ctqdd
VwaWA
tirl/-x.a -t PART I
Transmission Lines
Transmission Line
- a means of conveying electric energy / electromagnetic
waves
3t
various frequencies from one port to another pair of irnductor (two or more conduetor
.\
separated by an
insulator).
Wire
of copper that carries current at somevoltage
-
a strand
,
Cablo .= the complete wire asspmbly, which includes insulation, connectors,
-i l.
I
?rro-wire Farrllel lines arranged
sO
-
operated
in *re balanced
& uiy
mode,conductors being
thatthey presentequal capacitancestO ground 1-eAD
!...
'
:
I E
v
g1
H., f!ir ,::
,T.
ii -i,
:\ ..:
t
I2 For low
-
frequency applicatrons (e.g, telephone ckt)
::.,
.-a1-
-L
-
operated in the unbalanced mode, since the external capacitance
the outer conductor only
-
For high
Electromagnetic Fields
& ground.
frequency applications (e.g, relephone ckt.)
:
/l
r /l
ill t..'V Electomagnetic fields around a.coQxial line
Note:
E
- electic field
H-magnetic field
\ I
I
Erythnelirm:
The transmission of information as an electromagnetic signal always occurs as a TRAI{SVERSE ELECTROMAGNETIC (TEM) wave. The electric and magnetic field are at all points perpendicular to each other
cpen-wire between
&
&
& the signal is propagating into the page. For twin-lead transmission lines, the TEM wave propagates in the space
around the two conducting wires. The dielectric sheath "ribbon"
&
spacers
maintain a constant separation between the wires to maintain a balanced TEM freld for
&
best propagation characteristics. Although simplex structure, the ribbon results a less,structural integrity
since .the TEM fields are not a fined
&
less expensive than the coaxial
& more radio frequency interference
shielded the balanced coaxial line is use to
maintain radiation of the TEM flreld.
TWO WIRE, LIIIE:
i. Two Wire line: l. 'rwisted Pair -
made
fom two
insulated wires which are continuous
braided
(Typhical Wire Gauges : No. 2,A,22,& 24 AWG)
2.
Open Two Wire Parallel Line
- commonly
held apart by spacers every fcw inches.
separated only by
xc &
are
3. fwo-wire
Parallel (ribbon or
hvi' lead) - the same as the two,wire
open line, except that uniform spacing is assured by embedding the two wires in a low-loss dielectric, usualy polyethyrene. Eq. TV read in DtELEc'relc SUEATH
4.
Oval-Two-WireParallel
5.
Two-Wire-shielded other'
&
- consist of parallel conductors
separated from each
surrounded by, a solid dielectric. The conductors are contained
within a copper braid tubing that acts as a shield" The assembly is covered
with a rubber or flexible composition coating to protect the line from moisture or mechanical damage.
,*rfloo
Ro$06'
corl6
t
BBlrpm
sttraucr
B. Coaxial Line:
1. Rigid or Air coaxial Line
-
consist
of a wire mounted inside of &
coax:ally with, a tubular cuter conductor. The inner conductor is insulated
I
from the outer concluctor by insulating spacers, or beads, at regular iutervals. The spacers are marJe
uf Pyrex,
or some other & low loss at high
polystyrene,
rnaterial possessing good i,rsulating characteristics frequencies.
Chief Advantage : minimizes rr.,Jiation losses there are no electric or magnetic fields that extend outside of the outer (grounded) conductor. The fields are confined to the space between the two condrrctors; tlus, the coaxial
Iine is a perfectly shielded line. Noise pick up from other Iines is also prevented.
Disadvantages:
1.
expensive to construction
'2. it must be kept dry
to
present leakage b/n the trro
conductors
3. high frequency losses are still high such that the practical length of the line is limited.
,)
Flexible or solid Coaxial cable
-
concentric cables made with the inner
conductor consisting of flexible wire insulated from the. outer conductor
by a solid continuous insulating material. Flexibility is gained if the outer conductor is mad of branded wire.
*- csw(\ nRe,to aJrFR cTj.twctw f tsryf rUELyEaE
-,-,]
.i.
