Chapter 12

Metallic Cable Transmission Media

CHAPTER OUTLINE 12-1 12-2 12-3 12-4 12-5 12-6 12-7

Introduction Metallic Transmission Lines Transverse Electromagnetic Waves Characteristics o f Electromagnetic Waves Types o f Transmission Lines . Metallic Transmission Lines Metallic Transmission Line Equivalent Circuit

12-8 Wave Propagation on a Metallic Transmission Line 12-9 Transmission Line Losses 12-10 Incident and Reflected Waves 12-11 Standing Waves 12-12 Transmission-Line Input Impedance 12-13 Time-Domain Reflectometry 12-14 Microstripand Stripline Transmission Lines

OBJECTIVES ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■

Define the general categories of transmission media Define guided and unguided transmission media Define metallic transmission Unes Explain the difference between transverse and longitudinal waves Define and describe transverse electromagnetic waves Describe die following characteristics o f electromagnetic waves: wave velocity, frequency, and wavelength Describe balanced and unbalanced transmission lines Explain baluns Describe the following parallel-conductor transmission lines: open-wire, twin-lead, twisted-pair, unshielded twisted-pair, shielded twisted-pair, and coaxial transmission lines Describe plenum cable Describe a transmission-line equivalent circuit Define the following transmission characteristics: characteristic impedance and propagation constant Describe transmission-line propagation and velocity factor Explain what is meant by the electrical length of a transmission line

511

■ ■ ■ ■ ■ ■ ■ ■ ■ ■

Describefoe following transmission-line losses: conductor loss, dielectric Ue&ting loss, radiation loss, coupling loss, endcorona Define and describe incident and reflected waves Explain the difference between resonant and nonresonant transmission lines Describe the term reflection coefficient Describe standing waves and standing wave ratio Analyze startling waves on open and shorted transmission lines Define transmission line input impedance Describe how to match impedances on a transmission line Describe time-domain leflectometry Describe microstrip and stripline transmission lines

12-1

INTRODUCTION Transmission media can be generally categorized as either unguided ot guided. Guided transmission media are those with some form of conductor that provides a conduit in which electromagnetic signals are contained. In essence, the conductor directs the signal that is propagating down it. Only devices physically connected to the medium can receive signals propagating down a guided transmission medium. Examples of guided transmission media are copper wire and Optical fiber. Copper wires transport signals using electrical current,' tChereas optical fibers transport signals by propagating electromagnetic waves through a nonconductive material. Unguided transmission media are wireless systems ti e., those without a physical conductor). Unguided signals are emitted then radiated through air ora vacuum (or sometimes Water). The direction of propagation in.an ungnided transmission medium depends on die direction in which the signal was emitted and any obstacles the sig nal may encounter while propagating. Signals propagating down an unguided transmission medium are available to anyone who has a device capable of receiving them. Examples of unguided transmission media are air (Earth’s atmosphere) and free space (a vacuum).
12-2

METALLIC TRANSMISSION LINES A transmission line is a metallic conductor systemused to transferelectrical energy from one point to another using electrical current flow. M ore specifically, a transmission line is two or more electrical conductors separated by a nonconductiveinsulator fdielectrick suchas a pair of wires or a system of wire pairs. A transmission line can be as short as a few inches or span severanfiStEand miles. Transmission lines can be tried to propagate dc or low-frequency ac (suchas 60-cycle electrical power and audio siffials) or to propagatevery high frequencies (such as microwave radio-frequency signals). When propagating low-frequency signals, transmission line bdhavior is rather simple and quite predictable. However, when propagat-

512

Chapter 12

ing bigh-frequency signals, the characteristics o f transmission lines become more involved, and their behavior is somewhat peculiar to a student o f lumped-constant circuits and systems.

12-3

TRANSVERSE ELECTROMAGNETIC WAVES waves, There are two basic kinds o f waves: longitudinal and transverse. With ___________ the displacement (amplitude) is in the direction o f propagation. A surface wave o f water is a longitudinal wave. Sound waves are also longitudinal. With transvase waves, the direc tion o f ;displacement is perpendicular to the direction o f propagation. Electromagnetic wavesaretransverse waves. : Propagation o f electrical power along a transmission line occurs in the form of tmnsverseelectromagnetic CTEM>wgrgrrA wave is m oscillatory motion. The vibration o f a particle excites similar vibrations in nearby particles. A TE M wave propagates primarily in the nonconductor (dielectric) that separates the two conductors ofm e transmission line. Therefore, a wave travels or propagates itself through a medium. Electromagnetic waves ten produced by the acceleration ofan electric charge. In conductors, current and voltage are al ways accompanied by an electric (£) field and a magnetic (H ) field in the adjoining region of space. Figure 12-1 shows the spatial relationshipsbetween the£ and H fields o f an electromaghetic wave. From the figure, it can be seen that the fvand H fields are perpendicular to each other (at 90° ahgles) at all points. This is referred to as space Quadrature. Electromagnetic waves that travel along a transmission line from the source to the load are called incident waves, and those that travel from the load beck toward die source are called reflected waves.

12-4* CHARACTERISTICS OF ELECTROMAGNETIC WAVES Three o f the primary characteristics o f electromagnetic waves are wave velocity,frequency, and wavelength.

12-4-1

Wave Velocity

Waves travel at various speeds, depending on the type o f wave and tire characteristics o f the propagation medium. Sound waves travel at approximately 1100 feet per second in the normal atmosphere. Electromagnetic waves navel much faster. In free space (a vacuum), TEM waves travel at die speed o f light c = 186^83 statute miles per second, or 299,793.000

FIGURE 12-1 Transverse electromagnetic wave


513

meters per second, rounded o ff to 186,000 mi/s and 3 X 10s m/s. However, in air (such as Earth’s atmosphere), TEM waves travel slighdy slower, and along a transmission line, elec tromagnetic waves travel considerably slower.

12-4-2

Frequency and Wavelength

The oscillations of an electromagnetic wave are periodic and repetitive. Therefore, they are characterized by a frequency. The rate at which the periodic wave repeats is its frequency. The distance of one cycle occurring in space is called the wavelength and is determined from the following equation: ~' distance = velocity X time

(12-1)

If the time for one cycle is substituted into Equation 12-1, we get the length of one cycle, which is called the wavelength and representeid by the Gieek lowercase l&mbda (X). X = velocity X period ( 12- 2)

= v x r

where

X = wavelength v = velocity T = period

And because T = Ilf A ml

(12-3)

For ffee-space propagation, v = c; therefore, the length o f one cycle is 1_ ^ /

^ m^s _ meters /cycles/s cycle

’

(114!

To solve for wavelength in feet or inches, Equation 12-4 can be rewritten as ^ _ 118 X 109in/s _ inches /cycles/s cycle

(12-5)

x = 9.83 X 108ft/s _ feet / cycles/s cycle

( 12-6)

Figure 12-2 shows a graph of the displacement and direction of propagation of a trans verse electromagnetic wave as it travels along a transmission line from a source to a load. The horizontal (X) axis represents distance and the vertical ( ! ) axis displacement (voltage). One wavelength is the distance covered by one cycle of the wave. It can be seen thatthe wave moves to the right or propagates down the line with time, If a voltmeter were placed at any stationary point on the line, the voltage measured would fluctuate from zero to maximum positive, bade to zero, to maximum negative, back to zero again, arid then the cycle repeats.

12-5

TYPES OF TRANSMISSION LINES Transmission lines can be generally classified as balanced or unbalanced.

12-5-1

Balanced Transmission Lines

With two-wire balanced lines, both conductors carry current; however, one conductor carties the signal, and the other conductor is the return path. This type of transmission is called differential, or balanced, signal transmission. The signal propagating down the wire is measured as the potential difference between the two wires. Figure 12-3 shows a balanced 514

Chapter 12

.

t-T / 4

FIGURE 12-2 Displacement and velocity of a transverse wave as it propa gates down atransmissionline

Common-mode voltags (noise) Earth fjround

FIGURE 12-3 Differential, or balanced, transmission system

*

transmission line system. Both conductors in a balanced line carry signal currents. The two currents are equal in magnitude with respect to electrical ground but travel in opposite di rections. Currents that flow in opposite directions in a balanced wire pair are called metallic circuit currents. Currents that flow in the same direction are called longitudinal currents. A balanced wire pair has the advantage that mast noise interference (sometimes called common-mode interference) is induced equally in bothwires, producing longitudinal cur rents that cancel in the load. The cancellation ojf common modesignals is called commonmode rejection. Common-mode rejection ratios of 40 dB to 70 dB are common in balanced transmission lines. Any pair of wires can operate in the balanced mode, provided that nei ther wire is at ground potential. This includes coaxial cable that has two center conductors and a shield. The shield is generally connected to ground to prevent static interference from penetrating the center conductor. Metallic Cable Tran8mi8sion Media

*

515

FIGURE 12-4 Results of metejfic and longitudinal currents on a balanced transmission line;

(a] mstaffic currentsdue to Signal voltages; (t>) longitutfinal currents due to noise voltages Figure 12-4 shows the result o f metallic and longitudinal currents on atwo-wire bal anced transmission fine. Ftom the figure, it can be,seen that the longitudinal currents (of ten produced by static interference) cancel in the load.

1 2 -5 -2

Unbalanced Transmission Lines

„

With an unbalanced transmission line, one wire is at ground potential, whereas the other wire is at signal potential. This type o f transmission line is called single-ended, or unbalanced, signal transmission. With unbalanced signal transmission, the ground wire may also be tise reference for other signal-carrying wires. If this is die case, die ground wire must go wherever any o f die signal wires go. Sometimes this creates a problem because a length o f wire has resistance, inductance, and capacitance and, therefore, a small potential difference may exist between any two points on the ground wire. Consequently, the ground wire is hot a perfect reference point and is capable o f having noise induced into it. Unbalanced transmission ttaeshaiyp the advantage o f requiring only one wire for each signal,.a^d only one gpou ^j^p^reqm ted no matter how many signals are grouped into one Conductor. The prftH a^d iS & lw »ge o f unbalanced traissussioa Hues is reduced im munity to comraon-mode s%nalS,Mch as noise and other interference. Figure 12-5 shows tsyo unbalanced transmission systems. The potential difference on each signal wire is measured from that wire to a common ground reference. Balanced trans mission lines can be connected to unbalanced lines and vice versa with special transform ers called bpluns. *

1 2 -5 -3

Baiuns

A circuit device used to connects balanced transmission line to an unbalanced load is called i t, balun (balanced to unbalanced). Or more commonly, an unbalanced transmission line, such as a coaxial cable, can be connected to a balanced load, such as an antenna, using a special transformer with) an unbalanced primary and a center-tapped secondary winding. The outer conductor (shield) o f an unbalanced coaxial transmission line is generally con nected to ground. Atrelatively low frequencies, an ordinary transformer cad be used to iso late the ground from the load, as shown in Figure 12-6a. The balUn must have an electro static shield connected to earth ground to minimize the effects o f stray capacitances. For relatively high frequencies, several different kinds o f transmission-line baluns exist. The most common type is a narmwhnnfi balun, sometimes called a choke, sleeve; or bazooka balun,' which is shown in Figure 12-6b. A quarter-wavelength sleeve is placed around and connected to the outer conductor o f a coaxial cable. Consequently; the imped ance seen looking back into the transmission line is formed by the sleeve, and the outer conductor and is equal to infmity (i.e.i'tbeouterconduCtor no longer has a zero impedance to Chapter 13

'

,l .

....

Amplifier T

►V.

