
6.0
1.0
8.0
9.0 10
30
20
Ie. COLLECTOR CURRENT (mAde)
FIGURE 7 - INPUT ADMITTANCE versus FREQUENCY 0 8
10
r- V,~E : ~:~ ~!tdC
-g
6 4 u
z
«
::
~ «
2
~ ~
4.0
~
2.0
o
100
1E
0 5
u
0
::
5
:l
--
.....---
__
z
~
~
~
Oi.,/'
/
f/"
/
...-;1--'
E
V
z
~ ~
«
~ ~
~
,;'
.....---
150
200
~
300
400
500
600
800
1000
6.0~"
B.O
.1
7.0
+jb
6.0 4.0 3.0 2.0 1.0
o
--
100
01,
0
0
..............
V
'r-
300
200
400
500 600
SOD
..............
.........
.....- r--..
6.0 V" 4.5 f-VCE' Ie = 1.5 mAde 4.0
-
3.5 3.0 2.5 2.0
-jb'e
15
0.5 200
300
400
500 600
800
1000
o
100
150
--
-
200
-jbrt
........
./'
",
V
/'
-gre
300
400
500
600
BOO 1000
f. FREQUENCY (MHz)
f, FREQUENCY (MHz)
Fig. 5.6 Typical y-parameter representation of transistors. Copyright of Motorola, Inc. Used by permission_
and atlc
1000
FIGURE 10 - REVERSE TRANSFER ADMITTANCE ....... FREQUENCY
1.0
150
V .......
.--
f, FREOUENCY (MHz)
5. 0 0 100
150
,/
5.0
VeE = 6.0 'Ide Ie = 1.5 mAde
0
5
o7 /
",
5.0
f, FREQUENCY (MHz)
5
~0
V
L
~
9. o f-YCE' Ie = 1.5 mAde
FIGURE 9 - FORWARD TRANSFER ADMITTANCE verllls FREQUENCY
E
~«
+jb ie".,.
0 B. 0 0 6.
FIGURE 8 - OUTPUT ADMITTANCE ......s FREQUENCY
= 1.5mA d.c., F= 4dB (or 2.51) when Rs = 60Q (or 580Q)
rs =
60 - 50 60 + 50 = 0.0909.
Using these two sets of data in (5.6)
~IL _ _ _ _ _ _ _ _ _ _ _ _T_RA_N_S_IS_T_O_R_S_A_T_H_IG_H__F_RE_Q_U_E_N_C_IE_S__________~ FIGURE 2 - 500 MHz OSCILLATOR CIRCUIT
FIGURE 1 - 200 MHz AMPLIFIER POWER GAIN ANO NOISE FIGURE CIRCUIT TYPE lN31H
l'('~UT SEE NOTE 1
2200
OHMS Note 1- Couill·LintoutpulntlWOrkconlilliogol
V"
2 G__ R,dio Typt814·TEE or lquiVllen'
Vee
1 GtnerJlRldioTyprB14·020AdjllsllbirSIubOf.quiwalint
1 G,neral Rldio Type 81e·LA Adjumbl.line or equivalent 1 G,nml R.dio Typ. 874·WNJ Short~irtuillermln'lion or tquivale .. : Holt 2 - RFC = O.2J1H Ohmll' #2-460 or .qui~altftl
L 1 1·3/4 Turns, ~18 AWG, OS'l, O.S" Ol,mlter
II 2 Turns, .'6 AWG, 0.5" L, 0.5" Diameter LJ 2 Tllms. .13 AWG, 0.25" L, 0.5" Dllmellr (POSllion 1/4" from 121
Notl3 - Lnd Number 4 (ca.) flOiling
II - 2turmMl6AWGwirl, 318 Inch 00. 1114 inch long Q= 2NS119
FIGURE 4 - NOISE FIGURE _sus SOURCE RESISTANCE and COLLECTOR CURRENT
FIGURE 3 - NOISE FIGURE
¥enUsFREQUENCY
10 9. 0
f-f--
B.O 7.0
RS" Optimum
(% 250 Ohms@ 105 and 200 MHZ)
6.0
w
"" 100 Ohms@450MHz
u
z
'.0 2.0
*"
V
I---
3.0
600 500 400
-t-I-
200
r-- f-,.3.0dB
u
~
0
~
It
70
100
200
300
"
.........
VeE IE 6.0Vdc f
=
lOS MHz
r--.. 1-...."
'"
.-/
1.5~B
SO
f;::
II
40 SO
r-
-f-,.
r-. .......
100 ~ 90 80 ~ f-'.OdB 10 60 ~5dB
1.0
o
-
r--
300 Ir- f:::t:I-+.. ~ ~~91~
~
-
5. 0 ~.
I
VeE = S.OVdc Ie = 1.0 mAde
30 0.5
SOD
II
2.0
1.0
0.1
1. FREQUENCY (MHz)
3.0
5.0
7.0
Ie. COLLECTOR CURRENT (mAde)
FIGURE 5 - NOISE FIGURE ...... SOURCE RESISTANCE and COLLECTOR CURRENT 600 500
I . w u
z i;;
*" w u
~
51
It
-~
300 200
100
VeE'" 6.0Vdc f = 200 MHz
r-.
_..!;.2dB
..........
............
~"1·'~B
---
10 t-
r-.
./
'.0 dB
1"--1-.
SO
I--~ 30 40
0.5
0.1
I-2.0
1.0
3.0
5.0
7.0
10
Ie. COLLECTOR CURRENT (mAde)
Fig. 5.7 Noise figure of a transistor. Copyright of Motorola, Inc. Used by permission. 2.51
4Rn
2
+ To
11
10.0909 - 0.62961 2 - 0.0909 2 )
+ 0.62961 2 (1
or Rn = 57.85 Q (for Ic = 1.5 rnA d.c.) We can use this value of Rn to calculate the noise figure at Ic = 1.5 rnA d.c., V CE = 6.0 Vd.c. for other values of Rs and compare the calculated result with the values in Figure 4 (of Fig. 5.7).
10
~_________________B_~ __ IN_G__________________~I~ This latter method of noise-figure presentation is not as accurate as the former method because the optimum source impedance is not in general purely resistive. In actual circuit design, one should measure rs and Rn at the bias condition and frequency at which one intends the transistor to operate. The NF (Noise Figure) data given in the data sheet should only be used as a guide to transistor selection.
5.9 BIASING OF HIGH-FREQUENCY TRANSISTORS Good d.c. biasing for high-frequency circuits is essential because unstable d.c. bias will result in a set of actual s-parameters which is quite different from the set of values used for the design, and this affects the gain, noise figure and bandwidth, etc. Worse still, it may even render the transistor unstable. While purely resistive bias networks can be used over moderate temperature changes, active bias networks are usually employed for better temperature compensation. The function of the bias network, active or passive, is to maintain the values of VeE and Ie for the h.f. transistor (or Vos and los for h.f. FETs). The selection ofthe d.c. quiescent point for an h.f. transistor depends very much on the application. Figure 5.8 shows the transfer (d.c.) characteristic of a transistor (similar for a FET with VeE changed to Vos, Ie changed los and Is changed to Vos). In Fig. 5.8, point A corresponds to low noise and low power, point B corresponds to low noise and higher gain, point C corresponds to higherpower class A, whereas point D corresponds to higher output power and higher efficiency class AB or B.
IdmA)
40
IB
-B
=
800 flA
-C
/,-----~~--------~-------------600flA
20
/,-------------------------------- 400 fl A / , - - - - - - - - - - - - - - - - - 200 flA
-A
o Fig. 5.8
10
Bias point selection.
-D
20
30
VeE (V)
~ ~I____________T_RA __N_S_IS_T_O_RS__ A_T_H_IG_H__F_RE_Q_D_E_N_C_I_ES_____________ A popular active bias scheme for a microwave BJT is achieved by means of a pnp transistor stabilizer circuit as shown in Fig. 5.9. The idea of such a stabilizing network is to reduce Ie when Ie would otherwise increase due to temperature change at Q2. FETs can be biased similarly. However, it should be noted that FETs may be subject to damage if Vos is applied before the application of a reverse-biased Vos, or a reverse-biased Vos is taken away before Vos is switched off. Figure 5.10 shows five basic d.c. bias networks for microwave FETs. The power-supply connection and disconnection sequence must be observed in order to avoid device damage. The voltages indicated in all the circuit diagrams are for the bias conditions of Vos = 5 V and Vos = -2 V. Rs in Fig. 5.10 (d) and (e) provides transient protection as well as the negative voltage that Vos requires.
PROBLEMS 1. The available noise power from a thermal-noise source is given by Pa
= kTB,
where k is Bolzmann's constant = 1.3805 x 10- 23 J K- 1 . T is the absolute temperature of the noise source in kelvin and B is the system noise bandwidth (rectangular noise bandwidth). Calculate P a in dBm per hertz of system bandwidth at room temperature (290 K). Find the Thevenin's equivalent circuit of a noise source described by Pa. Repeat for the Norton's equivalent. Vee RI
CI
I
fBI
C2
QI
I R2
Fig. 5.9
RFC
Active biasing for a BJT.
Microwave transistor
~____________________P_R_O_B_LE_M_S____________________~I ~ Circuit Diagram
Jifll----o
(d) Vo=7V <>-l Vs=2V =IDsRs Rs
~
-: -
_
I
_
-::V0
Type of Application
Power Supply Requirements
Low noise High gain High PowerHigh efficiency
1 +ve and J-ve supply
Same as (a)
2+ve supply
Same as (a)
2-ve supply
Low noise High gain High Powerlower efficiency Variable gain adjust Rs
Same as (d)
Fig. 5.10
J +ve supply
J-ve supply
Five basic d.c. bias networks for FETs.
Connection Sequence
Disconnection Sequence
~
TRA __N_S_IS_T_O_R_S_A_T_H_IG_H__F_RE_Q_U_E_N_C_IE_S__________~
L !_ _ _ _ _ _ _ _ _ _ _ _
2. A noisy two-port network (between 1-1' and 2-2') is connected to a noise source of mean square current ~ and admittance Y s ' The noisy two-port network can be expressed by a 'two-generators' model as a noiseless two-port network connected to two noise generators as shown in Fig. P.S.l. Show that the noise figure F of a two-port is given by F = Fm + Rn G s IYopt - Ys12 4Rn
F = Fm + Zo
or
where
11 +
IFs - Foptl 2 Foptl 2 (1 - IFsI2)'
-,--_'--"--;""';"...,--"'t:..:..!...;---;-;;:-
is the minimum value of F, Yopt is the value of Ys which in Fm , Fopt is the value of Fs which results in Fm , G s = Re(Ys ), e2 = 4kToRnB, and Rn is the equivalent noise resistance. Note. The two noise generators ez. and P. are generally correlated. However, ~ is assumed to be uncorrelated to ;2 and fi. 3. From Figure S of Fig. S.7 calculate Rn for 2NS179 at 200 MHz, VCE = 6 V and Ic = 1 rnA. Use the value of Rn to calculate the noise figure when the source resistance as seen by the transistor is SO Q, 40 Q and SOOQ. 4. Design an active (with temperature compensation) unipolar biasing circuit for a GaAsFET to operate at Vos = 5 V, V GS = - 2 V and los = lOrnA. Use the bias configuration shown in Fig. S.lO(d). You have to modify it to allow temperature compensation. Use a pnp bipolar transistor with hFE = 100 and VBEsat = -0.7V, VCEsat = 0.2 V and a Vcc supply of 12 V for your design. S. For the bias network shown in Fig. P.S.2, calculate the value of Rs and Vo required to bias the MESFET (2SK609) to 10 = lOrnA and Vos = 3 V, given that Fm
~sults
?
+
a
2
Noiseless two-port
1'
Fig. P.S.l
a'
2'
~_________________P_R_O_BL_E_M_S________________~I~ r-----~--~~----~o
o
It--........- - H
Fig. P.S.2
where V p is the pinch off voltage which equals -1 V and I Dss is the zero-gate voltage drain current which equals 40 rnA. Estimate also the value of C1 and C2 if the transistor is to operate at 12GHz. 6. The operating point for the r.f. transistor shown in Fig. P.5.3 is VeE = 10 V and Ie = 5 rnA, and the power supply available is Vee = 20 V. Given that V BE = 0.7V and hFE = 50, calculate the values of R t and R 2. Ans: 1.96kQ, 90.9kQ 7. Determine the values of the resistors required to bias the transistor shown in Fig. P.5.4 to VeE = 15 V and Ie = 15 rnA. Given that hFE = 50, Vee = 20V, V BE = 0.7V and IBB = 1 rnA. Ans: Re = 316Q, RB = 4.22kQ, RBt = 1OkQ, RB2 = 1.96kQ 8. Figure P.5.5 shows the construction of the gate biasing circuit of a FET
r-~==r-~'-~==r------O
RFC
Fig. P.S.3
Vee
~
1L-___________T_RA__N_sI_s_To_R_s_A_T__H_IG_H__FR_E_Q_U_E_N_c_IE_S__________~ Re
Fig. P.S.4
+ Fig. P.S.S
Vee
F_U_R_T_H_ER__R_EA_D_I_N_G________________~I
L -_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
operating at 1.2 GHz. Assign a suitable value for ZOI and for Z02, then calculate the line lengths 11 and 12 , and the line widths WI and W2 , if the circuit is to be realized by microstrips on a PTFE printed-circuit board of lOr = 2.6 and of substrate thickness O.79mm (1132").
FURTHER READING Bowick, C. (1982) RF Circuit Design, Howard W. Sams. Danley, L., Mounting Stripline - opposed-emitted (SOE) Transistor, Motorola Application Note AN-555. Gardner, F.M. (1963) Optimum Noise Figure of Transistor Amplifier, Proc. IRE, pp. 45-8, March. Gonzalez, G. (1984) Microwave Transistor Amplifiers Analysis and Design, Prentice-Hall. Hewlett Packard, Diode and Transistor Designer's Catalogue 1984-85. Johnsen, R.J. Thermal Rating of RF Power Transistors, Motorola Application Note AN-790. Microwave Transistor Bias Considerations (1975) Hewlett Packard Application Note 944-1, April. Motorola Application Note AN-421, Semiconductor Noise Figure Considerations. Poole, c.R. and Paul, D.K., Optimum Noise Measure Terminations for Microwave Transistor Amplifiers, IEEE trans. MTI-33 , No. 11, November 1985, pp. 1254-7. RF Transistor Design (1986) Motorola RF Device Data, Technical Information Center, Motorola. Verdelin (1978) Five Basic Bias Design for GaAs FET Amplifiers, Microwaves, February.
~
Small-signal Amplifier Design
6.1 CHARACTERIZATION OF HIGH-FREQUENCY AMPLIFIERS
High-frequency amplifier design has traditionally followed the route of an art rather than a science. Engineers would carry out approximate calculations and then make the amplifier circuit work by means of tuning, shielding, grounding and the design of a good layout. Good shielding, grounding and layout are always essential in any highfrequency circuit. Tuning is getting more difficult as frequency increases. At HF and VHF bands circuits can still be adjusted by tuning ferrite-core transformers, for example the radio-frequency transformers (RFT) and intermediate-frequency transformers (1FT). At the lower UHF frequencies circuits can be adjusted by tuning variable capacitors. However, as frequency increases beyond, say, two or three gigahertz, capacitance values required in the circuits become too small to be tuned or even too small to exist in the form of a lumped element, and circuits have to be designed with distributed elements such as those realized by microstrips. Unfortunately, most distributed circuits are very difficult, if not impossible, to tune. Therefore, a more systematic approach to designing high-frequency amplifiers (as well as other circuits) is necessary. Scattering parameters are getting more popular in design work at frequencies above a few hundred megahertz because they are easier to measure and work with at high frequencies than are other kinds of parameters. They are conceptually simple, analytically convenient, and capable of providing a surprising degree of insight into a measurement or design problem. For these reasons, manufacturers of high-frequency transistors and other solid-state devices are finding it more meaningful to specify their products in terms of s-parameters than in terms of other parameters. However, for high-frequency small-signal transistors below 1 GHz, some manufacturers present their a.c. characteristics through y-parameters, whereas for power transistors below 1 GHz, their optimum input and output impedances are normally given. This chapter describes the important considerations and design procedures of high-frequency solid-state small-signal amplifiers using scattering-
L -_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
P_O_~ __R_G_A_IN________________~I ~
Zs at~
bt~
l
fs
Fig. 6.1
~b2
AmplifierS
~a2
r
l
fin
f out
r fL
Linear active transistor circuit.
parameter design techniques. Fundamental concepts of transmission lines and network theory are prerequisites in the understanding of this chapter. It is noted that the derivation and formulae used in the techniques described in this chapter can also be expressed in terms of y-parameters.
6.2 POWER GAIN Consider a single-stage amplifier shown in Fig. 6.1 and described by its sparameters. The amplifier can be characterized in terms of s-parameters by b!
=
sUa!
b2
=
s2!a!
+ S12a2 + S22a2'
(6.1) (6.2)
Also, the reflection coefficients of the load and the source are ZL -Zo FL = - ZL+ZO
(6.3)
Zs-Zo
(6.4)
Fs=--
Zs+Zo
bs
at S21
b2
Fig. 6.2 Signal-flow graph of a linear transistor circuit.
~I
SMALL-SIGNAL AMPLIFIER DESIGN
~----------------------------------------------------~
The four equations above can be represented graphically by means of a signal-flow graph as shown in Fig. 6.2, with 1bs 12 as the power coming from the source. In order to define the power gain of the amplifier, we need to know the power delivered to a load. As described in Chapter 3, the square of the magnitudes of the incident and reflected waves has the dimensions of power. The power delivered to the load is then the difference between the incident power and the reflected power at the output port
Ib 2 12 - la212.
Pout =
(6.5)
Likewise, the power delivered into the amplifier is
lad 2 - Ib 1 2 .
=
Pin
1
(6.6)
The power available from the source is the power delivered to a conjugately matched load connected directly to the source. This implies that the reflection coefficient of the input is the complex conjugate of the source reflection
a
b
Fig. 6.3
Signal-flow graph representing a conjugately matched source.
From Fig. 6.3 it can easily be shown that
a=
bs 1-
rsFS =
bs 1-
Ir sl2
(6.7)
I
bsr-s
(6.8)
A number of power ratios based on Fig. 6.2 can be derived using the same method employed in deriving (3.11); they are listed below
P_O_W_E_R_G __ A_IN_____________________~I
L -_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
b2
_
bs
-
S21
1-
Sl1TS -
S22TL -
1-
Sl1TS -
S22TL -
1-
sliTs -
S22TL -
1-
Sl1TS -
S21 T L
a2 _
bs
-
~
_
bs
-
-
S12S21TLTs
1 - T L s22
!!..L _ bs
S12S21TLTs
Sl1(1 -
S12S21TLTS
T Ls 22 )
S22TL -
+ SllS22TSTL
(6.9)
+ Sl1S22TSTL
(6.10)
+ SlIS22TSTL
(6.11)
+ S21 S 12T L
SI2S21TLTs
+ SlIS22TSTL'
(6.12)
6.2.1 Operating power gain The operating power gain (G p ) is defined as the ratio of the power delivered to the load (Pout of (6.5)) to the power input to the network (Pin of (6.6))
G _ Pout p -
Pin
Ib212 - la21 2 Ial 12 - Ib l 12 _ Ib212/1bsl2- la212/1b sl2 - lad 2 /1b sl2 - Ib l l2/lb sl2'
_ -
(6.13)
(6.14) (6.15)
6.2.2 Transducer power gain The transducer power gain of a two-port network is defined as the ratio of the power delivered to the load to the power available from the source, i.e.
Ib212 - la212 lal 2- Ibl 2 Ib212/1bsl2- la212/1b sl2 Ia 12/1 b s 12 - Ib 12/1 bs12 . On substituting (6.7-10) into (6.16)
(6.16)
~
~ ~[_____________S_M_A_L_L_-S_IG_N_A_L_A_M __ PL_I_F_IE_R_D_E_S_IG_N____________~ GT -_ IS211 2(1 - IrLI2)(1 - Ir sl2)2 11 - SllrS - S22 r L + LlrLrS I .
(6.17)
The transducer power gain is not equal to the operating power gain unless the transistor is conjugately matched to the source, i.e. r in = r~. Transducer power gain provides a measure of the advantages of using a transistor over driving the same load directly by the source. 6.3 UNILATERAL AMPLIFIER DESIGN
A unilateral device is one whose scattering parameter S12 (reversetransmission coefficient) is insignificant. This implies that the transistor network is assured to have virtually no internal feedback. When S12 = 0, i.e. r in = SII and rout = S22, the unilateral transducer gain, G TU , defined as G T when S12 = 0, can be written as
IS2112 (1 - IrLI2)(1 - Irsl2)
1(1 -
sllrs)(1 - S22rd 12
I 12 1 - Irs 12 1 - Ir d 2 S21 11 - sllrsl2 11 - s22rLI2
Gs =
Go
=
(6.18)
1 - Irsl2 11 - sllrsl2 1 - IrLi 2 11 - S22r d 2 IS211 2
and Go = 1s211 2 is a parameter of transistors usually given in the manufacturer's data sheet. Go (S21) is fixed once the bias conditions of the active device are chosen and remains invariant throughout the design. The term Gs is related only to the input network. Similarly, the term G L is related only to the output network. Hence it is seen that rs and r L are two quantities that designers are able to control in the design of an amplifier. In fact, the design of a unilateral amplifier (or even an amplifier in general) consists almost entirely of the design of the input and output matching networks. In most amplifier design the actual load impedance and signal source impedance are both equal to the system impedance (50 Q in most cases). These impedances, in general, do not match well with the active device represented by S, or in other words, these impedances do not produce the desirable rs and r L as seen by the active device. Hence a practical amplifier can be treated as a system consisting of an input matching network, an active device characterized by S and an output matching network as shown in Fig. 6.4.
U_N_IL_A_T_ERA __L__ A_M_P_LI_F_IE_R_D_E_S_IG_N________----~I
L -_ _ _ _ _ _ _ _ _ _ _ _ _
I""...--------- Amplifier ---------1·~1 Active device S
Input matching network
Output matching network
S22
z"
rL
(a)
z"
""1_.._-------Input matching network
Amplifier
--------t~~1 Output matching network
Active device S
(b)
Fig. 6.4 Block diagram for a single-stage amplifier: (a) unilateral amplifier, (b) general amplifier.
Maximum unilateral transducer gain G TU can be achieved if maximum power transfer occurs between the activ~"' device and both the input matching and output matching networks. This can be accomplished by designing the matching networks such that
rs
rL
=
Stl
=
S12
and then
GTUm."
=
1
S21
1-
1
2
1
1
IS111 2 1 - IS2212 1 1 12 1 ISII1 2 S21 1 - IS221 2' 1-
This is illustrated schematically in Fig. 6.5.
(6.19)
~
~I~
_____________S_MA__L_L_-S_IG_N_A_L_A_M__PL_I_FI_E_R_D_E_S_IG_N____________~ 1".." ..._------- Amplifier - - - - - - -..~~I Source Active device S
Input matching network
Fig. 6.5
Output matching network
Load
Maximum transducer gain of a unilateral amplifier.
6.4 NON-UNILATERAL AMPLIFIER DESIGN
When the reverse-transmission coefficient S12 of the transistor used is not insignificantly small, a more accurate design is required. The analysis is much more complicated. From Fig. 6.6 it can be written that b, =
slIa,
b2 =
s2,a,
rL
+ S12a2 + S22a2
(6.20) (6.21)
a2
(6.22)
= bz '
Dividing both sides of (6.20) by at gives (6.23)
Source
bl~
l Fig. 6.6
r
rs r i
Linear two-port network.
~b2
Linear active device
j+-a
l
r
r" r
L
ZL
2
N_O_N_-U_N_I_LA_T_E_RA __L_AM __P_L_I_FI_E_R_D_E_S_IG_N__________~I
L -_ _ _ _ _ _ _ _ _ _ _
On substituting (6.22) into (6.21) b2 = S21 al + S22 T L b 2 b2 (1 - S22Td = S21 a l b2 =
S21
1-
al
(6.24)
S22TL·
On substituting (6.22) and (6.24) into (6.23) (6.25)
Similarly r
_
10 -
S22
Sl2S21 TS
+1
-
Sl1
rS .
(6.26)
ro
By choosing a complex-conjugately matched load, i.e. r L = and complex-conjugagely matched source, i.e. rs = rt, maximum gain can be achieved. Hence for maximum gain T* S -
T* L -
Sll
S21 TL + 1SI2 T - S22 L
S22
+1-
(6.27)
S12S 21 T S
Sl1
r· S
(6.28)
Solving the last two equations simultaneously, we obtain the optimum source termination, r Sm , and the optimum load termination, r Lm , as
r _ CnB I Sm -
± V(Bt - 4Ict/z)] 21cl12
(6.29)
(6.30)
where: BI
= 1 + ISlliz - IS22lz - ILll z
B z = 1 - ISttl2 C t = Stt - LlS12 C2 = S22 - LlS(t
Ll
= Sl1S22
-
+
IS2212 -
ILll 2
St2S21.
B I and B2 are always positive for an unconditionally stable two-port device. Mathematically, it can be shown that the negative sign of the square root has to be taken in (6.29) and (6.30).
~
~I
S_MA __L_L_-S_IG_N_A_L_A_M __ PL_I_FI_E_R_D_E_S_IG_N____________~
L _____________
6.5 STABILITY CRITERIA
A primary concern in the design of an amplifier is to ensure that the active device does not oscillate at any frequency, especially at the frequency of operation. An active device (two-port network) is unconditionally stable if both its input and output impedances have a positive real part for any passive load and source terminations. In s-parameter terminology it is equivalent to saying that Ird and Iro I as defined in (6.25) and (6.26) are both less than unity. Alternatively, for the active device to be unconditionally stable, the simultaneous solution of (6.25) and (6.26) are required to satisfy (6.31) and
(6.32)
From (6.25) and (6.26) it is seen that for the particular cases when and rs = 0 (6.25) and (6.26) reduce, respectively, to
rL = 0
r = Sll ro = S22· j
and
Hence, for an unconditionally stable device,
ISIlI
< 1 and
Isd
<
(6.33)
1.
The expressions (6.33) are only necessary but not sufficient conditions for inherent stability of an active device characterized by a set of s-parameters. The stability criteria given in (6.31) and (6.32) have to be expressed in a convenient form, a form which can readily be used (in terms of the device s-parameters) to test the inherent stability of the device. To this end we consider (6.31), i.e.
Ir Sm I = =
thus
Ict[B I ± V(Br - 4 IC I 1 2 )]1
21Cd2
IIct 1 cdI 21 Cd By V(Br
~I[ B[2+ B[2-
<
1
4ICl 12 ± 2B[
V(Bl2-
41C[1 2 ± B[ V(Br - 4 IC[1 2 ) < 0 - 41Cd 2 ) [VcB[2 - 4IC[1 2 ) ± Bd <
4IC[ 12)] <
O.
1,
(6.34)
Since the first term is positive, the second term must be negative. A necessary condition for this to be true is that the negative sign must be taken when Bl > 0, and vice versa. Another necessary condition for (6.31) to hold is that
Br > 41Cd
2•
A similar result can be derived from (6.32) and is given as
(6.35)
ST_A_B_IL_I_T_Y_C_R_IT_E_R_IA________________~I
L -_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
B~ > 41c212.