V
hrtrc&
Line Fundamentr.
ts
:
I
r Eniralent ckt. of Transmispion line
le-
t uttlr LEN
I
"il1
1
uurT
LrNerH
I
-l
Primar
Constants:
- series resistance (O/unit length) l L - series inductance (p iunit length) J G - shunt conductance (u/unit length)
R
A cc'o r,tvTs
fir tt, ingl,l / J>-"/, n
- since the wires are separated by a medium called the dielectric, w'c cannot be perfect in its insulation, the current leakage, throught it can be represented by a shunt conductive C
-
shunt capacittrnce ( F/unit
length)
'
'i
- accounts for the capacitance b/n the two wires separated from each other. :
Zo
: charQcteristiryas the
characteristic.' impedance
surse imnedance)
of a transmission lirrel Zo, is the
.impedance measur'ed at iis input when its length is infinite.
-
the ratio of voltage to current at any point along the line on which no reflected waves exists.
-
Impedance that would pro, ide a perfect match termination
Note: The Zo is the most important specification of a transmission line. This impedance & its value relative to ottLer impedances at the source & the receiver determineshow well the the signal enr)rgy will propagate through the line.
Ilerivation of Zo:
'
-
&e characteristic impedance of a line will be measured at its input when the line
is terminated at the far and with an impedance equal to Zo, no matter what length the line has:
'rLI', 4'zo
zn
,o
z=zo=#
=EI
=,
Z=R,- jwl, y = G.r jwC
* -*-- (r) :
E = IIZ
*!ttZo1
=
/r(h)t Itz ' +
,/r,
= Itz' +Uz'v/;l Y, + zo'
1,;qat,:
r: =
*
7.o+ZozY =Z+ZZoY +Zo
zozY=ZZoYtz ;,.:Li:,
F:.F-,r;;.!.
ZozY-hYZ=z
..: i,
,
:
r.
11:ti
a.:
lr:r Zo 772 I
ZoY(Zo-Z)=Z
;
'
.{}.ir i:
ZoY(Zo\= Z ZozY = Z tfr
Zo =
-la
\lr
But:
Z;'R+ jwl Y
=G+ Jwe.
4_@ h =
lG'* i*c
1
General Equation of Zo(O) '
At Low IrooBep0ies
zo=^lL Yq
6&tl
6 dtjwl
j*c I-
"P tl
6lt tl€;! :
lZ 11 JUL :''...,....
RIV iu'r, ,r7
&A 0 t B,
t.fu
*a:{T;.,u
Characteristic impe{ance of lossless or dissip6ionless lines
:
Propagation Coeffi cient, (6) Determines the variation of current or vottage e---with distance
alongatransmissionline.
Like Zo, the 6 arso depends on the primary constants and the angular velocity of the signal.
S -Ihis
=
is a complex
per unit length
quantity, so 6 may be written
as:
6 = q + jg; per unit tength
where: ;
q,
= attenuation coefficient (neper per unit lenth) = determines how the vortage & current decreases
with distance along the line B
-
phase-shift coefficient (racrians per unit tength)
variations with distance..
g=!-=3ooo ' ' shift of 2n rad occurs over a (phase -
of
1
l wavelength)
1
l, = wavelength ;
I =-9
f
C= speed
of light
f = frequency,
f,=
3x108
im
Hz
distance
USEFULL EQUIVALENTS:
'1.
1 NEPER = 8.585 dB
z. o=
-
o'toz
tr'rz3t"E6'
; Lncn
Where: O = dian:lter of AWG wire n = AWG no.
3. 1 inch = 1000 mils 4. Area (cir. mits) = (dialrirt)2 5. l\=E
f
where: Vp = vetocity of propagation
f = frequency, Hz (if Vp is not given, Asuume: Vp = C = vel. Of light)
,
C=3
x 108 m/sec = 186,000 miles/sec
LOSSLESS IN TRANSMISSION LINES
1. Radiation
Loss
- arises because a transmission line may act
as an antenna if the separation of the conductors is an a
Z'
ppreciable fraction of wavelength.