. Circuit 1 signs! wbe

Amplifier 2

T

1 • Signal 1 voltage

v .- v b vM - v .- v b- o Circuit 2 signal wiia

AmpHflar

Amplifier

;,

3

-

4

Signal 2 voltage jnced transmission line: to noise voltages

Ground reference

rats on a two-wire baligitudinal currents (of-

itial, whereas the other filed single-ended, or ision, the ground wire e case, the ground wire ». a problem because a efore, a small potential msequently, the ground nduced into it. only one wire for each gnals are grouped into bn fines is reduced imice. potential difference on trence. Balanced transwith special transform-

ibalanced load is called teed transmission line, as an antenna, using a fd secondary Winding, fine is generally conmer can be used to isomust have an electrotay capacitances, rasmission-line baluns lied a choke, sleeve, or ength sleeve is placed hsequently, the imped«ve, and die outer con es a zero impedance to

FIGURE 12-5

Single-ended, or unbalanced, tranarfuasion system

Balanced diode antenna Balanced Una

< Transformer

Unbalanced input

Sleeve

Shield ■

Center conductor

UnbalaruHicosxia! line lb) FIGURE 12-6

Baltina: (a) transformer baiun; (b) bazooka balun

ground). Thus, one wire o f the balanced pair era be connected to the Sleeve without short-* circuiting the signal. The second conductor is connected to the inner conductor o f the coax ial cable. ■. ■ ■

1 2 -6

M E TA L U C TR A N S M IS S IO N U N E S A ll data communications systems and Computer networks are interconnected to some de gree or another with cables, which are all or part o f the transmission medium transporting signals between computers. Although there is an enormous variety o f cables manufactured today, only a handful o f them are commonly used for data communications circuits and 1 computer networks| sdmiMwesnnect


517

Conductor!

FIGURE 12-7

Open-wire transmission line

1

■V Conductors

Solid dielectric

FIGURE 12-8 Twin lead towowire transmission line

12-6-1

Paraltel-ConduGtor Transmission Lines

wiw lemu*up Jiggt,' ii ' comprised o f two or more metallic conductors (usu ally copper) separated by a nonconductive insulating material called a i Com mon dielectric materials air pap^rfttten md Taflnw The most common parallel-conductor transmission lines areflfl8 V v,lft!fn foa 4 and f l M f l i r [including
Chapter 12

Conductor*

FIGURE 12-9 Twistedpair two-wire transmission line

Four insulated twisted-pair cables

*% } Sheath

FIGURE 12-10 Unshielded twisted-pair (UTP) cable

t tC-H ,

etallic conductors (usukUed a iie k c tiic , CommteargUssv and Toflon. m ire, twin lead, and twisted-patr (STP)]. tm m & fies are two-wire

5sm m *& m oftyo spacers are placed at pebetween the conductors tetweenlihchesand 6 ■cooduetoreVwhicluhe asmission line is its simligh, and the cable is susuces crosstalk. Crosstalk acent cable. The primary phone applications. wire parallel-conductor pially the same as openjlactocs are replaced with he entire cable. Uniform ater in this chapter. Twino connect tekwiaiev>iMh pad cable are Teflon

12-6-1-3 Twisted-pair transmission lines. A twisted-pair transmission line (shown in Figure 12-9) is formed by twisting two insulated conductors around each other. Twisted pairs are often stranded in units, and the units are then cabled into cores containing up to 3000 pairs o f wire. The cores are then covered With various types o f sheaths forming cables. Neighboring pairs are sometimes twisted with different pitches (twist length) to reduce the effects o f electromagnetic interference (EMI) and radio frequency interference (RFI) from external sources (usually man-made), such as fluorescent lights, power cables, motors, re lays, and transformers. Twisting the wires also reduces crosstalk between cable pairs./ The sire o f twisted-pair wire varies from 16-gauge (16 AWG [American Wire Gauge]) to 26 gauge. The higher the wire gauge, the smaller the diameter and the higher the resis tance. T wisted-pair cable is used for both analog and digital signals and is the most commnniv used transmission1medium fortelephone networks and building cabling syiteiris. Twisted-pair transmission lines are also the transmission medium o f choice for most focal area networks because twisted-pair cable is simple to install and relatively inexpensive when compared to coaxial and optical fiber cables. There are two basic types o f twisted-pair transmission lines specified by the EIA/llA 568 Commercial Building Telecommunications Cabling Standard for local area networks: 100-ohm unshielded twistedpa ir and 150-ohm shielded twisted pair. A typical network utilizes a variety o f cabling technologies’ depending on the network’s size, its topology, and technologies by dividing network-wiring systems into six subsystems: horizontal cabling, backbone cabling, work area, telecommunications closet, equipment room, and building entrance. The six subsystems specified in the 568 standard are described in more detail in a later chapter. ' Unshielded twisted-pair (UTP) cable consists o f two copper wires where each wire is separately encapsulated in PVC (polyvinyl chloride) insulation (see Figure 12-10). Be cause a wire can act like an antenna, die wires are twisted two or more times at varying lengths to reduce crosstalk and interference.'By carefully controlling the number o f twists per foot and the manner in which multiple pairs are twisted around each otljer, manufac turers can improve the bandwidth (i.e., bit rate) o f the cable pair significantly. The mini mum number o f twists for UTP cable is two per foot. Most telephone systems use UTP cable, and themajority o f new buil<|iBgs art prewired with UTP cable. Generally* more cable is installed than is initially needed, providing room


519

Table 12-1

EIA/TIA 568 UTP and STP Levels and Categories

Level 1 (UTP) Level 2 (UTP) Category3 (UTP/Sl?) Category 4 , (UTP/STP) Category 5 (UTP/STP) Bnhanced category} (UTP/STP)

Standard voice and low-speed data Standard voice and low-speed data Low-speed local area network!

•

Low-speed local area networks High-speed local area networks High-speed local area networks and asynchronous transfer mode (ATM)

2400 bps 4 Mbps 16 Mbps and all level 2 applicatioas 20 Mbps and all category 3 applications 100 Mbps

18,000 feet 18,000 feet 100 meters

330 Mbps

100 meters or more

550 Mbps

100 meters or mote 100 meters or mote

100 meters . . 100 meters

Proposed New Categories Category 6 (UTP/STP) Category 7 shielded screen twisted pair (STP) Foil twisted pair (STP)

Shielded foil twisted pair (S IP )

Very bigh-speed local area networks and asynchronous transfer mode (ATM) Ultra-high-speed local area networks and asynchronous transfer mode (ATM)

Ultra-higbTspeed local area networks and asynchronads transfer mode (ATM); designed to minimize EMI susceptibility and maximize EMI immunity Ultra-high-speed local area networks and asynchronous transfer mode (ATM); designed to minimize EMI susceptibility and maximize EMI immunity

1 Gbps-

>lGbps

7

>1 Gbps

7

for orderly growth. This is one o f the primary reasons why UTP cable is so popular. UTP cable is inexpensive, flexible, and easy to install. It is die least expensive transmission medium, bat it is also the most susceptible to external electromagnetic interference. To meet the operational requirements for local area networks, the EIA/TIA 568 stan dard classifies UTP twisted-pair cables into levels and categories that certify maximum data rates and recommended transmission distances for both UTP and STP cables (see Ifeble 12-1). Standard UTP cable for local area networks is comprised o f four pairs o f 22- or 24-gauge copper wire where each pair o f wires is twisted around each other. There are seven primary unshielded twisted-pair cables classified by the EIA/TIA 568 standard: level 1, level 2, category 3, category 4, category 5, enhanced category 5, and category 6. 1. Level 1. Level 1 cable (sometimes called category 1) is ordinary thin-copper, voicegrade telephone wire typically installed before the establishment o f the 568 standard. Many o f these cables are insulated with paper, cord, or rubber and are, therefore, highly susceptible to interference caused by insulation breakdown. Level 1cable is suitable only for voice-grade telephone signals and very low-speed data applications (typically under 2400 bps). 2. Level 2. Level 2 cable (sometimes called category 2) is only marginally better than level 1£able but well below the standard’s minimum level o f acceptance. Level 2 cables are also typically old, leftover voice-grade telephone wires installed prior to die establish ment o f the 568 standard. Level 2 cables comply with IBM’s Type 3 specification GA273773-1, which was developed for IEEE 802.5 Token ring local area networks operating at transmission rates o f 4 Mbps. , 3, Cateeorv 3. Category 3 (CAT-3) cable has more stimgent requirements than liw ell same

Chapter 18

|

cable can have the same number of turns per inch. This specification provides the cable mote immunity to crosstalk. CAT-3 cable was'designed to accommodate the requirements for two local area networks: IEEE 802.5 Token Ring (16 Mbps) and IEEE 802.3 lOBase-T Ethernet (10 Mbps). In essence, CAT-3 cable is used for virtually any voice or data transmission rate ' up to 16 Mbps and, if four wire pairs are used, can accommodate transmission ratesupto

j

I level

18.000 feet 18.000 feet lOOmeten

Icategory

100 meters

j

' 100Mbps. '■

4. Category 4. Category 4 (CAT-4) cable is little more than an upgraded version of CAT-3 cable designed to meet tighter constraints for attenuation (loss) and crosstalk. CAT-

lOOmeten lOOmeten or moee

lOOmeten or more lOOmeten or move

?

? x

4 cable was designed for data transmission rates up to 20 Mbps. CAT-4 cables can also han dle transmission rates up to 100 Mbps using cables containing four pairs of wires. 5. Category 5. Category 5 (CAT-5) cable is manufactured with more stringent design specifications than either CAT-3 or CAT-4 cables, including cable uniformity, insulation type, and number o f turns per inch (12 turns per inch for CAT-5). Consequently, CAT-5 ca ble has better attenuation and crosstalk characteristics than the lower cable classifications. Attenuation in simple toms is simply the reduction of signal strength with distance, and crosstalk is the coupling of signals from one pair of wires to another pair. Near-end crosstalk refers to coupling that takes place .when a transmitted signal is coupled into the receive signal at the same end of the cable. CAT-5 cable is the cable of choice for most modern-day local area networks. CAT-5 cable was designed for data transmission rates up to 100 Mbps; however, data rates in ex cess of 500 Mbps are sometimes achieved. CAT-5 cable is UTP cable comprised of four pairs of wires, although only two (pairs 2 and 3) were intended to be used for connectivity. The‘other two wire pairs are reserved spares. The following standard color code is specified by the ELA for CAT-5 cable:

r teis so popular. UTP cae transmission medium, frenee. tthe EIA/TIA 568 stanit certify maximum data ►cables (see Table 12-1). airs of 22- or 24-gauge

| f

j

ssified by the EIA/TIA ihanced category 5, and nary thin-copper, voicethe 568 standard. Many fore, highly susceptible ble only for voice-grade ier 2400 bps). i marginallybetter than ptance. Level 2 cables 1prior to the establish3 specification GA27,networks operating at ]uirements than level 1 o pairs within the same

i

Pair 1: blue/white stripe and blue Pair 2: orange/white stripe and orange Pair 3: green/white stripe and green Pair 4: brown/white stripe and brown

Each wire in a CAT-5 cable can be a single conductor or a bundle of stranded wires re ferred to as C at-5 s o lid or CAT-S fle x , respectively. Whenbothcable types are usedin thesame application, the solid cable is used for backbones and whenever the cable passes through walls or ceilings.The strandedcable istypically usedfor patchcablesbetweenhubsandpatch panell and for drop cables that are connected directly between hubs and computers. 6. Enhanced ca tegory 5. Enhanced category 5 (CAT-5E) cables are intended for data transmission rates up to 250 Mbps, although they often operate successfully at rates up to 350 Mbps and higher. , , 7. C ategory 6. Category 6 (CAT-6) cable is a recently proposed cable type comprised of four pairs o f wire capable of operating at transmission data rates up to 550 Mbps. CAT6 cable is very similar to CAT-5 cable except CAT-6 cable is designed and fabricated with closer tolerances and uses more advanced connectors. Shielded twisted-pair (STP) cable is a parallel two-wire transmission line consisting of two copper conductors separated by a solid dielectric material. The wires and dielectric are enclosed in a conductive metal sleeve called zjoil. If the sleeve is woven into a mesh, it is called h braid. The sleeve is connected to ground and acts as a shield, preventing sig nals from radiating beyond their boundaries (see Figure 12-11). The sleeve also keeps elec tromagnetic noise and radio interference produced in external sources from reaching the signal conductors, STP cable is thicker and less flexible than UTP cable, making it more difficult andexpensive to install. In addition, STP Oable requires an additional grounding connector and is more expensive to manufacture. However, STP cable offers greater secu rity and greater immunity to interference.