(6.36)
On substituting the expressions for B2 and C2 into (6.36) (1 - IS1112 + Iszzl 2 - ILlI 2? > 4(S22 - LlSI'I)(s12 - Ll*Sl1) ~ [(1 - ISl112 - IS2212 + ILl 12) + (21szzl 2 - 21Ll1 2)F > 41s2212 + 41Lll 2 ISl112 - 81s1112 IS2212 - 41Lll 2 - 41s1112 IS2212 + 4Is12s2t12 ~ (1- IS1112 - IS2212 + ILl 12)2 + 4(ls2212 - ILl 12)2 + 4(1- IS1112 - Iszzl 2 + ILlI 2)(lszzl 2 - ILl 12) > 41s12S2112 + 4(1 - ISlt!2)ls2212 - 4(1 - IS1l12)ILl12 ~ (1 - IS1112 - IS2212 + ILl 12)2 + 4(ls2212 - ILl 12)(1 - ISl112 - Iszzl 2 + ILll 2 + Iszzl 2 - ILl 12) > 41s12S2t!2 + 4(1 - ISII 12)(lszzl 2 - ILl 12) ~ (1 - ISll12 - Iszzl 2 + ILlI 2? + 4(lszzl 2 - ILl 12)(1 - IS1l12) > 41s12S2112 + 4(1 - ISIl12)(lszzl2 - ILlI 2) ~ (1 - ISII 12 - IS2212 + ILl 12)2 > 4Is12S2112. (6.37)
The inequality (6.37) has two possible roots 1 - IS1112 - IS2212 + ILll 2 > 21s12S211 1 - ISII 12 - Iszzl 2 + ILll 2 < -2Is12S2t!.
and
(6.38)
Define the stability factor K, known as the Rollet stability factor, as K
= 1 - ISl11 2 -
Isnl 2 21 Sl2S211
+ ILll 2
(6.39)
The solution to (6.37) is thus or
K>1
(6.40)
K < -1.
(6.41)
However, it can be shown that the negative root for K violates the basic stability criteria that Islll < 1 and Is221 < 1. Most literature states that the criteria for inherent stability, i.e. the device is unconditionally stable, are K > 1 and one of the following conditions: 1. 1 -
IS1112 > IS12s2ti
> 0, 3. B2 > 0, 4. ILlI < 1. 2. Bl
and 1 -
IS2212 > IS12s2d,
In fact, 1. can be derived from K > 1 by assuming Isill < 1 and Iszzl < 1, 2. and 3. can be derived from 1. with no further assumption and 4. is derived by adding 2. and 3. with no further assumption. Hence the necessary and sufficient conditions for inherent stability are > 1 ISIII < 1 IS221 < 1.
K
and
1
(6.42)
~
C§JI
SMALL-SIGNAL AMPLIFIER DESIGN
L -_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
~
In designing an amplifier the s-parameters of the active device are first measured and the device is tested against the stability criteria of (6.42). If (6.42) is fulfilled, the device is unconditionally stable, otherwise it is potentially unstable (also called conditionally stable). For an unconditionally stable device, Bl and B2 are both positive. The optimal load and source terminations are equal to the complex conjugate of the input and output impedances, respectively, and are given by (6.43) (6.44)
where:
+ ISl112 - Iszzl 2 - ILlI2 > 0, 1 - ISI\ 12 + Iszzl 2 - ILlI2 > 0,
BI = 1
B2 = C I = Sl1 - LlS!2, C2 = S22 - LlStl, LI = Sl1S22 - S12S21.
(6.45)
The transducer gain of an amplifier with the active device conjugately matched at both ports can be found by putting r L = r Lm and rs = rSm into (6.17).
6.6 LOAD AND SOURCE STABILITY CIRCLES When an active device is potentially unstable i.e. either one or more of the following expressions or their equivalent statements is not valid: K> 1, IS111 < 1 and Iszzl < 1.
The device can still be used to perform as an amplifier provided the regions of source and load termination that cause instability are avoided.
6.6.1 Load stability The input reflection coefficient of a loaded two-port network is given by
One of the criteria for the loaded network to be unconditionally stable is that Ird < 1. We are now going to find out what kind of load (rd will push the network to the boundary of stability represented by Ir j I = 1.
I~
LOAD AND SOURCE STABILITY CIRCLES
I
Sl1
+
S12S21rL 1 - r LSZ2
1--
1
S = 1 I 1l1- - r LJrLI LS Z2 (Sl1 - LJrd(Sl1 - LJrd* = (1 - r LS2Z )(1 - rLS ZZ )* * SZ2 - LJSj1 S2Z - LJ*Sl1 * _ IS11lz - 1 rLr L - Isnl 2 _ ILJI 2rL - IS221 2 _ ILJlzTL - Isd z _ ILJI2'
(6.46)
Consider the equation of a circle in the complex Z-plane with the centre at A and a radius of R,
IZ-AI
=R = R2 ZZ* - AZ* - A*Z + AA* = R2. (Z - A)(Z - A)*
(6.47)
Comparison of (6.46) with (6.47) shows that there is a circle in the r Lplane, points on which represent load reflection coefficients or load impedances which will push the input stability to its limit corresponding to
Ird
= l.
The centre (r2S ) and radius (Q2S) of the load stability circle are thus given by
r -
(S22 - LJSj1)* _ q 2S - 1s221 2 - 1LJI2 - 1S221 2 - 1LJ1 2'
(6.48)
The radius of the load stability circle is given by IS12S211
(6.49)
An example of the load stability circle is shown in Fig. 6.7. The centre of the Smith chart (rL plane) in Fig 6.7 corresponds to r L = 0. When r L = 0, r j = S11' If IS111 < 1, the centre of the Smith chart corresponds to a stable load. Since in this case, the Smith chart centre is outside the load stability circle, the shaded region corresponds to values of r L which, when used in terminating the two-port (at port 2), will render net power flowing out from the input port. If IS11 I > 1, the shaded region corresponds to loads granting input stability. Examples of load stability plots are shown in Fig. 6.8. In Fig. 6.8 the Smith chart corresponds to the rL-plane, and the shaded regions are regions of stable load terminations. If IsIll > 1 in Fig. 6.8 the shaded regions represent regions of unstable terminations.
6.6.2 Source stability Following exactly the same arguments as in the derivation of the load stability circle, the source stability circle can be derived by letting ro in
~ ~I_____________S_MA __L_L_-S_IG_N_A_L_A_M __ PL_I_FI_E_R_D_E_S_IG_N____________~ Load stability circle \ r.l = I
Unstable region if \SII\ < 1
r L -plane Smith chart
Fig. 6.7
Load stability network.
(6.26) equal unity. The centre (TIs) and radius (QIS) of the source stability circle are thus given by centre:
(6.50)
radius:
(6.51)
The source stability circle is a locus on the Ts-plane Smith chart. The centre of the Smith chart represent Ts = 0 or To = S22' Hence, if Is221 < 1 the centre lies in the stable region, and vice versa.
6.7 CONSTANT POWER-GAIN CIRCLES An amplifier can be designed using a stable active device (i.e. K > 1, ISIII < 1 and Isni < 1) by selecting the conjugately matched input and output terminations, i.e. Tsm and TLm . The gain of such an amplifier is the highest gain that can be derived from the active device. In this case, the operating gain of the amplifier is the same as the transducer gain of the amplifier and is given by (6.17). For designs using potentially unstable active devices, the maximum operating gain corresponds to a source and/or load termination outside the Ts and/or TL Smith chart. Hence, it is not achievable and is thus meaningless. In designing with potentially unstable devices, one has to give
CONSTANT POWER-GAIN CIRCLE_S_ _ _ _ _ _--'1
Smith chart
Smith chart
Smith chart
(c)
Stable region
Smith chart
Stable region
(d)
Smith chart (f)
Fig. 6.8 Load stability circle: (a) conditionally stable (K < 1, ISlIl < 1); (b) conditionally stable (K < 1, ISlIl < 1); (c) conditionally stable (K < 1, ISlll < 1); (d) conditionally stable (K < 1, ISlll < 1); (e) unconditionally stable (K > 1, ISlll < 1); (f) unconditionally stable (K > 1, ISlll < 1).
up looking for the maximum gain (conjugate input/output matching), but rather one should be looking for realizable source and load terminations by defining a lower (than maximum) gain. The operating power gain of an active device for any arbitrary terminations is given by (6.15), i.e.
~
~
S_MA __L_L_-S_IG_N_A_L_A_M __ PL_I_FI_E_R_D_E_S_IG_N____________~
L I_ _ _ _ _ _ _ _ _ _ _ _ _
IS2d 2 (1 - IrLI2) I1 - rLS 22 12 - ISlI - rL.1 12'
G _ P -
Note that G p is independent of rs by virtue of its own definition. Defining the normalized operating gain, g2, as
Gp
g2 = IS2ti2 1- Ihl 2 The locus of equation
Ir L-
rL 1
for a constant value of
g2q
+ g2(lszz\2 -
1.112)
12 =
(6.52) g2
is expressed by the circle
g21 s 12S2I1 2 - 2g 21s 12S2dK + 1 [1 + g2(ls2212 - 1.112)F .
(6.53)
The centre and radius of the circle are
r2C --
Centre:
1
g2q
+ g2( IS22 12 -
(6.54)
I.1 12)
_ V(g~IS12S2112 - 2g21s12S2ti K + 1) 1 + g2(l s221 2- 1.112)
Radius:
Q2C -
(6.55)
The circle is known as the constant power-gain circle. The constant powergain circle is on the Smith chart of the r L plane. It is the locus of the load terminations (rLS) which, when connected to port 2 of the active device, will provide an operating gain of IS211 2 g2. For an inherently stable device, K> 1 and ILl I < 1, conjugate matching at both ports is possible and this leads to the maximum gain, i.e. G p ~ G pmax .
Q2C ~
and
On putting
0 (the locus being a point)
r2C~
(}2C
r Lm .
= 0 into (6.55)
iz
or or
IS12S2ti 2 - 2g2 IS12S2ti K + 1 = 0 1 g2 = -I- - I [K ± V(K2 - 1)] = g2max S12S21 G pmax
=
I s 21II [K ± V(K2 - 1)] S12
= GTmax ,
(6.56) (6.57)
where the minus sign is used for inherent stability. It is noted that r Lm = r 2C when g2 = g2max and that r 2C and r 2S lie on the same radial line independent of whether the device is unconditionally stable or potentially unstable. The angle of the radial line is defined by the angle of C1. In amplifier design using a potentially unstable device, a set of constant
CONSTANTPOW_E_R_-G_A_I_N_C_I_RC __ LE_S____________~I power-gain circles is generated first. A load reflection coefficient r L, in the stable region of the r L-plane Smith chart with as high a gain as desired is selected. The source reflection coefficient rs can be selected by choosing the complex conjugate of the input reflection coefficient, i.e.
or
(6.58)
The value of rs as given by (6.58) is usable if it falls well inside the stable region of the rs-plane Smith chart. If not, a different r L has to be selected and the procedures are repeated until both r Land rs are on the stable region.
Example 6.1 The s-parameters of a GaAs MESFET measured at 6GHz are given as
s" s12
= 0.614L - 167.4° = 0.046L65°
S2' S22
= 2.187L32.4° = 0.716L - 83°.
Design a narrow-band amplifier at 6 GHz using the device for maximum gain. Calculate the transducer gain and the operating gain.
Solution First, check the stability of the device.
Is,,1 Isd
= 0.614 < 1 = 0.716 < 1
also: Ll =
S"S22 -
S'2S21
= 0.614 x 0.716L - 250.4° - 2.187 x 0.046L97.4° = 0.3420L113.16°.
The stability factor K can be computed using (6.40), i.e. K
= 1 - 0.3770 - 0.5127 + 0.1169 = 1.1292> 2 x 0.1006
l.
The device is unconditionally stable at 6 GHz. Hence, maximum gain can be achieved by simultaneously conjugate matching the source and load. The required terminations are given by
r.
_ Ci[S, - V(BT - 4IC,1 2)]
Sm -
21 C l l2
~
~
S_MA __L_L_-S_IG_N_A_L_A_M __ PL_I_FI_E_R_D_E_S_IG_N____________~
L I_ _ _ _ _ _ _ _ _ _ _ _ _
Tsm = 0.8682LI69.76°
T
_
q [B2 -
V,.,.,(B:1~-_-4""'I-:CC2"12""')]
21 C2 12
Lm -
TLm = 0.9055L84.48°.
The maximum transducer gain is equal to the maximum operating gain for a conjugately matched device. It is given by (6.1S) or (6.17) as GTmax = G pmax = 28.7 = 14.5 dB.
Exampie6.2
Given the s-parameters of a silicon bipolar transistor 2N6603 (MRF 902) at Ic = SmA, VCE = SV at 1 GHz
Sll = O.64L
S21
S12
S22
=
- 158° 0.087L28°
= 4.13L88° = 0.39L - 68°.
Design a narrow-band 1 GHz amplifier. Solution LI = 0.2496L - 266° - 0.3593L116° ILl 1 = 10.1443L - 96.33°1 = 0.1443 < 1 K
= 1 - 0.4096 - 0.1521 + 0.02082 = 0 6389 2 x 0.3593
.
1
<.
Although both 1s II 1 < 1 and 1s221 < 1, K < 1 still implies that the device is potentially unstable. The source and load stability circles can be computed as
ISllI2C~ ILlI2 = 1.7421LI61.67° IS 12S 211
TIS
=
QIS
= IIStd2 _ ILlI21 = 0.9241.
The stability circles are shown in Fig. 6.9. The next step is to select a value for TL in the stable region of the TLplane. Since the required gain is not specified, TL can be chosen arbitrarily provided it stays clear of the stability circle in the stable region. Ts can then be found from (6.64) using source conjugate match. If TL = 0.71SL4So (arbitrary) is chosen, then
LO_W __ -N_O_IS_E_AM ___ PL_I_FI_E_R_D_E_SI_G_N____________~I ~
L -_ _ _ _ _ _ _ _ _ _ _ _ _
/
Load stability circle Source stability circle
Stable region (a)
(b)
Fig.6.9 Stability circles for Example 6.2 (a) rL-plane, (b) rm-plane.
rs
=
rt;.
=
0.901L175°.
This solution is not acceptable since rs lies inside the source stability circle. If r L = 0.41L77° (arbitrary) is chosen, then
rs = rt;. = 0.8137L
- 159.57°.
This solution is in the stable region but only marginally, and hence it is not recommended. If r L = O.22L77° (arbitrary) is chosen, then
rs
=
rt;.
=
0.7256L158.97°.
This solution is slightly better than the previous one, and can be accepted for the purpose of this example. The operating gain G p and the transducer gain GT can be found by using (6.15) and (6.17), respectively. They are G p = GT = 40.91 = 16.12dB.
G p and GT are the same in this case because complex-conjugate matching at the input has been employed.
6.8 LOW-NOISE AMPLIFIER DESIGN The noise figure of a two-port active device was quoted in Chapter 5 and is repeated here (6.59)
~ ~~~ ~~_S_MA~L_L_-S_IG_N_A_L_A_M~PL_I_F_IE_R_D_E_S_IG_N~~~__~~~ LI
__
where Fm is the minimum noise figure of the two-port active device when its source termination is at an optimum value, rapt. Rn is the equivalent noise resistance and Zo is the system impedance to be taken as SO Q for all practical purposes. Equation (6.S9) can be derived using the 'twogenerator' model on a two-port network. The noise performance of an active device such as a BJT or FET is completely described by F m , R n , Irapt I and Lropt . These parameters can either be measured or taken from the data sheet of the active device such as those given for HXTR 6102 in Fig. S.S(c). The measurement of noise in general will be discussed in Chapter 11. The source termination rs can be chosen to be rapt if the amplifier is to be designed for the lowest noise figure. This is always possible if the device is unconditionally stable. The output port is normally conjugately matched for maximum power transfer to the load, i.e.
or
rL
-
rL
-
_
_
(
S22
(S22 -
1-
S12S 21 r opt )
+1-
Sl1
r.opt
LlroPt)* roptSll
*
.
(6.60)
For potentially unstable devices, it is necessary to ensure that rs = r opt lies in the stable region of the source stability circle plot. If input-port stability is not satisfied by rs = r opt ' then a value other than r opt must be chosen for r5 at the expense of having a noise figure higher than Fm. After selecting r s , whether it is equal to rapt or not, conjugate matching at the output is normally assumed. The r L so calculated must also lie within the stable region of the load stability circle plot. Example (6.3)
A low-noise amplifier at 6 GHz is to be designed using a GaAsFET whose a.c. parameters at 6GHz under the low-noise bias condition of Vos 3.5 V and los = 0.15/055 are given below Sl1
= 0.674L - 152° S21 = 1. 74L36.4°
S12
= 0.075L6.2°
S22 = 0.60L - 92.6° Fm = 2.2dB Rn= 6.64Q r opt = 0.575L138°.
Solution Stability analysis gives Is,,1 IS221
10.6741 < 1 10.61 < 1
~____________L_O_W__-N_O_IS_E_A_MP __L_IF_I_E_R_D_E_SI_G_N____________~I = 0.3865L134.22° ILl I < 1 K=1-l s llI 2 - l sd 2 + ILlI2 =1.284>1. 21 s 12S21I LI = SllS22 -
so
S12S21
Hence the device is inherently stable. Therefore we may choose rs = r opt = O.575L138° to achieve the lowest noise figure of F = Fm = 2.2dB. The output is assumed to be conjugately matched, i.e.
rL
=
( s22 -LITopt 1 - ropts ll
)* = 0.601L104
0•
The power gains are C2 =
S22 -
LIsT! = 0.3633L - 105.97°.
The operating gain G p is given by IS2t1 2 (1 - IrLI2) 1 - ISlll2 + Ihl 2 (IS2212 - ILl 12) - 2Re(r LC2) = 10.43 = 1O.18dB.
G _ p -
The transducer gain G is given by IS2l1 2(1 - IrLI2)(1 - Ir optl 2) suropt - szzrL + LlrLropt 12 = 7.97 = 9.01 dB.
G _ T -
I1 -
For the sake of comparison the gains for conjugate matching design, i.e. the case when rs = rSm and r L = r Lm , are calculated to be G p = GT = 10.45 dB. It is seen that while the operating gain of the low-noise design is only marginally lower, the transducer gain is more than 2dB below what the FET can achieve with conjugate matching. This is due to the mismatch loss in the input circuit.
6.8.1 Noise circles
ropt is in the unstable region or is too near to the source stability circle, we may design for a prescribed noise figure F = F j • To this end we define a parameter N j such that
If the design using rs = r opt is not possible because
(6.61)
Then from (6.59), we have N
[D2J
~I
SMALL-SIGNAL AMPLIFIER DESIGN
L -_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
~
(6.62)
It follows from (6.61) that
(rs - ropt)(TS - npt) = N j - N j Ir sl2 Irsl2 + Iropt l 2 - 2Re(rs r opt) = N j - N j Ir sl2 (1 + N j ) Irsl2 + Iropt l 2 - 2Re(rs r opt) = N j • On multiplying throughout by (1 + N j )
Irs where r lF
=
rop/(1
rlFI = QIF,
+ NJ and
QIF = 1 : N Jl[Nr
+ N j (1 -
Iropt I2)].
(6.63)
I
Equation (6.63) is a circle equation known as the noise circle with its centre at r lF and a radius of !?IF. The noise circle is the locus of all values of rs having the same noise figure F j • By varying the value of F j , a family of noise circles each signifying a specific noise figure can be plotted.
Example (6.4)
Design an amplifier with a noise figure F j = 3 dB at 6 GHz using the same FET under the same bias conditions as that used in Example 6.3.
Solution Fj Fm
= 3 dB = 1.9953 = 2.2 dB = 1.6596
6.64 50 = 0.1328 N = (F F m) 11 + rop, 12 = 0.3009 Tn
=
j-
4rn
I
ropt
r lF = 1 + N
= 0.4420L138° I
QIF = 0.4154.
The noise circle for F j = 3 dB is plotted in Fig. 6.10. The conjugate source termination rSm is calculated to be O.7974LI61.35°, and rSm is also entered into the plot in Fig. 6.10. It is found that it almost lies on the noise circle for F j = 3 dB. Hence we may choose rs = rSm and r L = r Lm ·
B_R_O_A_D_BA __ ND __C_O_N_S_ID_E_RA __T_IO_N_S____________~I
L -_ _ _ _ _ _ _ _ _ _ _ _ _
Fm=2.2 dB (data sheet)
Fig. 6.10
Design for Example 6.4.
6.9 BROADBAND CONSIDERATIONS The design of broadband amplifiers is more critical than that of narrowband amplifiers because both the s-parameters of the active device and the reflection coefficient looking into the matching networks vary with frequency. Hence conjugately matched conditions fail to exist over a band of frequencies. Bearing in mind that both the s-parameters of the active device and of the matching network vary with frequency, the transducer gain at three different frequencies within the band, namely the lowest frequency, the centre frequency and the highest frequency, can each be calculated using their respective s-parameters and the reflection coefficient for fixed matching networks. The transducer gains so calculated over the band of frequencies will show, in most cases, a flatness of not more than 3 dB variation. If the variation is too large to be accepted, a different matching network can be tried and the process continued until a satisfactory combination of gain flatness and transducer gain is obtained. Another method is to forget about the conjugate source matching at the centre frequency as in the narrowband amplifier design and choose arbitrarily the input and output matching networks. The transducer gain at the three frequencies mentioned above are then calculated. Since the process is merely a trial and check method, it is best done with the aid of a microcomputer. The versatility of using a microcomputer is that various choices of matching networks can be tried until the maximum transducer gain with an acceptable degree of flatness is obtained.
~
[IBJ I
SMALL-SIGNAL AMPLIFIER DESIGN
L -_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
~
6.10 SUMMARY OF DESIGN PROCEDURES In all small-signal design the initial step is to check for inherent stability of the active device using
IS111 2- IS221 2+ ILll 2> 2Is'2s21 I ISIlI < 1 and IS221 < 1.
K = 1-
1
For inherent stability any load and source reflection coefficient within the Smith chart represents a stable termination. For a potentially unstable device, the load stability is calculated and plotted on the Smith chart using (6.48) and (6.49).
Cz
(6.48)
IS2212 _ ILll 2 _ IS!2s2d Q2S -ll s221 2- ILl121·
Centre:
T 2S =
Radius:
(6.49)
Likewise the source stability circle was given in (6.50) and (6.51). Centre:
ISlll2 _ ILll 2 _ IS'2s211 -ll sul 2- ILl121·
(6.50)
TIS =
Radius:
QIS
(6.51 )
The load and source terminations selected, i.e. r L and r s , must fall within the stable region as defined. Design can normally be classified as either maximum-gain design or low-noise design. Broadband design adds additional constraints to, but is basically the same as, the maximum-gain or the low-noise design.
6.10.1 Maximum-gain design An inherently stable device can achieve a maximum gain of
Gpmax = Il s2'11 S'2
[K - V(K2 - 1)]
with conjugately matched source and load using (6.29) and (6.30), i.e.
r.
_ CnB, - V(BI - 4IC,1 2)]
Sm -
T
21C,12
_ q[B2 - V(B~ Lm -
21 C212
41C212)]
,
PROBLEMS L -____________________________________________
I [DB
~
where: Bl=1+ ISl1I2_lsd 2 -1L112 B2 = 1 - ISl112 + IS2212 - 1.,11 2 C1 = Sl1 - L1S!2 C2 = S22 - L1Sl'1
.,1
= Sl1S22 -
S12S21'
For potentially unstable devices, constant power-gain circles are plotted, with Centre: Radius:
r2C --
1+
_
V(g~
Q2C -
g2q
1 12
1.,1 12) IS12S2d2 - 2g2 IS12S2d K + 1 + g2( Isd 2 - 1.,112)
g2( S22
-
1)
r L is selected on the stable region for a chosen power gain. rs is then calculated by using
The calculated
rs must fall on the stable region of the rs-plane.
6.10.2 Low-noise amplifier A set of noise-figure circles is generated using (6.63) with Centre:
r lF
-~
-
1 + N1
Radius:
rs is selected to lie on the required noise-figure circle and r L is calculated using
rL
= (S22 -
1-
L1rs) *. rSs l1
Both r L and rs must lie in the stable region. The transducer gain is then checked using (6.17). If rL does not lie in the stable region or the transducer gain is not high enough, the process of selecting rs can be repeated until a suitable noise-gain compromise is achieved. PROBLEMS
1. The small-signal behaviour of a microwave transistor (bipolar or MESFET) is described by a set of two-port parameters. Show that for
~
S_MA __L_L_-S_IG_N_A_L_A_M __ PL_I_FI_E_R_D_E_S_IG_N____________~
L I_ _ _ _ _ _ _ _ _ _ _ _ _
maximum operating gain of the transistor, the source and load termination are given by
r.
-
Ci[BI - V(Bi - 4I C
21CI12
Sm -
r
_ CHB 2
-
Lm -
d2)]
V(B~ - 4IC2 12 )] 21C212 ,
where: BI = 1
+ ISlll2 - IS2212 - ILlI2 ISlll2 + Isnl 2 - ILlI2
B2 = 1 -
CI = Sll - LlS12 C2 = S22 - Llstl LI = SllS22 - S12S21'
2. For an unconditionally stable transistor the Rollet stability factor K > 1. Show that it is necessary that BI
> 0 and
B2
> 0,
where: BI = 1 B2
+ ISlll2 - IS2212 - ILlI2
= 1 - ISlll2 + IS2212 - ILlI2
LI =
Sl1S22 -
S12S21'
3. Show that the expressions given below are necessary consequences of the conditions K > 1, Is 11 I < 1and IS221 < 1(which are the necessary and sufficient conditions for inherent stability): 1.
1-l sS22lll 2z>>
2. 1 -
IS12S211 S12S Z1 •
4. The a.c. parameters of the bipolar junction transistor Motorola MRF571 at 1 GHz under the bias conditions of VCE = 6.0 V and Ic = 5.0mA are given by Sll = 0.61L178° S21 = 3.0L78° S12 = 0.09L37° S22 = 0.28L - 690 Fm = l.5dB = 0.48L134° Rn=7.5Q
r opt
1. Show that the device is unconditionally stable at 1 GHz under the above stated bias conditions. 2. Calculate the required source and load terminations for maximum gain design. 3. Calculate the maximum operating gain and the maximum transducer gain.