COwOUCTOR HEATING LOSS
proportional to current
&
tt erefore inversely proportional
to
characteristic
impedance
it lower frequencies, this loss is ieCuced simply by using thicker wire with less resistance. At higher frequencies more & more of the signal energy stays clo3er i'.r the outer surface,
or "skin, of the conductor, so loss increases with frequency
due to the skin effect."
3. Dielectric Heating
Loss
-
Proportional to the voltage across the dielectric
hence inversely proportional to the curve impedance
-
comes from the leakage current that flows through the dielectric of the coax.
Important Formulas:
t
A. Two-Wire Lines
(at H.F)
l-D where: d = wire wirediameter diameter D
l.
=
center to center spacing
Characteristic lmped ance, Zo
Zo=
276
"ler
rcr,Tt
or
.
Er '
Zo -r2o h?D
'
rr)
where: Er = dielectric constant = relative permittivity
Er = I forair
&
-,for high impedance applications -
R;
typhical :
1s0o - 600C,
3000
Series Resistance
Rl
16.8J7
where:
Rl = pCUm
f = fi:q.rHz d ,= wire diamerer, cm R2 =
CY100ft.
f
= freq., MHz d = wire diameter, inch.
3. L, $erieg Inductancc
7=t-6fl);s1tm
r'd
F = dielectric permeability [r = Fo for a transmission
h=
4rcx l0'7 tr/n for free space !
4.
C, Shunt Capacitance
line ; p-po
B '!
,a
u=ft;F/m d.:.. where:
Er= relative permittivity, ( dielectric constant)
5.
G, Shunt Conductance
G=O'atRF 6.
d=..W/unitlength @RF:
:
a;J@ 6=iw,{E=j0
7. Attenuation constani
d,L
=
I
07
dZ=
wnere:
4.35^R
Zo
a
.E
a?
=. clB
,
E .-a :::
lunitlanght
rrx. R=Qlm a*d*'lm' ' ' ,'. B. Coaxial
Lines :
at HF
D= lrqrgB pliilrgrH oriTtr, hxrFF coNtuCTDR
d, l.
Cn
pthl,lElER trTtF THNEF CDitDtr6,Sr
14,
Charaeteristic Impedaauguh
Zo
:ffir'{1o
or i
zn=ftry$,st where: ,'
Er*
,
,.t
I :
'.,
t.'
dielecuic constsst
]r i:i'ioi
air ", .
."'
appiicatioari',.t' -: :,,i;,, 4m _ lJgg,:r,.. i:'
For low impeciaace
.::
Typical:
i:.
,.
,
75O
''.,.''i:.,''.i..'1.,-
,i;., ,t
. ,..,.
-. ,
l
l
*
j:i=
E
fil
= 41.tiJ -) t7+
f
or R2
*0,1f,4,*
Rl=
nCt
) o)
j)
,,
l cm
f = freq,,Hz
:'
:
:t '
3.
'
! r,
CrnAAfr. -
R2 =
Series Inductance,
.
L,
L=!n(y,! Zn d-m
. r
., .r
:
.
_::
:.
where: p = Permeability
note:
for a transmission lile, p may be assrrrued equal to the free space valuo
lt= ln =4rl0a
::.'
:
aii:i;
#:
ffi |:?,:6a
r-:5i1.' t
ffi;*:,' -;r
.,
L. m
General Equations for Zoz
Two Wire Lines:
h=J-Eb#)
Coorial Lines:
^=*E^rrf Dieler.tric e po+tap$ of ,MnF,thl.t,
Material
ffietectric Constant ( APPror)
Air
1,0
Glass ( electrical)
3.8 - 14.5
- . - l-Mica(deceiPal) ,
I
r
I
!l
4.0 - 9.0
Paper (dry)
1.5
Flexiglass
2.6 -3.5
- 3.0
polyethylene Polystyrene
2.4
-3.0
Quartz
5.0
Styrofoam
1.03
Teflon
2.t
Line Perfurmance Calculation:
Consider a section of an infinite line with fundamental constant ( concentates in very small units of length
dL
& L, G & C)
d/.