521

t

Sheath

t

Fbilshielding

FIGURE 12-11 Shielded twistattpairfSTP) cable

Table 12-2 'Attenuation and Crosstalk Characteristics of Twistedpair Cable

Attenuation (dB per 100 meteis) 1 4 16 25 100 300

2.6 5.6 13.1 — —

2.0 4.1 8.2 10.4 22.0 — -

41 32 23

62 53 44 41 32

1.1

2.2

4.4 6.2 12.3 21.4

Near-End Crosstalk (dB) 1 4 16 25 100 300

y'

' ——

: —

58 58 50.4 47.5 38.5 31.3

There are seven primary STPcables classified by the EIA/TIA 568 standard: category 3, category 4, category 5, enhanced category 5, category 7, foil twisted pair, and shielded-foil twistedpair. 1. Category 7. Category 7 shielded-screen twisted-pair cable (SSTP) is also called PiMF (pairs in metal foil) cable. SSTPcable is comprised of four pairs of 22 or 23AWG cop per wire surrounded by a common metallic foil shield followed by a braided metallic shield. 2. Foil twisted pair. Foil twisted-pair cable is comprised of four pairs of 24 AWG copper wire encapsulatedin a common metallic-foil shield with a PVC outer sheath. Foil twisted-pair cable has been deliberately designed to minimize EMI susceptibility while maximizing EMI immunity. 3. Shielded-foil twisted pair. Shielded-foil twisted-pair cable is comprised of four pairs of 24 AWG copper wires surrounded tty a common metallic foil shield encapsulated in a braided metallic shield. Shielded-foil twisted-pair cables offer superior EMI protection.

12-6-2 Attenuation and Crosstalk Comparison Table 12-2 shows a comparison of the attenuation and near-end crosstalk characteristics of three of die most popular types of twisted-pair cable. Attenuation is given in dB of loss per 100 meters of cable with respect to frequency. LowefdB values indicate a higher-quality cable, and the smaller the differences in the dB value for the various frequencies, the bet ter die frequency response. Crosstalk is given indBof attenuation between the transmit sig nal and the signal returned due to crosstalk with higher dB values indicating less crosstalk.

12-6-3

ed-Pair Cable

1.1 22 4.4 6.2 12.3 21.4

38 38 30.4 47.5 38.5 31.3

568 standard: category 3, d pair, and shielded-foil

le (SSTP) is also called Irso f 22 or 23 AWG copi braided metallic shield, f four pairs o f .24 AWG PVC outer sheath. Foil M I susceptibility while tie is comprised o f four foil shield encapsulated superior EMI protection.

psstalk characteristics o f is given in dB o f loss per indicate a higher-quality bus frequencies, the betbetween the transmit sigindicating less crosstalk.

Plenum Cable

JPlenum is the name gi ven to the area between the ceiling ami the roof in a single-story build ing or between die ceiling and die floor o f the next higher level in a multistory building. Metal rectangular-shaped air ducts twere traditionally placed in the plenum to control airflow in the building (both for heating and for cooling). In more modern buildings, the ceiling itself is used to control airflow because the plenum goes virtually everywhere in the building. This presents an interesting situation i f afire should occur in die plenum because die airflow would not be contained in a fire-resistant duct system. For ease o f installation, networking cables are typically distributed throughout the building in the plenum. Traditional (nonplenum) ca bles use standard PVC sheathing for the outside insulator; such sheathing is highly toxic when ignited. Therefore, if a fire should occur, the toxic chemicals produced from the burn ing PVC would propagate through the plenum, possibly contaminating the entire buildinjg. The National Electric Code (NEC) requires plenum cable to have special fire-resistant insulation. Plenum cables are coated with Teflon, which does not emit noxious chemicals when ignited, or special fire-resistant PVC, which is called plenum-grade PV C. Therefore, plenum cables are used to route signals throughout die building in the air ducts. Because plenum cables are considerably more expensive than nonplenum cables, traditional PVC coated cables are used everywhere else.

12-6-4

Coaxial (Concentric) Transmission Lines

In the recent past, parafiel-conduCtor transmission lines were suited only for low data trans mission rates. At higher transmission rates, their radiation and dielectric losses as well as itheir susceptibility to external interference were excessive. Therefore, coaxial cables were often used for high data transmission rates to induce losses and isolate transmission"paths. However, modem UTP and STP twisted-pair cables operate at bit rates m excess o f 1 Gbps at much lower costs than coaxial cable. Twisted-pair cables are also cheaper, lighter, and easier to Work with than coaxial cables. In addition, many extremely high-speed computer networks prefer optical fiber cables to coaxial cables. Therefore, coaxial cable is seeing less and less use in computer networks, although they are still a very popular transmission line for analog systems, such as cable television distribution networks. The basic coaxial cable consists o f a center conductor surrounded by a dielectric ma terial (insulation), then a concentric (uniform distance from the center) shielding, and fir nally a rubber environmental protection outer jacket. Shielding refers to the woven or stranded mesh (or braid) that surrounds some types o f coaxial cables. A coaxial cable with one layer o f foil insulation and one layer o f braided shielding is referred to asdual shielded Environments that are subject to exceptionally high interference use quad shielding, which consists o f two layers o f foil insulation and two layers o f braided metal shielding. The center conductor o f a coaxial cable is the signal wire and the braider outer conductor is the signal, return (ground). The center conductor is separated from the shield by a solid dielectric material or insulated spacers when air is used for the dielec tric. For relatively high bit rates, the braided outer conductor provides excellent shield ing against external interference. However, at lower bit rates, the use o f shielding is usu ally not cost effective. Essentially, there are two basic types o f coaxial cables: rigid a ir filled and solid flexi b le . Figure 12-12a shows a rigid air coaxial line. It can be seeriTEat a tubular outer conductor surrounds the center conductor coaxially and that the insulating material is air. The outer con ductor is physically isolated and separated from the center conductor by space, which is gen erally filled with Pyrex, polystyrene, or some other nonconductive material. Figure 12-1% shows a solid flexible coaxial cable. The outer conductor is braided, flexible, and coaxial Ip the center conductor. The insulating material is a solid nonconductive polyethylene material that provides both support and electrical isolation between the inner and outer conductors. The inner conductor is a flexible copper wire that can be either solid or hollow (cellular).


583

Outer. conductor

(a)

Braided shielding (outer conductor

(b) ■

'

'

FIGURE 12-12 Coaxial or concentric transmission line: (a) rigid airfilled; (b) aolid flexible

Rigid air-filled coaxial cables are relatively expensive to manufacture, and to minimize tosses, tne air insulator must be relatively free o f moisture. Solid coaxial cables have lower losses than hollow cables and are easier to construct, install, andtnaintain. Both types o f coaxial cables are relatively immune to external radiation, radiate little themselves, and are capable o f operating at higher bit rates than their parallel-wire ' counterparts. For these reasons, coaxial cable is more secure than twisted-pair cable. Coaxial cables can also be used over longer distances and support more stations on a shared-media network than twisted-pair cable. The primary disadvantage o f coaxial transmission lines is their poor cost-to-performance ratio, low reliability, and high maintenance. The RG numbering system typically used with coaxial cables refers to cables ap proved by the U.S. Department o f Defense (DoD). The DoD numbering system is used by most cable manufacturers for generic names; however, most manufacturers have developed several variations o f each cable using their own product designations. For example, the Belden product number 1426A cross-references to one o f several variations o f die RG 58/U solid copper-core coaxial cable manufactured by Belden. Tablel2-3 lists several common coaxial cables and several oftheir parameters. Keep in mind that these values may,Vary slightly from manufacturer to manufacturer and for dif ferent variations o f the same cable manufactured by the same company. 12-6-4-1 Coaxial cable connectors. There are essentially two types o f coaxial ca ble connectors: standard BNC connectors and type-N connectors. BNC connectors are sometimes referred to as “bayonet mount,” as they can be easily twisted on or off. N-type connectors are threaded and must be screwed on and off. Several BNC and N-type con nectors are shown in Figure 12-13.

T

Table 18-3 Coaxial Cable Characteristics

RG-8/A-AU RG-8/U foam RG-58/A-AU RG-58foam RG-59/A-AU RG-59 foam

52 50 5? 50 73 75

0.66 0.80 0.66 0.79 0.84 0.79

-

0.585 / 0.405 ’ 1.250 1.000 0.800 0.880

12 12 20 20 20 20

29.5 25.4 28.5 25.4 16.5 16.9

(a)

(c) manufacture, and to ture. Solid coaxial cauct, install, and mainmal radiation, radiate nan their parallel-wire an twisted-pair cable, prt more stations on a âdvantage o f coaxial reliability, and high

(a) FIGURE 18-13 Coaxial cable connectors: (a) BIMC connector; (b) BNC barrel; (c) BNC T; (dj Type-N; and (e) Type-N barrel

les refers to cables apering system is used by icturers have developed ions. For example, the nations o f the RG 58/U :their parameters. Keep anufacturer and for difjany. two types o f coaxial caBNC connectors are visted on or off. N-type BNC and N-type con-

12-7 METALLIC TRANSMISSION UNE EQUIVALENT CIRCUIT 12-7-1

Uniformly Distributed Transmission Lines

The characteristics o f a transmission line are determined by its electrical properties, such as wire conductivity and insulator d ielectric constant, and its physical properties, such as wire diameter and conductor spacing. These properties, in turn, determine die prim ary elec trica l constants: series dc resistance (/?), series inductance (L ), shunt capacitance (Q , and shunt conductance (G ). Resistance and inductance occur along die line, whereas capaci tance and conductance occur between the conductors. The primary constants are uniformly distributed throughout the length o f the line and, therefore, are commonly called distributed parameters. To simplify analysis, distributed parameters are commonly given for a unit length o f cable to form an artificial electrical model o f the line. The combined parameters


523

•*— ---------— 1 Section (unit length) R

Jo>L

'----------V----------' z, G « capacitance - two conductor* separated by an insulator R ■ reaiatance - opposition to current flow L maalf inductance 1A? ■ leakage raafatanca of dielectric Ra• shunt leakage raaiatance FIGURE 12-14 Twowire parallel transmission line, electrical equivalent circuit

Source

Load Z i -Z .

Iflec: Zj

FIGURE 12-15 Equivalent circuit for a single section of transmission line terminated in a load ; equal to Z ,

are called lumpedparameters. For example, sejie&rfesistance is generally given in ohms per unit length (i.e., ohms per meter or ohms per foot). Figure 12-14 shows the electrical equivalent Circuit for a metallic two-wire parallel transmission tide showing the relative placement o f the various lumped parameters. For convenience, the conductance between the two wires is shown in reciprocal form and given as a shunt leakage resistance (Rs).

12-7-2

Transmission Characteristics

The transmission characteristics o f a transmission line are called secondary constants and are determined from the four primary constants. The secondary constants are characteristic impedance and propagation constant. 12-7-2-1 Characteristic Impedance. For maximum power transfer from the source to the load (i.e., no reflected power), a transmission line must be terminated in a purely resistive load equal to the characteristic impedance o f the transmission line. The characteristic impedance (Za) o f a transmission line is.a complex quantity thalis expressed in ohms, is ideally independent o f line length, and cannot be directly measured. Character istic impedance (sometimes called surge impedance) is defined .asthe impedance seen look ing into an infinitely long line or the impedance seen looking into a finite length o f line that is terminated in a purely resistive load with a resistance equal to the characteristic imped ance o f the line. A transmission line stores energy in its distributed inductance and capaci tance. If a transmission line is infinitely long, it can store energy indefinitely; energy from the source enters the line, and none 6f it is returned. Therefore, the line acts as a resistor that dissipates all the energy. An infinitely long line can be simulated if a finite line is termi nated in a purely resistive load equal toZ„; all the energy that enters the line from the source is dissipated in the load (this assumes a totally lossless line). Figure 12-15 shows a single section o f a transmission line terminated in a load ZL equal to Z„. The impedance seen looking into a line o f n such sections is determined from the following equation: zk 4 = Z.Z2 + n

(12-7)

where n is the number o f sections. For an infinite number o f sections, Zjjn approaches 0 if = 0 526

Chapter 1 2

Metallic Cable Transn

( 12- 8)

Then,

z0 = v m

where

Z, = R + j < a L 1 R,

iv = 4Zj

mit length) -

'1 1/jtaC

— G + jtoC

*

1 G + ywC

Zl -Z, Therefore,

t circuit for a single a terminated in a load

1 G + Ja>C

•= V (* 7

or

=

(12-9)

[ * + -! jL •

( 12- 10)

V G +-j/&C

°

For extremely low frequencies, the resistances dominate and Equation 12-10 simplifies to anally given in dims per tetallic two-wire parallel lumped parameters. For eciprocal form and given