~_________________F_U_R_T_H_ER__R_EA_D_I_N_G________________~I ~ 5. Implement the design in Problem 4 by using single-stub microstrip matching circuits for both the input and output port, or other microstrip matching circuits of similar complexity. The minimum circuit Q is to be greater than 5. 6. Calculate the noise figure of the circuit in Problem 4. 7. Using the same device under the same bias conditions as in Problem 4, design a low-noise amplifier of noise figure F = 2.2 dB and implement it with the simplest possible microstrip circuit. 8. The a.c. parameters of the Motorola BJT MRF572 at 1 GHz under the bias conditions of VCE = 6.0V and Ic = 5.0mA are given by Sll
= 0.66L
S12
= 0.lOL22°
- 167°
S21
= 3.3L79°
S22
= 0.29L - 77°
Fm = 1.5dB Fopt = 0.56L116° Rn = 6.0Q
1. Plot the constant-power circles in the rL-plane for G p = 12, 13 and 14dB. 2. Plot the noise circles in the rs-plane for noise figure F = 1.5,2.0 and 2.5dB. 3. Design an amplifier for the best compromise of gain and noise figure by giving your choice of rs and r L with reasons. 9. The s-parameters of a Motorola MRF966 (N-channel dual-gate GaAs MESFET) under the bias conditions of V DS = 5.0V and IDs = lOrnA were measured to be as in Table P6.1. Table P6.1. Frequency (GHz)
SlI
S21
S12
S22
0.8 1.0 1.2
0.950L - 24° 0.928L - 30° 0.893L - 36°
1.58L153° l.57L146° l.55L139°
0.005L91° 0.006L95° 0.006L101°
0.96L - 15° 0.96L - 19° 0.95L - 22°
Write a computer program (or use software such as Touchstone, Supercompact or MDS) to design a wideband amplifier covering the range 0.8 GHz to 1.2 GHz with highest possible gain and a gain flatness of better than 3 dB. The matching circuits are to be implemented by microstrips.
FURTHER READING Bodway, G.E. (1987) Two port power flow analysis using generalized scattering parameters. Microwave Journal, May.
~ I~
____________S_MA__L_L-_S_IG_N_A_L_A_M_P_L_I_FI_E_R_D_E_SI_G_N____________~
Carson, R.S. (1982) High-Frequency Amplifiers, 2nd edn, Wiley. Froehner, W.H. (1967) Quick amplifier design with scattering parameters. Electronics, October. Gonzalez, G. (1984) Microwave Transistor Amplifiers Analysis and Design, Prentice-Hall. Ha, T.T. (1981) Solid-State Microwave Amplifier Design, Wiley. Hejhall, R., RF Small Signal Design Using Two-port Parameters, Motorola Application Note AN-2I5A. Hewlett Packard Application Note 154 (1972) S-Parameter Design, April. Kurokawa, K. (1965) Power Waves and the Scattering Matrix, IEEE Transactions on MTT, March. Linvill, J.G. and Gibbon, J.F. (1961) Transistors and Active Circuits, McGraw-Hill. Pengelly, R.S. (1982) Microwave FET Theory, Design and Application, Wiley. Poole, C.R. and Paul, D.K. (1985) Optimum Noise Measure Terminations for Microwave Transistor Amplifiers, IEEE Transactions on MTT, Vol 33, November. Vendelin, G.D. (1978) Five basic bias designs for GaAs FET amplifiers. Microwaves, February. Vendelin, G.D. (1982) Design of Amplifiers and Oscillators by the SParameter Method, Wiley.
~___p_o_w_e_r_A__m_p_li_fi_er_s____~1 ~ 7.1 INTRODUCTION The y- or s-parameter representation of a high-frequency transistor operating under small-signal conditions provides a very convenient linear circuit model for the purposes of analysis and design. For power transistors, however, linear models are not applicable in describing the terminal behaviour of the transistors. The behaviour simply varies with the power level. Some transistors, known as 'linear power transistors', such as Hewlett Packard HXTR-5103 (2N6741), when operated in class-A mode, are lowlevel power-amplifier transistors capable of delivering only hundreds of milliwatts. Design methods using small-signal s-parameters can still be used. This kind of amplifier can be considered as 'high-power' small-signal amplifiers. Small-signal transistors are all class-A biased. Power amplifiers can nevertheless be biased in class-A, class-B or class-C mode, and at frequencies below, say, a few hundred megahertz in class-D, class-E or class-S mode employing VMOS or TMOS devices. The most commonly used mode for power amplifiers is class-C because of its high efficiency. All power amplifiers are fairly non-linear, but a class-C is more non-linear than is a class-B which in turn is more non-linear than is a class-A amplifier. A class-C amplifier is most non-linear in the sense that it generates more harmonics than do other modes. The efficiency of a class-A amplifier is less than 50% while that for a class-B amplifier, where the conduction angle is 1800 , is less than 78.5%. For a class-C amplifier the theoretical limit of efficiency is 100% as the conduction angle approaches zero; however in practice it does not normally exceed 80%. Bipolar transistors can be biased in all of class-A, B or C modes, whereas GaAs MESFETs are normally limited to class-A operation, and the power output is usually lower than a few watts. For relatively low-power applications of BJTs, common-emitter mode is popular while for applications above, say a few watts common-base operation is quite common. Most power BJTs have either their base or
~I
POWER AMPLIFIERS
~------------------------------------------------------~
emitter internally grounded to a substantial mass of metal which serves as both the circuit and thermal connection to the chassis. For transistors with this kind of package the choice of common-emitter or common-base operation is predetermined by the transistor manufacturer. The base or emitter lead inductance, which most often is the source of instability, is greatly reduced with internal grounding. The transfer characteristic of a power amplifier with harmonic distortion can be expressed as where Po and Pi are the output and input power, respectively, and aj is the coefficient for the jth harmonic term. When the linear term is equal to the second-harmonic term, i.e. when alPj = a2P? Pi = al/a2
or
(7.1) I
al/a2 is called second-order intercept. Similarly, Pi = (al/a n )n-l is known as the nth-order intercept. The nth-order intercepts are figures of merit, the larger the values of the intercept the better. The second- and third-order intercepts are usually the most important. The linearity of a power amplifier can also be expressed in terms of the 1 dB compression defined in Fig. 7.1. The higher the value of the 1 dB compression the larger is the linear dynamic range of the amplifier. Pi
Power gain in dB
o
I.. Fig. 7.1
Input power in watts Dynamic range of amplifier
Definition of the 1 dB compression.
~____________B_IA_S_I_NG__O_F_P_O_W__ER__T_RA_N_S_I_ST_O_R_S____________~I ~ 7.2 BIASING OF POWER TRANSISTORS Class-A biasing of BJTs and FETs has been encountered briefly in Chapter 5. Techniques for biasing class-A small-signal and class-A power transistors are similar except that, for maximum output voltage swings, the V CE for power transistors tends to be fixed at V cc (through an RFC without the collector resistor). For class-C operations, the base-emitter junction of the transistor must be reverse-biased to ensure that the transistor conducts for less than 180°. Figure 7.2 shows a popular self-biased common-emitter class-C configuration. The base-emitter junction is biased by the input (sinusoidal) voltage applied between point X and ground. Due to the p-n junction effect of the BE junction, the base current iB (and hence ic) will assume a waveform similar to that shown in Fig. 7.3. By applying Fourier series analysis to i B, it is easily seen that iB contains a d.c. current in addition to various harmonic current terms. The d.c. current IB varies with the magnitude of the input a.c. signal and must be returned to ground. Hence an RFC between point Y and ground is necessary to provide a d.c. path for the BE junction current. The base-spreading resistance Tbb' is the parasitic between the base of the transistor and the base contact of the transistor package. Reverse bias of the BE junction is achieved by the d.c. current passing through TW. If this voltage drop is not enough to reverse bias the BE junction, an external resistor can be connected between point Y and ground in series with the RFC.
/---
X
I 0-------1
I
//
........
"-
\
\
t - - -.......-'--\---i
\
\
"- ""........
Fig. 7.2
--- -
--------/
/
/
/
\
I
A self-biased common-emitter amplifier suitable for cIass-C operation.
~
IL-_________________P_O_W_E_R_A_M_P_L_IF_I_E_RS________________~ - iB (orie)
o
Time or conduction angle (8)
I~ < 180~1 Fig. 7.3
Base current in class-C operation.
A class-C common-base self-biased circuit is shown in Fig. 7.4. The resistor R is a very small-value high-power resistance in the range of an ohm. It serves many purposes. It tends to increase the reverse bias of the BE junction to ensure class-C operation. R decreases the Q-value of the RFC and makes low-frequency oscillation less likely to occur. R also limits the extent of Ie. 7.3 POWER TRANSISTOR DESIGN DATA
Figure 7.5 shows the manufacturer's data sheets for the Motorola microwave power transistor MRF2001M. Out of the vast amount of data given, some are more pertinent to power amplifier design than others.
RFC
R
Vee
Fig. 7.4 Self-biased common-based class-C operation (rbb' omitted; C 1 and C2 are feedthrough capacitors).
MOTOROLA
• SEMICONDUCTOR TECHNICAL DATA
MRF2001M
The RF Line 1.0 W
2 GHz
NPN SILICON MICROWAVE POWER TRANSISTOR
MICROWAVE POWER TRANSISTOR
· .. designed for Class Band C common base broadband amplifier applications in the 1.7to 2.3 GHz frequency range.
NPN SILICON
• Internal Input Matching for Broadband Operation • Guaranteed Performance @ 2 GHz, 24 Vdc Output power = 1.0 Wan Minimum Gain = B.5 dB • 100% Tested for Load Mismatch at All Phase Angles with 10: 1 VSWR • Hermetically Sealed Industry Standard Package • Gold Metallized, Emiller Ballasted for Long Life and Resistance to Metal Migration • Silicon Nitride Passivation • Characterized for Operation from 20 V to 28 V Supply Voltages
r-Ni~C
H -f
r==:==J
~L..I.... i~_ _ _ _
MAXIMUM RATINGS Symbol
Value
Unit
Collector-Emitter Voltage
VCEO
20
Vdc
Collector-Base Voltage
VCBO
45
Vdc
VEBO
4.0
Vdc
Rating
Emitter-Base Voltage Collector-Current -
Continuous
Total Device Dissipation @ TC
=25°C (1)
IC
250
mAde
Po
7.0 40
Watts mW/oC
Tstg
-65 to +200
°C
Symbol
Max
Unit
R8JC
25
°C/W
Derate above 25°C Storage Temperature Range
THERMAL CHARACTERISTICS Characteristic Thermal Resistance, Junction to Case (2)
(1) These devices are designed for RF operation. The total device dissipation rating applies only when the devices are operated as RF amplifiers. (2) Thermal Resistance is determined under specified RF operating conditions by infrared measurement techniques.
-I
-
~
STYLE L PIN t. EMITTER 1. COLLECTOR 3. BASE NOTES. t. DIMENSIONS WAND ARE DATUMS. 1. POSITIONAL TOLERANCE FOR MOUNTING HOLES.
W
1*19. t31O.0051@ITII'@IB@1
rn
3. IS SEATING PLANE. 4. DIMENSIONING AND TOLERANCING PER ANSI
Y14.5.1973.
MILLIMETERS DIM MIN MAX A 10.07 10.57 6.22 6.48 I 4.06 C 3.68 1.29 1.79 0 E t.41 t.73 t4.171se G H 2.29 2.79 4.t9 K 3.43 7.87 8.38 N Q 3.05 3.30 7.49 R 7.24
INCHES MtN MAX 0.790 0.810 0.245 0.155 0.t45 0.t60 0.090 0.110 0.056 0.068 0.5608SC 0.090 O.lto 0.t35 0.165 0.3tO' 0.330 0.t20 0.t30 0.285 0.295
CASE 337·02
Fig. 7.5(a) Motorola MRF2001M data sheet. Copyright of Motorola, Inc. Used by permission.
E
I
ELECTRICAL CHARACTERISTICS ITC = 25°C unless otherwise noted)
I
Symbol
Min
Typ
Max
Unit
VIBR)CEO
20
-
-
Vde
Collector-Emitter Breakdown Voltage (lC = 5.0 mAde, VBe = 0)
VIBRICES
45
-
-
Vde
Collector-Base Breakdown Voltage IIC = 5.0 mAde, IE = 0)
VIBR)CBO
45
-
-
Vde
Emitter-Sa.. Breakdown Voltage liE = 1.0 mAde, IC = 0)
VIBR)EBO
4.0
-
-
Vde
ICBO
-
-
0.5
mAde
GpB
8.5
9.5
-
dB
~
35
40
Characteristic OFF CHARACTERISTICS Collector-Emitter Breakdown Voltage IIC = 5.0 mAde, IB = 0)
Collector Cutoff Current IVCB = 2B Vde, IE = 0)
ON CHARACTERISTICS DC Current Gain IIC = 100 mAde, VCE = 5.0 Vde)
DYNAMIC CHARACTERISTICS Output Capacitance IVCB = 24 Vde, IE = 0, f = 1.0 MHz)
FUNCTIONAL TESTS Common-Base Amplifier Power Gain IVCC = 24 Vde, Pout = 1.0 W, f = 2.0 GHz) Collector Efficiency IVCC = 24 Vde, Pout = 1.0 W, f = 2.0 GHz) Load Mismatch IVce = 24 Vde, Pout = 1.0 W, f = 2.0 GHz) VSWR = 10:1 All Phase Angles)
.
No Degradation
In
Power Output
FIGURE 1 - 2.0 GHz TEST CIRCUIT
Vcc= +24 Vde
Z1-Z12 - Microstrip, See Photo master Cl - 0.6-4.5 pF Johanson 7271 C2, C3 - 56 pF Chip Capacitor C4-0.1 ~F C5-10"F,35V Board Material- 0.0312" Teflon Fiberglass ,,= 2.5 ± 0.05
Fig. 7.5(b) Motorola MRF2001M data sheet. Copyright of Motorola, Inc. Used by permission,
FIGURE 2 2.0
~
::::>
20 V
I--
/ ~V
h
1.0
24 V
J
51
~ ~
.... ::>
::::>
II' r-
<:>
_r-
ii)
'"~
I/, V
OUTPUT POWER versus INPUT POWER (f = 2.0 GHz)
2.0
Vee - 28 V
-
/'./
'"~
.... ::>
f...f...-
./
i
FIGURE 3 -
OUTPUT POWER versus INPUT POWER (f=1.7GHz)
/
10
/ V / V
<:>
J
V V V
./
/V
,,- I-""
Vee
=28 V
-
24 V 20 V
..... 1--
V
V/ V
100
120
140
160
180
200
220
240
060
260
80
100
FIGURE 4 -
OUTPUT POWER versus INPUT POWER (f = 2.3 GHz)
FIGURE 5 -
1.6 ii)
t=
!
.... ::>
::::>
1.4 1.2
0.8
<:>
0.6
~
0.4
140
160
180
200
220
240
POWER GAIN versus FREQUENCY
./
/'
,/'
.,/
V
,L/ V
V
. . .V
---.....
~ ~ ~
!----
Vee = 28 V
I-
24 V
f...-
20 V
--- ---
12
~ z
~
r--,.....
10
'"~
V
~ 8.0
1e
<..0
,/'
Pin= 140mW
t---
r-- J Vee
28 V
r- -.... r-.. ----
24 V
l -I--
r--......
6.0
........... 20 V
O. 2V 060
260
14
'" ~ 1.0 ~
120
P,n' INPUT POWER ImWI
Pin. INPUT POWER ImW)
80
100
120
140
160
180
200
220
240
260
4.0
17
2.0
Pin. INPUT POWER ImW)
FIGURE 6 -
f. FREQUENCY IGHz)
SERIES EQUIVALENT INPUTIOUTPUT IMPEDANCE
+j50
Vee = 24 V. Pin = 140' mW f 6Hz 17 2.0 2.3
lin Ohms 15.5 + j 3.0 7.5 + j11.0 10.0+j100
ZOl' Ohms 4.5 - j15.0 4.0 - j12.0 30-,7.0
·ZOL = Conjugate of the optimum load impedance
into which the device output operates at 8 given
output power, voltage and frequency.
-j&O
Coordinates in Ohms
Fig. 7.S(c) Motorola MRF2001M data sheet. Copyright of Motorola, Inc. Used by permission.
2.3
~
IL-_________________
P_O _W_E_R_A_M_P_L_IF_ I_ER _ S________________
o
~
0 00
~00
..... 100 "
" , 0 _ _' _ _ 0
o
0
o Denote Eyelet o 4-40 crcw Placcmcnt Fig, 7.S(d) Motorola MRF200lM data sheet. Copyright of Motorola , Inc. Used by permission .
It is noted in the first page of the data sheets that MRF200lM is designed for common-base class-B or class-C operation over a frequency range of 1.7 -2.3 GHz. Some special features of the transistor are also stated in the same page; they include a load mismatch ruggedness of VSWR = 10 and a maximum output power of no less than 1.0 W. From Figure 3 in Fig. 7.5(c) it is seen that a power amplifier using MRF2001M can deliver an output power of 1.2 W at 2.0 GHz for an input power of 140mW, i.e. it has a power gain of 9.3 dB provided the input and output ports of the transistor are conjugately matched. Figures 2, 3 and 4 give the same input-output power relationship at different frequencies .
PO __ W_E_R_T_RA __N_S_IS_T_O_R_D_E_S_IG_N_D_A_T_A______------~I
L -____________
Figure 5 provides the maximum achievable gain as a function of frequency at various values of V cc at 140 mW input level. Figure 6 in Fig. 7 .5(c) shows the most important data for power amplifier design. It gives the input impedance (Zin) and the 'output impedance' (Z(,)d as functions of frequency at Vcc = 24 V and Pin = 140mW. If variable loads (stub tuners) are placed at both the input and output of the transistor, and are tuned for zero reflected power and an optimum gain/efficiency compromise, then the complex conjugate of the impedance seen by the transistor input is defined as the input impedance Zin of the transistor. The complex conjugate of the optimum load impedance as seen by the transistor output, i.e. Z(')L, is defined as the 'output impedance' of the transistor. Strictly speaking, the output impedance of a transistor operated in class-B and class-C is undefined, however the conjugate of Z2k, i.e. ZOL, does represent the optimum load impedance required for optimum performance, hence for the purpose of output matching network design, Z(')L may be regarded as the equivalent series output impedance of the transistor. The input impedance and the 'output impedance' of the MRF200lM at Vcc = 24V, Pin = 140mW andf= 2.0GHz are given by Zin = 7.5 + jl1.0 Q ZbL = 4.0 - j12.0Q.
Note that both impedances are series impedances. Some manufacturers give these impedances in parallel form and some give this information in both forms. The series and parallel equivalent circuits of the MRF2001M at 2.0 GHz under the conditions described are shown in Fig. 7.6. For power transistors intended to be used at lower frequencies, say below a few tens of megahertz, the parallel output resistance Ro is roughly equal to the parallel loading resistance RL in order to deliver a maximum output power of Po according to R _ (Vee - Vsat )2 L 2Po '
(7.2)
where V cc is the supply voltage to the output port and V sat is the saturation VCE or V CB voltage for CE and CB mode, respectively. In the case that V CEsat or V CBsat is not given, they can be taken as zero for the purpose of using (7.2). The expression (7.2) is valid for class-A, class-B and class-C (Smith, 1987) at low frequencies where the a.c. impedance of the transistor output port is neglected. It is derived with the assumption that the supply voltage Vcc is connected to the collector through an RFC (without collector resistor). However, (7.2) becomes increasingly inaccurate as frequency increases. Certainly at upper UHF frequencies or above (7.2) is not meaningful. Expression (7.2) is historically used to calculate the loading resistance required for a given output power when the parallel
~
~
IL__________________P_O_W_E_R_A_M_P_L_I_FI_E_R_S________________~ MRF2001M
,----------------,
E
I I I I I
+jll.O
n
-j1.2
I
I
n
I I I I
I I
L_________ ________
I
0 C
I I I
~
B Series
MRF2001M
,-----------------, I
E
I
I
I
11.28 nH
I I I
23.6
4.36
n
n
I I
C
I
I I I
L _______ _
--------~
B
Parallel
Fig. 7.6 Series and parallel equivalent circuits for MRF2001M.
output resistance of the transistor is not given. The use of this equation at higher frequencies should be avoided.
7.4 POWER AMPLIFIER DESIGN The first step in the design of a high-frequency power amplifier is to determine the number of stages (transistors) required to deliver the output power intended for the input power provided. The design of a multi-stage amplifier is the same as that for a single-stage except for the inter-stage matching requirements.
P_O_W __E_R_A_M_P_L_IF_IE_R__ D_ES_I_G_N______________~I
L -_ _ _ _ _ _ _ _ _ _ _ _ _ _
When a transistor is chosen, the input and output impedances for the required power level can be read from the transistor data sheet as discussed in the previous section. It is noted that the impedances read from the data sheet only serve as a rough guide because the actual values vary from device to device. Also, the data sheet may not give the input and output impedances at the power level and the frequency required. For these reasons, as an alternative, a test setup as shown in Fig. 7.7 can be used to measure the input and output impedances of the transistor at the intended power level and operating frequency. The input and output power levels are monitored by the two power meters. When the input power level is properly set, the two stub tuners are tuned until the output power is maximum. The input power is adjusted until the maximized output power is at the required operating power level. The circuit is then disconnected at PP' and QQ', and the transistor holder is removed by disconnecting AA' and BB'. The triple stub tuners are terminated with 50-ohm loads at PP' and QQ'. The impedance at AA' looking into the input tuner can be measured using a vector network analyser; the complex conjugate of the measured sit is the rin looking into the transistor holder but
rin
is only
r AA',
not
r l1 ,.
They are related by
Transistor Holder
B I-~":""-+""-j
50 ohm
load
B'
Fig. 7.7
Measurements of input and output impedance.
~
~
IL-_________________P_O_W_E_R_A_M_P_L_I_FI_E_RS________________~ Similarly, the output reflection coefficient of the transistor can be obtained by measuring the sh at BB' looking into the output tuner with QQ' loaded by a 50-ohm termination, then Once the input impedance and the 'output impedance' of the transistor at the proper power level are read from the data sheet or measured, power amplifier design is reduced to the design of two impedance matching networks matching the input and output impedances to the source and load, respectively. Techniques of impedance matching thoroughly discussed in Chapter 4 can be applied to the design of matching networks for power amplifiers in exactly the same way as for small-signal amplifiers. Example 7.1 The input impedance and the 'output impedance' of a common-base BJT 2N6267 were measured at 2.3 GHz under aVec of 24 V and an input power of 1.5W. They are Zin = 1.8
+ j12Q
Zout = 1.6 - j8 Q.
Design a power amplifier for maximum power gain with no less than 10% bandwidth. Note: The manufacturer suggests an emitter resistance of 0.43Q.
Solution For maximum gain design the input and output ports of the transistor must be conjugately matched. In order to design an amplifier with a bandwidth greater than 10%, all the impedance (admittance) points for the input and output matching circuits as indicated on the Smith chart must have a Q-factor smaller than 10 (in fact, smaller than, say,S or 6). The major task ofthis design is thus the design of the input and output matching circuits with Q-values low enough to provide 10% bandwidth. A constant-Q curves chart is shown in Fig. 7.8 for reference purposes.
Input Matching A simple matching network as shown in Fig. 7.9 is proposed for the input circuit. The purpose of the microstrip section of characteristic impedance ZOl is to transform Zin (point AI) to (point A 2 ) an impedance of 50 Q plus a position reactance (inductive). The inductive component of the impedance at point A2 is then neutralized by the negative reactance provided by CI to give a purely resistive impedance of 50 Q at point A 3 .
PO_W __E_R_A_M_P_L_IF_IE_R__ D_ES_I_G_N________________I ~
L -_ _ _ _ _ _ _ _ _ _ _ _ _ _ _
Smith chart
Fig.7.8 Constant Q-curves chart.
The choice of ZOl affects the overall Q-value of the input circuit. Since is very small, if ZOl is relatively large, say 50 Q, Zin will be located very near to the edge of the Smith chart after being normalized by 50 Q. This will produce a fairly high Q-value at point A 2 • To this end, a smaller value of ZOI = 12.5 Q (say) is chosen. Point AI. i.e. Zin = 0.144 + jO.96 Q, and a 50-ohm circle, i.e. R = 4 are entered into a Smith chart as shown in Fig. 7.10. The microstrip of length II transforms point Al through a constantVSWR circle to point A2 where it meets the 50-ohm circle (R = 4). The length II required is It = 0.109 wavelength (A g l)' The Q-values at points Al and A2 are 6.7 and 1.5, respectively. Since the Q-value of A2 is much lower
Zin
Microstrip section (ZOl) 11
E
2N6267
B
(A2 )
Fig. 7.9 Input circuit.
(A 1)
Z = 1.8 + j12
~ ~[__________________ P_O_W __E_RAMP_L_IF_I_ER_S___________________~
50 n circle
Fig.7.10
Input matching for Example 7.1 (impedance coordinates).
than that of AI> the transformation does not reduce the bandwidth. The reactance at A2 is jX2 = +j5.8, hence the capacitance required of C 1 is 1 2n x 2.3 X 109 x C1 = 5.8 x 12.5
or
C1
=
0.95pF.
Therefore, a variable capacitor of 0.3 - 3.5 pF can be used as CI . Note: ZOI = 12.5 Q, 11 = 0.109A g i. Output Matching The series equivalent output impedance is Zout = 1.6 - j8Q
and the parallel output admittance is Y out = 0.024
+ jO.12mhos.
An output matching network as shown in Fig. 7.11 is proposed. The Zout (or Y out ) has a relatively large reactance (or susceptance), hence if it is transformed by a transmission-line (microstrip) section, the Q-value will be quite high even if Z02 is small. Therefore, a pair of balanced short-circuit stubs can be added to shift Y out to a better starting
~______________P_O_W__ER__A_M_P_L_lF_lE_R_D_E_S_l_G_N______________~I ~
c
1
2N6267
(A7)
50
B (A4)
n
(A6)
Microstrip short-circuit stubs
Fig. 7.11
Output matching circuit.
lis
Fig. 7.12
=
1
Output matching chart for Example 7.1 (admittance coordinates).
~
P_O_W_E_R_A_M_P_L_IF_I_ER_S________________~
L I_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
point before it is transformed by the microstrip section of length 12 , Since we are using shunt stubs to add susceptance, we enter YOU! into a Smith chart as point A4 in Fig. 7.12 by normalizing YOU! by Z02 = 25 Q, i.e. -Y out
=
YOU!
1125 =
06 .
+ J'3 .
In order to move from point A4 to point A 5 , a susceptance Ii 1 - 3 = -2 is added.in parallel with YOU!' Hence B
1
= 25 x
B mho
=
= B5 - B4 =
-0.08 mho.
Note that the choice of point A5 is fairly arbitrary; as long as point A5 has a Q-value low enough for our purposes, it is acceptable. This susceptance, B = -0.08 mho, is provided by a pair of balanced short-circuit stubs, each contributing BI2mho or -0.04 mho. The required stubs may have a Z03 different from Z02 (25 Q). 13 will be determined later. the impedance looking into the transistor and the shunt stubs, Point is then entered into Fig. 7.13. Point is then transformed by the microstrip section of length 12 (in Ag2) to point A6 through the constanttowards 'generator' until it meets the 50-ohm VSWR circle from point circle (R = 2) at point A 6 • The second intersecting point is not taken because it represents a capacitive point, the capacitance of which cannot be neutralized by adding a series capacitance as scheduled. The reactance at point A6 is X 6 • In order to neutralize the inductance X 6 , the required capacitance C2 is thus given by
As,
As
As
25
X
X6
1
=
2n x 2.3 x 109 x Cz
1 Cz = 2n x 2.3 x 109 x 25 x 1.7 = 1.63pF.
Hence a 0.3 - 3.5 pF variable capacitor can be used as C2. 12 is read from the chart as 0.318A g2' We now go back to find the length 13 of the short-circuited shunt stubs. The required susceptance for each stub was previously found to be -0.40 mho. The choice of short-circuit stubs rather than open-circuit stubs for 13 is because of the inductive nature of required 13, For an open-circuit stub to be inductive, 13 is necessarily greater than one quarter of a wavelength and Z03 needs to be fairly high in order to keep 13 reasonably short, but a high Z03 will ma~e the stub width too small to be accurately fabricated; hence short-circuited stubs are chosen. Zo3 is arbitrarily chosen to be 75 D. The normalized susceptance (by 75 Q) of each stub is thus
I~
POWER AMPLIFIER DESIGN
~-----------=~~~------------~
50n
circle
Fig. 7.13
Output matching chart for Example 7.1 (impedance coordinates).