dR
l+ i:
If the current flows through an impedance the voltage drop is IZ & therefore the voltage drop along the element of line dl will be:
dE
=*IZdl
or
#=-IZ +(t)
{
i'{ffi
':r:lli::i::iffi
As.qqrent progresses down the line, a cenain amount is lost thru the leakage & ca^riacitance of each elernent. This current loss at ea;h element is Ey:
I
*i:l ::'
w
dI = -,Eydl dI DI -=-Ey+(2)
:
Differentiating (1) & Q)tc, eliminate E & I with respect to l:
F-+r' ES.
9+,
:=.
#=-z#-+i:r dE d2I _= -+(4) '\/ 'dl dlz Eliminating undesired variables by r;ubstituting (3)
e $) with (l)
e-nd
(2) respectively:
d2E _=_Z(_Ey)=Zy+(5) dl2 ;
d2I
:
dlz
-E
(5) & (6) are sirnpte
-y(-IZ)=.ZyI -+(6)
diffl
equati,:
;
of the
2nd ord,.:r
fomrs are:
E*Ae-t@ +Be'ffi +(7) 1
.
t:.4
=1Ae-'@ -
ur'*
lh-+
(B)
whose general solution.for these
Nois U) A.(8) are General Solutions/ Equations : rr-.'i:.';'..';::l:i.Il :,::..-:-: li.-"
.
I
l,
',r,
AITALYSIS OF TRANSMISSION LINE: Reference.Transmission Line
:
eg Sworrc h
€
LR
s)uRcE
Wglurz,-
+ z =E Js,
ZS
+,-
L0ao
X..0 r' "rr':
7s ,E T
Zg = internal irapedanco oftho genersior
Eg= generator voltage .:;:r:
Zs = sending end / souree lapq{agcc
:::a.:'i-::
r'.:i::. ?;,,1:,
ttl:1:
Is = sending end./ source cun€-nl
,: j j:..i.:
;;irrf..
Es = sending end / source voltage
Zr = receiving end llodimpedance'
E = voltage
,l
N
tl,l a x 'Sottrco \ fto'* ?/','!""!' I=current I or d'l*ru' tl'w /oa./ Z: impedancf S:
I
total length of the transmission line
General Equations:
E=Ae-e+Bee 1
J
=L1Ae-* - Be*l LO
0
fzy =JZY
Where:
Note: A & B are arbitrary constant whose values depend on the Load
Case
I : AC
SteaCy State
-
Liues with No Reflections
"'1
Basically, a reflection less line is defined by an infinitely long transmission line (No reflections could possibly occur from the far end to the load). No reflections would also be possible
if a transmission tine is terminated at its own characteristic impedance (le.
Match Condition ; Zr
= Zo)
To Derive the General Equation for lines with no reflections, sblve for A & B from the general Equations.
As
?.,
increases, the second term for either the voltage or current equations tend to increase
to infinity. This is practically impossible. Therefore we can assume that B =0.
If
B=0:'then:
E=Ae-tu+(1) t
I -Lr-u +Q\ 'Zo
'
Solving For A:
tr. in terms of gending e-nd: set: x=0 in (l)':
I-Zo =!!e-*'
L
solution in terms of senfir9 errd
in terms of receiving end:
set x=s .: fhen: "-." 1
E=Er=Aea \
.Er,-ofu'',,'' uR-
A=-=
e-6
lial:.)' !l*:i: i:r'.l1:: i
Subttituting A ia (1) &
fii,= Ereee&
bufi
*
d=s-r.
AISOI
l.
1,.:,.
,.tr,'
,'
::i..'
,
| = +.rnr'o Zo
J
,* =E* fut
(2):
::
,
'
,
.
::
Ei6.l
e
seen at any point along the line is always,equal: tor
h:for,transrnission
E
lineswith no reflections. This can !s prov€n by dedvir.rg,Z =4: ,t
. 1. Using Sending end Solution: E'=
f
.: .
Es.e*d
I =4 ,-* Zo
ryE
Es.ea
I
4,r* Zo
:.2
2.