Zo-yfe

For extremely high frequencies, the inductance and capacitance dominate and Equation 12-10 simplifies to Z„ =

secondary constants and instants are characteristic ower transfer from the must be terminated in a le transmission line. Hie quantity that is expressed ;tly measured. Characterthe impedance seen looka finite length of line that the characteristic impedid inductance and capaciindefmitely; energy from ;line acts as a resistor that d if a finite line is termi tethe line from the source

d m )

[jiiiL jtoC

( 12- 12)

From Equation 12-12, it can be seen that for high frequencies, thecharacteristic im pedance o f a transmission line approaches a constant, is independent of both frequency and length, and is determined solely by Re distributed inductance and capacitance. It can also be seen that the phase angle is OVIherefoie, Z„ looks purely resistive, and all the incident energy is absorbed by the lihe. From a purely pssistive approach,it can easily be seen that the impedance seen look ing into a transmission line made up o f an infinite number o f sections approaches the char acteristic impedance. This is shown in Figure 12-16. Again, for simplicity, only the series resistance R and the shunt resistance R, are considered. The impedance seen looking into, the last section o f the line is simply the sum of R and R,. Mathematically, Z, is \

Z, = R + R s = 10+ 100= 110

Adding a second section, Z^, gives

^ ^ R

. + Z,

= 10 + I ” ? X ? j» = 10 + 52.38 = 62.38 100 + 110

and a third section, Zj, is i terminated in a load ZL etions is determined from

Z, = R +

= 10 +

(12-7) ions, Z\Jn approaches 0 if

R ,z2 R, + Zj 100, X 62.38 = 10 + 38.42 = 48.32 100 + 62.38

A fourth section, Z4, is

z < - ,0 4 - r o Metallic Cable Transmission Media

f f i - 10 + 32®

- 42® 527

FIGURE 12-16 Characteristic impedance of a transmission line of infinite sections or termi nated in load equal to Za

It can be seen that after each additional section, the total impedance seen looking into the line decreases from its previour'value; however, each time the magnitude of the de crease is less than the previous value. I f the process shown above were continued, the im pedance seen looking into the line will decrease asymptotically toward 37 ft, which is the characteristic impedance o f the line. If die transmission line shown in Figure 12-16 were terminated in a load resistance ZL = 37 ft, the impedance seen looking into any number o f sections would eqttal 37 ft, the characteristic impedance. Fbr a single section o f line, Zais ~ r , R' X Z l Z i - z \- R r^ P zl

in . 100 X 37 , n . 3700 + io o + '3 7 = 10 + W

„ n 370

Adding a second section, Zj, is „

^

P

t o - 2* - n +

«.XZ,

100 X 37

1f. . 3700

~ 10 + 1 0 0 T 3 7 ” 10 + W

„ n “ 37n

Therefore, if this line were terminated into a load resistance Z£ = 37 ft, Z0 = 37 ft no mat ter how many sections are included. The characteristic impedance o f a transmission cannot be measured directly, but it can be calculated using Ohm’s law. When a source is connected to an infinitely long line and a voltage is applied, a current flows. Even though the load is open, the circuit is com plete through the distributed constants of die line. The characteristic impedance is simply the ratio of the source voltage (E0) to the line current (/„). Mathematically, Z„ is Zo = T

•O

where

(12-13)

Z„ = characteristic impedance (ohms) E0 = source voltage (volts) /„ - transmission line current (amps)

The characteristicimpedance o f a two-wireparallel transmission linewith an air dielectric can be determined fromits physical dimensions (see Figure12-17a) and the formula Z„ = 276 log where

528

Z„ = characteristic impedance (ohms) D - distance between the centers of die two conductors (inches) r — radius of the conductor (inches)

Chapter 12

(12-14)

lance seen looking into magnitude o f the defere continued, the imard37Q> which isthe ed in a load resistance would equal 37 fi, the

la) FIGURE 12-17 Physical dimensions of transmission Unas: (a) two-wire parallel transmission tine; (b] coaxial-cable transmission line

Example 12-1

.

:

: :

Determine the characteristic impedance for an air dielectric two-wire parallel transmission line with a DAr ratio of 12.22. .

Solution Substituting into Equation 12-14, we obtain 2^ = 276 k>g(12.22) * 300 ohms The characteristic impedance of a concentric coaxial cable can also be determined from its physical dimensions (see Figure 12-17b) and the formula

- 37 n Q ,ZD= 37£lnomat-

138 sasured directly, but it an infinitely long line en, the circuit is comimpedance is simply ically, Za is (12-13)

(12-15)

VT, where

Za = characteristic impedance (ohms) D = inside diameter of the outer conductor (inches) er * relative dielectric constant of the insulating material (unitless)

Example 12-2 Determine the characteristic impedance for an RG-59A coaxial cable with the following specifica tions: 0.025 inches, D * 0.15 inches, and er *2.23.

Solution Substituting into Equation 12-15, we obtain tsion line with an air Jure li-17a) and the

„ 138 ( , OJSrnA _ in . z° ^ V g o ^ j = 7I-9ohms For extremely high frequencies, characteristic impedance can be determined from the induc tance and capacitance of the cable using the following formula:

(12-14)

Khes)

(12-16)

Example 12-3 Determine the characteristic impedance for an RG-59A coaxial cable with the following specifica tions: L = 0.118 pH/ftand C = 21 pF/ft.


529

Solution Substituting into Equation 12-16 gives us fi

/0.118 X 10_#H/ft

Z°~yc ~yj

21 X 10"12F/ft 3=75 0

Transmission lines can be summarized thus far as follows: 1. The input impedance o f an infinitely long line at radio frequencies is resistive and equal to Z0. 2. Electromagnetic waves travel down the line without reflections; such a line is called nonresonant. y . 3. The ratio o f voltage to current at any point along die line is equal to Z0. 4. The incident voltage and current at any point along die line are in phase. - 5. Line losses on a nonresonant line are minimum per unit length. 6. Any transmission line that is terminated in a purely resistive load equal to Z0 acts as if it were an infinite line. a . Z t ^ Z 0. b. There are no reflected waves. " c. V and / are in phase. . . d. There is maximum transfer of power from source to load. 12-7-2-2Propagation constant Propagation constant(sometimes calledpropagation confident) is used to express die attenuation (signal loss) anddie phase shift per unit length o f a transmission line. As a signal propagates down a transmission line, its amplitude de creases with distance traveled, The propagation constant is used to determine the reduction in voltage or current with distance as a TEM wave propagates down a transmission line. For an infinitely long line, all the incident power is dissipated in the resistance of the wire as the wave propagates down the line. Therefore, with an infinitely long line or a line that looks in finitely long, suchas a finite line terminatedin a matched load( 4 = Z j), no energy is returned or reflected back toward the source. Mathematically, the propagation constant is Y =«+;P where

(12-17)

y = propagation constant (unitless) a = attenuation coefficient (nepers per unit length) P = phase shift coefficient (radians per unit length)

The propagation constant is a complex quantity defined by y = V (R + j(a L)(G + ;o»C)

(12-18)

Because a phase shift o f 2jcrad occurs over a distance of one wavelength, P= y

(12-19)

At intermediate and radio frequencies, rnL > A and (DjC> G; thus,

+

and

p = (oVLC

(«-»>

(12-21)

The current and voltage distribution along a transmission line that is terminated in a load equal to its characteristic impedance (a matched line) are determined from the formulas

Chapter 12

/ = IjT * 1

(12-22)

V = V f ~ tt

(12-23)

where

lencies is resistive and

/, = current at me source end of the line (amps) Vj = voltage at die source end o f the line (volts) y = propagation constant I — distance from the source at which the current or voltage is determined

Fbr amatched load ZL = Z„ and for a given length o f cable /, the loss in signal volt age or current is the real part of yl, and the phase shift is the imaginary part

sctions; such a line is equal to Z0. : ' are in phase, th. load equal to Z„ acts

12-8 W AVE PROPAGATION O N A METALLIC TRANSMISSION LINE Electromagnetic waves travel at the speed o f light when propagating through a vacuum and nearly at the speed of light when propagating through air. However, in metallic transmis sion lines, where the conductor is generally copper and the dielectric materials vary con siderably with cable type, an electromagnetic wave travels much more slowly. 1 ^2-8-1 8-

descalledpropagation ine, its amplitude de termine the reduction transmission line. For tnce of the wire as the nr a line that looks inno energy is returned xmstantis

Velocity Factor and Dielectric Constant

Velocity factor (sometimes called velocity constant) is defined simply as the ratio of the ac tual velocity o f propagation o f an electromagnetic wave thtough.a given medium to the ve locity of propagation through a vacuum (free soacei Mathematically, velocity factor is Vf = where

(12-24)

Vf = velocity factor (unidess) Vp = actual velocity o f propagation (meters per second) c = velocity o f propagation through a vacuum (3 X 108m/s) 12-24 gives

(12-17)

VfX c = Vp

(12-25)

The velocity at which an electromagnetic wave travels through a transmission line depends on the dielectric constant o f the insulating material separating die two conductors. The velocity factor is closely approximated with the formula V = — I— -v r .

(12-18)

(12-19)

( 12- 20) ( 12- 21) hat is terminated in a iedfrom die formulas ( 12- 22) (12-23)

(12-26)

where er is the dielectric constant o f a given material (die permittivity of the material rela tive to the permittivity of a vacuum - the ratio e/e,,, where e is the permittivity of the di electric and e0 is the permittivity o f air). Velocity factor is sometimes given as a percentage, which is simply the absolute ve locity factor multiplied by 100. For example, an absolute velocity factor Vf = 0.62 may be stated as 62%. Dielectric constant is simply die relative permittivity o f a material. The relative di electric constant o f air is 1.0006, However, the dielectric constant of materials'commonly used m transmission lines ranges from 1.4872 to 7.5. giving velocity factors from 0.3651 to 0.82. The velocity factors and relative dielectric constants Bfseveral insulating materi als are listed in Table 12-4. Dielectric constant depends on the type o f insulating material used. Inductors store magngtjaenergy, and capacitors store electric energy. It takes a finite amount of time for an inSuctor or a capacitor to take on or give up energy. Therefore, the velocity at which anelec tromagnetic wave propagates along a transmission line varies with the inductance and Metallic Cable Tranamission Media

531

Table 12-4 Velocity Factor and Dielectric Constant

.1.0000 0.9997 0.8200 0.6901 0.6637 0.6325 0.6325 0.5505 0.5774 0.4472 0.3651

Vhcuum Air Teflon foam Teflon Polyethylene Paper, paraffined Polystyrene Polyvinyl chloride Rubber Mjca Glass ^

1.0000 1.0006 1.4872 2.1000 2.2700 2.5000 2.5000 3.3000 3.0000 5.0000 7.5000

capacitance o f the cable. It can he shown that time T — V Z Z '. Therefore, inductance, ca pacitance, and velocity o f propagation are mathematically related by die formula . velocity X time = distance ™ , Therefore

»,

distance = " t o T

D

w (12*27)

T

Substituting V L C fo r time yields

( 12- 28)

y/LC

I f distance is normalized to 1 meter, the velocity o f propagation for a lossless transmission line is

.

;

iu *> where 1

K = velocity o f propagation (meters per second) V Z C = seconds L = inductance per unit length (H/m)

C - capacitance per unit length (F/m)

Example 1 2 -4 For a given length o f RG 8A/U coaxial cable with a distributed capacitance C = 96.6 pF/m, a dis tributed inductance L = 241.56 nH/tn, and a relative dielectric constant er = 2.3, detennine die ve locity o f propagation and the velocity factor.

Solution From Equation 12-16, V„ =

p

■■■■-.— . , 1 .. V(96.6 X 10->2)(241.56 X 10"»)

From Equation 12-24,

v •

’

2.07 x 10® m/s --------J— = o69 3 X 10® m/s

From Equation 12-26, .