Fig. 7.14 Output matching chart for Example 7.1 (admittance coordinates).
~
L I_
_
_
_
_
_
_
_
~
-_--_---_~~
P_O_W_E_R_A_M_P_L_I_FI_E_RS_ _ _ _ _ _
_
RFC
Vcc=2SV
ZOI = 12.5 n Z02=25n Z03 = 75 n
(Zm)
L-L---_l
-~~-----~J 12
Fig. 7.15 Overall circuit for Example 7.1. T, and T2 are microstrips on Teflon glass board: Cr = 2.55, thickness = 0.79 mm = 1/32". C, and C2 are 0.3 to 3.5 pF (Johanson 1700 or equivalent). C3 and C4 are 470 pF (Allen-Bradley FAC5 or equivalent). Cs and C6 are 220 pF chip capacitors. RFC is No.2 wire, three turns, ID = 1/8". R, is 0.43 Q (recommended by manufacturer). All dimensions in mm. I, = 1O.62(0.109).g')' W, = 13.10, 12 = 31.59(0.318).g2), W2 = 5.56, 13 = 5.45(0.052).g3), W3 = 1.11.
B' = -0.04 = -3
1/75
.
B' = -3 is entered into an admittance chart as shown in Fig. 7.14. The short-circuit point, SIC, is also noted on the chart. The required stub length 13 is found by reading the arc length from SIC to B' towards 'generator', i.e 13 = 0.052).g3'
The overall circuit of the power amplifier is shown in Fig. 7.15.
P~R~O~BL~E~M~S____________________~I ~
L -_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
PROBLEMS 1. Explain why common-base operation is commonly employed for higher-power d. amplifiers.
2. For the c1ass-A biased power amplifier shown in Fig. P.7.1, show that the power-conversion efficiency YJ is smaller than 50%. 3. Even if the VCEsat of the circuit shown in Fig. P.7.1 is zero, the coil resistance of the collector RFC is zero and the base circuit does not consume any power, the conversion efficiency for a c1ass-A RF power amplifier is still normally much smaller than 50%. Explain why this is so. Note that the answer can be extended to c1ass-B and c1ass-C operations. 4. Show that the theoretical power-conversion efficiency for a non-linearly biased power transistor (class-Band c1ass-C) is given by YJ =
Po 0 - sin 0 Pi = 4 sin 012 - 20 cos 012 '
Vee
o~----~~----e-------;-.
Fig. P.7.1
~
P_O_W_E_R_A_M __ PL_I_FI_E_RS________________~
L I_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
where: () is the conduction angle of the transistor in radians, Po is the a.c. power output, and Pi is the power supplied to the amplifier. 5. Refer to the design data of MRF2001M in Fig. 7.5. Use the input and output impedances at 2.0 GHz given in Fig. 7 .5(c) to design a 2.0 GHz test circuit with a similar microstrip configuration to that shown in Fig. 7.5(d). Note that the design given in the data sheet contains a number of arbitrary choices, hence it is virtually impossible to design a circuit which is exactly the same as that given unless these arbitrary choices are specified. Note that the circuit shown in Fig. 7.5(d) is the microstrip implementation of the schematic diagram shown in Fig. 7.5(b). 6. The input and output impedances of the class-C operated transistor, Motorola MRF233, at 100 MHz under aVec of 12.5 V and at an output power level of 15 Ware given by (common-emitter mode) Zin = 1.7 - j2.7Q Zout = 5 - j5.6 Q.
Use simple L-matching networks with discrete components to complete the power amplifier design. Draw the circuit diagram and give component values including those of the biasing circuit.
FURTHER READING Bowick, C. (1982) RF Circuit Design, Howard W. Sams. Miceli, M., A i-Watt, 2.3GHz Amplifier, Motorola Application Notes, EB89. Moline, D., 800 MHz Test Fixture Design, Motorola Application Notes, An-923. Motorola RF Device Data (1986) Motorola Inc. Power Circuits, DC to Microwave (1970) RCA Electronic Components. RF Power Transistor Manual (1972) RCA Solid State Division. Smith, J. (1987) Modern Communication Circuits, McGraw-Hill.
~______o_S_C_il1_a_to_r_s______~1 ~ 8.1 GENERAL OVERVIEW OF OSCILLATOR DESIGN
One way of classifying transistorized oscillators is according to whether their design is based on linear device (BJT or FET) parameters such as the s- and y-parameters, or based on large-signal behaviour of the transistors. The former category is sometimes known as the linear oscillator whereas the latter is sometimes referred to as the power or large-signal oscillator. Power oscillator design utilizing non-class-A biasing is a trial-and-error art. Very little about it in any form has been reported. Power oscillators normally make use of low-frequency techniques in oscillator design, such as the Colpitt's, Hartley's and Clapp's, the design procedures are not as well defined as those of low-frequency amplifiers and they normally require a substantial amount of after-design trimming. The design of oscillators at high frequencies using linear device sparameters will be discussed in this chapter. At the outset it is assumed that the s-parameters of a transistor biased and operated in a particular mode, most often the common-emitter mode, have already been measured or read from the databook. We shall then go through some matrix algebra to allow the 2 x 2 common-emitter s-matrix to be converted, first, to a 2 x 2 common-emitter z-matrix, which enables the designer to connect reactive components to the transistor so as to provide more feedback (positive in sense!) in order to make the device more inclined to oscillate. The feedback is normally added in the form of a capacitor from emitter to ground or an inductor from base to ground. The loaded device can then be described by a 2 x 2 common-emitter or common-base s-matrix depending on the terminal to which the reactive element is connected. All this matrix algebra is necessary if the device on its own represented by the original measured s-matrix is stable (K> 1). These matrix operations enable the addition of reactive elements to the device resulting in a 'loaded' s-matrix which represents an oscillating or potentially unstable device.
~
O_S_C_IL_L_A_T_O_RS__________________~
L I_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
With a potentially unstable device represented by a 2 x 2 s-matrix, the design of an oscillator is fairly similar to the design of an amplifier using sparameters. In the output port of the device, one has to design a matching network to match the device to the load (usually 50 Q). The only difference is in the design of the input network. The input network of an oscillator is a network which when connected to the input port of the device will form a resonator resonating at the desired oscillator frequency. 8.2 CONVERSION OF THE s-MATRIX
Suppose the s-parameters of a transistor to be used for an oscillator design are measured in common-emitter mode and are represented by Sm = (SlIm Sl2m) , S21m S22m
(8.1)
where the subscript m stands for measured values (or values from databook). Note that Sm could also be measured from common-base mode. For an active device to oscillate, its stability factor K must be less than unity. If this condition is not satisfied, either the common terminal should be changed, e.g. from common emitter to common base, or some kind of feedback is necessary. In this section some expressions enabling the addition of positive feedback and the change of common terminals (i.e. either common emitter or common base) are listed. The measured common-emitter s-matrix Sm is represented as in (8.1). In this case, port 1 is the base-to-ground port and port 2 is the collector-to-ground port. Since the actual transistor configuration used in the oscillator may be different from that of the measurement, conversion is necessary. The two-port common-emitter z-matrix (2 x 2), Z, can be obtained from the measured common-emitter s-matrix S (2 x 2) by the expression
z = (ZII ZI2) Z21
=
Z22
Ro(I + Sm) (/ - Sm)-I,
(8.2)
where I is a 2 x 2 identity matrix, and Ro is the characteristic impedance from which the s-parameter is defined. In high-frequency linear oscillator design, two feedback models are commonly used to make the transistor potentially unstable, or in other words, to decrease the value of K. They are shown in Fig. 8.1. Model A is a common-emitter topology with 'positive' feedback and model B is a common-base topology with 'positive' feedback. The reactive element shown in Fig. 8.1 can be a capacitor, an inductor, a series LC network or a parallel LC network. These reactive networks can either be discrete or distributed.
C_O_N_V_E_R_SI_O_N_O_F__T_HE__~_M_A_T_R_IX____________~I
L -_ _ _ _ _ _ _ _ _ _ _ _ _
~
.-----02
)r----o 2
Model A
ModelB
3 Reactive element
Fig. 8.1
Two models showing transistor feedback.
2
1'
l'
~ 2'
2'
Fig. 8.2 Series connection using z-parameters.
Model A
r-----,
0
I I I
or
I I ZB I IL ____ ...JI 0
I
Fig. 8.3 Series connection using z-parameters.
ModelB
~ I~
___________________O_S_C_IL_L_A_TO__RS__________________~
In circuit theory it is well known that when two two-port networks are connected in series as shown in Fig. 8.2 the overall z-matrix Z is equal to the sum of the individual z-matrices, i.e. (8.3)
Hence models A and B of Fig. 8.1 can be redrawn as the circuits shown in Fig. 8.3 with the reactive elements taken as a capacitor and an inductor, respectively, for model A and model B. The overall s-parameter representation of a transistor with series feedback can easily be obtained from the conversion formula
s=
(Z
+ RoI)-1 (Z - RoI).
(8.4)
8.3 THEORY OF OSCILLATION A simple and popular method in oscillator design using s-parameters is to place a tuning resonator across the input terminals of the transistor (considered as a two-port network) and to place the load at the output terminals of the two-port network as shown in Fig. 8.4. We will show that the conditions for oscillation can be expressed as K
(8.5) (8.6)
if and only if (8.7)
The stability factor K of the active network should be less than unity in order for the device to oscillate. If this condition is not satisfied, either the common terminal should be changed or positive feedback should be added. The passive termination r 1 and r z must be so designed that the input and output ports are resonating simultaneously at the same frequency.
Resonator
~
-
."
,...
~
Two-port active network
s
~
fout = S~2'
Fig. 8.4
Block diagram of an oscillator.
,...
Load
r+'
~_______________TH__EO_R_Y__O_F_O_S_C_IL_L_A_T_IO_N______________~I ~ It shall be shown in Theorem 1 that when port 1 is in resonance, the following expression holds
(8.8) In Theorem 2, it will be shown that if the circuit is oscillating at port 1, it must simultaneously be oscillating at port 2, i.e.
(8.9) if and only if (8.10)
It shall also be shown in Theorem 3 that in order for oscillation to start and
be sustained, it is necessary that
ITII
>1+1·
(8.11)
Sll
Theorem 1 The condition for port 1 to be in resonance is given by
Proof For oscillation to occur, the real and imaginary parts of the input impedance of the network must be equal to zero. Referring to Fig. 8.5, the conditions for oscillation can be expressed as Rin
i.e.
+ R I = 0 and jXin + jXI = 0 R in = -RI and X in = -XI'
The reflection coefficient network, is given by
Tin,
-
-"
~
Fig. 8.S
looking into the input port of the active
Port 2
Two-port active network
Port 1
RdJ
(8.12)
jX1
=~ jXin
1""1
,..
~
Oscillation in port 1.
-
,...
JRm. ~
-
'"
Load ~
~
O_S_C_IL_L_A_T_O_RS__________________~
L I_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
F; = Zin 10 Zin R in R in
The reflection coefficient
r
1,
- Zo + Zo
+ jXin - Zo + jXin + Zo·
looking into the resonator, is given by
F _ ZI - Zo 1 - ZI + Zo
Rl + jX1 - Zo Rl + jX1 + Zoo
Now that
SIt
= rin , therefore silFI = FinF l
_ (Rin + jXin - Zo) (Rl + jXl - Zo) - R in + jXin + Zo Rl + jX1 + Zo
(8.13)
On substituting (8.12) into (8.13) 'F Sll 1
= (-R 1
-
jX1
Zo)(R 1 + jX1 - Zo) R I+Jl+ ·X Z ' 0
-
-R 1 - J·X l+ Z0
Theorem 2
When the system is oscillating at port 1, it must simultaneously be oscillating at port 2, i.e. if and only if Proof
From the theory of two-port networks, it may be written that (8.14)
or
(8.15)
Similarly, I
S22
=
S22
S12S 21 F l F Sll 1
+1-
_ S22(1 -
SUFI)
-
1-
+ S12S21 F l
SUFI
T_H_E_O_R_Y_O_F_O_S_C_IL_L_A_T_IO_N______________~I
L -_ _ _ _ _ _ _ _ _ _ _ _ _ _
S22 - S2zSUr i + SlzS21r l 1 - SUrl _ S22 - (SUS22 - S12S21)rl 1 - surl
(8.16)
On subsituting (8.15) into (8.16) I
S22 - {[sh - (SI2S2Ir 2)/(1 - S22r 2)]S22 - S12S21}rl 1 - [sh - (S12S2Ir2)/(1 - S22r 2)]rl
_
S22 I
_
Szz -
szz(l - r lS1l] + (slzS2I r 2)/(1 - S22r 2) [1 - sllrd + (S12S2Ir lr 2)/(1 - S22r 2) .
(8.17)
At the threshold of oscillation, it is required at port 1 that hence (8.17) reduces to Conversely,
Sll
or
rin can be expressed in terms of sh and r2 • By putting
shr2 = 1 into the expression for sil, we obtain the consequence that sitrl = 1. Hence, the necessity and sufficiency of shrl = 1 shr2 = 1
and
are proved. Theorem 3 Under stable oscillating conditions, both Theorems 1 and 2 are valid, i.e. shrl
=1
if and only if shr2 = 1.
However, in order for oscillation to start, the condition required is Irlshl > 1
or
Irll>I+I· Su
(8.18)
Oscillations will start to build up until device non-linearities, or voltage or current limitations cause it to reach a steady state. The oscillation frequency, rb r 2 and the device s-parameters reach a stable state when
Proof
Suppose that unit power is flowing into port 1 of the two-port active network shown in Fig. 8.6. The power reflected is given by Isil1 2 x 1 unit. The reflected power travels towards the source termination and is reflected
~
~
IL-___________________O_S_C_IL_L_A_TO__RS__________________~
Resonator
I+-
r+
Two-port active network S
-
~
-
r+
Matching network and load
Fig. 8.6 Two-port oscillator. Copyright of Motorola, Inc. Used by permission.
again by the source impedance Zl' Since the reflection coefficient of Zl is
r1 , the power reflected back from the source termination is Is 1112 X Ir1 12
units. This reflected power again travels towards port 1 of the active network. This power should be greater than unity in order for oscillation to build up. This condition can be expressed by or
Isill z x Ird z > 1 ISitrll > 1.
If this power is less than unity, the oscillation will eventually be damped
off. Normally, the resonator connected to port 1 is a passive device with a value of Ir11 slightly less than unity, so therefore in order for oscillation to be sustained ISI11 of the active two-port device should be greater than unity, such that Isitr1 1> 1.
8.4 OSCILLATOR DESIGN Based on the theory of oscillation discussed in the previous section, the design of an oscillator is reduced to the problem of finding the loads at port 1 and port 2 that will cause oscillation simultaneously at both ports at the desired frequency using the requirements
sitr1 = 1 and
sizrz
= 1.
To find the suitable terminations at port 1 and port 2, we first attempt to determine the termination required at port 2 in order to cause oscillation when port 1 is arbitrarily terminated. Assume that only passive terminations are placed in both ports. Passive terminations in port 1 are represented by Irtl < 1, i.e. loads located inside the area of Irtl = 1 circle in the rrplane Smith chart. The binding circle Irtl = 1 can be mapped into the SZ2 plane by
O_SC_I_LL_A_T_O_R_D_E_S_IG __ N________________~I
L -_________________
(8.19)
can further be mapped into a circle in the r 2 plane using the relationship of r2 = 1IS 22 . The mapping of the Iril = 1 circle into the rz-plane is a circle described by
S22
Ir2 -AI
=b,
where A -
(S22 -
- Isd 2
LlStl)' (centre of circle) ILlI2
-
IS12 S 21 I
(radius of circle).
(8.20)
However, the area inside the Iril = 1 circle, which represents all passive source terminations, may be mapped either inside or outside the Ir2 - A I = b circle on the rz-plane. Hence it is essential to determine which the case is by locating the r l = 0 point on the rz-plane. For the purposes of illustration, the area inside the Iril = 1 circle is assumed to be mapped outside the Ir 2 - A I = b circle, and is illustrated in Fig. 8.7. The shaded region represents the intersection between Iril < 1 and Ir21 < 1. If the chosen source termination r l is mapped into a point within this region, the required load termination may be realized by a passive network. 1 ---;- mapping 522
/
/'
-
Ir2 - AI = b ( Ir11 = 1)
Intersection of
Ir11 < 1 and Ir2 1< r1 Fig. 8.7
r 2 - plane
plane
Mapping of
Ir1 I =
1 circle into the rrplane.
1
~
~
O_S_C_IL_L_A_T_O_RS__________________~
L I_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
Similarly, the Ir21 = 1 circle, which represents all passive terminations at port 2, can be mapped into a circle in the rt-plane using I
Sll = Sll
r r
S12S 21 2
+ 1-
=,1
S22 2
and
r1
This results in a circle Ir t - C I
= d in the rrplane with
Sll
C - (Sll - L1S!2)* -I S llI2_1L112
(centre of circle)
d IS 12S 2t1 (radius of circle). -ll slll 2 - 1..1121
(8.21)
It is also necessary to test whether the area inside or outside the circle Irt - CI = d represents Ir21 < 1 by using the point r2 = O. Figure 8.8 illustrates this mapping. With the aid of Fig. 8.7 and Fig. 8.8, we may find out whether or not a certain source termination crt) that we chose will map on to the shaded region in the r 2-plane. If not, this means that the load required to be placed at port 2 for the particular source termination at port 1 that we chose cannot be realized by a passive network, and a different source termination has to be chosen until the mapping of this r t value in the r2 plane lies inside the shaded area. The same argument applies to choosing a
if) - CI ( If21
=
= d
1)
/
/-
-" .....
\
~I I I
1
.
,mappmg Sl1
on
If21
=
1
Intersection of
If)l < 1 and If21 < 1 f) - plane
Fig. 8.8 Mapping of Ir21 = 1 circle into the r1-plane.
f2 - plane
~_________________O_SC_I_LL_A_T_O_R_D_E_S_IG_N________________~I ~ load r 2 to see whether the required source termination falls inside the shaded area in the rl-plane or not. In Fig. 8.9 it is assumed that the mapped areas in both the rr and r 2-planes which correspond to passive terminations are outside the Ir2 - A I = b and Ir l - CI = d circles in the rr and rz-planes, respectively. Suppose that a source termination r l is chosen and is denoted by point Al in the rrplane. Point Al is mapped by r2 = 1Is22 into point A2 on the r 2-plane. Point A2 must lie in the shaded region, otherwise the required port-2 termination for the source r l cannot be realized by a passive load. A load termination (r2 ) slightly different from that
,---" /
I
/'
" '\
I
t\
\
) \
"
fl - plane
f2 - plane
/-.........-
If1
I
I
\
If2 -AI=b
~
'"
\
/
)
-
Ci =
d
/ / - - ........
I
I
\ I
(
I
\
\
"\ If2 -AI=b ~
"- .......
fl - plane
f2 - plane
Fig. 8.9
r 1 and r 2 required for simultaneous oscillation at both ports.
~
O_S_C_IL_L_A_T_O_R_S__________________~
L l_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
represented by A z should be chosen, and this termination is represented by point A z on the rrplane. Point A z is then mapped by r1 = lIs 11 back into point A3 on the r 1 plane. By theorem 3, if Ir11 > IlIs 11 I, i.e. if point A3 is nearer to the centre of the chart than is point A], then oscillation can start. If the load at port 2 is taken as point A z, then it will simply map back to point Al and Ir11 = IlIs 11 I, which is the static condition for oscillation. Choosing a load at port 2 different from point A2 is sometimes necessary because if Al is a locus instead of a point (AI represents a tuning element, e.g. a trimmer capacitor), then A2 is also a locus, and this locus of A z may not be easily realized. If the chosen r 1 and r2 can follow the entire procedure, then oscillation can occur and sustain itself for these terminations. 8.5 SUMMARY OF DESIGN PROCEDURES
Based on the previous discussions, a set of procedures can be drawn up for the design of oscillators using s-parameters. A transistor with sufficient gain and output-power capability for the desired frequency range is selected. A configuration for the transistor (CE, CB) is fixed and the amount of feedback, if required, to make the transistor more unstable is chosen. The transistor (with feedback, if necessary) will form a new active two-port network. A resonator is placed at port 1. For example, the resonator can be a tunable capacitor with a small amount of parasitic resistance and is represented in the rrplane as a locus as shown in Fig. 8.10. This locus represents the change of r 1 when the capacitor is tuned or when the frequency is varied. The resonator locus is mapped into the rrplane using the relationship
and
The locus of r1 mapped by lIsh is plotted on the rz-plane. The rrplane Smith chart is separated into two regions: one represents a passive source with Ir 1 I < 1 and the other an active source with Ir1 I > 1. The identity of these two regions can be determined by mapping a test point from inside the Ir11 = 1 circle, usually r 1 = 0, into the rrplane. If the mapped locus of Zl or r1 in the rrplane falls in the shaded region corresponding to Ir1 I < 1 and Irzl < 1, then the chosen Z1 is acceptable. If the locus falls in the unshaded region, there is no passive load termination that will cause oscillation with this chosen r l , and then either Zz has to be changed or the feedback network has to be altered. In the case where the mapped locus of Z1 in the rrplane is acceptable,
~____________S_U_M_MA __R_Y__ O_F_D_E__ SIGN_P_R_O_C_E_D_U_R_E_S____________~I ~ Mapping of Ir2 1= 1 (source stability circle)
Mapping of Z2 over a frequency range 1',[2
Z2
Fig. 8.10
=
actual load chosen at operating frequency 10
Resonator ZI (e.g. a trimmer capacitor or a varactor diode)
1
--;- mapping of ZI
522
Mapping of Ir,1 = 1 (load stability circle)
•
Oscillator design procedures.
then the next task is to tailor-design a load termination r 2 such that when r2 is mapped back to the rl-plane by lIS!I, the oscillation start-up condition of Irls!ll > 1 is satisfied, or in other words, the mapped value of r2 is closer to the centre of the rl-plane than is the chosen rl. The ideal Z2 (or r2 ) should be an impedance function of frequency which, when mapped into the rl-plane, shows a frequency characteristic that has a minimum value of IlIs!11 at the intended oscillating frequency, so that signal strength at this particular frequency can be built up much faster than at other frequencies. Example 8.1
The common-emitter s-parameters of a BJT, HXTR3102, measured at 1.2 GHz under the biasing condition of V CE = 15 V and Ic = 30 rnA are SUm Sl2m
== O.5262L - 141.86° == O.089L33.15°
S21m S22m
== 3.812L85.56° == O.555L - 47.36°.
~
O_S_C_IL_L_A_T_O_RS__________________~
L I_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
Using this transistor to design an oscillator at 1.2 GHz for a 50-ohm load. Solution The Rollet's stability factor of the transistor at 1.2 GHz is given by 1 - ISllml2 - IS22ml2 - ILlI2 K = ----'--'=:.:..:...,.--=--==::,'------'--'21 s 12mS21ml '
where L1
= SllmS22m
-
S12mS21m'
Hence
K = 0.727
< 1.
The device is potentially unstable at 1.2 GHz. However, K is not very much smaller than unity, and this means that the choice of input and output impedances that will make the device oscillate is limited. In order to make the transistor more liable to oscillate at 1.2 GHz, a capacitive feedback from emitter to ground is added. This will make K smaller. The feedback is shown schematically in Fig. 8.11. C is normally chosen to have a reactance of the same order of magnitude as the input impedance of the active device. For a transistor such as HXTR3102, there are two emitter pads in its package, hence C can be conveniently realized by putting the emitter pads on a pair of open-circuit microstrip stubs as shown in Fig. 8.11. Once the reactance of C is chosen the required microstrip stubs can easily by calculated using techniques described in Chapter 3. Open-circuit microstrip implementation ofC
I (a)
(b)
Fig.8.11 Series capacitive feedback: (a) schematic and (b) microstrip realization of the capacitor C.
S_U_M_MA ___ RY__O_F_D_E_S_IG_N__ PR_O_C_E_D_U_R_E_S____------~I
L -_ _ _ _ _ _ _ _ _ _ _
r2 -
plane
Load stability circle
Fig. 8.12 Load plane realization.
Using (8.2) the z-matrix Zm of the device before adding feedback is given by where Ro
= 50 Q or ZUm = ZI2m =
22.94LI7.01° 8.62L5.27°
Z21m = Z22m
369.35L57.68°
= 93.I5L - 22.88°.
Since the input impedance of the device is of the order of Zllm = 22.94 Q, the impedance of the feedback capacitance is arbitrarily taken as -j18 Q. The feedback network can be represented by the z-matrix Zf, given by Zf =
( -jI8 -j18
-jI8) -j18 .
The overall z-matrix Z, of the transistor with the series capacitive feedback is thus given by
~
~
IL-___________________O_S_C_IL_LA_T_O_R_S__________________~ Z = Zm
or
Z11
=
Z12
=
24.73L - 27.47° 19.34L - 63.64°
+ Zf 354.17L56.11° - 32.34°.
Z21
=
Z22
= 101.58L
The s-matrix corresponding to Z can be obtained using (8.3). The sparameters of the transistor with feedback are given by S11
= 2.192.d21.8°
S12
=
0.377L - 1°
S21
=
S22
=
6.897L118.75° 1.153L - 82.1°.
The stability factor of the modified device, i.e. the transistor with feedback, is calculated to be K = -0.44 < l. The load stability circle Ir2 - A = b can be found by using (8.20)
i'
A = 2.341L - 9.18°, b = 1.72.
The load stability circle is shown in Fig. 8.12 on the rz-plane. The arrows on the load stability circle are pointing at the region where Irll < 1 and
Ir2 1<
l.
The resonator in the base-to-ground circuit is chosen to be a capacitor with capacitance C1 and it is assumed to have a small resistance of 2.5 Q. The resonator circuit is shown in Fig. 8.13. A locus of the resonator capacitor at 1.2 GHz for various values of C1 corresponding to ZI = 2.5 - j100 Q to ZI = 2.5 - jO Q is plotted on the rr plane as shown in Fig. 8.14. The locus of ZI for various values of C1 is then mapped into the r 2-plane by 1IS 22 . This locus on the rz-plane represents the load terminations required for their corresponding source terminations ZI in order for the transistor to oscillate in the steady state.
s
r------,
f
I I
f
I I
f
I
I
I
I I
I I
I f
IL Fig. 8.13
-
_ _ _ _ _ ...1f
Resonator circuit of the oscillator.
c===___________SU_M__M_A_R_Y__OFD_E_SI_G_N_P_R_O_C_E_D_U_R_E_S __________~I ~ fl - plane
21 for various
C1
Source stability circle
Fig. 8.14
Source plane considerations.