= Zo
Using Receiving End Solution:
E =,E^ed
'
D
I - "R-oil ZO
uD ,=-=.+
p-d
un€
I L"*
;.2
= Zo
If atransmission line has a finite length & if cbamcteristic impedancc (Z a
the line is not terminated at its own
*Jo) thcn reflections wlll occur, Reflections
tlutgobacktothosource(i.e.thesaaresignalsnotabeorbedbytheload),
are signalsi
:,,
:
',
",' i,
2,,
*
Zo
TheGeneral Solution for lines with reflections can be derived in the following manner. :;.:;i::
Fi.i Li:r'i' 1.::
l.
Determine the value of A & B frorn the General Equations
2.
Substitute A & B in the General Equarions
:,:,
1
i
General Equations:
E=Aea+Bza'
I =Ll.e"-e - Betul 2o,"". In terms of Sending End: Set:
x=0
Es=A+^B+(l) Z.ols=A-B+(2)
# .,.
add:(t)'&(2) Es + IsZo
=ZA;Es = IsZs
:, trZs + IsZo =2A Is(Zs + Zo)
=Ztl fiz _g &
(t' t z;1
Sub: (1) & (2)
i Es=lsk :. IsZs - IsZo =28 i Es-IsZo=)B Is(Zs
-
Zo)
=28
'n=!< 2'
.t i
-zo)
Sub: A & B in the General Equations:
li
ru
n = lyZs + Zo)e-e :'-. 2" '..,.. -, . :
+ (Zs
- Zo)ea' f
:
1
2.
=
Lylzs
2Zo-' -_:
solutions in the terms ofsending end
+ Zo)e-e
- (h -
Zo)ee I
in terms of Receiving En.1:
Set:
x=s :
:
Eo=Ae&+Bee+(3) ^
t
I*=L1Ae-e -8etu1 ZO I^,Zo=Ae-tu,*Beb
'
i&)
Add: ($) +'(4) E
^+I*Zo=ZAca
! r;Zr + I *Zo *ZAe*& I
*(Z*
+ Zo)
,r , , Sub. (3)
-
*2Ac'e
l=$
(4)
ER-I RZn =ZB.ee
I*Zr-I*h=ZBe-e' i I n(Z
=!;l!:r - Zo)ea
: ,',,i::s{tbstitufingA & B
a =$uzr+za)etue'*
,dL.t
* lxZn
* - Zo) =ZBc-*
e
,
En
ilths
Genenrl Equations:
+(z^ . zopeatul
"'
r'
E1:
ill
't-:
ffi
.
Case
III.
...
:
Dissipatiion Less /Loss Less Lines
dffilT
Zo= I ' = -l\G + jwc llc
j
tia
;
d=J(X + jwl)\i+ jwC) = jwlLC
Genoral Equations:
;'E=Ae-iB +Berfi 1
:
I
=!1tle'rr 7n.
. .i ..
Beiftl
:
,
At tte $endlng End: ( x* 0)
=
jB
!
',',.i-,?i-ffiF'
Es = Ar * B, =
t,
I
=
)(AZO
/rZ, -+ (l)
B)
+
(2)
Add (1) a Q)z
-Za)(wsft -sinpr)]'-,, Sinplifvinei
g
T
ft - jltusia Bxl
r cos ' = f-122 o
2'
.:
::i
E = Escos
p*-
i*zosinfu
''
I
T
\ i:l
i
P.
E =,-Er[cos Px-
F'
iffsinpxl Sending:End Solutions
I = Islcostx- iffsapxl t,
In torms of Receiving End : ( x=s )
,
.,'
E* = ls-itu + pblfi = IaZn+ G)
4=)Udin Solve (3)
,e
Bein)+ (4)
& (4) Simultaueourly:
=,,
$tz rl
zo)eik
': T B=t@R-Zo1e-$
f
E=+lG*+zo)eiu +(zn-zo'1e-tu1 ; d=s-x f
4= *Ilz.u +
Zo)(cos pd+
i:--!-.