Chapter 12

V, = ~ = = 0.66 ’ V23

= 2.07X 10* m/s

Because wavelength is directly proportional to velocity and the velocity of propagation o f a TEM wave varies with dielectric constant, the wavelength of aTEM wave also varies with dielectric constant Therefore, for transmission media other thanflee space, Equation 12-3 can be rewritten as (12-30)

12-8-2

Electrical Length of a Transmission Line

The length o f a transmission line relative to the length o f die wave propagating down it is an important consideration when analyzing transmission-line behavior. At low frequencies (long wavelengths), the voltage along die line remains relatively constant. However, for highfremay be present on the line at the same time. Thercs. several wavelengtns o f the signal sis quencies, fore, thevoltagealong the line may vary appreciably. Consequently, (he length o f a transmis sion line is often given in wavelengths rather thaii in linear dimensions. 'Ihmsmission-line phenomena apply to long lines. Generally, a transmission line is defined as long i f its length exceeds l/16th o f a wavelength; otherwise, it is considered short. A given lengdi o f transmis sion line may appear short at one frequency and long at another frequency. For example, a 10m length o f transmission line at 1000 Hz is short (A. = 300,000m; 10 m is only a small fraction o f a wavelength). However, the same line at 6 GHz is long (X. - 3 cm; the line is 200 wavelengths long). It will be apparent later in this chapter, in Chapter 9, and in Appendix A that electrical length is used extensively for transmission-line calculations and antennadesign.

efore, inductance, cathe formula.

(12-27)

12-8-3

Delay Lines

In the previous section, it was shown that the velocity o f propagation o f an electromagnetic wave depends on the media in which it is traveling. The velocity o f an electromagnetic wave in free space (i.e., a vacuum) is the speed o f light (3 X 10s m/s), and the velocity is slightly slower .through the Earth’s atmosphere (i.e., air). The velocity o f propagation through a metallic transmission line is effected by die cable’s electrical constants: inductance and ca pacitance. The velocity o f propagation o f a metallic transmission line is somewhat less than the velocity) o f propagation through either free space or the Earth’s atmosphere. Delay lines are transmission lines designed to intentionally introduce a time deity in the path o f an electromagnetic wave. The amount o f time delay is a function o f the transmission line’s inductance and capacitance. The inductance provides an opposition to changes in current, as does die charge and discharge times o f die capacitance. Delay time is calculated as follows:

(12-28) i lossless transmission

(12-29)

td = L C (seconds) ' where

td = time delay (seconds) L — inductance (henrys) C - capacitance (farads)

(12-31)

.

I f inductance add capacitance are given per unit length o f transmission line (such as per foot or per meter), die time delay will also be per unit lengdi (i.e., 1.5 ns/meter). The time delay introduced by a length o f coaxial cable is calculated with the follow ing formula: f td = 1.016 e (12-32)

t C = 96.6 pF/m, a dis2.3, determine the ve-

m/s

where e is the dielectric constant o f cable.

12-8

TRANSMISSION LINS LOSSES For analysis purposes, metallic transmission lines are often considered to be totally lossless. In reality, however, there are several wavs in which signal power is lost in a transmission line. They include conductor loss, radiation loss, d ielectric heating loss.


533

Current density

Moderate ■frequency Higher frequency Radial position

FIGURE 12-18 Isolated round conductor showing magnetic lines of flux, current distributions, and the skin effect

coupline loss, and corona. Cable manufacturers generally lump all cable losses together and specify them as attenuation loss in decibels per unit length (e.g., dB/m, dB/ft, and so on).

12-9-1

Conductor Losses

Because electrical current flows through a metallic transmission line and the line has a fi nite resistance, there is an inherent and unavoidable power loss. This is sometimes called conductor loss or conductor heating loss and is simply an 1ZR power loss. Because resis tance is distributed throughout a transmission line, conductor loss is directly proportional to the square o f the line length. Also because power dissipation is directly proportional to the square o f the current, conductor loss is inverselyproportional to characteristic imftedance. fo reduce conductor loss, simply shorten the transmission lifle or use a lareer-diameter wire (Tel, onelvith less resistance). Keep in mind, however, that changing the wire diameter dso cKangesthe characteristic impedance and, consequently, the current. Conductor loss depends somewhat on frequency because o f a phenomenon called the skin effect. When current flows through an isolated round wire, the magnetic flux associ ated with it is in the form o f concentric circles surrounding die wire core. This is shown in Figure 12-18, From the figure, it can be seen that the flux density near the center o f the con ductor is greater than if is near the surface. Consequendy, the lines o f flux near the center o f the conductor encircle the current and reduce the mobility pf the encircled electrons. This is a form o f self-inductance and causes the inductance near the center o f the conductor to be greater than at the surface. Therefore, at high frequencies, most o f the current flows along the surface (outer skin) o f the conduct rather than near its center. This is equivalent to reducing the cross-secdonal area o f the conductor and increasing the opposition to cur rent flow (i.e., resistance). The additional opposition has a 09 phase angle and is, therefore, a resistance and not a reactance. Therefore, the ac resistance o f the conductor is proportional to the square root o f the frequency. The ratio o f the ac resistance to the dc resistance o f a conductor is called the resistance ratio. Above approximately 100 MHz, the center o f a conductor can be Chapter 12

completely removed and have absolutely no effect on the total conductor loss or EM1 wave propagation. Conductor loss jn metallic transmission lines varies from as low as a fraction of a decibel per 100 meters for rigid air dielectric coaxial cable to as high as 200 dB per 100 me ters for a solid dielectric flexible coaxial cable. Because both I2R losses and dielectric losses are propoitipnal to length, they are generally lumped together and expressed in decibels of loss per unit length (i.e., dB/til).

12-9-2

Dielectric Heating Losses

Adiffcrence of potential between two conductors o f a m etallic transmission line causes di electric heating. Heat is a form o f energy and must be taken from the energy propagating down the line; For air dielectric transmission lines, the heating loss is negligible. However, for solid-core transmission lines, dielectric heating loss increases with frequency.

12-9-3 Isolated round ing magnetic lines of tributions, and the

til cable losses together dB/m, dB/ft, and so on).

ne and the line has a fihis is sometimes called wer loss. Because resisi directly proportional to ectly proportional to the laractenstic impedance. le a larcer-diameter wire g the wire diameter also it iphenomenon called the ie magnetic flux associe core! This is shown in iarthe center o f the coni of flux near the center incircled electrons. This liter of the conductor to 1st of the current flows tnter. This is equivalent g the opposition to cur! angle and is, therefore, Si to the square root of >f a conductor is called of a conductor can be

Radiation Losses . ^

If die separation between conductors-in a metallic transmission line iân appreciable frac tion of a wavelength, the electrostatic andelectromapneticfields that surround the conduc tor cause the line to~acfas if it were ah antenna and transfer energy to any nearby conduc tive material The enerfly raHintpH is called radiation loss, and depends on dielectric material, conductor spacing, and length o f the transmission line. Radiation losses are reduccdby properly shielding the cable. Therefore, shielded cables (such as STP and coaxial cable) have less radiation loss than unshielded: cables (such as twin lead, open wire, and UTP). Radiation loss is also directly proportional to frequency.

12-9-4

Coupling Losses

Coupling toss occurs whenever a connection is made to or from a transmission line or when two sections of transmission line are connected together. MechanicaTconnections ate dis continuities, which are locations where dissimilar materials meet. Discontinuities tend to heat up, radiate enetgy, and dissipate power.

12-9-5

Corona

Corona is a luminous discharge that occurs between the two conductors of a transmission line when the difference of potential between them exceeds the breakdown voltage o f the dielectric insulator. Generally, when corona occurs, the transmission tine is destroyed. ■ “

12-10

INCIDENT A N D REFLECTED W AVES An ordinary transmission line is hirfiiwtiâ|- power can propagate equally well in both di rections. Voltage that prbpagates from the source toward the load is called incident voltage. and voltage that propagates from the load toward the Source is called reflected voltage. Sim ilarly, there are incident and reflected currents. Consequently, incident power propagates toward the load, and reflected power propagates toward the source. Incident voltage and current are always in phase for a resistive characteristic impedance. For an infinitely long line, all the incident power is store? by the line, and there is no reflected power. Also, if the line is terminated in a purely resistive load equal to the characteristic impedance of the line, the load absorbs all the incident power (this assumes a lossless line). For a more practical definition, reflected power is the portion of the incident power that was not absorbed by the load. Therefore, the reflected power can never exceed the incident power.

12-10-1

Resonant and Nonresonant Transmission Lines

A transmission line with no reflected power is called a flat at nonresonant line. A trans mission line is nonresonant if it is o f infinite length or if it is terminated with a resistive Metallic Cable Transmission Media

535

Transmission line

Source

Load

FIGURE 12-19 Source, load, transmission line, and their corresponding incident and reflected waves

load equal in ohmic value to the characteristic impedance o f the transmission line. On a flat line, the voltage and current are constant throughout its length, assuming no losses. When the load is not equal to the characteristic impedance o f the line, some o f the incident power is reflected back toward the source. I f the load is either a short or an open circuit, all the incident power is reflected hack toward the source. I f the source were replaced with an open or a short and the line were lossless, energy present on the line would reflect back mid forth (oscillate) between the source and load ends similar to the way energy is transferred back and forth between the capacitor and inductor in an L/C tank circuit This is called a resonant transmission line. In a resonant line, energy is alternately transferred between the magnetic and electric fields o f the distributed inductance and capacitance o f the line. Figure 12-19 shows a source, transmission line, and load with their corresponding incident and reflected waves. '

1 2 -1 0 -2

Reflection Coefficient

The reflection coefficient (sometimes called the coefficient o f reflection) is a vector quan tity that represents the ratio o f reflected voltage to incident voltage or reflected current to incident current. Mathematically, the reflection coefficient is gamma, T , defined by (12-33)

where

T = Et = Er “ /( = lr =

reflection coefficien' . incident voltage (volts) reflected voltage (volts) incident current (amps) reflected current (amps)

;

From Equation 12-33 it can be seen that the maximum and worst-case value for V is 1 (Er —E^, and the minimum value and ideal condition occur when T = 0 (Er — 0).

1 2 -11

S TA N D IN G W A V E S When Z a = Z 0 all the incident power is absorbed by the load. This is called a matched line. When Z a * Z u some o f the incident power is absorbed by the load, and some is returned (reflected) to the source. This is called an unmatched or mismatohed line. With a mismatched line, there are two electromagnetic waves, traveling in opposite direc tions, present on the line at the same time (these waves are in fact called traveling waves). The two traveling waves set up an interference pattern known as a standing wave. This is shown in Figure 12-20. As the incident and reflected waves pass each other, stationary patterns o f voltage and current are produced on the line. These sta-

536

Chapter 12

Seurea

M

(b)

nsmission line. On a flat turning no losses. When ae of the incident power an open circuit, all the were replaced with an would reflect back and ay energy is transferred circuit This is called a transferred between the apacitance o f the line, corresponding incident

(c l

FIGURE 12-20 Developing a standing waVe on a transmiasionline: (a] incident wave; (b) reflected wave; (c) standing wave

tionary waves are called standing waves because they appear to remain in a fixed posi•tion on the line, varying only in amplitude. The standing wave has m inim a (nodes) sep arated by a half wavelength o f the traveling waves and maxima (antinodes) also sepa rated by a half wavelengths

12-11-1 :turn) is a vector quanor reflected current to la, T, defined by

Standing-Wave Ratio

The standing-wave ratio (SW R) is defined as the ratio o f the maximum voltage to the min imum voltage or the maximum current to the minimum current o f a standing wave on a transmission line. SWR is often called the voltage standing-wave ratio (VSWR). Essen tially, SWR is a measure o f the mismatch between the load impedance and the characteristic impedance o f the transmission line. Mathematically, SWR is

(12-33) SWR = ■^H i[unitless)

(12-34)

Ute voltage maxima (Vm„ ) occur wheB the incident and reflected waves are in phase (i.e., their maximum peaks pass die same point on die line with the same polarity), and the voltage nnnima ( V „in) occur when the incident and reflected waves are 180° out o f phase. Mathematically. V „.. and are 1 due for T is 1(Er = Et),

V in a * ~

; ^’ E r

(12-35)

0).

'

Vnin

(12-36)

Therefore, Equation 12-34 can be rewritten as s is called a matched

SWR =

he load, and some is iismatohed line. With ig in opposite direcfact called traveling mown as a standing Eed Waves pass each the line. These sta-

(12-37)

From Equation .12-37, it can be seen teat when the incident and reflected waves are equal in amplitude (a total mismatch), SWR = infinity. This is the worst-case condition. Also, from Equation 12-37, it can 6e seen that when there is no reflected wave (Er - 0), SWR = Ej/E„ or 1. This condition occurs when Z „ = ZL and is the ideal situation.

Metallic Cable Transmiseion Media

537

The standing-wave ratio can also be written in terms o f I\ Rearranging Equation 12-34 yields TE, = Er (12-38) Substituting into Equation 12-37 gives us

SWR =

E + ET

(12-39)

Factoring out E, yields

swr -W

T^T) =T^ T

(1M0)

Cross multiplying gives

SWR(1 - D = 1 + r SW R-SW Rr=^T

(12-41) (12-42)

SWR = 1 + r + (SWR)F SWR - 1 = T(1 + SWR)

(12-43) (12-44)

„ SWR - 1 r = swR+T

(12'45)

Example 12-5 For a transmission line with incident voltage a. Reflection coefficient b. SWR.

E, = 5 V and reflected voltage Er = 3 V, determine ■■■'

*

Solution a. Substituting into Equation 12-33 yields

r * f - f - M b.

Substituting into Equation 12-37 gives us

Substituting into Equation 12-45, we obtain

r _±r_! = 2 4+ 1

,

5-

When the load is purely resistive, SWR can. also be expressed as a ratio o f the char acteristic impedance to the load impedance or vice versa. Mathematically, SWR is

2

2.

SWR = — or ^(w hichever gives an SWR greater than 1)

(12-46)

The numerator and denominator for Equation 12-46 are chosen such that the SWR is always a number greater than 1 to avoid confusion and comply with the convention estab lished in Equation 12-37. From Equation 12-46, it can be seen that a load resistance ZL = 2Z0 gives the same SWR as a load resistance ZL - ZJ2; the degree o f mismatch is the same. The disadvantages o f not having a matched (flat) transmission line can be summa rized as follows:

\

1. One hundred percent o f the source incident power is not absorbed by the load. 2. The dielectric separating the two conductors can break down and cause corona as a result o f the high-voltage standing-wave ratio. 3. Reflections and re-reflections cause more power loss.

Chapter.' 12

r. Rearranging Equation (12-38)

(12-39)

I

4. Reflections cause ghost images. 5. Mismatches cause noise interference.

Although it is highly unlikely that a transmission line w ill be terminated in a load that is either an open or a short circuit, these conditions are examined because they illustrate the worst-possible conditions that could occur and produce standing waves that ate represen tative o f less severe conditions.

12 -11 -2 (12-40)

(12-41) (12-42) (12-43) (12-44) (12-45)

age Er = 3 V, determine

ed as a ratio o f the charlaticaUy, SWR is erthanl)

Standing Waves on an Open Line

When incident waves o f voltage and current reach an open termination, none o f the power is absorbed; it is all reflected back toward the source. The incident voltage wave is reflected in exactly the same manner as if it were to continue down an infinitely long line. However, tire incident current is reflected 180° reversed from how it would have continued if the line were not open. As the incident and reflected waves pass, standing waves are produced on the line. Figure 12-20 shows the voltage and current standing waves on a transmission line that is terminated in an open circuit It can be seen that the voltage standing wave has a max imum value at the open end and a minimum value one-quarter Wavelength from the open. The current standing wave has a minimum value at the open end and a maximum value onequarter wavelength from the open. It stands to reason that maximum voltage occurs across an open and there is minimum current. The characteristics o f a transmission line terminated in ah open can be summarized as follows: 1. The voltage incident wave is reflected back just as if it were to continue (i.e., no phase reversal). 2. The current incident wave is reflected back 180° from how it would have continued. 3. The sum o f the incident and reflected current waveforms is minimum at the open. 4. The sum o f the incident and reflected voltage waveforms is maximum at the open. From Figure 12-21, it can also be seen that the voltage and current standing waves repeat every one-half wavelength. The impedance at die open end Z = and is maximum. The impedance one-quarter wavelength from the open Z - V— and is min imum. Therefore, one-quarter wavelength from the open an impedance inversion occurs, and additional impedance inversions occur each one-quarter wavelength. Figure 12-21 shows the development o f a voltage standing wave on a transmission line that is terminated in an open circuit. Figure 12-22 shows an incident wave propagat ing down a transmission line toward the load. The wave is traveling at approximately the speed o f light; however, for illustration purposes, the wave has been frozen at eighthwavelength intervals. In Figure 12-22% it can be seen that the incident wave has not reached the open. Figure 12-22b shows the wave one time unit later (for this example, the wave travels one-eighth wavelength per time unit). As you can see, the wave has moved

(12-46)

en such that the SWR is tit the convention estabt a load resistance ZL of mismatch is the same, ion line can be summa-

ibsorbed by the load, iwn and cause corona as FIGURE 13-81 Voltage and current standing waves on a transmission line that is terminated in an open circuit


539

FI I f H -X/4 » +«

I

x/4 » |< x/4

I

tr

AM »*H

X/4— H

FIGURE 12-22 Incident and reflected waves on a transmissiCn line terminated in an open circuit [Continued^

one-quarter wavelength closer to the open. Figure 12-22c shows the wave just as it arrives at the open. Thus far, there has been no reflected wave and, consequently, no standing wave. Figure 12-22d shows the incident and reflected waves' one time unit after the inci dent wave has reached the open; the reflected wave is propagating away from the open. Figures 12-22e, f, and g show the incident and reflected waves for the next three time units. In Figure l2-22e, it can be seen that the incident and reflected waves are at their maximum

P< al su in

ui

in ,a i m

th lu

re 540

Chapter 12 Metallic Cable Transmissn

I I/

I \ ;r; ■'

I -i/v.

XAt ►}< - lt/4 ►j< »■; —

I i : XM

I i t/4 > )<

.

I i

'

X/4— ^ j

Incident Reflected

FIGURE 12-22 (Continued) Incident and reflected waves on a transmission line terminated in an open circuit

I

he wavejust as it arrives msequently, no standing ( time unit after die inciing away from the open, the next three time units, res are at their maximum

positive values at the same time, thus producing a voltage maximum at the open. It can also be seen that onerquarter wavelength from the open end of the transmission line the sum o f the ihcident and reflected waves (the standing wave) is always equal to 0 V (a min imum). Figures 12-22h through m show propagation o f the incident and reflected waves until the reflected wave reaches the source, and Figure 12-22n shows the resulting stand ing wave. It can be seen that the standing wave remains stationary (the voltage nodes and antinodes remain at the same points); however, the amplitude o f the antinodes varies from maximum positive to zero to maximum negative and then repeats. For an open load, all r the incident voltage is reflected (£ , = E,Y. therefore. V „ „ = E, + E„ or2Eh A similar illustration can be shown for a current standing wave (however, remember that the current reflects back with a 180° phase inversion). > Metallic CableTransmission Media

541

Current

is terminated in a short circuit

1 2 -1 1 -3

4

Standing W aves on a Shorted tine

As with an open line, none o f the incident power is absorbed by the load when a transmis sion line is terminated in a short circuit However, with a shorted line, the incident voltage and current waves are reflected back in the opposite manner. The voltage wave is reflected 180° reversed from how it would have continued down an infinitely long line, and the cur rent wave is reflected in exactly the same manner as if there were no short. Figure 12-23 shows the voltage and current standing waves on a transmission line that is terminated in a short circuit. It can be seen that the voltage standing wave has a min imum value at the shorted end and a maximum value one-quarter wavelength from the short, The current standing wave has a maximum value at the short and a minimum value one-quarter wavelength back. H ie voltage and current standing waves repeat every onequarter wavelength. Therefore, there is an impedance inversion every quarter-wavelength interval. The impedance at the short Z = - minimum, and one-quarter wave length back Z = Vmax//mifI = maximum. Again, it stands to reason that a voltage minimum w ill occur across a short and there is maximum current. The characteristics o f a transmission line terminated in a short can be summarized as follows: *■ 1. The voltage incident wave is reflected back 180° reversed from how it would have I continued. \ 2. The current incident wave is reflected back the same as if it had continued. I 3. The sum ofthe incident and reflected current waveforms is maximum at the short. 4. The sum o f the incident and reflected voltage waveforms is zero at the short. For a transmission line terminated in either a short or an open circuit, the reflection coefficient is 1 (the worst case), and the SWR is infinity (also the worst-case condition).

12-12 TR A N S M IS SIO N -LIN E IN P U T IM P ED A N C E In the preceding section, it was shown that, when a transmission line is terminated in either a short or an open circuit, there is an impedance inversion every quarter-wavelength. For a loss-less line, the impedance varies from infinity to zero. However, in a more practical sit uation where power losses occur, the amplitude o f the reflected wave is always less than that o f the incident wave except at the termination. Therefore, the impedance varies from SO|Hf maximum to some minimum value or vice Versa, depending on whether the line is terminated in a short or an open. The input impedance for a lossless line seen looking into a transmission line that is terminated in a short or an dpen can be resistive, inductive, or capacitive, depending on the distance from the termination.

1 2 -12-1

i

Phasor Analysis of Input Impedance: Open Line

Phasor diagrams are generally used to analyze the input impedance o f a transmission line because they are relatively simple and give a pictorial representation o f the voltage and cur542

Chapter 12

Metallic Cable Transmis

90* phase delay from load to source

Shorted I end

It

180° phase reversal at the open 90" phase delay from load to source

90° phase delay from source to load

90° phase delay from source to load

3n line that

FIGURE 12-24 Voltage and current phase relationships for a quarter-wave line terminated in an open circuit: (a) voltage phase relationships: (b) current phase relationships B load when a transmisne, the incident voltage oltage wave is reflected y long line, and the cur io short. on a transmission line ending waVe has a minwavelength from the t and a minimum value raves repeat every onerery quarter-wavelength and one-quarter wavethat a voltage minimum

it can be summarized as from how it would have it had continued, s maximum at the short, is zero at the short. in circuit, the reflection vorst-case condition).

e is terminated in either arter-wavelength. For a in a more practical sitave is always less than impedance varies from on whether the line is is line seen looking into sistive, inductive, or ca-

jne e of a transmission line i of the voltage and cur-

rent phase relationships. Voltage and current phase relations refer to variations in time. Figures 12-21 to 12-23 show standing waves o f voltage and current plotted versus distance and, therefore, are not indicative o f true phase relationships. The succeeding sections use phasor diagrams to analyze the input impedance o f several transmission-line configurations. 12-12-1-1 Quarter-wavelength transmission line. Figure 12-24a shows the phasor diagram for the voltage, and Figure 12-24b shows the phasor diagram for the current at the input to a quarter-wave section o f a transmission line terminated in an open circuit. I, and V, are the in-phase incident current and voltage waveforms, respectively, at the input (source) end o f the line at a given instant in time. Any reflected voltage (£ r) present at the input o f the line has traveled one-half wavelength (from the source to the open and back) and is, conse quently, 180° behind the incident voltage. Therefore, die total voltage (£,) at the input end is the sum o f £, and Er E, = £, + Er /180°, and, assuming a small line loss, E, = £, - Er The reflected current is delayed 90° propagating from the source to the load and another 90° from the load back to the source. Also, the reflected current undergoes a 180° phase re versal at the open. The reflected current has effectively been delayed 360°. Therefore, when the reflected current reaches the source end, it is in phase With the incident current, and the total current /, = /, + Ir By examining Figure 12-24, it can be seen that £, and /, are in phase. Therefore, the input impedance seen looking into a transmission line one-quarter wave-length long that is terminated in an open circuit Zin = £ ,/0°//,/0° = Z'„ /0°..Zinhas a 0° phase angle, is resistive, and is minimum. Therefore, a quarter-wavelength transmission line terminated in an open circuit is equivalent to a series resonant l 6 circuit. Figure 12-25 shows several voltage phasors for die incident and reflected waves on a transmission line that is terminated in an open circuit and how they produce a voltage standing wave. 12-12-1-2 Transmission line less than one-quarter wavelength long. Figure 12-26a shows the voltage phasor diagram, and Figure 12-26b shows the current phasor diagram for a transmission line that is less than one-quarter wavelength long (A/4) and terminated in an open circuit. Again, the incident current (/;) and voltage (£,) are in phase. The reflected volt age wave is delayed 45° traveling from the source to the load (a distance o f one-eighth wavelength) and another 45° traveling from the load back to the source (an additional oneeighth wavelength). Therefore, when the reflected wave reaches the source end, it lags the incident Wave by 90°. The total voltage at the Source end is the vector sum o f the incident and reflected waves. Thus, £, = ■V li? + E2 r — E, / -4 5 °. The reflected current wave is delayed 45° traveling from the source to the load and another 45° from the load back to the source (a total distance o f one-quarter wavelength). In addition, the reflected current wave has undergone a 180° phase reversal at the open prior to being reflected. The reflected cur rent wave has been delayed a total o f 270°. Therefore, the reflected wave effectively leads


543

0/8

M2

3M8

M4

MB

Load

Dfetanoe from toad

FIGURE 12-25 Vector addition of incident and reflected waves producing e standing wavs 46* phasedelay 1to source

i i i j 45° phase daisy from source'to toad

45*phaso delay from toed to source

45* phase (May fromsource to toed

180° phase reversal at the open

(a)

FIGURE 12-26 Voltage and current phase relationships for a transmission line less than one-quarter wavelength terminated in ah open circuit: (a) voltage phase relationships; (b) current phase relationships

the incident wave by 90°. The total current at the source end is the vector sum o f the pres ent and reflected waves. Thus, I, = V / f + 1, - /„ /+45°. By examining Figure 12-26, it can be seen that E, lags /, by 90°. Therefore, = £ ,/ -4 5 °/f, /+45° = /-9 0 °. Z * has a —90° phase angle and, therefore, is capacitive. Any transmission line that is less than one-quarter waveiehgth and terminatetfin an open circuit is equivalent to a capacitor. The amomtt o f capacitance depends on the exact electrical length o f the line. 12-12-1-3 Transmission line more than one-quarter wavelength long. Figure 12-27a shows the voltage phasor diagram and Figure 12-27b shows the current phasor diagram for a transmission line that is more than one-quarter wavelength long and

544

Chapter 12

terminated in an open circuit. For this example, a three-eighths wavelength trans mission line is used. The reflected voltage is delayed three-quarters wavelength or 270°. Therefore, the reflected voltageêffectively leads die incident voltage by 90°. Consequently, the total voltage E, ~ V fif + £? /+45° * E ,/ + 45°. The reflected cur rent wave has been delayed 270° and undergone an 180® phase reversal, Therefore, the re flected current effectively lags the incident current by 90°. Consequently, the total current I, - V/,2 + I? /—45° = /,/ —45°. Therefore, Z,„ = & /+45®//,/-45® * Z * /+90®. has a +90° phase angle end is therefore inductive. The magnitude o f the input imped ance equals the Characteristic impedance at eighth wavelength points. A transmission line between one-quarter and one-half wavelength that is terminated in an open circuit is equivalent to an inductor. The amount o f inductance depends on the exact electrical length o f the line.

'

12-12-1-4 Open transmission line as a circuit element. From the preceding dis cussion and Figures 12-24 through 12-27, it is obvious that an open transmission line can behave as a resistor, an inductor, or a capacitor, depending on its electrical length. Because standing-wave patterns on an open line repeat every half-wavelength interval, the input im pedance also repeats. Figure 12-28 shows the variations in input impedance for an open transmission line o f various electrical lengths. It can be seen that an open line is resistive and maximum at the open and at each successive half-wavelength interval and resistive and minimum one-quarter wavelength from the open and at each successive half-wavelength interval. For electrical lengths less than one-quarter wavelength, the input impedance is*capacitive and decreases with lengdi. For electrical lengths between one-quarter and one-half wavelength, the input impedance is inductive and increases with length. The capacitance and inductance pattems also repeat every half-wavelength.

12-12-2

Phasor Analysis of Input Impedance: Shorted Line

The following explanations use phasor diagrams to analyze shorted transmission lines in the same manner as with open lines. The difference is that with shotted transmission lines, the voltage waveform is reflected back with a 180®phase reversal, and the current wave form is reflected back as if there were no short. 12-12-2-1 Quarter-wavelength transmission line. The voltage and current phasor diagrams for a quarter-wavelength transmission line terminated in a short circuit are iden tical to those shown in Figure 12-24 except reversed. The incident and reflected voltages are in phase; therefore, E, = E t + Er and maximum, The incident mid reflected currents are 180° out o f phase; therefore, /, = / ,- Ir and minimum. = E, /0°//, /Q®and maximum.

■''

(«) .

V";

FIGURE 12-27 Voltage and current phase relationships for a transmission line more than onequarter wavelength terminated in an open circuit: (a) voltage phase relationships; lb] current phase relationships Metallic Cable Transmission Madia

348

FIGURE 12-28

Input impedance variations for an open-circuited transmission line

FIGURE 12-29 Input impedance variations for a short-circuited transmission line

Zin has a 0° phase angle, is resistive, and is maximum. Therefore, a quarter-wavelength transmission line terminated in a short circuit is equivalent to a parallel LC circuit. 12-12-2-2 Transmission line less than one-quarter wavelength long. The volt age and current phasor diagrams for a transmission line less than one-quarter wavelength long and terminated in a short circuit are identical to those shown in Figure 12-26 except reversed. The voltage is reversed 180° at the short, and the current is reflected with the same phase as if it had continued. Therefore, the total voltage at the source end o f the line leads the current by 90°, and the line looks inductive. 12-12-2-3 Transmission line more than one-quarter wavelength long. The voltage andcurrentphasor diagrams for a transmission line more than one-quarter wavelength long and terminated in a Short circuit are identical to those shown in Figure 12-27 except reversed. The total voltage at the source end o f the line lags the current by 90°, and the line locks capacitive. 12-12-2-4 Shorted transmission line as a circuit element. From the preceding discussion, it is obvious that a shorted transmission line can behave as if it were a resistor, an inductor, or a capacitor, depending on its electrical length. On a shorted transmission line, standing waves repeat every half-wavelength; therefore, the input impedance also re peats. Figure 12-29 shows the variations in input impedance o f a shorted transmission line for various electrical lengths. It can be seen that a shorted line is resistive and minimum at the short and at each successive half-wavelength interval and resistive and maximum onequarter wavelength from the short and at each successive half-wavelength interval. For electrical lengths less than one-quarter wavelength, the input impedance is inductive and increases with length. For electrical lengths between one-quarter and one-half wavelength, the input impedance is capacitive and decreases with length. The inductance and capaci' tance patterns also repeat every half-wavelength interval. 12-12-2-5 Transmission-line input impedance summary. Figure 12-30 summa rizes the transmission-line configurations described in the preceding sections, their input

Chapter 12

Input Imptdanco

4 » max. resistive

Short Parallel LC circuit (resistive and maximum)

V4

Load j Z|n ■ min. resistive

ransmission line

I Open

r ........

-'7WPSeriea LC circuit (resistive and mlnimuml

1 1

Max. | 4

«

-n tm -

1 S h o rt

In d u c tiv e

Inductor 1

1 1 4

W4

> C a p e c tiv e

-u-

1 Open

Load

msmission line

Capacitor 1 • J 4 , -

ore, a quarter-wavelength arallelLC circuit elength long. The voltmone-quarter wavelength m in Figure 12-26 except tis reflected with the same rarce end of the line leads length long. The voltage Barterwavelengthlong and 12-27except reversed. The d the line looks capacitive.

n t From the preceding ive as if it were a resistor, )n a shorted transmission tinput impedance also re shorted transmission line resistive and minimum at istive and maximum onewavelength interval. For ipedance is inductive and and one-half wavelength, ie inductance and capaci-

r. Figure 12-30 summading sections, their input

C e p e c t iv e

j

Short Capacitor

1

.

1

1 [ 4

1 ■ In d u c tiv e

j

Open Inductor

L I Input end

■X/4-

FIGURE 1 2 -3 0

j I load end

Transmission-line summary

impedance characteristics, and their equivalent LC circuits. It can be seen that both shorted and open sections o f transmission lines can behave as resistors, inductors, or capacitors, de pending on their electrical length.

12-12-3

Transmission-Line Impedance Matching

Power is transferred most efficiently to a load when there are no reflected waves, that is, when the load is purely resistive and equal to Z9. Whenever the characteristic impedance of a transmission line and its load are not matched (equal), standing waves are present on the line, and maximum power is not transferred to die load. Standing waves cause power loss, dielectric breakdown, noise, radiation, and ghost signals. Therefore, whenever possible, a transmission line should be matched to its load. Two common transmission-line techniques are used to match a transmission line to a load having an impedance that is not equal to Z„. They are quarter-wavelength transformer matching and stub matching. 12-12-3-1 Quarter-wavelength transformer matching. Quarter-wavelength transformers are used to match transmission lines to purely resistive loads whose resistance is not equal to the Characteristic impedance of the line. Keep in mind that a quarter-wavelength transformer is not actually a transformer but rather a quarter-wavelength section of Metallic Cable Transmission Media

547

Quarter-wavelength transformer

Z0 (transmission tine) Zm* ZD

-load

FIGURE 12>31 Quarter-wavelength transformer

sion line varies from some maximum value to some minimum value or vice versa every quarter-wavelength. Therefore, a transmission line one-quarter wavelength long acts as a step-up or step-down transformer, depending on whether ZL is greater than or less than Ze. A quarter-wavelength transformer is not a broadband impedance-matching device; it is a quarter-wavelength at only a single frequency. The impedance transformations for a quarterwavelength transmission line are as follows: 1. Rl = Z0: The quarter-wavelength line acts as a transformer with a 1:1 turns ratio. 2. Rl >Z„: The quarter-wavelength line acts as a step-down transformer. 3. Rl < ZQ: The quarter-wavelength line acts as a step-up transformer. As with a transformer, a quarter-wavelength transformer is placed between a trans mission line and its load. A quarter-wavelength transformer is simply a length o f trans mission line one-quarter wavelength long. Rgure 12-31 shows how a quarter-wavelength transformer is used to match a transmission line to a purely resistive load. The character istic impedance o f the quarter-wavelength section is determined mathematically from die formula. ' (12-47)

ZL =

load im pedance

Example 12-6 Determine the physical length and characteristtc imipedance for a quarter-wavelength transformer that is used to match a section ofRG-8A/U (Z a ~ 50 t l ) to a 150-Q resistiveload. The frequency o f op eration is 150 MHz, and the velocity factor V f— I. Solu tion The physical length o f the transformer depends on die wavelength o f the signal. Substituting into Equation 12-3 yields c X

— ■ -J =

f

3 X 10® m/s i e i\ —1 — 2m 150 M H z

The characteristic impedance o f diet).5-m transformer is determined from Equation 12-47:

Z0 = VZ^L - V(50)(l50) = 86.6 ft Chapter 12

X « di*t»nc*fromto»d

Load

input impedance to a transmistimum value or vice versa every [uarter wavelength long acts as a ZL is gteater than or less than Z„. pedance-matching device; it is a ince transformations for a quarterransformer with a 1:1 turns ratio, itep-down transformer, itep-up transformer. ’ormer is placed between a transrmer is simply a length o f trans shows how a quarter-wavelength ely resistive load. The charactersrmined mathematically from the

(12-47) elength tr nsformer bn line. . is being matched

l quarter-wavelength transformer that resistive load. The frequency o f op-

FIGURE 12-32 Shorted stub impedance matching

12-12-3-2 Stub matching. When a load is purely inductive or purely capacitive, it absorbs no energy. The reflection coefficient is 1, and the SWR is infinity. When the load is a complex impedance (which is usually the case), it is necessary to remove die reactive component to match the transmission line to the load. Transmission-line stubs are com monly used for this purpose. A transmission-line stub is simply a piece of additional trans mission line that is placed across die primary line as close to the load as possible. Thesusceptance of the stub is used to tune out the susceptance of the load. With stub matching, either a shorted or an open stub can be used. However, shorted stubs are preferred because open stubs have a tendency to radiate, especially at the higher frequencies. Figure 12-32 shows how a shorted stub is used to cancel the susceptance of the load and match the load resistance to the characteristic impedance of the transmission liqe. It has been shown how a shorted section of transmission line can look resistive, inductive, or ca pacitive, depending on its electrical length. A transmission line that is one-half wavelength or shoiter can be used to tune out the reactive component of a load. The process o f matching a load to a transmission line with a shorted stub is as follows: 1. Locate a point as close to die load as possible where the conductive componentof the input admittance is equal to the characteristic admittance of the transmission line: Ym = G - JB, where G = ~ o

on the wavelength o f the signal.

2. Attach the shorted stub to die point on the transmission line identified in step 1. 3. Depending on whether the reactive component at the point identified in step 1is inductive or capacitive, the stub length is adjusted accordingly: Yin ~ G o - JB +JBtoa> mined from Equation 12-47:

86.6 n

if

~ G„ B = Bstub

For a more complete explanation of stub matching using the Smith chart, refer to Ap pendix A.

propagation is 3 X 10® m/s, or approximately 1 ns/ft. Consequently, a pulse width of sev eral microseconds would limit the usefulness o f TDR only to cable impairments that oc curred several thousand feet or farther away. Producing an extremely narrow pulse was one o f the limiting factors in die development of TDR for locating cable faults on short cables. imunications system, can id. Cable problems often stances, sometimes as far luted to chemical erosion ecurs in a cable, it can be letermine the type and ex-

Example 12 -7 A pulse is transmitted down a cable that has a velocity o f propagation o f 0.8 c. The reflected signal is received l ps later. H ow far down the cable is the impairment?

Solution

Substituting into Equation 12-48,

d= tallic cable is called timerments can be pinpointed well-established theory use a portion o f the inciiignal returns depends on Sere the impairment is lo! causes a portion of the cable. If no energy is rethe line is either infinitely the characteristic imped-1 tort duration pulse with a on of that signal to return in echo. Knowing the vethe impairment and the

0.8 X (3 X 108m/s) X I X 10‘ s

“

ransmitted pulse to the reortant that the transmitted ocatedclose to the source, ismitted (Figure 12-33b), •light (c), the velocity of

2

~ r~

120 m

Example 1 2 -8 agation o f 0.9c, determine the time elapsed from the beginning o f the pulse to the reception o f the echo.

Solution

Rearranging Equation

12-48 gives t= — = v

2d kx c

2(3000 m ) 0.9(3 X 10* m/s)

- 22.22 ps

M ICROSTRIP A N D STRIPLINE TR ANSM ISSIO N U N E S At frequencies below about 300 MHz, the characteristics o f open and shorted transmission lines, such as those described earlier in this chapter, have litde relevance. Therefore, at low frequencies, standard transmission lines would be too long for practical use as reactive components or tuned circuits. For high-frequency (300 MFlz to 3000 MHz) applications, however, special transmission lines constructed with copper patterns on a printed circuit (PC ) board called microstrip and stripline have been developed to interconnect components on PC boards. Also, when die distance between the source and load ends of a transmission line is a few inches or less, standard coaxial cable transmission lines are impractical be cause the connectors, termina|ions, and cables themselves are simply too large. Both microstrip and stripline use the traces (sometimes called tracks) on the PC board itself. The traces cqn be etched using the same processes as the other traces on the board; thus, they do not require any additional manufacturing processes. If the lines are etched onto the surface of the PC board only, they are called microstrip lines. When the lines are etched in the middle layer of a multilayer PC board, they are called striplines. Microstrip and stripline can be used to construct transmission tines, inductors,- capacitors, tuned cir cuits, filters, phase shifters, and impedance matching devices.

12-14-1

► Time

!

Using TD R , a transmission-line impairment is located 3000 m from the source. For a velocity o f prop

1 2 -1 4 (12-48)

(0.8c) X 1 ps

Microstrip

Microstrip is simply a flat conductor separated from a ground plane by mi insulating di electric material. A simple single-track microstrip tine is shown in Figure 12-34a. The ground plane serves as the circuit common point and must be at least 10 times wider than the top conductor and must be connected to ground. The microstrip is generally either onequarter or one-half wavelength long at the frequency o f operation and equivalent to an un balanced transmission tine. Shorted tines are usually preferred to open tines because open Metallic Cable Transmission Media

551

Rat conductor

Twin flat conductors

Dielectric PC material

Qround plane

Rat conductor —

i w

ftis tn r lr ir

V W W nv

(c) FIGURE 12-34

Micrastrip transmission line: (a] unbalanced; (b) balanced; and (c) dimensions

lines have a greater tendency to radiate. Figure 12-34a shows a two-wire balanced microstrip transmission line. As with any transmission line, the characteristic impedance o f a microstrip line is de pendent on its physical characteristics. Therefore, any characteristic impedance between 50 ohms and 200 ohms can be achieved with microstrip lines by simply changing its dimen sions. The same is true for stripline. Unfortunately, every configuration o f microstrip has its own unique formula. The formula for calculating the characteristic impedance o f an un balanced microstrip line such as the one shown in Figure 12-34c is 87 V e + 1.41

where

Z„ = e= w= t= h=

, f 5.98ft \ \0.8w + t j

(12-49)

characteristic impedance (ohms) dielectric constant (FR-4 fiberglass e = 4.5 and Teflon e = 3) width o f copper trace* thickness o f copper trace* distance between copper trace and the ground plane (i.e., thickness o f di electric)*

'The dimensions for w, t, and A can be any linear unit (indies, millimeters, and so on) as long as they all use the sameunit.

552

Chapter 12

-(b). FIGURE 12-35

1 2 -1 4 -2

Stripline transmission line: (a) end and side views; (b) dimensions

Stripline

Stripline is simply a flat Conductor sandwiched between two ground planes, as shown in Figure 12-35. Although stripline is more difficult to manufacture than microstrip, it is less likely to radiate; thus, losses in stripline are lower than with microstrip. Again, the length o f a stripline is either one-quarter or one-half Wavelength, and shorted lines are used more often than open lines. The characteristic impedance o f a stripline configured as shown in Figure 12-35 is

lanced; and (c) dimensions

a two-wire balanced misof a microstrip line is detic impedance between 50 mply changing its dimen;uration o f microstrip has istic impedance o f an un is

60 where

(12-49)

In

Ad 0.67jcw(0.8 + t/h)

(12-50)

Z„ = characteristic impedance (ohms) e= dielectric constant (FR-4 fiberglass e = 4.5 and Teflon e = 3) d — dielectric thickness* w = width o f conducting copper trace* t = thickness o f conducting copper trace* h = distance between copper trace and the ground plane*

*The dimensions for d, w, t, and A can be any linear unit (inches, millimeters, and so on) as long as they all use the same unit.

iflon e = 3)

Q U E S T IO N S me (i.e., thickness o f di-

12-1. D efine transm ission line.

-

12-2. Describe a transverse electromagnetic wave. >on) as long as they all use the

12-3. D efine wave velocity. 12-4. Define frequency and w avelength for a transverse electromagnetic wave. 12-5. Describe balanced and unbalanced transmission lines.

Metallic Cable Transmissiofi Media

553

12-6. Describe an open-wire transmission line. 12-7. Describe a twin-lead transmission line. 12-8. Describe a twisted-pair transmission line. 12-9. Describe a shielded-cable transmission line. 12-10. Describe a concentric transmission line. 12-11. Describe the electrical and physical properties of a transmission line. 12-12. List and describe the four primary constants o f a transmission line. 12-13. Define characteristic impedance for a transmission line. 12-14. What properties of a transmission line determine its characteristic impedance? 12-15. Define propagation constant for a transmission line. 12-1& Define velocityfactor for a transmission line. 12-17. What properties of a transmission tine determine its velocity factor? 12-18. What properties of afrtmsmission line determine its dielectric constant? 12-19. Define electrical length for a transmission line. 12-20. List and describe five types of transmission-line losses. 12-21. 12-22. 12-23. 12-24. 12-25. 12-26. 12-27. 12-28.

Describe an incident wave; a reflected wave. Describe a resonant transmission line; a nontesonant transmission line. Define reflection coefficient. Describe standing waves; standing-wave ratio. Describe the standing Waves present on an open transmission line. Describe the standing waves present on a shorted transmission line. Define input impedance for a transmission line. Describe the behavior of a transmission line that is terminated in a short circuit that is greater than one-quarter wavelength long; less than one-quarter wavelength. 12-29. Describe the behavior of a transmission line that is terminated in ah open circuit that is greater than one-quarter wavelength long; less than one-quarter wayelength long. 12-30. Describe the behavior of an open transmission line as a circuit element 12-31. Describe dm behavior o f a shorted transmissionline as a circuit element 12-32. Describe the input impedance characteristics of a quarter-wavelength transmission line. 1243. Describe the input impedance characteristics of a transmission line that is less thanOne-quarter wavelength long; greater than one-quarter wavelength long. 12-34. Describe quarter-wavelength transformer matching. 12-35. Describe bow stub matching is accomplished. 12-36. Describe time-domaih reflectometry.

PRO BLEM S 12-1. Determine the wavelengths for electromagnetic waves in free space with the'following fre quencies: 1 kHz, 100-kHz, 1 MHz,'"and 1 GHz. 12-2. Determine the frequencies for electromagnetic waves in free space with the following wave lengths: 1 cm, 1 m, 10 m, 100 m, and 1000 m. , 12-3. Determine the characteristic impedance for an air-dielectric transmission line with D/r ratio of 8.8. 124. Determine the characteristic impedance for an air-filled concentric transmission line with D/d ratio of 4. 12-5. Determine the characteristic impedance for a coaxial cable with inductance L = 0.2 pH/ft and capacitance C = 16 pF/ft. 12-6. For a given length of coaxial cable with distributed capacitance C = 48.3 pF/m and distrib uted inductance L = 241.56 nH/m, determine the velocity factor and velocity of propagation.

554

Chapter 12

12-7. Determine the reflection coefficient for a transmission line with incident voltage E, = 0.2 V and reflected voltage Er = 0.01 V. 12-8. Determine the standing-wave ratio for the transmission line described in problem 12-7. 12-9, Determine the SWR for a transmission line with maximum voltage standing-wave amplitude Vmax = 6 V and minimum voltage standing-wave amplitude V^n = 0.5. ine.

12-10. Determine the SWR for a 50-12 transmission line that is terminated in a load resistance ZL = 75 Q. 12-11. Determine the SWR for a 75-12 transmission line that is terminated in a load resistance

impedance?

Z ^S O Q .

12-12. Determine the characteristic impedance for a quarter-wavelength transformer that is used to match a section o f 75-fl transmission line to a 100-fl resistive load. «■?

tstant?

12-13. Using TDR, a pulse is transmitted down a cable with a velocity of propagation o f 0.7c. The reflected signal is received 1.2 ps later. How far down the cable is the impairment? 12-14. Using TDR, a transmission-line impairment is located 2500 m from the source. For a veloc ity o f propagation o f 0.95c, determine the elapsed time from the beginning of the pulse to the reception o f the echo.

line.

12-15. Using TDR, a transmission-line impairment is located 100 m from the source. If the elapsed time from the beginning o f the pulse to the reception o f the echo is 833 ns, determine the ve locity factor. 12-16. Determine the wavelengths for electromagnetic waves with the following frequencies: 5 kHz, 50 kHz, 500 kHz, and 5 MHz. *• 12-17. Determine the frequencies for electromagnetic waves with the following wavelengths: 5 cm, 50 cm, 5 m, and 50 m. ;

short circuit that is greater th.

12-18. Determine the characteristic impedance for an air-dielectric transmission o f 6.8.

open circuit that is greater thlong.

12-19. Determine me characteristic impedance for an air-filled concentric transmission line with a D/d ratio o f 6.

nent.

12-20. Determine the characteristic impedance for a coaxial cable with inductance L = 0.15 pH/ft and capacitance C = 20 pF/ft.

sment. jth transmission line, bat is less than one-quarter

» with a D/r ratio

12-21. For a given length o f coaxial cable with distributed capacitance C = 24.15 pF/m and distrib. uted inductance L = 483.12 nH/m, determine the velocity factor and velocity o f propagation. ^12-22. Determine the reflection coefficient for a transmission line with incident voltage £, = 0.4 V and reflected voltage Er = 0.002 V. 12-23. Determine the standing-wave ratio for the transmission line described in problem 12-22. ,-42-24. Determine the SWR for a transmission line with a maximum voltage standing-wave ampli tude Vmajt = 8 V and a minimum voltage standing-wave amplitude = 0.8 V. ^ i2 -2 5 . Determine the SWR for a 50-12 transmission line that is terminated in a load resistance ZL = 60 0 .

be with the'following fre-

12-26. Determine the SWR for a 60-12 transmission line that is terminated in a load resistance , ZL = 5012.

•with the following wave-

12-27. Determine the characteristic impedance for a quarter-wave transformer that is used to match a section o f 50-12 transmission line to a 60-12 resistive load.

lission line with D/r ratio ransraission line with D/d Ktance L = 0.2 pH/ft and = 48.3 pF/m and disttib1velocity of propagation.

Metallic Cable Transmiesion Media

55S

Chapter 12

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