The next task is to choose a load impedance Z2 close to the mapped Z, locus on the Tz-plane such that when Z2 is mapped back to the T,-plane by lIs!1, it satisfies the 'start oscillation' criterion of IT,I > IlIs!]I· Z2 (normalized by 50 Q) is chosen to be 1 + j3.2 Q because it lies on the R = 1 circle on the Tz-plane. Z2 is then mapped by lis!] as shown in Fig. 8.14. The 'start oscillation' criterion is satisfied. There are many ways to realize Z2 = 1 + j3.2 or Z2 = 50 + j160 Q. Note that Z2 is what it would be best for the transistor output port to see, hence for the sake of simplicity, an inductor L in series with the 50-ohm output load is taken as the Z2 as shown in Fig. 8.15. The value of L is given by 2n x 1.2 x 1Q 9 L = 160
L = 21.2nH.
If a discrete inductor is not desirable, the output impedance Z2 can be
realized by a simple stub matching network, single or double. Design
Output matching network
r--------, L
50n
L________ .-J Fig. 8.15
Output matching circuit.
To base supply
HXTR3102
50n
C,
(3 - 15 pF)
I
Fig. 8.16 Circuit for Example 8.1.
i.!~~8 n) open-circuit stubs
~____________________P_R_O_BL_E_M_S____________________~I ~ techniques for stub matching networks were discussed in some detail in Chapter 4. The value of the resonator capacitance C1 can be found in Fig. 8.14 by noting that the mapped value of Z2 is near to the section of the locus of Zl where the normalized reactance is about -0.3 to -0.4, hence the required reactance of Zj is about -0.35 x 50Q = -17.5Q, or 1 2n x 1.2 x 109 x C 1 C1 = 7.6pF.
=
175 .
Therefore a 3-15 pF trimmer capacitor can be used for C 1 . The overall circuit is shown in Fig. 8.16. PROBLEMS 1. A 2 x 2 s matrix Sm representing a transistor (BJT or FET) with one of the terminals taken as common can be converted to a 2 x 2 y-matrix Y m' Show that Ym = (Yll Y21 1
Y1Z) Y22
= Ro (/0 - Sm) (/0
+ Sm)
-1
,
where 10 is a 2 x 2 identity matrix and Ro is the system impedance (usually 50 Q) with reference to which Sm is defined. 2. A 3 x 3 y-matrix Y representing a transistor in terms of a three-port network with an arbitrary common terminal usually taken as the circuit ground can be obtained from the 2 x 2 y-matrix Y m as defined in Problem 1. Show that
where Y13 = -(Yll + Y12) Y23 = -(Y21 + YZ2) hI = -(Yll + Yzd Y32 = -(Y1Z + Y2Z) Y33 = (Yll + Y1Z + Y21 + yn).
3. Show that the conversion from the y-matrix Y to the s-matrix S, both describing the same n-port network (n = 2,3, ... ), is given by
~
O_S_C_IL_L_A_T_O_RS__________________~
L I_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
2
2
3 x 3
S
3
I
Ground
Ground
Fig. P.S.l
-
52 x 2
r---------,
I
I
-
~
jX
I I
I I
.
~
.....
-.....
Ym
I _ I L ____-___ -1
Fig. P.S.2
from 5 m
.....
.....
--
Parallel feedback for P.8.S.
.,
II
C
ZOI
1 --
--
(T I)
T --
Output
500. load
--
Fig. P.S.3 Common-base oscillation for P.8.6. Note: Tl and T2 are microstrip lines of characteristic impedance ZOI and Z02, respectively.
L-_________________F_U_R_T_H_ER __R_EA_D_I_N_G________________
~I ~
S = (Go/o + y)-I. (Go/o - Y),
where 10 is an n x n identity matrix and Go = lIRo is the system admittance with reference to which S is defined. 4. A transistor is described by a 3 x 3 matrix S. Show that if port 3 is terminated by a load of reflection coefficient r3 , the resulting 2 x 2 s-matrix S with respect to ground is given by (refer to Fig. P8.1)
5. The common-emitter s-parameters of Motorola BJT MRF901 at 500 MHz under the biasing condition of V CE = 10 V and Ic = 15 rnA are given by Slim
= O.50L - 166° S12m = O.05L57°
S21m
= 6.81L93°
If the feedback is to be added from collector to base as shown in Fig.
P .8.2 in order to make the transistor more liable to oscillate, derive the necessary matrix formulae and write a computer program to evaluate the stability factor K and to plot the load stability circle as a function of the feedback reactance X from X = -100 Q to X = 100 Q in steps of lOQ.
6. The s-parameters of a common-base transistor at 2 GHz under a certain biasing condition are given by Slim
= O.94L174°
SI2m
= O.013L98°
Complete the oscillator design at 2.0 GHz by calculating the values of the circuit elements, C, 110 Zo1o 12 and Z02 as shown in Fig. P.8.3. 7. Derive an approximate formula for estimating the output power of an oscillator based on a class-A biased device. State the assumptions made in your derivation. 8. Using the Motorola r.f. transistor MRF901 (see p. 160) design an oscillator at 912.5 MHz (actually tunable over 905-920 MHz) for use in the mobile telephone receive-end of the CSL-T ACS system of Hong Kong (receive range 835-850 MHz). The output power of the oscillator should be no larger than 23 dBm and no less than 20dBm. Choose your bias accordingly.
~
IL-___________________O_S_C_IL_L_A_TO__RS__________________~ Table I Ie
(mA)
(MHz)
5.0
5.0
S'I
S"
f
VeE (Volts)
Szz
SI'
IS'II
Lq,
Isul
Lq,
Iszzl
Lq,
-38 -75 -141 178 130
11.30 9.48 5.40 2.93 1.51
153 133 100 76 48
0.03 0.05 0.07 0.09 0.16
68 55 43 48 62
0.92 0.76 0.48 0.40 0.35
-17 -29 -44 -56 -85
0.57 0.51 0.52 0.52 0.59
-58 -103 -161 166 125
16.95 12.61 6.24 3.24 1.66
145 123 93 73 47
0.03 0.04 0.06 0.09 0.17
63 53 50 61 67
0.85 0.64 0.38 0.33 0.29
-23 -35 -45 -54 -84
100 200 500 1000 2000
0.48 0.47 0.53 0.53 0.60
-75 -121 -170 162 123
20.08 13.89 6.44 3.33 1.70
139 117 91 72 46
0.02 0.04 0.05 0.09 0.18
61 53 56 66 68
0.80 0.57 0.34 0.31 0.28
-27 -38 -44 -52 -82
20
100 200 500 1000 2000
0.44 0.47 0.53 0.53 0.61
-88 -132 -175 159 122
21.62 14.33 6.45 3.31 1.69
136 114 89 70 45
0.02 0.03 0.05 0.09 0.18
60 54 60 68 70
0.76 0.53 0.32 0.31 0.28
-28 -38 -41 -50 -80
30
100 200 500 1000 2000
0.43 0.50 0.57 0.57 0.65
-112 -148 178 156 121
21.45 13.38 5.82 2.99 1.50
130 109 86 68 42
0.02 0.03 0.05 0.08 0.18
58 57 65 73 74
0.72 0.51 0.35 0.35 0.33
-28 -33 -34 -46 -78
Is,,1
Lq,
100 200 500 1000 2000
0.71 0.62 0.54 0.53 0.59
10
100 200 500 1000 2000
15
Table II Ie
(mA)
(MHz)
10
5.0
ISZlI
Lq,
-35 -69 -135 -177 132
11.32 9.69 5.65 3.11 1.58
0.59 0.52 0.49 0.50 0.57
-52 -95 -156 170 126
100 200 500 1000 2000
0.51 0.47 0.50 0.50 0.58
100 200 500 1000 2000 100 200 500 1000 2000
Szz
Isul
Lq,
ISzzl
Lq,
154 135 101 77 48
0.03 0.05 0.07 0.08 0.14
69 57 43 50 66
0.93 ·0.79 0.54 0.47 0.41
-14 -25 -38 -48 -75
17.06 13.06 6.58 3.44 1.75
147 125 95 74 47
0.02 0.04 0.05 0.08 0.16
64 54 51 62 70
0.87 0.69 0.45 0.41 0.36
-19 -30 -37 -45 -72
-66 -112 -166 164 124
20.36 14.48 6.81 3.54 1.78
141 119 92 72 46
0.02 0.03 0.05 0.08 0.16
63 54 57 67 72
0.83 0.63 0.41 0.39 0.35
-22 -31 -35 -43 -70
0.47 0.46 0.50 0.51 0.59
-78 -123 -171 162 123
22.08 15.07 6.84 3.51 1.77
138 116 90 45
0.02 0.03 0.05 0.08 0.17
61 55 60 69 73
0.80 0.60 0.40 0.39 0.35
-23 -30 -32 -41 -68
0.44 0.47 0.53 0.54 0.62
-98 -139 -177 158 122
22.70 14.47 6.33 3.26 1.61
133 111 87 69 42
0.02 0.03 0.04 0.07 0.16
59 55 65 74 77
0.76 0.57 0.43 0.43 0.39
-23 -27 -28 -39 -68
Is,,1
Lq,
100 200 500 1000 2000
0.73 0.63 0.53 0.51 0.57
10
100 200 500 1000 2000
15
20
30
Su
SZl
S"
f
VeE (Volts)
71
Motorola r.f. transIstor MRF901.
F_U_R_T_H_ER__R_EA_D_I_N_G________________~I
L -_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
FURTHER READING Abe, H., et al. (1978) A highly stabilized low-noise GaAs FET integrated oscillator with a dielectric resonator in the C-band, IEEE Transaction on MIT, vol. MIT-20, March. Alley, G.D. and Wang, H. (1979) An ultra-low noise microwave synthesizer, IEEE Trans. MIT, vol. MIT-27, No. 12, December. Basawapatna, G.R. and Stancliff, R.B. (1979) A unified approach to the design of wide-band microwave solid-state oscillators, IEEE transaction on MIT, vol. MIT-27, no. 5, May. Ha, T.T. (1981) Solid State Microwave Amplifier Design, Wiley. Ishihara, O. et al., (1980) A highly stabilized GaAs FET oscillator using a dielectric resonator feedback circuit in 9-14 GHz, IEEE Trans. MIT, vol. MIT-28, No.8, August. Johnson, K.M. (1979) Large signal GaAs MESFET oscillator design, IEEE Trans. MIT, vol. MIT-27, No.3, March. Kurokawa, K. (1976) Microwave solid state oscillator circuits in Microwave Devices (ed. M.J. Howes and D.V. Morgan), Wiley. Murphy AC. and Murphy P.J. (1988) Computer program aids dielectric resonator feedback oscillator design. Microwave Journal, September. Pengelly R.S. (1984) Microwave Field-effect Transistors - Theory, Design and Applications, Wiley. Vendelin G.D. (1982) Design of Amplifiers and Oscillators by the SParameter Method, Wiley.
~
9
The Spectrum Analyser and its Applications
9.1 INTRODUCTION A spectrum analyser is basically a test instrument for displaying signals in the frequency domain, normally over a wide range of frequencies. It is capable of displaying the magnitude spectrum of signals, that is, the magnitude versus frequency characteristics of signals, in a similar way that Fourier series or Fourier transforms 'display' the Fourier coefficients or the Fourier integral as a function of frequency. Spectrum analysers are perhaps the most popular pieces of equipment in r.f. engineering because of their wide range of application in this area. The primary area of applications is in the analysis of signals from a source such as the output of an oscillator, the output of a modulator and the input to a receiver. With a reasonable degree of accuracy they can also be used to perform noise measurements on signals such as the carrier-to-noise ratio of a received signal and the noise figure of a two-port circuit such as an amplifier. Together with a matched tracking generator, a spectrum analyser is capable of measuring the frequency response and VSWR of two-port networks such as amplifiers and filters. A modern spectrum analyser normally consists of a superheterodyne receiver which captures an input signal and converts it to an intermediate frequency by means of an up or down frequency converter. The front end of a basic spectrum analyser is shown in Fig. 9.1. Suppose that the signal to be analysed by the spectrum analyser can be decomposed by Fourier series into four frequency components as shown in Fig. 9.2. This composite signal consisting of four sinusoidal signals of various amplitudes is fed to the input of the spectrum analyser and the spectrum analyser should be able to display on its CRT a plot very similar to that shown in Fig. 9.2. The intermediate frequency (IF) of the filter shown in Fig. 9.1 is set outside the measuring frequency range, so that input signals cannot pass directly through the IF filter without being transformed into an IF signal. Most spectrum analysers have their IF set at a frequency higher than the measuring frequency range.
~__________________IN_T_R_O_D_UC_T_I_O_N__________________~I ~ Mixer up or down converter Input -----II~ signal
IF bandpass filter
IF signal to detection I-----I~ and display L....-_ _---l circuitry
Local signal
Fig. 9.1
Front end of a spectrum analyser.
Amplitude
Frequency
Fig. 9.2 Spectrum of a test signal.
The local signal in Fig. 9.1 comes from a swept oscillator which when sweeping through a certain range will convert each frequency in the intended measuring range to a fixed IF according to mixer output frequency = fIF =
fLO ± fs,
(9.1)
where fIF is the intermediate frequency, fLo is the local oscillator frequency, and fs is the input signal frequency. For example, a certain spectrum analyser has a measuring range from 0 Hz (theoretically only) to 1.7 GHz. The intermediate frequency fIF is chosen to be 2.3 GHz. In order to cover the whole measuring range from 0 to 1.7 GHz, the local oscillator (LO) must also be sweeping over a range of 1.7GHz. There are two possible choice for the LO sweep range, namely 0.6 to 2.3 GHz and 2.3 to 4.0 GHz as shown as band 1 and band 2, respectively, in Fig. 9.3. Ifthe LO is sweeping through band 1 (0.6-2.3 GHz) , only the fLO + fs will be translated to anfIF = 2.3 GHz, but band 1 coincides in part with the
~
IL________________TH_E__SP_E_C_T_R_U_M_A_N_A_L_Y_S_E_R______________~
I~
Ib't Ib
f~'
=======B=a=n=d=1=======IIIr-----B-an-d-2-----. 0.6 GHz
10 o Fig. 9.3
2.3 GHz
4.0 GHz
Input measuring range
~-------~~-----~~~h
1.7 GHz
Relationship between input and local oscillator frequencies.
input range, hence in order to provide better isolation between the mixer ports and to avoid harmonics of the input from hitting the IF port, band 2 (2.3-4.0 GHz) is chosen to be the LO sweep range, i.e.
hF
=
fLO -
fs·
(9.2)
Thus the output of the IF filter in response to the composite input signal as shown in Fig. 9.2 is non-zero when the LO sweep is at 2.3 GHz + Ii> 2.3 GHz + Iz, 2.3 GHz + /J and 2.3 GHz + h The magnitudes of the IF filter output at these instances are Ai> A 2 , A3 and A 4 , respectively. 9.2 OPERATING PRINCIPLE OF A SPECTRUM ANALYSER
The block diagram of a practical spectrum analyser covering a measuring range of 0-1. 7 GHz is shown in Fig. 9.4. As mentioned in the last section, the intermediate frequency is chosen to be 2.3 GHz. The high value for IIFI minimizes image interferences by shifting the image frequencies well above the IF passband. The second intermediate frequency !IF, is set at a relatively low value of 70 MHz. This implies that the second local oscillator frequency lLO, is fixed at 2.23 GHz. The third mixer receives a signal frequency of 70 MHz and a third LO frequency (fLO) of 68.5 MHz in order to produce a very low third intermediate frequency I IF3 of 1.5 MHz. A low !IF3 is desirable because narrow (absolute) IF bandwidth of the filter can be more easily achieved if the centre frequency is low and the detector can be more sensitive and cheaper at a lower frequency. 1. The horizontal axis of the CRT is swept by a saw-tooth signal. In Fig. 9.5: A to B is the sweep time; B to C is the return time, which does not appear on the CRT because of blanking; and C to A' is the time during which the sweep is stopped and the next sweep begins at A'. 2. The output of the saw-tooth generator is also fed to the first local oscillator which is often a YIG-tuned oscillator. This oscillator is a
Fig. 9.4
(850 MHz)
1st mixer
2nd LO
Aoz
Sawtooth generator
3rd mixer
r)
~
rXJ
3rd IF 1.5 MHz BPF
T
CRT
IF amp.
-----------j
1-1
r)<.J
r)
rp
2nd IF 70 MHz BPF 0....)
~
2nd mixer
0....)
f)<;
1st IF 2.3 GHz BPF
Block diagram of a spectrum analyser.
0....)
()
1.7 GHz LPF
h
*-
Detector
~I
THE SPECTRUM ANALYSER
L -_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
~
wide band voltage-controlled oscillator (VCO) and sweeps in the frequency range from 2.3 to 4.0 GHz. For example, let us consider the sweep which is set over the entire range of 0 to 1.7 GHz. The frequency of the local oscillator at the left edge A of the CRT (Fig. 9.5) becomes 2.3 GHz, and, at the right edge B, 4.0GHz, and the frequencies between these edges are linearly swept in proportion to the inclination of A ---'» B. 3. The input signal and the first LO signal are converted by the mixer to a frequency of fLO - fs. Only when this difference in frequency is equal to 2.3 GHz (fIF) would the output of the mixer be picked up by the first IF filter of centre frequency equal to 2.3 GHz and fed to the second mixer and the subsequent circuits. For example, if the input signal frequency fs is 850 MHz and the first LO begins to sweep from 2.3 GHz: at the beginning of the sweep the mixer output frequency is 2.3 GHz and no signal can get through the 2.3 GHz IF filter; only when fLO, is at 2.3 GHz + 850 MHz would a signal pass through the first IF filter. In other words, when the LO sweep which began at 2.3 GHz reaches 3.15 GHz, the output of the mixer becomes 2.3 GHz and passes the 2.3 GHz band-pass filter. As pointed out in (2), the local oscillator is linearly swept from 2.3 to 4.0 GHz, so that the mid-point of the CRT horizontal axis corresponds to 3.15 GHz. 4. We have explained that the first local oscillator frequency is varied from 2.3 to 4 GHz and the signal is picked up when the mixer output becomes 2.3 GHz. However, a problem arises when this local frequency itself becomes 2.3 GHz. The local signal output is at a considerably higher power level than the input signal so as to enable the mixer output to have a response in proportion to the input power level. Consequently, part of this local signal will pass directly to the 2.3 GHz band-pass filter as a leakage of the first mixer even when no signal of 'zero frequency' is present at the input. Hence in a spectrum analyser display which covers
B
B'
~ A
CN
C'
Scan signal
Fig. 9.5
Scanning the horizontal axis.
A'
A
B
~C----CRT beam horizontal movement
~_ _ _ _ _ _O_P_E_R_A_T_IN_G_A_S_P_EC_T_R_U_M_A_N_A_L_YS_E_R_ _ _ _ _~I ~ zero hertz, there is always a fairly substantial spectral component at the origin. 5. The signal converted to 2.3 GHz is mixed with the 2.23 GHz signal of the second local oscillator to become a 70 MHz signal, and it is further mixed with the 68.5 MHz signal of the third local oscillator to become the final IF signal of 1.5 MHz. 6. The third IF signal is detected by a peak detector after it has been passed through the third IF filter of bandwidth selectable from the front panel. The last IF bandwidth of a spectrum analyser is normally selectable over the range from 100 Hz to 300 kHz in steps. The filtered third IF signal is then amplified by a logarithmic or linear (selectable) amplifier before being fed to the vertical-axis input of the CRT. Figure 9.6 shows the spectrum analyser display of an input sinusoidal signal of Is = 850 MHz when the first local oscillator sweep range is set from 2.3 GHz to 4.0 GHz which is equivalent to setting an input measuring range of 0-1.70GHz. Figure 9.7 shows the graphical representation of the spectrum analyser display when the control settings are varied. The diagram is self explanatory and is not further elaborated here.
I
I
I
Input frequency LO frequency
Fig. 9.6
~ ~ ~ ~ ~ lNvJ 'Wvvi VNw ~ ~ 0
850 MHz
1.7 GHz
2.3 GHz
3.15 GHz
4 GHz
An example of full scanning.
HH_I
THE SPECTRUM ANALYSER
- - - - - - - - - - - - -----------
Setting of control Freq. span (wide)
----1
B
T~
A
-
/
Fig. 9.7
/
/
/
/
/ /
/
/
\
\
Freq. span (narrow)
Centre freq.
Sample display with wide span (waveform A)
\
\
\
\\
\
\
Sample display with narrow span (waveform B)
Frequency span and centre frequency controls.
9.3 CHARACTERISTICS OF A SPECTRUM ANALYSER A spectrum analyser is characterized by a number of important parameters, an understanding of which is essential for proper use of the equipment. These parameters are discussed in this section.
9.3.1 Frequency range The highest frequency that a spectrum analyser can measure is limited by the range of the sweep of the first local oscillator (a yeO). For example, the spectrum analyser used as an illustration in the last section has a yeO tuning range of 2.3 to 4.0 GHz; the upper frequency limit for the input signal is therefore equal to 4.0 GHz-fIF, = 4.0 GHz-2.3 GHz = 1.7 GHz. The lower frequency limit is determined by the sideband noise of the local oscillator leakage. When the local oscillation frequency fLO, = 2.3 GHz, this LO signal will leak through the mixer and be picked up by the first IF stage, which will eventually appear on the display as a 'signal' at zero
~________C_H_A_R_A_C_TE_R_I_ST_I_C_S_O_F_A_S_P_E_C_T_R_UM __A_N_A_L_Y_S_ER________~I ~ frequency. The noise sideband is defined as the undesirable response caused by noise generated inside the spectrum analyser (mainly due to the local oscillators) and it appears on the display in the vicinity of a desirable response. The LO leakage and its associated sideband noise are shown in Fig. 9.8. The lower frequency limit depends on what sensitivity is required. With reference to Fig. 9.8, if the required sensitivity is -90dBm, the limit is about 7 kHz, but if the required sensitivity is -100 dBm, the lower frequency limit must be greater than (roughly) 14 kHz.
9.3.2 Resolution The frequency resolution of the spectrum analyser is its ability to separate signals closely spaced in frequency. The frequency resolution is determined by the bandwidth and shape factor of the last IF filter and by the sideband noise of the spectrum analyser. The IF bandwidth (last filter before detection) is normally specified in terms of its 3-dB bandwidth. A narrower IF bandwidth provides better resolution but requires more poles to implement the filter, hence a
-20 dBm 10 dB/div
/
/ \
\~
I
.~,. I I
I I
o
~ ~A .l~ AA.I~ I 'Vl'
10 kHz
5 kHzldiv
.l~ j A.l~ AA Jf AA.l I~~ il~~ I~~Y' IfV~' IVY~
20 kHz
IFBW:
300 Hz
AA
!
30 kHz
Fig. 9.8 LO leakage and lower limit of measurable frequency.
~
IL-______________T_H_E__SP_E_C_T_R_U_M_A_N_A_L_Y_S_ER______________~ narrower IF filter has a longer propagation delay and thus imposes a limit on the sweep rate. For example, in order to measure an AM or FM signal of a modulating frequency of 1 kHz, the IF bandwidth has to be at most 300 Hz, preferably 100 Hz. In measuring closely spaced frequency components, the characteristic of the filter shape (skirt) is important. The filter skirt is normally characterized by the ratio of LJh dB /LJf60dB (the 3-dB to 60-dB bandwidth), which is known as the 'IF shape factor'. The shape factor for most commercially available spectrum analysers ranges from 1:15 to 1:5, the larger the ratio the better, i.e. 1:5 is better than 1:15. Figure 9.9(a) shows the bandpass characteristic of an IF filter of shape factor approximately equal to 1:7. Figure 9. 9(b) shows the bandpass characteristic of another IF filter of shape factor approximately equal to 1: 15. A small signal of power 50 dB below that of a stronger signal and at a frequency 600 kHz from it will not be detected by the spectrum analyser of an IF shape factor of 1: 15 as illustrated in Fig. 9.9(b). However, if the same signals are input into a spectrum analyser whose IF shape factor is 1:7 such as the one shown in Fig. 9.9(a), the smaller signal can be detected. In order to make an IF filter with a sharper roll-off characteristic, the number of poles in the filter has to be increased, and this will increase the response time of the filter. Thus the response-time requirement imposes a limit on how sharp an IF filter can be.
9.3.3 Stability It is important that a spectrum analyser be more frequency stable than the signals it measures. The stability of the srvctrum analyser depends on the frequency stability of its local oscillators. Stability is usually categorized in terms of short-term or long-term properties. Residual FM is a measure of the short-term stability which is similar to having low-frequency signals frequency modulating the local oscillator and hence the displayed signals. Residual FM is usually specified in terms of the peak-to-peak deviation in hertz. Residual FM is illustrated in Fig. 9.1O(a). Short-term stability is also characterized by its sideband noise, usually specified in terms of: 'X dB below the carrier when measured with an IF resolution (bandwidth) of B Hz at a frequency offset of YHz from the carrier'. The sideband noise specification is an indication of the spectral purity of the local oscillators of the spectrum analyser. The sideband noise of a spectrum analyser is illustrated in Fig. 9.1O(b). Long-term stability is characterized by the frequency drift of the local oscillators of the analyser. Frequency drift is usually specified by the drift within a certain duration, i.e. in hertz per minute or hertz per hour.
C_H_A_RA __C_T_E_RI_S_TI_C_S_O_F_A __S_PE_C_T_R_U_M__ A_NA_L_Y_S_E_R________~I
L -_ _ _ _ _ _ _
tJ.Fi 3 dB)
-ll-
/ \
II \
!
/
/"
/
t
-10
tJ.F(60 dB)
/
-1000 kHz
-3JL
\,
\
V
tJ. Ref.
-20
\
/
0
t3JB
-40 -50
"Xl\
L
o G
dB
-60
/ ~ -70
~ +1000 kHz =100 kHz =200 kHz/div
IFBW Sweep rate (a)
f...--"
/
/
/
I
0
I\\ /
-10
\
-20
'" " /\
;1
/ -1000 kHz
IF BW Sweep rate
-40 -50
r'-
~
Undetected sig~al
-30
L
o G
dB
-60
-70
I, +1000 kHz = 100 kHz =200 kHz/div
(b)
Fig.9.9
tJ.
Ref.
Filter shape: (a) 1:7 IF filter and (b) sideband noise.
~
0 --10
1--
Ref.
-20
1_ 30 1_ 40
o
I
G dB
.:'-"-.-
-l-:-~
t---
",
.'" "" 'If'
'l1' '11' I''fl' ''1'
,
~
99.5 MHz
L
r 60 I
-70 --'-"---
,.I.
['1",", yr"If' i""
,.I
' 'rr'VIY' 100.5 MHz
100 MHz (a)
0 -10 -20
( "'\.
--
-30
--
/ \
/
IldliL lil,ll
99 MHz
XdB ---r~l
down-e
1\
J IlIfJflrfHlII
-40
\
I
Ref.
L
o
G
dB
! 1-70
---''-
lilu 111I11l~AH'lI
IAIIH11I
In"T
\Til
~
YHz away
IIIlfII III \111
~
101MHz
(b)
Fig, 9,10 Stability of a spectrum analyser: (a) residual FM, and (b) sideband noise.
T_RA __C_K_I_N_G_G_E_N_E_RA __ TO __ R______________~I
L -_ _ _ _ _ _ _ _ _ _ _ _ _ _ _
9.3.4 Dynamic range The dynamic range of a spectrum analyser is determined by the noise level of the analyser and the linearity and input power rating of its front-end amplifier. For most spectrum analysers the input power rating is limited to +13dBm (to 30dBm exceptionally), above which damage to the equipment may occur. At high input power levels, say above OdBm, the system is not linear due to saturation of its input amplifier. In this case, an input (r.f.) attenuator may have to be used. The lower input power limit is governed by the system noise level. A signal to be measured must have a power higher than the average noise power of the system in order for it to be displayed. The average noise level, known as the noise floor, is roughly equal to (kTB x amplification factor
+ noise added by the analyser),
where k is the Boltzmann constant, T is the system's absolute temperature and B is the IF bandwidth. kT is equal to -174dBm per hertz at room temperature. A typical value for the noise floor is -120 dBm to -130 dBm for 1 kHz IF bandwidth. 9.4 TRACKING GENERATOR
A tracking generator (see Fig. 9.11) is a swept oscillator running at the observation frequency of its matched spectrum analyser. The trackingIF amplifier
Is
CRTn
VCO
T Ivco
SPECTRUM ANALYSER TRACKING GENERATOR
Fig. 9.11
A spectrum analyser with its matched tracking generator.
~
~
TH_E__SP_E_C_T_R_U_M_A_N_A_L_Y_S_E_R______________~
L I_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
generator output signal is generated by mixing the swept oscillator signal of the spectrum analyser and a fixed frequency oscillator signal at the spectrum analyser IF frequency. The tracking generator has a stable and fixed local oscillator of frequency ifF' ifF is chosen to be equal to the IF frequency IIF of the spectrum analyser, so that
Is Is
and
=
Ivco - !IF
(9.3)
=
Ivco - fiFo
(9.4)
Hence, the tracking generator output frequency (fs) precisely tracks the tuning frequency (fs) of the spectrum analyser since both are effectively tuned by the same VCO. This precision tracking exists in all analyser scan modes. Thus, in full scan, the tracking generator output is a start-stop swept oscillator, in 'per division' scan the output is a LJF swept oscillator and in 'zero' scan the output is simply a continuous sinusoidal (CW) signal.
9.5 APPLICATIONS OF SPECTRUM ANALYSERS In this section a few examples showing how spectrum analysers can be used in some basic applications will be briefly illustrated.
9.5.1 Amplitude modulation A tone-modulated AM signal can be written in the form IAM(t) = Ae(1 + mcoswmt)coswet or
IAM(t) = Ae cos wet
+ m:e cos(We + Wm)t + m;c cos (We
- Wm)t,
(9.5)
where m is the AM modulation index, and We and Wm are the carrier and the modulating tone (angular) frequencies, respectively. The ratio of the carrier power to the power of one sideband is (~A~)! (kA~m2), or 41m2. Figure 9.12 shows the time-domain (CRO display) and frequency-domain views of a 2 % AM. From the time-domain display on the CRO, it is virtually impossible to measure the modulation index. On the other hand, the frequency-domain display on a spectrum analyser shows clearly that the carrier power is 40 dB above that of its sideband power, hence 4 40 = 1OloglO--2
m
or
m
=
0.02.
J@l
APPLICATIONS OF SPECTRUM ANALYSERS Spectrum analyser display
CROdisplay
I0 I
-10
~20 I
-30
~40
L
o
G
I
-50
~60
dB
I
,~
2% AM time domain
Fig. 9.12
.
•••
""n~
tl.al
""
I.
~[9I
2% AM frequency domain
Amplitude modulation.
9.5.2 Frequency modulation Tone-modulated PM and FM have the same spectral distribution. A tonemodulated FM or PM is represented by (9.6) n=-oo
0.8
0.6 0.4
0.2
o -0.2
2
Fig. 9.13
3
456
7
8
Plot of a Bessel function of the first kind In(J3).
9
10
11
12
~
T_H_E__ SP_E_C_T_R_U_M_A_N_A_L_Y_S_E_R______________~
L I_ _ _ _ _ _ _ _ _ _ _ _ _ _ _
where J n(f3) is the ordinary Bessel function of the first kind and of the nth order, and f3 is the modulation index. The plot of I n(f3) as a function of f3 with n as a parameter is shown in Fig. 9.13. For distortion-free detection of the modulated signals, all sidebands must be transmitted. This implies a transmission bandwidth of infinity. In practice, only the significant sidebands are transmitted, so that the transmission bandwidth is finite. By the word 'significant' it is usually meant all those sidebands which have a voltage of at least 1% (-40 dB) of the voltage of the unmodulated carrier. Measurement of FM (or PM) bandwidth with a spectrum analyser is done by simply counting the significant sidebands. Table 9.1 shows the value of f3 where Jo(f3) (the carrier term) vanishes. Table 9.1 Zeros of carrier amplitude Order of carrier zero
Modulation index (fJ)
1 2
3 4 5 6
2.40
n(n > 6)
18.07
5.52 8.65 11.79 14.93 18.07
+ .n(n -
6)
Table 9.1 can be used in conjunction with the spectrum analyser to find the important parameters of an FM or PM signal, namely the FM peak frequency deviation L1f(Hz) and the FM modulator sensitivity mf(Hz V-I), the PM peak phase deviation f3 (rad) and the PM modulator sensitivity mp(rad V-I). These parameters are related by fJ = mfAm for FM fm fJ = mpAm for PM,
where Am andfm are the amplitude and frequency of the modulating tone. When f3 is increased from zero by increasing the modulating voltage, the magnitude of the carrier of the FM or PM signal displayed on the spectrum analyser will vanish at f3 = 2.40, and then at f3 = 5.52, etc. At the points where the carrier vanishes, the value of f3 can be read from Table 9.1, fm can be read from the spectrum analyser display, and Am can be measured at the modulator input, hence the parameters L1f, mf and mp can be found.
A_P_P_L_IC_A_T_IO_N_S__ O_F_S_PE_C_T_R_U_M__ A_N_A_LY_S_E_R_S_________~~
L -_ _ _ _ _ _ _ _ _
9.5.3 Incidental FM (AM plus FM) Although AM and angular modulation are different modulation methods, they have one property in common: they always produce a spectrum with symmetrical sideband amplitudes, although the odd-order sidebands in FM and PM do differ in sign. Figure 9.14 shows a modulated carrier with asymmetrical amplitude sidebands. The only way that we can have one sideband larger than the corresponding one on the other side of the carrier is for both AM and FM or phase modulation to exist simultaneously and at the same modulating frequency. The reason is that the odd-order sidebands in FM and PM differ in sign. Since the sideband components of both modulation types add together vectorally, the resultant amplitude of one sideband is reduced, whereas the amplitude of the other is increased. The spectrum analyser does not retain any phase information and thus displays the absolute magnitude of the resultant. When a relatively small amount of FM exists simultaneously with AM, the modulation index of the AM component can be calculated with acceptable accuracy by taking the average amplitude of the first sideband pair. The amount of incidental FM can also be calculated from the magnitudes of the first sideband pair of the display (Problem 3). Figure 9.14 shows a carrier with its asymmetrical sidebands. The first pair of sidebands are -16 dBc and -12 dBc (dB below carrierlevel). It can be shown that the amplitude modulation index m is 0.42.
d
0
Ref.
-10 -- -20 ~
-40 -50 -60 -70 AIl}.h 1M\, IlAAI' .,\ 1" "f 'I'
Fig. 9.14
,MA iyMi
"'-»
42% AM with incidental FM.
h
\Hea, "1
~IAAIIJ
'r
l.oIA.. , "Y''''
L
o
G dB
~
~
TH_E__ SP_E_C_T_R_U_M_A_N_A_L_Y_S_E_R______________~
L I_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
9.5.4 Frequency measurement of small spectral components A composite signal (e.g. a modulated carrier) has many spectral components. If a frequency counter is used to measure this signal, it will normally capture the spectral component with the largest magnitude and leave the other components undetected. However, the same signal can easily be decomposed into its constituent spectral components and displayed on a spectrum analyser. If the frequency of any of these components, of large amplitude or not, is to be accurately measured, the arrangement shown in Fig. 9.15 can be used. The frequency of any signal or spectral component displayed on a spectrum analyser can be measured simply by stopping the automatic scan and manually scanning the trace. The trace can manually be stopped at any signal displayed and a CW signal of the same frequency will be generated by the tracking generator. The amplitude of the tracking generator output is adjustable and is independent of the original signal amplitude. The tracking generator output is then of sufficient amplitude and spectral purity for the frequency counter to determine accurately the frequency of the signal.
9.5.5 Frequency response of devices The insertion loss or transmission response and the return loss of a device (DUT) can be measured by using a spectrum analyser with either its own tracking generator or an independent swept generator with the sweep controlled by the spectrum analyser, or vice versa. The insertion-loss or transmission-response measurement is set up as shown in Fig. 9.16. A return-loss measurement set-up is shown in Fig. 9.17. The transmission measurement of a DUT as shown in Fig. 9.16 is very straightforward. Before the DUT is inserted, a 'through' connection is
lJl Spectrum analyser
Frequency counter
)
Tracking generator
.... "..
"..
r.f.outpu Input signal
Fig. 9.15 Frequency measurement of spectral components.
A_P_P_L_IC_A_T_IO __ N_S_O_F_S_P_EC_T_R_U_M__A_N_A_LY_S_E_R_S________~I
L -_ _ _ _ _ _ _ _ _
Tracking generator
~
r.f. output
Short circuit for calibration
Calibration
Transmission response Spectrum analyser
Input
0---+---------.---------'
Fig. 9.16 Transmission measurement.
made to establish a reference, and a (theoretically) straight horizontal trace appears on the analyser, labelled as 'calibration' in Fig. 9.16. The calibration trace is stored and compared with the trace when the DUT is connected to eliminate the uneven frequency response of the tracking generator and the cables. The return-loss measurement as shown in Fig. 9.17 employs a directional coupler where the signal reflected from the DUT is picked up by the sampling arm and is displayed by the analyser. A short circuit is used for calibration in order to provide total reflection or 'zero dB' return loss.
9.5.6 Random- (thermal-) noise measurements A spectrum analyser can be used to measure noise as well as signal, or the noise that comes with signals, provided the noise power to be measured is Directional coupler
Tracking 0-+--...-----. generator
L..----r<-,---J
Calibration
Sampling arm
l I I I
I
Spectrum Input analyser
o-...,----..,-----.J
Fig. 9.17
Return loss measurement.
Return loss reference plane
b I
Return loss
Matched termination
I
I
Short circuit for calibration
]
~ ~~~~~ ~~T_H_E~SP_E_C_T_R_U_M ANALYSER LI
_ _
higher than the noise floor of the analyser at a particular IF bandwidth setting. Thermal-noise power is directly proportional to the measurement (last IF filter) bandwidth. The noise-power bandwidth of a spectrum analyser is defined as the ideal rectangular filter bandwidth with the same power response as the actual IF filter of the analyser. This is illustrated in Fig. 9.18. For a spectrum analyser (SA) with a shape factor of 1:15, the noisepower bandwidth BWn is approximately equal to 1.2 times that of the 3-dB bandwidth (BW3dB ). BW 3dB is adjustable on the SA front panel. For SAs with a better shape factor, say, 1:5, BWn < 1.2 X BW3dB . Figure 9 .19( a) shows a display of thermal noise when the IF bandwidth is set at a relatively high value of 100 kHz. It is seen that the noise display occupies many vertical divisions (lOdB/div), hence it is difficult to determine the exact noise level of the input (noise). However, all spectrum analysers are equipped with a video filter of adjustable bandwidth. The video filter is a post -detection lowpass filter used before feeding the detected signal to the CRT display. It tends to give on the display an average value of the noise 'signal'. When used for thermal-noise measurements, the video bandwidth is normally set below 100 times that of the IF bandwidth (BW3db). Figure 9.19(b) shows the same noise input when the video filter is switched on with a 10 Hz bandwidth. The detector used in a spectrum analyser is usually a peak detector, it does not respond well to random noise and hence it does not give the true r.m.s value of the noise power. The detector together with the logarithmic display of the analyser tends to give a lower reading than the r.m.s. noise power, and the combined effect is such that approximately 2.5 dB has to be added to the average value displayed with the video filter switched on. Power
Noise power bandwidth BW n
Equal area under those curves
Ideal rectangular filter response Spectrum analyser IF filter response Frequency
Fig.9.18
Definition of noise power bandwidth BWn-
----.JI ~
APP_L_IC_A_T_I_O_N_S_O_F_S_P_E_C_TR_U_M_A_N_A_L_YS_E_R_S_ _ _ _ 6 Ref.
1 0
1_ 10
1-20
~30
I (a)
IIJ'I IAj
~ijr~\ I
'I I
IABI
ijijl III
AU
IVI
'"
L
o
-40
fl •.
G dB
-50
11101
1-60
ml
-70
IFBW = 100 kHz
Video filter OFF
0 -10
6 Ref.
-20 -30 -40
(b)
-50 A verage noise power
L
o
G dB
-60 -70
IF BW = 100 kHz Video filter ON
Fig. 9.19 Thermal noise measurement.
As an example, suppose that the average noise level (with video filtering) is measured at - 35 dBm in a 10 kHz (IF) bandwidth. The noise in dBm per hertz is desired. 2.5 dB is added to the measured value to account for the detector and log display effect, and the correct noise power over the 10 kHz IF bandwidth is thus -32.5 dBm. The thermal-noise bandwidth BWn = 1.2 X BW3dB , or BW n
= 1.2 x 10kHz = 12kHz.
~
T_H_E__ SP_E_C_T_R_U_M_A_N_A_L_Y_S_ER____________~
L I_ _ _ _ _ _ _ _ _ _ _ _ _ _ _
The noise power per hetz is thus given by 12kHz N = -32.5dBm - 101oglO 1Hz = =
-32.5dBm - 40.8dB -73.3dBm HZ-I.
PROBLEMS 1. The average noise floor of a spectrum analyser, as seen on the display is -110 dBm when the video filter is switched on and the IF bandwidth is set at 1 kHz. The thermal-noise bandwidth factor is 1.1. Determine the noise figure of the spectrum analyser. Calculate also its sensitivity when the IF bandwidth is switched to 3 MHz Ans.: 36dB, -75dBm Note: The noise figure is defined as the ratio of the signal-to-noise at the system input to that at the system output. 2. The noise floor of a spectrum analyser at 1 kHz IF bandwidth is -llOdBm. In order to achieve the stated (by manufacturer) dynamic range of 70 dB at an input level of -30dBm, what is the maximum IF bandwidth setting allowed? What are the implications of the stated dynamic range on the performance of the spectrum analyser in terms of its residual spurious components and higher harmonics? Ans.: 10kHz, -100dBm 3. An AM modulator also produces a small amount of FM. The display of the modulator output on a spectrum analyser shows that the first sideband above the carrier is at -12 dBc and that the first sideband below the carrier is at -16 dBc. Calculate the AM modulation index and the FM modulation index. Ans.: 0.42, 0.09 4. Most spectrum analysers have a built-in comb generator which produces all the harmonics of a fundamental oscillator (e.g. fundamental at 100 MHz and harmonics at 200 MHz, 300 MHz, ... ) for the purpose of frequency calibration. Explain how the comb generator can be ultilized to measure the frequency of a signal with a degree of accuracy better than that of the tuning display. A comb generator of a fundamental frequency of 100 MHz is built in to a certain spectrum analyser. The stability of the comb fundamental oscillator is ± 1 in 108 . If the spectrum analyser is used to measure the frequency of a signal at about 1.5 GHz using the comb-generator method, determine the uncertainty of the most accurate measurement, given that the analyser display has 2048 horizontal points. 5. The carrier level of a composite TV signal (6 MHz transmission bandwidth) is read from the spectrum analyser as -25 dBm with the
~____________________P_R_O_BL_E_M_S_____________________~I ~ video filter switched off. The noise floor of the display with the video filter switched on is read as -95 dBm at an IF bandwidth of 10 kHz. Calculate the carrier-to-noise ratio of the TV signal. Ans.: 40.5dB 6. A log-periodic antenna is intended to be used over a range from 30 MHz to 1000 MHz. The antenna has an N-type female connector. Propose a measurement scheme ultilizing a spectrum analyser with its tracking generator to measure the VSWR of the antenna with respect to the plane of the connector over the frequency range of interest. 7. The total harmonic distortion, THD(%), of a signal is defined as THD(%) = 100 x
V(A~ + A~:l
... +
A~),
where AI is the fundamental amplitude (volts), and Aj is the jth harmonic amplitude (volts); j = 2,3, .... The spectrum analyser display of a certain signal is shown in Fig. P.9.1. Determine the THD(%) of the signal. Ans.: 1.28% 8. In performing antenna measurements, electric field strength is usually expressed in terms of dB!!V m - I (dB with respect to 1 microvolt per metre). Derive an expression to convert dBm readings from a spectrum analyser to dB!! V for 50-ohm and 75-ohm systems. 9. The antenna factor (K) of an antenna is related to the field strength (E) and its receiving voltage (Vr ) by E = KVr . The receiving area of an
0
-10
Ref.
-20 -30 -40 -50
V II 1/ ~ V
0.2 GHz
Fig. P.9.1
Display for Problem 7.
'1 HI ~~~~ II ~~, ~~I
L
o
G dB
-60
AH
-70
~~'I 2.2 GHz
~
TH_E__ SP_E_C_T_R_U_M_A_N_A_L_Y_S_E_R______________~
L I_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
antenna is defined as AT = GA 2/(4.n), where G is the antenna gain (power ratio) and A is the wavelength in metres. Derive an expression to relate the electric field strength and the voltage measured by a spectrum analyser (Vr ) in terms of the antenna gain and the frequency of the received signal. Ans.: K = 2010gf - 10 log G - 29.8dB/m (fin MHz)
FURTHER READING Ai/tech 757 spectrum analyser product note, Eaton Corp. Benedict, B. (1983) Fundamentals of Spectrum Analysers, Tektronix. Hewlett Packard Application Note 150, 150-1, 150-2, 150-7, 150-8, 150-9, 150-10 and 150-11. Noise measurements using the spectrum analyser, Tektronix, Inc., 1975. Spectrum analyser - MS62 series, Anritsu Application Note NO. 4-E, 1978-9.
Microwave Frequency Counting
10.1 BASICS OF DIGITAL FREQUENCY COUNTERS A modern frequency counter is basically a digital instrument capable of converting an input sinusoidal signal or pulse train into digital pulses and counting the number of pulses within a specified duration of time. The counted data is then converted into a frequency reading to be displayed digitally. Being a digital instrument, the frequency range that a frequency counter can measure is limited by the speed of its logic circuitry. A stateof-the-art counter with emitter-coupled-logic (EeL) components can count directly up to around 1 GHz. Even a 1 GHz counting capability is hardly adequate for applications in circuits and systems in the upper UHF and microwave frequency range. For applications up to about 21 GHz, there are basically two techniques, the heterodyne conversion and the transfer oscillator technique, available to extend the range of a low-frequency digital counter. In this chapter we will initially look at the basic operating principles of typical low-frequency counting and will then go into the two conversion techniques used to extend the counter to the microwave range. A basic low-frequency counter capable of counting up to, say, 200 MHz is shown schematically in Fig. 10.1. The input amplifier has a limiter incorported in it in order to protect the subsequent stage from being overloaded. The Schmitt trigger converts the input signal into rectangular pulses of the same repetition frequency. The pulses are then counted by a decade counting assembly and displayed. The number of pulses allowed to be counted depends on the time for which the main gate is opened by the time-base circuitry. The ratio of the number of pulses counted to the known maingate-oN time is the frequency of the input signal.
10.2 MICROWAVE FREQUENCY COUNTING
In extending a basic frequency counter to above 21 GHz, it is required to translate the frequency to be measured into a lower frequency within the bandwidth of the counter. Two commonly used techniques are heterodyne conversion and transfer oscillator.
10
MICROWAVE FREQUENCY COUNTING Input signal conditioning
r------------,
I
I
I
I L __________ -.l Schmitt trigger
I
Decade <;ounter
Main gate flip flop
Time base oscillator Decade divider
Fig. 10.1
Gate time control
Basic frequency counter.
10.2.1 Heterodyne conversion Figure 10.2 shows the schematic of a heterodyne converter which extends the measuring range of a counter from 0-200 MHz to 200 MHz-12.4 GHz. The clock signal of 10 MHz from the counter is applied to the converter and is multiplied by 20 to 200 MHz. This signal is then amplified to a sufficient power level to drive a harmonic generator. This harmonic generator is usually made of a step-recovery diode and it generates a comb of frequencies at 200MHz intervals from 200MHz up to, say, 12.4GHz with approximately 60 discrete frequency components. This 'picket fence' of comb-frequency components is then applied to the tunable filter where only one of these frequency components (a harmonic of 200MHz) is selected to mix with the input signal to be measured at the mixer, which is an untuned wideband device. When an d. signal of the range of 200 MHz to 12.4 GHz is applied to the he-terodyne converter, the tunable filter (usually a tunable cavity) is tuned from low to high frequencies until a signal representing the difference in frequency between the r .f. input and the filter output appearing at the mixer output falls within the passband of the lowpass amplifier (1-220 MHz in this example). A detector circuit is built in at the lowpass video amplifier to give an indication if a measurable signal is emerging from the amplifier. When an indicating signal emerges the filter tuning is stopped and the output of the video amplifier is fed to the counter and its frequency Is is read from the counter. From the filter
MICROWAVE FREQUENCY COUNTING
I
~~~~~---~~====~~~~~~~
Lf.
/'=fx-fn
input fx
Low-pass amplifier 'video amp.'
To counter input /, =
fx - fa
fn=nXfo
From counter clock
200 MHz
~~
+++ ++
---'f
Fig. 10.2
Harmonics 'comb' generator
fa
=
200 MHz
Block diagram of a heterodyne counter.
tuning the order of the harmonic (n) is known, hence the input frequency ix can be found according to:
Ix
=
I,
(from counter reading)
+n
x 200 MHz.
For example, if the frequency of an r.f. signal (fx) to be measured is 4.45 GHz, a measurable signal is emerges at the output of the video amplifier when the filter is tuned to the twenty-second harmonic (n) of 200 MHz. is is read by the counter to be 50 MHz, and hence the 'measured' ix is given by
Ix
=
50MHz + 22 x 200MHz
= 4.45 GHz.
It is noted that the harmonic-selecting filter is tuned from low to high frequencies, so that when a measurable is first occurs the frequency in is always lower than ix, therefore the frequency to be measured is always equal to the counter-read frequency plus the frequency of the harmonic chosen. A disadvantage of this method is that one must always add the counter's digital readout to the harmonic reading of the converter. In order to overcome this disadvantage the manually tuned heterodyne converter is automated by replacing its tunable cavity filter by an electronically tuned yttrium-iron-garnet (YIG) filter. While the technical performance is equivalent to that of the manually tuned converters, the operation could be made fully automatic and provides a direct digital readout. Figure 10.3 shows the coupling structure of a single-stage YIG bandpass filter. Yttrium iron garnet (YIG) in the form of single crystals has become an important material in magnetically tunable devices such as filters and circulators. A highly polished YIG sphere or disc has a high Q-value and is electrically equivalent to a metallic cavity. However, the resonant frequency of a YIG sphere or YIG disc can be determined by an externally
[i87]
1ls8l1----~~CROWAVE FREQUENCY COUNTING -~--~
l,____._"________~ _.
_______________ .__ , _
--J __
Coupling loops YIG sphere
~ Output
Input
s Fig. 10.3 YIG bandpass filter.
applied static magnetic field, and is independent of the size of the YIG sample. Hence, a YIG sphere or disc can function as a tunable cavity by varying the applied voltage which generates the static magnetic field. The tuning characteristic for YIG spheres is given by the simple relationship of fr = yH, where fr is the resonant frequency in megahertz, H is the applied magnetic field in oersted and y is the gyromagnetic ratio in megahertz per oersted. Pure YIG has a gyromagnetic ratio of y = 2.8 megatertz per oersted. A YIG sphere can typically be tuned over a range of more than a decade of frequency. A simple YIG bandpass filter is shown in Fig. 10.3. The input and output loops without the insertion of the YIG sphere are decoupled due their orthogonal orientation. However, with the YIG sphere installed as shown, coupling between the loops will be greatly enhanced at the resonant frequency of the YIG sphere, thus making the structure shown in Fig. 10.3 function as a bandpass filter. With the aid of a YIG filter, an automatic version of the heterodyne converter can be constructed as shown schematically in Fig. 10.4. The manually tunable cavity filter is replaced by the electrically (magnetically) tuned YIG filter in the automatic version of the heterodyne converter. The passband frequency of the YIG filter is determined by an initially freerunning ramp signal from the control circuitry. When a measurable output from the video lowpass amplifier is detected, the control circuitry will stop the ramp signal at the voltage of that particular instant. Knowing the YIG control voltage, and with an advance knowledge of the resonant frequency versus tuning voltage characteristic of the YIG filter, the harmonic number of the comb frequency chosen can be calculated.
~___________M_I_C_RO_W __A_V_E_F_R_E_Q_U_E_N_C_Y_C_O_U_N_T_IN_G__________~I ~ Is = Ix - nlo r.f. input
Ix
To counter input
Video amplifier (1-220 MHz)
nlo Detector YIG control circuitry
Harmonic number to counter
r--_L-----.:fo (200 MHzr-)_ _ _----,
x20 multiplier
Fig. 10.4
10 MHz clock from counter
Automatic heterodyne counter.
10.2.2 Transfer oscillator A basic building block of a transfer oscillator is shown in Fig. 10.5. Similar in principle to the heterodyne converter, the transfer oscillator, normally abbreviated as TO, has the input (unknown) signal mixed with an internally generated signal. The main difference between the two methods is that while for the heterodyne converter the internal signal is a standard signal of known frequency and the mixer product is measured by the digital counter, the internal signal in the TO is of variable frequency and it is this internally generated signal that is being measured by the counter.
d. input Vi(t) - - - - . . - I
Sampler V,(t)
. . . - - - - - - - To counter
veo Fig. 10.5
Principle of a transfer oscillator.
Zero beat
~ IL-~~~~~_M_I_C_RO_W~A_V_E_F_R_E_Q_U_E_N_CY~C_O_U_N_T_IN_G~~~~~~ The sinusoidal signal generated by the veo is turned into a sampling train, which can be expressed mathematically as
2:
Vo(t) =
o(t - mT),
(10.1)
m=-'JC
where T = 2nlw and w is the frequency of the veo (much lower than the r.f. frequency to be measured). The pulse train is then used to sample or to mix with the r.f. input signal. The r .f. signal may be written as (10.2)
where Wx is the unknown frequency to be measured and rp is the phase between Vj(t) and the sinusoid generated by the yeo. The sampled output vs(t) is then equal to vs(t) = Vj(t)* = cos(wxt + fjJ)
2:""
o(t - mT)
m=-oo
Vs(t)
If the
= Vj(t)* =
veo frequency
2:
m=-oo
cos(wxmT + fjJ)o(t - mT).
(10.3)
w is a sub-multiple of w x , i.e. w =
Wx
N'
(lOA)
where N is a large integer (say in the order of 100), then vs(t) =
2:""
cos(2mNn + fjJ)o(t - mT)
m=-oo
Vs(t) = cos fjJ
2:
m=-oo
o(t - mT).
(10.5)
That is to say, if the veo frequency is a sub-multiple of the frequency to be measured, the sampler output is a constant-amplitude pulse train of magnitude cos rp. If rp is n/2, the output would be zero; this is known as the zero beat. Or if a holding device (e.g. a shunt capacitor) is connected to the sampler output the result will be a d.c. (zero-frequency) voltage when Wx = Nw. For Wx =1= Nw, the sampler output, even with a holding device, will give an output which varies with time. When zero beat occurs, the veo frequency can be measured by the counter, and the r.f. input frequency Ix can be found by multiplying the counter frequency Ivco by N. Hence Ix is found if N is known. To find Nit is noted that the input frequency Ix and the veo frequency 11 are related by Ix = Nil at zero beat. If the veo frequency is increased so that the next higher frequency fz causes another zero beat to occur, thenlx = (N - l)fz; this imples that
M __ IC_R_O_W_A_V_E_F_R_E_Q_U_E_N_Cy __C_O_U_N_TI_N_G__________~I
L -_ _ _ _ _ _ _ _ _ _
Nfl = (N - l)fz
N=~
or
fz -
(10.6)
It"
Therefore N, and hence lx, is determined by counting two consecutive zero-beat frequencies. The basic idea of a transfer oscillator type of frequency counter can be made use of in the implementation of an automated version of the TO frequency counter as shown in Fig. 10.6. The input signal of frequency Ix (to be measured) is divided by the splitter. On the upper signal path the input signal of frequency Ix is beated with a harmonic of the veo of frequency II by the sampler (mixer 1), and the sampler output is fed into a bandpass 'video' amplifier tuned at, say, 20 MHz. The output of mixer 1 has the frequency of nil - fx. No signal at the video amplifier output is registered and the veo keeps on sweeping until nfl - fx =
fIF!
(20MHz).
(to.7)
is maintained accurately at 20 MHz with the aid of a reference frequency IIFref of 20 MHz from the counter and a phase detector which provides the frequency locking. The veo frequency It can then be measured by the counter. The locked
JrFI
1m =
20 MHz
'Video'IF! amplifier (20 MHz)
IIF .(20 MHz) " From counter
Sweep generator 'sawtooth' e----i~
Frequency translation
Fig. 10.6 Automatic TO counter.
To counter
10 (20 KHz) From counter
~
~I
MICROWAVE FREQUENCY COUNTING
~~~~~~~~~~~~~~~~~~~~~~~~~~~~
VCO frequency It is also frequency translated to h by another reference signal of frequency fo, say, of 20 kHz such that
h
= 11
+ 10.
(10.8)
h
is then sampled with fx in mixer 2 and passed through the lowpass IF amplifier to provide a signal of frequency fIF2' so that fIF2
=
nh - Ix + 10) - (nIl nlo + 20 MHz.
= n(fl =
- 20 MHz)
By further mixing this IF amplifier output signal with the fIFref = 20 MHz signal from the counter and rejecting the higher-frequency components at mixer 3, the output frequency is nfo. Since fo is a reference frequency of 20kHz sent by the counter, by counting nfo n is known. Thusfx is virtually determined according to fx = nfl'
10.3 PERFORMANCE OF DOWN-CONVERTED FREQUENCY COUNTERS
There are a number of parameters commonly used to describe the performance of microwave frequency counters where down-conversion techniques are employed. These parameters are now briefly described. (a) Frequency range The upper frequency limit for both the heterodyne-conversion (HC) and the transfer-oscillator (TO) counter is about 21 GHz. This is basically limited by the frequency response of the front-end amplifier and mixer. Higher-frequency counting is possible with additional down-converting mixing of the input signal before it is applied to the microwave counter. This will substantially lower the sensitivity and the measurement speed of the whole operation. (b) Measurennentspeed Measurement speed is normally specified in terms of the acquisition time of a counter reading. An HC or TO counter has a typical acquisition time of 100 to 300ms. (c) Sensitivity The sensitivity of a counter is the smallest signal level that can accurately be counted. Since both the He and TO counters have a wideband frontend mixer, the sensitivity is typically limited to about - 35 dBm.
D_O_W_N_-C_O_NV __E_R_T_E_D_F_R_E_Q_U_E_N_C_y_C_O_U_N_T_E_R_S________~I
L -_ _ _ _ _ _ _ _ _
(d) Signal-to-noise ratio
The signal-to-noise ratio of a frequency counter is the minimum difference (in dB) allowable between the power of the carrier whose frequency is to be measured to the level of the noise floor. A typical value for both the He and TO counters is 20 dB. (e) Amplitude discrimination
When more than one spectral component is present at the input of a frequency counter, the counter will correctly count the spectral component of the highest amplitude if the amplitude difference between this component and its nearest (in frequency) component exceeds a certain value, and this value in dB is the amplitude discrimination of the counter. However, the amplitude discrimination of a microwave counter is not a constant, but is an increasing function of the frequency separation between the spectral components concerned. For example, a typical He or TO counter has a amplitude discrimination of 2 dB when the separation is 20 GHz and 10 dB when the separation is 20 MHz. (f) FM tolerance
When a signal to be measured is frequency modulated or contains a fair amount of incidental FM, the amount of FM in terms of the peak-to-peak frequency deviation that a counter can tolerate without giving erroneous readings is called the 'FM tolerance'. Typical He counters can normally tolerate up to 50 MHz peak-to-peak whereas TO counters can normally tolerate up to 10 MHz peak-to-peak. (g) AM tolerance
Very rarely would a microwave carrier or r.f. signal be amplitude modulated in practical applications, but most frequency-modulated r.f. signals have a certain amount of incidental AM due to the system nonlinearities and power-supply fluctuations. The size of AM modulation index that a counter can tolerate without giving erroneous readings is called the AM tolerance. A typical value for both types of counter is about 90%. PROBLEMS
1. With reference to the block diagram of the automatic TO counter as shown in Fig. 10.5, design the system by specifying the following: 1. The range of veo frequency ([1), 2. The input and output frequency relationship of mixer 2, 3. The frequency and bandwidth of the IF2 amplifier;
~
~
1L-___________M_I_C_RO_W__A_V_E_F_R_E_Q_U_E_N_Cy__C_O_U_N_T_IN_G__________~ given that: 1. The low-frequency counter has a counting range of 0 to 200 MHz, 2. The required counting range (fx) is 220 to 1000 MHz, 3. The low-frequency counter provides the TO counter with two sinusoidal reference signals at 20 MHz and 20 kHz. 2. A heterodyne frequency-counting system is shown in Fig. P.lO.1. If the low-frequency counter has an accuracy of 40 Hz at 100 MHz and the crystal oscillator has a time base error of 4 x 10- 7 , calculate the measurement error (%) when the system is used to measure a 5.1 GHz signal. 3. If the accuracy of 10- 7 is required of a digital counter with a crystal oscillator having an aging rate of 3 x 10- 9 per day, how often should this crystal oscillator be calibrated? Ans.: 1 month 4. The timebase accuracy of a lO-digit counter is 10- 9 and the display accuracy is ± 1 digit. If the counter is used to measure a 400 Hz signal with a time base setting of 10 seconds, what is the measurement accuracy? Ans.: 2.5 x 10- 4 5. If the counter in Problem 4 with the same time base setting is used to measure a 400 MHz signal, what is the measurement accuracy? Ans.: 1.025 x 10-8 6. An automatic heterodyne-conversion counter has an AUTO mode with selectable preset value for the YIG filter. The YIG filter is then swept upwards from this preset value. The comb-frequency distance of the system is 200 MHz. Two frequencies of 1.92GHz and 9.05GHz are applied to the input of this counter when the YIG preset is set at 2.00 GHz. What should be the counter reading? 7. In the process of determining the harmonic number of an input of frequency Ix using a manual TO counter, the two beat frequencies were Mixer (5.1GHz)
To low-frequency counter
Ix 10 x 500
Fig. P.IO.I
10 MHz crystal oscillator
F_U_R_T_H_ER__R_EA_D_I_N_G________________~I
L -_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
found to be 168 MHz and 174 MHz. Determine the harmonic number andfx. Ans.: 29, 4.872GHz 8. A 10 MHz crystal oscillator is to be calibrated to an accuracy of 10- 7 with the aid of a nine-digit 500 MHz counter. What is the gate time (measuring time) required in order to achieve this accuracy? Ans.: >1 second
FURTHER READING Automatic Frequency Measurement of Millimeter Waves, EIP Application Note 20I. Bouwens, A.I., Digital Instrument Course, Part 2 Digital Counters and Timers, N. V. Philips' Gleoilampenfabrieken. Fundamentals of Microwave Frequency Counters, Hewlett Packard Application Note 200-I. Source Locking the EIP microwave source, EIP Application Note 902.
~
G
N_O_iS_e_M_e_a_S_u_r_e_m_e_n_t_-----'
1L....-_ _
11.1 NOISE AND NOISE FIGURE Noise is a phenomenon inherent to any electronic system. It is of particular concern especially in electronic communications where small signals are received in the midst of noise. In any electronic system there are noises generated within and outside the system. In most modern electronic systems the noise generated within the system itself contributes a significant proportion of the total noise, and in most cases, the internally generated noise is virtually the total noise. For the sake of the noise characterization of devices and systems, only the combined effect of all internally generated noise will be considered. This combined noise effect is often referred to as the thermal noise of the system. In this chapter the measurments of thermal noise of systems and the required instrumentation will be discussed. Any impedance expressed in the form of Zs = R
+ jX
(11.1)
can be considered as a noise source when it is connected to a load ZL = + jXL if Zs is at a temperature Ts , where Ts > 0 (kelvin). If ZL = Zs, the amount of noise power generated by Zs, which is available to ZL, is equal to kTsB, where k is the Boltzmann's constant, equals to 1.380 x 10- 23 J K- 1 and B is the bandwidth in hertz of the path connecting Zs and ZL (e.g. a doubly conjugate matched bandpass filter). This noise power is known as the available noise power of the impedance Zs. For a noise source at room temperature (290 K or 17 0c), the available noise power is -174dBm per hertz of the system bandwidth. A concept created to provide a standardized means of characterizing the internally generated noise of a system is known as the noise figure. The noise figure F of a system is defined as
RL
with
F- S/Nil - So/No Ts=To (290 K) FdB = 1OioglOF (in dB),
(11.2) (11.3)
~_______________N_O_IS_E_A_N_D__N_O_IS_E_F_IG_U_R_E______________~I ~ where SJNi and So/No are the signal-to-noise ratios of a system at the input and output, respectively. In defining the noise figure, a standard temperature of To = 290 K was adopted, originally by the IRE (forerunner of the IEEE), and this standard is widely accepted. Figure 11.1 shows the display of the input and output signals (plus noise) of an amplifier on a spectrum analyser. The input signal-to-noise ratio is seen from Fig. 1l.I(a) to be 40dB. In Fig. 11.1(b) it is seen that the input signal is amplified by 20 dB, or the system (amplifier) gain is 20 dB. While
-20
-120L-____________ ____________ 2.6 2.65 2.7 ~
~
Frequency (GHz) (a)
-20 03 :-
~
... E
-40
~
'l.l
:::o:l
8.~ -60
:;
..s
0-
-80 -100 -12 2.6
Fig. 11.1
2.65 Frequency (GHz) (b)
2.7
Typical signal and noise level plotted against frequency.
~ I~_________N_O_IS_E_M_E_A_SUREM._E_N_T_ _ ..__~_.==~ the signal is amplified by 20 dB, the noise level is increased by 30 dB. The system amplifies equally both the signal and the noise available at its input, and the output noise level is expected to be -80dBm. The fact that the output noise level is -70 dBm instead of -80 dBm shows that the system (amplifier) not only amplifies the noise available at its input, but it also adds to the output some internally generated noise. The amount of noise added by the system itself, usually denoted by No (watt), is thus equal to 9 x 10- 11 watt or -70.46dBm. The noise figure of the system (amplifier) is thus equal to 10 dB according to (11.2) and (11.3), assuming all events took place at 290 K. The noise figure of a system can also be written as F - S;lN; I So/No at 290 S;/N;
K (To)
(11.4 )
where G a is the gain of the system (in ratio) and Na is the noise added by the system itself, or F = Na
+ kToBG a kToBG a '
(11.5)
where N j = kToB is the available input noise power. The magnitude of kTo (or power spectral density of the available input noise) is equal to 4.00 x 10- 21 watt per hertz of system bandwidth (B) or -174 dBm per hertz. The noise figure of the system is the degradation in the SIN ratio as a signal passes through the system. The system noise figure is independent of the signal level as long as the system is levelled (constant gain).
11.2 EFFECTIVE INPUT NOISE TEMPERATURE The noise added by a system is sometimes described by the effective input noise temperature Te. For low-noise systems such as the front-end amplifier of a satellite receiver, that is when Na is small, Te is a more sensitive indicator than is the noise figure because the 290 K standard temperature necessary for the definition of the noise figure is not a good reference for low-noise systems, and Te has no need for such a reference. Te is the temperature of a fictitious additional source resistance at the system input that produces the same noise power at the system output as does the system to be characterized assuming that the system itself is noiseless. Te is defined by the equation (11.6)
The noise figure (F) and the noise temperature (Tc) of a system are related by
~__________E~F~F~E~CT~I~V~E~IN~P~U~T__N~O~IS~E~T~E~M~P~E~RA__TU~R~E__________~I ~ or
F
= SJNj = Sj(Na + kToGaB)
F
= Na + kToGaB = kGaB(Te + To)
So/No
GaSj(kToB)
kToGaB
kGaBTo
F = Te + To To
(11.7)
The relationship between Te and F is shown schematically in Fig. 11.2. Figure 11.3 shows the typical noise figure and noise temperature values for bipolar transistors, FETs, low-noise HEMTs and broadband amplifiers. Data shown in Fig. 11.3 are typical state-of-art values at the time of writing. This diagram is intended to give a rough idea of the relative noise performance of various devices.
Na = 0 Ga,B
Z,at Te
o Temperature of source impedance
Fig. 11.2
Definition of effective input noise temperature.
NOISE MEASUREMENT @Ji~_______________J 1500
8 7
1000
6
700
$' 5 "0 '-'
...::I
4
500 400
3
300
<1)
01)
.~ 0
:z:
g ~.
200
2
100 1 0
8
4
16
20
Frequency (GHz)
Fig. 11.3 Typical values of noise figure and noise temperature.
11.3 MEASUREMENT OF NOISE Noise figures and noise temperatures are usually measured by injecting noise power from a noise source over a broad band covering the entire bandwidth of the system to be measured at two temperatures, Tc (cold) and Th (hot). The output noise power corresponding to Tc and Th , respectively, are Nt and N 2. If Nt and N2 can be measured accurately by a power-detecting device, then the noise power Na added by the system to be measured (DUT) can be found by extrapolation as shown in Fig. 11.4. Noise ouput power
N. '-0_ _.L...-_ _---'L.-_ To Th
Fig. 11.4 Measurement of noise.
___.
Ts (Source impedance temperature)
M_E_A_S_U_R_EM __E_NT__O_F_N_O_IS_E______________~I
L -_ _ _ _ _ _ _ _ _ _ _ _ _ _
The noise power output Nl (at Tc) and N2 are related to the effective noise temperature Te of the system by NJ = kGaB(Tc + Te)
(11.8)
N2 = kGaB(Th + Te)·
(11.9)
The ratio of N2 to Nl is defined as the Y-factor of the DUT (device under test), i.e. (11.10)
Te can be related to Y by T = Th - YTc
(11.11)
Y _ 1 '
e
F = Te + To To ' F = (ThlTo - 1) - (TJTo - 1).
since therefore
(11.12)
Y-l
So it is seen that by measuring Nl and N2 at Tc and at T h , Te and F can be found. This is, the basis for almost all noise measurements. However, it is unfortunate that all measuring equipment used to measure the noise power Nl and N2 will generate its own internal noise, hence the values of noise power shown by these equipment are not the noise power of the DUT, but are the sum of the DUT noise and the noise generated by the measuring equipment. Before we go to the details of the measurement methods, it is appropriate to look at the effect of noise propagation in a system containing more than one stage, in order to isolate the DUT noise and the noise of the measuring equipment. Figure 11.5 shows the noise power at various points of a two-stage system. It can be shown that the overall noise figure of a system of n stages is given by F tolal
=
FI
For a two-stage system
F2 - 1
+ -G + ... + 1
Flolal
Fn - 1
G G 1
2···
G· n-l
(11.13)
= F12 reduces to
F12 =
FI
F2 - 1
+ ----c;-.
(11.14)
It can be seen in the last two equations that for a system processing very weak signals, Flolal must be small in order to allow a high enough SIN ratio (called the system margin) at the system output for the output signal to be detected. It is also observed that Flolal is mainly contributed by F l , the
~
~ I~
_________________N_O_I_SE__M_E_A_SU_R_E_M_E_N_T________________~ First
Second
Input noise kToB Na2
...- ...-"'Nat
r-----r--
_-t-----+ kToBG t
---
N at
G2
kToBG t G 2
f
Total noise added
i
r
Total noise power out ut
Noise input x system gain
Fig. 11.5 Noise in a two-stage system.
noise figure of the first stage (front end) of the system, provided the gain G I in this stage is high. Hence it is necessary to have a low-noise and highgain device as the first stage of a system processing weak signals. The last two equations can also be applied to noise measurements. The system to be measured (DUT) can be considered as the first stage and the measuring equipment can be considered as the second stage of a two-stage system. The noise figures and the noise temperatures for these two stages are (Flo Tel) and (F2' Te2 ), respectively. The noise figure measured is not FI but F 12 , therefore in order to find Flo a knowledge of the noise figure of the equipment F2 and the gain G I of the DUT is necessary. A calibration procedure where the noise source is fed directly into the measuring equipment is performed and the result is shown in Fig. 11.6(b). The noise N~ added by the measuring system itself can be deduced from the calibration as shown in Fig. 11.6(b), from which the system noise figure F2 can be found. From the results of the DUT measurement and the system calibration as shown in Fig. 1l.6(a) and Fig. 11.6(b), it is seen that the gain G I of the DUT is equal to the ratio of the slopes in these plots, i.e. _ (N2 - Nl)/(Th - Tc) Gl
-
(Nz - Ni)/(Ti, -
T~)·
(11.15)
By measuring at the same 'cold' and 'hot' temperatures (11.15) reduces to N2 - Nl G l =N'2 - N'· 1
(11.16)
Since both F2 and G 1 are found, the required noise figure Fl for the DUT is
~_______________M_E_A_S_U_R_EM__EN_T__O_F_N_O_I_SE______________~I ~ (a)
N2
Output noise power ----------
slope =kGpj3
-N = N_,_
T-T h c
I
(b)
Input noise power
N; N; N~
O L---~--------------~----~~~
Th
T~
slope
N~ - N' =kGj3 = _,_-___ ,1
Th - Tc
Fig. 11.6 Gain measurement; (a) DUT measurement (DUT noise system noise); (b) calibration (measuring system noise).
+ measuring
[
~~-I
~~
,--------
L _ ______
~-~-~~--~~~~-
-----
NOISE MEASUREMENT
obtained. The method just described for measuring the noise figure forms the basis for all system noise characterization.
11.4 NOISE SOURCE The measurement of two points along a straight line described in the last section requires two different temperatures for the source impedance connected to the input of the DUT. The devices used to achieve this effect (not necessarily at two physical temperatures) of two source temperatures include temperature-limited vacuum diodes (thermionic diodes), physically heated and cooled terminations (commonly used in standards laboratories), and avalanche diodes (see Fig. 11. 7). The avalanche diode (also known as the solid-state noise source) has gained more popularity in recent years and is useful for most noisemeasuring purposes because it is broadband and can conveniently be switched between Tc and Th under electronic control. When a sufficiently large positive bias is applied, the avalanche diode is reverse biased. Avalanche action occurs where a large d.c. plus random current of all frequencies flows through the diode. The d.c. current is returned to the bias supply whereas the high-frequency components are prevented from doing the same by the choke (inductor) and are forced to develop their 'noise' voltage in the matching resistors. Hence, looking from the output port into the device one sees a source impedance (say, 50 Q) with plenty of noise power available to a load if connected to the output port. Therefore, depending on the noise power available, the equivalent source impedance seems to have been heated up to a certain temperature T h . When a negative bias is applied, the diode is in normal conducting mode with no random high-frequency components (noise) generated, and the diode end of R J is almost at ground potential. Therefore, looking from the output port into the device one sees a source impedance of R3 in series with Matching pad
,------------1
Bias input
Noise output
L___________ ..J Fig. 11.7
Avalanche noise source.
N_O_I_S_E-_F_IG_U_R_E_M __ EA_S_U_R_E_M_E_N_T________----~I
L -_ _ _ _ _ _ _ _ _ _ _ _ _
a parallel combination of Rl and R2 , which should also be 50 Q (say). Since no 'noise' of substantial quantity is available, the equivalent source impedance seems to be maintained at a certain low temperature Te. The 'hot' temperature Th of a noise source is normally described by its excess noise ratio (ENR) in decibels, defined as ENR(dB) = lOloglO
(Th ;0290 )
(11.17)
Th = 290 [lOENR(dB)/10 + 1].
or
(11.18)
A noise source usually comes with a calibration report of Th or ENR (dB) versus frequency within its band of operating frequencies. Te is the actual temperature of the source's output matching pad, usually at room temperature.
11.5 NOISE-FIGURE MEASUREMENT (SINGLE FREQUENCY) The noise figure of a OUT is measured by measuring the ratio of the OUT noise power N2 and Nl (the Y-factor), at two known temperatures Th and Te, respectively. Assuming that Nl and N2 have been corrected by the calibration procedures to take away the noise contribution of the measuring system, noise-figure measurement at a single frequency can be performed using a simple system consisting of a filter tuned to the frequency of interest and a detector that has a true Lm.s. response (e.g. a power meter) as shown in Fig. 11.8. The most serious limitation of the measuring system shown in Fig. 11.8 is that measurements are restricted to a single frequency determined by the centre frequency of the bandpass filter and the frequency resolution of the measurements is restricted by the bandwidth and shape factor of the filter.
Power meter Solid-state noise source
N2 (noise source ON)
N J (noise source OFF)
Noise source Bias voltage
Fig. 11.8 Single-frequency noise figure measurement.
~
~
N_O_I_SE__M_E_A_SU_R_E_M_E_N_T________________~
L I_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
11.6 WIDEBAND NOISE-FIGURE MEASUREMENT An obvious improvement to the single-frequency measurement arrangement shown in Fig. 11.8 is to use a heterodyne front end to permit measurements at any frequency within a frequency range determined by the range of the local oscillator. The arrangement is shown in Fig. 11.9. For example, if the measuring range is to be 0-1.5 GHz, the low-pass filter will have a cutoff at 1.5 GHz and the bandpass IF filter will be tuned to an IF frequency (e.g. 2GHz) in order to avoid r.f. to IF feedthrough. The local oscillator should then be made tunable over the range from 2.0 to 3.5 GHz in order to chanJ~~ the measurement frequency fM from 0-1.5 GHz to the IF frequency of .0 GHz according to fM = fLO - fIF' The arrangement shown in Fig. 11.9 can be automated by injecting a sawtooth (in steps) to sweep the local oscillator (in steps) through its tuning range. At each step a synchronous pulse is sent to drive the noise source so that the cold and hot noise powers, Nl and N2 can be measured at that particular frequency (step) before the local oscillator is swept to the next step. An instrument based on the arrangement shown inside the dotted box is usually known as a noise figure meter. With the addition of sweep oscillator and noise-source drive capabilities, the equipment is known as an automatic noise figure meter.
11.7 NOISE-FIGURE MEASUREMENT AT MICROWAVE FREQUENCIES The noise-figure meter discussed in the last section functions up to a certain frequency, say 1.5 GHz as described. If the same equipment, whether the manual or the automatic version, is to be used for higher frequencies, a down-conversion stage has to be incorporated. A noise-measurement system with down conversion is shown in Fig. 11.10. The little block labelled as the noise-figure meter in Fig. 11.10 is itself a system similar to that shown inside the dotted box of Fig. 11.9. A noise-measurement system with down conversion requires a broadband mixer and a signal generator or synthesizer covering roughly the same band as the intended measuring range. For most applications, a low-noise IF amplifier is required to produce a large enough signal for the noise-figure meter to measure. Since the mixer produces both the sum and difference frequencies between its inputs, the actual input being measured at any time is equal to fM = FLO ± fIF' i.e. there is noise at the intended frequency as well as at the image frequency. This is called the double-sideband (DSB) noise measurement. For example, the noise-figure meter (NFM) in a measurement system with down conversion has a measuring range of 0-15 GHz, and the whole system is used to measure noise input over the range of 1.5-3.0 GHz with a fixed local oscillator of 1.5 GHz. Suppose at a certain instant the swept LO
S_I_D_EB_A_N_D__ M_E_A_SU_R_E_M_E_N_T_S____________~ __~
L -_ _ _ _ _ _ _ _ _ _ _ _ _ _
r--------------I
I I
Power meter
IL ______________ fM =
I I I I I I I
~
fLO - flF
Fig. 11.9 Tunable noise-figure meter.
of the NFM is such that it allows a noise input of 1.0 GHz to be measured by the NFM. There are two noise frequencies at the input of the whole system, 0.5 GHz and 2.5 GHz, that when mixed with the system LO of 1.5 GHz, will produce a 1.0 GHz noise signal for the NFM to detect. Therefore the NFM will given a noise power reading for fM = 2.5 GHz that is actually the sum of the noise power for fM = 2.5 GHz and fM = 0.5 GHz. Hence the noise power detected by a noise-measuring system with down conversion is always higher than the actual value. This is a consequence of the double-sideband measurement technique.
11.8 SINGLE-SIDEBAND (SSB) AND DOUBLE-SIDEBAND (DSB) MEASUREMENTS If the frequency response of the DUT of the example in the last section is
flat over the entire frequency range of 0-3.0GHz, the noise powers from Measurement system
Solid-state noise source
Noise figure meter Noise source drive _
Fig. 11.10 Measurement system with down conversion.
I I
I
~oltag~
~
~
[-_____ ___ ~=~
NOISE~~~~URErV1ENT---'----'-"-'-
J
the OUT at 0.5 GHz and at 2.5 GHz are the same. Hence the noise power shown by the NFM for 1M = 2.5 GHz is twice as much as it should be, and therefore the correct reading should be 3 dB below the reading shown by the NFM in a DSB noise measurement. However, a 3 dB adjustment to the noise power measured by the DSB method is only valid if the OUT has a constant response at both the intended measuring band and its corresponding image band. If a OUT does not have a constant response at both the measuring and the image bands, a high-pass filter (e.g. cutoff at 1.5 GHz for the system in the previous example) has to be inserted between the OUT and the system mixer so that the noise power contributed by the image band (0-1.5 GHz) can be eliminated. Sometimes, a lowpass filter instead of a high-pass filter is required in order to prevent noise at the image band affecting the system. Whether a low-pass or a high-pass filter is required depends on whether the image band is above or below the measuring band frequencies, respectively. Down-converted measurements with an image-rejection filter are known as single-sideband (SSB) measurements. A noise figure measured by the down-conversion technique has therefore to be qualified by specifying whether it is obtained by the DSB or the SSB method.
11.9 SUMMARY There are essentially two kinds of noise-figure measuring systems. One is for single-frequency operation like that shown in Fig. 11.8, which consists of a bandpass filter and a detector-meter. This type of system is known as the system noise monitor. A system noise monitor is particularly useful when installed permanently into a communication link (e.g. a satellite receiver) or a radar system so that the noise performance of the system can be checked either continuously or when required. The other kind of noisemeasuring system is the tunable type like those shown in Fig. 11.9 and Fig. 11.10. This type of instrument is particularly useful for characterizing devices such as amplifiers, filters and mixers, where the noise-figure or the noise-temperature frequency response is desired.
PROBLEMS 1. The Ailtech (Eaton) 757 spectrum analyser has an IF output (to external connector) at 21.4 MHz. Design a noise-measuring system using the Ailtech 757 and a fixed-tuned noise-figure meter (tunable to 21.4 MHz) that is capable of measuring the noise figure of devices from 10 MHz to 18 GHz (the entire input range of Ailtech 757), in sweep mode. 2. Show that for a two-stage system with noise figures FI and F 2 , effective noise temperatures Tel and Te2' and gains G I and G 2 , for the first and the second stages are, respectively,
P_R_O_BL_E_M_S~------------------~I ~
L -____________________
Te12 = Tel + Te2 G1 and
F2 - 1 F12 = Fl + 0;-'
where Te12 and F12 are the effective noise temperature and the noise figure, respectively, of the whole system. 3. A 12 GHz direct-broadcast satellite TV down link has the following characteristics. Transmitter power Transmitter antenna gain Bandwidth of transmission Feeder loss Transmission path Path loss Atmospheric attenuation: Receiver Antenna gain Calculate: Transmitter
lOOW 39dB 27 MHz 3dB 206 dB (38000 km) 2dB 38dB
1. Power to the receiver (after the receiving antenna), 2. The maximum noise figure of the receiver in order to have a link margin (C/N carrier to noise ratio) of 12 dB. Ans.: -84dBm; 2.7 dB 4. The maximum noise power P n generated by a resistance and made available to a load resistance R( = l/G) is equal to kTB watts, where k is Boltzmann's constant, T is absolute temperature and B is the bandwidth of measurement from which P n is measured. Show that the equivalent noise voltage (en) and the equivalent noise current (in) of the source are given by ~ = 4kTBR ~ = 4kTBG,
where the bars respresent 'the mean value of'. 5. A swept measuring system with down conversion, like that shown in Fig. 11.10, can be modified such that the swept LO inside the NFM is replaced by a fixed LO and the fixed system LO can be replaced by a swept LO. Devise a system (in block diagram form) capable of swept noise measurements over the frequency range of (a) 0.01-3.0 GHz; and (b) 0.01-18.0 GHz. State whether the system you devised is of the SSB or DSB type. 6. In the calibration of a noise-figure meter, the noise figure of the NFM is found to be 8 dB. If the NFM is used to measure the noise figure of a certain device of gain 11.8 dB, it gives a noise-figure reading of 2.23 dB. What is the contribution of the NFM Ans.: 0.23 dB
~ ~I
N_O_I_SE__M_E_A_SU_R_E_M_E_N_T________________~
_________________
7. If the effective input noise temperature (Te) of a mixer, the conversion loss of the mixer at the signal frequency (LSig) in ratio and the conversion loss of the mixer at the image frequency (Lim) in ratio are given, shown that the noise figure of the mixer measured by the SSB method (FssB ) and that measured by the DSB method (FDsB ) are given by
LSig) ( 1 + 1: Te ) FssB = ( 1 + L 1m
and
F DSB = 1
+
0
~~.
8. A certain mixer is specified as follows. r.f/LO range: 1.0-2.0 GHz
Conversion loss: 20 dB Noise figure: 8 dB Image rejection: lOdB. Assuming that all ports are perfectly matched, calculate the noise figure measured (i) by the SSB method, and (ii) by the DSB method. Ans.: 8.4dB; 8.0dB
FURTHER READING Fundamentals of RF and microwave noise figure measurements (1983) Hewlett Packard Application Note 57-1, July. Measuring noise performance factors (1983) published by Ailtech Instruments. W.E. Pastori (1983) Image and second-stage corrections resolve noise figure measurement confusion. Microwave Systems News, May. W.E. Pastori (1983) Transform noise figure to noise temperature. Microwaves and RF, May. J.B. Winderman (1980) Perform true DSB to SSB noise figure conversions, Microwaves, July.
Swept Measurements and Network Analysers
12.1 NETWORK ANALYSIS Network analysis is the process of finding the response of a network to a certain input waveform. It is well known that the behaviour of a linear network can be totally specified by its response to a unit impulse; this is known as the impulse response of the network. Since an impulse has infinitely many frequency components at all frequencies from zero to infinity with equal magnitude, it is therefore equivalent to sending a signal of continuously changing frequency of equal magnitude at all frequencies into the input port of the network. The response at the output port is known as the frequency response of the network. If the input signal can be swept from zero to infinity, the frequency response of the network is the same as the Fourier transform of the impulse response. Networks or circuits at high frequencies, say above a few tens of megahertz, are normally described by their s-parameters if they are linear or piecewise linear. These s-parameters are generally frequency dependent, so in order to find out the frequency response of a network in terms of its transmission and reflection characteristics expressed by its s-parameters, swept frequency measurements over the frequency range of interest have to be made. Equipment catering for this kind of measurement at high frequencies are normally called network analysers.
12.2 SIGNAL SOURCE FOR SWEPT MEASUREMENTS The first thing required of a high-frequency network analyser is a signal source covering the frequency range of interest. In the manual type of frequency-response measurements where each frequency is tuned manually with the aid of a frequency counter, an ordinary signal generator would suffice. However, if a 'real-time' display of the frequency response is required (e.g. in tuning a filter), swept oscillator (generator) is necessary. In the case where the frequency response is displayed from a set of measured data contained in the memory of a computer/controller (i.e. not real-time in the absolute sense of the word), a programmable synthesizer can also be used. Synthesizers are more stable and accurate CW sources
12
~
s_W_E_P_T_M_~ASUREMENTS--AND NETWORK ANALYSERS--~
L I_ _ _ _
than are swept oscillators in general. Synthesizers are required to test narrowband devices, devices whose characteristics change rapidly with frequency and low-noise devices because synthesizers offer much lower phase noise than swept oscillators and much better frequency resolution. In most other frequency-response measurements a swept oscillator would suffice. The block schematic diagram of a typical swept oscillator is shown in Fig. 12.l. The output frequency of the voltage-controlled oscillator (YCO) is proportional to the instantaneous voltage of the sweep generator output. By altering the sweep voltage, the YCO frequency range is broadened or narrowed. By varying the d.c. offset of the sweep, the YCO start and stop frequency can be changed. A small sample of the output signal to the directional coupler taken from the sampling arm is converted to d.c. which acts as a negative feedback by varying the attenuation of the PIN attenuator. The feedback circuit is known as the automatic levelling control (ALC). The ALC is essential in order to produce output magnitudes which are constant with frequency as required by the swept measurement technique.
12.3 VECTOR OR SCALAR MEASUREMENT? Characterization of a two-port network (which can be extended to an S21 in the forward configuration where the power is fed into port 1 and of S22 and S12 in the
n-port) is essentially the measurement of Sl1 and
ALe
PIN attenuator
t-------l~ Sweep output
Sweep generator
Fig. 12.1
Swept oscillator.
Lr. output Directional coupler
SC_A_L_A_R__ N_ETW ___ O_R_K_A_N_A_L_Y_SI_S___________~ __~
L -_ _ _ _ _ _ _ _ _ _ _ _ _ _
reverse configuration where the power is fed into port 2 of the network. In both configurations the reflection coefficient (Sl1 or S22) is measured at the point where the input power is applied to the network. From these measurements of SII, SI2, S2I and S22, a whole list of other parameters, such as return loss, input and output VSWR, insertion loss, and gain and input/ output impedances, can be obtained. The reflection and transmission coefficients (Sl1' S12, S21 and S22) are, in general, complex quantities with phase as well as magnitude. It is obvious that measuring magnitude ratios is simpler than measuring both the magnitude ratios and phase differences of two quantities. Therefore, a class of equipment, known as scalar network analysers, is specifically designed to measure the magnitude of the transmission coefficient and the return loss of a network. Although a scalar network analyser cannot be used to specify fully a network due to the lack of phase information, it is still very useful in checking whether a network, a device or a system is functioning properly, and in the preliminary tuning of the network. In some research and development applications it is sometimes required to find out why a network does not function, and then both phase as well as magnitude measurements are necessary. These measurements require a class of equipment known as vector network analysers (VNA). 12.4 SCALAR NETWORK ANALYSIS A scalar network analyser (SNA) system (see Fig. 12.2) normally consists of a swept oscillator, a signal-separation device and a power-detection system. A fairly popular way of constituting a SNA system is to use a swept oscillator, a signal-separation device and a frequency-response test set (also known as a scalar network analyser). The test set comes with three crystal detectors which provide a d.c. or low-frequency output voltages proportional to the input power. The Lf. signal from the swept oscillator is fed into the signal-separation device. A small portion of this incident signal power is sampled by a coupling arm to be detected by detector R whereas the rest of the incident power goes through the separation device and acts as the input signal power to the DDT. The d.c. voltage detected by the detector R is proportional to the Lf. power sampled by the detector and hence it is also proportional to the actual power input to the DDT. A portion of the signal power fed to the DDT is reflected by the DDT to the separation device if the input impedance of the DDT is not equal to the system impedance Zoo A small portion of the reflected power is sampled by another coupler arm and is detected by detector A. The d.c. outputs of detector R and detector A represent the incident and reflected power of the DDT, hence the ratio of A and R is proportional to the square of the input reflection coefficient
~
~
C
SWEPT MEASUREMENTS AND NETWORK ANALYSERS Frequency-response test set
t~'l\:PI" Swept oscillator ----. Freq.
Sweep output
Lf. output )
R ()
A
c:>
B
:>
IDet~ctor I Det~ctor
•)
+l
DUT
Signal-separation device
~
~
Fig. 12.2 Scalar network analyser system.
of the D UT. Since the test set takes the ratio of V A/VR and displays it in decibels against frequency, the vertical axis of the display is equal to the return loss (L R ), i.e. LR
2
VA
= -lOlogQ = -lOlog V R (dB).
(12.1 )
It is noted that (12.1) is only correct if the coupling arms to A and R have
the same coupling coefficients. Up to now the term 'signal separation device' has been used to describe the circuit block which samples both the incident and reflected power. The most commonly used separation devices are dual-directional couplers and
~______________SC_A_L_A_R__NE_T_W__O_RK__A_N_A_L_Y_SI_S____________~I ~ reflectometer bridges. The schematic of a dual-directional coupler is shown in Fig. 12.3. The main advantage of using a dual-directional coupler as the signalseparation device is that theoretically any variation of coupling factor CR with frequency will be the same as that for CA , and since it is the ratio of the R and A outputs which is measured, any frequency variation due to the coupler will be cancelled. Another commonly used signal-separation device is the reflectometer bridge as shown schematically in Fig. 12.4. The two detectors R and A are normally built into the bridge. They produce a d.c. or low-frequency (if the swept oscillator is amplitude modulated by a low-frequency signal in the order of kilohertz) voltage which is proportional to the power dissipated in the 50 Q loads that they are connected to. It is not difficult to show that the voltage across the balancing 50 Q resistor (reflection) is directly proportional to the reflection coefficient of the device connected to the test port and that the voltage across the 'incident' 50 Q resistor is directly proportional to the input power. The reflectometer bridge can also be represented schematically by a circuit symbol similar to that of a dualdirectional coupler, as shown in Fig. 12.5. Coming back to the scalar network analyser system shown in Fig. 12.2, the d.c. or low frequency output voltage of detector B is proportional to the output power of the OUT. The transmission coefficient of the OUT can be displayed as the ratio of VBIVR in decibels on the frequencyresponse test set. The frequency-response test set has three d.c. (or low-frequency) inputs, R, A and B. It normally has two traces on its CRT display. The test set can display the absolute power of either two of R, A and B in dBm or the ratio of V AIVR and VBIVR in decibels on the vertical axis. The input to the horizontal axis of the display is the sweep signal (time base) of the swept oscillator, which corresponds to the frequency axis. Since d.c. voltages are more difficult to amplify and compare accurately, almost all frequency-response test sets provide a low-frequency square-
t
-20 dBm (0.01 mW)
Matched termination
I
X
CA = 20 dB A
t
Matched termination
I
o dBm ~~------t~ (1 mW) Fig. 12.3 20 dB dual-directional coupler.
-0.044 dBm (0.99 mW)
~
_A_N_D_N_E_T~WORK ANALYSERS
I
S_W_E_P_T_M_E_A_S_U_R_E_M_E_N_T_S
L I_ _ _ _
~
r.f.
input
50
n
(Reflection)
Short for fioIl-/+-- calibration /
<
"
///~ DUT
",/ /
Fig. 12.4
Reflectometer bridge.
Incident (R)
Reflection (A)
r.f.
input
Test port
Fig. 12.5 Schematic diagram ot a reflectometer bridge. Incident (R) has r.t. input coupling -6dB; Reflection (A) has test port coupling -9dB; and the test port has r.t. input coupling -9 dB.
--'I ~
O_T_H_E_R_SC_A_L_A_R_N_E_TW_O_RK_-_A_N_A_LY_S_E_R_S_Y_S_T_EM_S_ _ _
L - -_ _ _ _
wave signal (e.g. 1kHz or 27.8 kHz) to modulate the swept oscillator into pulsed r.f. mode, so that the outputs of the detectors are square waves of 1 kHz or 27.8 kHz. 12.5 OTHER SCALAR NETWORK-ANALYSER SYSTEMS
The spectrum analyser/tracking generator system together with a directional coupler acting as the signal-separation device as described in Section 9.5 form a complete scalar network-analyser (SNA) system. This type of SNA is accurate, easy to use and easy to calibrate. It does not rely on expensive crystal detectors which are non-repairable. In performing transmission measurements they offer a large dynamic range, which is basically the dynamic range of the spectrum analyser. The only commonly encountered problem with this type of SNA is in the measurement of small reflection signals because of the coupling factor (say 20 dB) of the reflection sampling arm of the coupler. The cost of an SNA system based on a spectrum analyser is slightly more than that of a system based on a frequency-response test set. However, it is noted that a spectrum analyser is a general instrument with many application areas. This fact should also be taken into account in comparing the costs of various SNA systems. A cheaper SNA system can be configured using power meters as shown in Fig. 12.6. It is noted that power meters used for this purpose must have fast acquisition time and that analogue outputs must be available in order for the output voltage to be displayed on the oscilloscope.
meter
Swept oscillator Sweep
r.f.
output
Scope
~ ChI
L.......j-Oll'O
Fig. 12.6 Scalar network analyser with power meters.
Ch2
meter
@DI
SWEPT MEASUREMENTS AND NETWORK ANALYSERS
L -_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
~
12.6 VECTOR NETWORK ANALYSER
When the phase as well as the magnitude of the s-parameters of a DUT is required, it is necessary to have a heterodyne system converting the signals to be measured to a low frequency where phase comparison is possible. This forms the basis of all modern vector network analysers. A basic vector network analyser (VNA) connected in the transmission mode is shown in Fig. 12.7. The test signal from the swept oscillator is divided equally into a test and a reference channel. The test signal is 'transmitted' through the DUT before it is down converted in a mixer by a local oscillator signal (LO). The reference signal is directly down converted in an identical mixer by the same LO signal under strict phase lock. The mixers convert these two signals to a low frequency (IF) where comparisons of phase and magnitude between signals from the two channels are made. Because of the provision for the insertion of the DUT in the test channel, the electrical lengths of the two channels are different, hence even if the two DUT ports are connected directly together, the phase and magnitude of the signals on the two channels are different. In order to account for the phase and magnitude offset of the two channels due to different physical lengths and hence electrical lengths, a transmission-line section of variable length known as the line stretcher is inserted in the reference channel as shown in Fig. 12.8. In the calibration for transmission measurements, the DUT ports are connected directly and the line stretcher is mechanically tuned until the phase difference between the two channels is as close to zero as is possible over the frequency range of interest. The magnitude offset between the two channels is also affected by the difference in the path lengths, but to a much smaller extent.
IF. BPF IF (fixed) '------' Reference
Reference channel Sweptosc.
r----, I
I I
I
l-+-~
I
~-r-....J
I I I
IL ___ ..JI
IF (fixed) Test channel
Mixer
Fig. 12.7 Basic vector network analyser.
'------' Test IFBPF
~______________S_O_U_R_C_E_S_Y_N_C_H_RO__N_IZ_A_TI_O_N______________~I ~ IF Test
LO (yeO)
Swept osc.
r.f. reference channel
IF Reference Line stretcher
Fig. 12.8
Phase equalization.
Modern vector network analysers do not need phase equalization by the line stretcheL In the calibration of these VNAs, the phase and magnitude offsets at a number of points in the frequency range of interest are recorded in an externally connected controller or built-in microcomputer and the stored data are used to correct the measured data before they are displayed.
12.7 SOURCE SYNCHRONIZATION The intermediate frequencies (IF) of the test and the reference channel must be of the same fixed frequency, independent of the frequency of the Lt. input at any instance. This requires a swept local oscillator sweeping over a bandwidth equal to the swept bandwidth of the r.f. input. The local oscillator is also required to have a frequency which exceeds the Lf. input frequency by the amount of the intermediate frequency at all times. The establishment of a fixed relationship between the Lf. and the LO signals is known as source synchronization. Figure 12.9 shows an example of how source synchronization can be achieved. In this example, the Lf. sweep range is 0.1 to 1.5 GHz and the 100 kHz IF is maintained by means of an accurate reference oscillator of the same frequency. It is noted that both the phase detector and the reference oscillator are normally part of the analyser.
[j~~
L I_ _ _
~S~W~E~P~T_M~E~A~S~U~RE~M_ENTS AND NE._T~W~O~R~K_A~N~A_L_Y~S~E~R~S_~ __
J
Phase To power splitter
Sweep osc.
0.1 to 1.5 GHz
Reference oscillator fixed at fIF = 100 kHz
fIF =
100 kHz
Reference channel From r.f. test and reference channel 0.1 to 1.5 GHz
LO 0.1001 (yeO) To 1.5001 GHz
To phase and magnitude comparator and display
Test channel fIF
=
100 kHz
Fig. 12.9 Source synchronization.
12.8 POWER-SPLITTER CIRCUIT
The basic operating principles of a vector network analyser have been illustrated with reference to its transmission mode in Fig. 12.7. In reflection measurements a signal-separation device to sample the reflected signal is necessary. One possible way of achieving this purpose using a dual-direction coupler is shown in Fig. 12.10. The circuit shown inside the dotted box, which includes the power-splitter function, is sometimes known as the transmission and reflection test set of a vector network analyser.
PROBLEMS 1. For the reflectometer shown in Fig. 12.4, show that the output voltage
of detector A is proportional to the square of the voltage-reflection coefficient as seen by the test port and that the output voltage of detector R is proportional to the power coming into the bridge, assuming that the output of the detectors is proportional to the power dissipated in the resistors across which they are connected.
___________________P_RO_B_L_EM__S________________~I ~
~C===
DUT output to test channel for transmission measurements
Output DUT Input
r--I I I I I I
---------------,I
I I
I
I I
I
I
I
To test channel for reflection measurements
I I
I I
I
To reference channel
I
I
I I
I Line I IL __ ~ ______________ stretcher I ~
r
Signal source from swept oscillator
Fig. 12.10 Signal separation - transmission and reflection test.
2. With reference to the schematic diagram of the reflectometer bridge shown in Fig. 12.5, show that: (a) the power coming out from the test port is 9 dB below the power coming into the bridge; (b) the power coming out from the incident (R) arm is 6 dB below the power coming into the bridge; and (c) the power coming out from the reflection (A)
[222 I [-
.=-
S~EPT MEASUREMENTS~~D NETWORK ANALYS_E_R_S_ _-==_l---"
t t-----------~I.- Pi (b)
Fig. P.12.1
arm is 9 dB below the power reflected back into the bridge at the test port. 3. The coupling factor C of a directional coupler as shown in Fig. P.12.l(a) is given by P C = 10 iog lO p;
and the directivity of a directional coupler is given by
Pf
D = lOiog lO Ph'
where P b is defined in Fig. P.12.l(b). If the coupling factor and the directivity of a directional coupler are 10 dB and 24dB, respectively, find PI and P2 which are defined in Fig. P.12.2. 4. The input impedance of an amplifier was measured via an adapter by a vector network analyser to be 61- jS Q at 1 GHz. The s-parameters of the adapter at 1 GHz were measured in a separate measurement to be
F_U_R_T_H_ER__R_E_A_D_IN_G__________________~I
L -_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
o dBm
o dBm
Short circuit
------II [J
~ ~I
Fig. P.12.2
ISI1I IS121
= =
-27 dB
-0.05dB
LS I1 LS 12
8° = 30° =
IS211
Iszzl
=
-0.1 dB
= -30dB
LS 21 = LS22 =
30° _12°.
Find the 'true' input impedance of the amplifier. Ans.: 52 - jO.005 n 5. A detector/frequency-response test set type of scalar network analyser is to be configured to measure the gain of a 5 W power amplifier. Draw a possible configuration for this measurement and state the major disadvantage of the system with respect to such a measurement. 6. An open-circuited air line (coaxial with air as dielectric) is connected to a network analyser (either scalar or vector) for reflection measurement over the frequency range of 1.0-10.0 GHz. Sketch the magnitude display (in dB) against frequency for the entire range. Note: Similar shapes are frequently encountered in swept reflection measurements. Explain why. FURTHER READING High Frequency Swept Measurement, Hewlett Packard, Application Note 183. Laverghetta, T.S., (1984) Practical Microwaves, Howard W. Sams. Vector measurements of High Frequency Networks (1987), Hewlett Packard.
~
Index
Active biasing 86 Admittance chart 53 Alumina 26, 27 AM 1, 173, 176, 192 Amplifier small-signal 91-108 broadband 113 low-noise 109-13, 115 high-gain 114 power 119-37 Associated gain 77, 78 Attenuation constant 8, 10 Available power power available 95 noise 195, 197 Avalanche diode 203 Balanced stubs 134, 136 Bandwidth 43, 70, 197, 208 IF 168,169,179,181 BJT3, 71,119,139 biasing 86 data-sheet 77-85 design 116, 117 Boltzmann's constant 86, 172, 195, 197 Broadband 113 Characteristic admittance 10 impedance 9, 22 Clapp 139 Class A 85,119,137 ClassAB 85 Class B 85,119,137 Class C 119,137,138,139 biasing, 121, 122 Class D 119 ClassE 119 Class S 119 Colpitt 139 Conjugate matching 104,107-11,114, 126 Constant power gain circles 104, 106
Constant Q curves 57 Complex conjugate 129 Coupling factor 214,221,222 Dielectric resonator oscillator (DRO) 2 Direct broadcasr satellite (DBS) 2 Directional coupler 178, 217, 220-2 Dissipation factor 27,28 see loss tangent Dynamic range 120, 172 Effective input noise temperature 197 Effective permittivity 22-6 Electrical length 216 Equivalent noise resistance 77,80,81 FAX 2 Feedback 36, 73, 75, 96, 140,211 FET 139,198 biasing, 85 Frequency counting 184-92 FM 1, 174 residual 169 incidental 176 GaAs FET 3,71 Gain maximum available 79, 82 operating power 95 transducer power 95 unilateral transducer 97 Gain compression 120, see 1 dB compression Harmonic 43,119,163,185,188 distortion, 120 Hartley, 139 HBT71 (heterojunction bipolar transistor) HEMT71,198 (high electron mobility transistor) Heterodyne conversion, 185-8, 191, 206,216
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Identity matrix 140, 157 IFA3 IFbandwith 168, 169, 179, 181 Incidental FM 176 Indefinite matrix 140, 157 Input impedance (of power transistors) 127 Input stability circle 102-4 Instability, see stability Intermediate frequency (IF) 3, 4 Image 163, 207, 209 Immittance 14, 15, 36 Immittance chart 54, 55, 57,58 Impedance chart 52, 53 Impulse response 210 Linvill 51 LNA (low noise amplifier) 3,109-13 Load stability circles 102,103,105 Local oscillator (LO) 162,216 Loss tangent 27, 28 MAG (maximum available gain) 79,82 Matching 42-63, see also conjugate matching MESFET88, 107, 115, 117, 119 Microstrip 18-27 Minimum noise figure 110 Mobile Radio Phone (MRP) 3 MOSFET3 Network analyser 210-19 Noise circles 111, 112, 113 Noise figure 77, 78, 80, 81,111 meter 205, 207 Noise measurement 178-81, 199-203 Noise parameters 80, 81 Noise power bandwidth 179, 181 Noise temperature 197-99 Non-woven 27 One dB compression 120 Operating power gain 95,106,109,114 Open-circuit stub 60, 63, 64 Optimal load reflection coefficient 99, 100, 102, 114 Optimal source reflection coefficient 99, 100, 102, 114 Oscillation conditions 142 Output impedance (of power transistor) 127 Output stability circle 102-4
Phase constant 8, 9 PIN diode 211 Potentially unstable 82, 102, 104, 106, 110, 114, 139 Power meter 204, 216 Propagation coefficient 8 Propagation velocity 22 PTFE (polytetrafluorethylene) 20, 26 Q curves 56,131 Quality factor (Q factor) 42, 43, 46, 57 Quarter wave transformer 14, 68 Quasi-TEM 21 Radio frequency choke (RFC) 121, 122, 127, 136, 137 Reflection coefficient 11, 13 Reflection loss 18 Residual FM 169 Resolution 168 Return loss 18 Rollet, see stability factor s-parameter 31-41 Short-circuit stub 60, 63, 64 Smith chart 15, 17,51 Source stability circles 103, 104, 105 Spectrum analyser 161-83 Stability factor (Rollet's) 101, 106, 107 Stub matching 60 single 60, 61, 62 double 64, 65 SUPERCOMP ACT 117 Swept oscillator 162, 172,211 Synthesizer 210,211 Teflon (PTFE) 20, 26 Thermal noise 179, 180, 195 measurement 178-181 TOUCHSTONE 117 Tracking generator 172 Transducer power gain 82, 95-8, 109,
111
Transmission coefficient 98, 212, 214 reverse 96 Transmission line 7-15, 60 Transmission measurement 177,216, 217 Transmission transducer gain 82 Unconditionally stable 82, 99, 100 Unilateral amplifier 96, 98 Unilateral transducer gain 96, 97,98
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Voltage controlled oscillator (VeO) 2, 3,189,211 Voltage standing wave ratio (VSWR) 8, 12,13,18
Woven 27
Wavelength 12,24,68
Z-parameters 140,142
Y-parameters 76, 77 Yttrium-iran-garnet (YIG) 163, 186