Slilq/tFl/tl6 .:,
jsii fd) + (Z o -
Zo\(cos Bd
- j sn
Bd)j
:
"
r
:
:
:r 'r:i::': '! i:1; i.:iir: i
::
=lVzrcosfr r ih{nfil E * Encos B{ + grZosinfd : ;,, n
: i,'. r.
,'
':
tl
i
.
E=Enlcos
I
=
I
N*i{"i"pd))
nlcosN + i
$sin
, -- lrr,,rZ*+ ,
,ar
I
-lJvt-
prJ
Recei.rirrg End Solutions
jzoTanPd-,
'Zo+ jZ*Tanpd'
Note: Wave Length (1,) = The distance between poinrs that have corresponding phase in two consecutive cycles in a periodic wave.
7-e ^ ft--
3xlos
f
f
m/ Is
), =360' =2trRAD
Special Ca,$es: trmpedance of Loss Lcss Llnes
l,
When;
Zr=Zo zsn=h
(matched line & frequency impedance)
Frt'l
..
r:,.
-
!ri:l
,
.''. i:.r.
ii
iliii
#i:.; b'
2. When
s=L2
:
(or,rlven rnultiples)
z* + jzoffon(41*l
*
=
zoE-----------4'Y Zo+
I
1Z*Tan(+X*) AI
7 Zs = Zo("R,.) ZO
Zs*Za
'3. When
: S =L 4
(orodd multiples)
z^+jzoran<4><$ t(
'
LR
k=zol
zo+
!
?-1.#
iz.ran(+xi;
Yk
:l
Let: *=ranffn*r= '
r
,
'
a , zn/ + i?n
h=- z
\::,.
4. When
if,:i'
i
l
/k ' !'R
.uRi ii1."it
'' ri
=q
k=Zol#1 . &ot.i7
'
iE:;ri
Tatt))o
(
Z
*
u shurt ciresit
*0 J.1.
,,
,
i:-
;r :.r:arll
l.ra :
r
:
il.r' .
7a+ JZ.xTwNo.,,,,,,,!
-, :,, .,.,,,,,.,,:ri..
... .t:
,:'
7.s=+jZoTanff ,:
:.:
(inductive)
.l
5.rVhen:Zp=OpenCircuit=o
I
,.,
..
;i;W
t., ,,
I
h
'Zo+ jZ*TanftS' I a
'LR
)T@fi :.,
,:
b*)*
,
i
Tanfit -l
pa
7a --
jTanfi
/6 = -JtuCotg ( eapacirive) :
6.zo-
$,i
l:'
Jm
&
Ir F
I
.:
ii;
,
Zsc=InFs impedfficc ( seudiqgcndimpedarrce) Withtherreceiving erd
*o{t eircEited (21= 0)
It
'ra
With the receiving end open circuircd (Zn
g*". ,,,,:,,,'':. &6.
.
fi*F ':"t.: ,:''
' r' '
,
r,:';::;6fidsenfre ffite&atTyhves (toss Lcat Llaffi)
d)
'
,
E=EI+E-
I=1"+IWhc:
qf, ' :
,, . ,
"'i
-.,:: .i
..
t
:,
:.
,
" .r. ,..,'.t
= Incident Waves '=
uaves tavsliflg from the sending end to ttlc lpad
E, I'= R€fltcted waves
i'
= rvaves fraveling from the load to the source
i;'.':.:l]..:. ,:;.
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GeneralEquations in Terms of Receiving End F::''
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*
t
=
E
= En* e*iFd + Ex'gj9a
|=
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(zn
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zo)e+i\a *
z,) e tpd
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lx'g
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{z*
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REFLECTION COEFFICIENT
'
(r)
i
i
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,
The negative indicated a dlrection r€versal took place
,=5{=-I*ForVoltage: Tv=Tt I
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T=!e-
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t,.,,,
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For Current i .a
..'..
t:
filQTe r is a complex guanttty, lt csri
r*lrl & MNGE: 0 TO 1 !F:
r =E perfect match r
i 0; mhmatch
r=lPeifegtmatch
be.ffiae:
