transline_handout1

7/28/2019 transline_handout1

http://slidepdf.com/reader/full/translinehandout1 1/32

ctqdd VwaWAtirl/-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 - a strand of copper that carries current at somevoltage

.\

Cablo .= the complete wire asspmbly, which includes insulation, connectors, & uiy

-i

l. ?rro-wireFarrllel lines

-operated in *re balanced mode,conductors being

I arranged sO thatthey presentequal capacitancestO ground

For low - frequency applicatrons (e.g, telephone ckt)

::.,

1-eAD

,T.t

I2



- operated in the unbalanced mode, since the external capacitance

the outer conductor only & ground.

For high - frequency applications (e.g, relephone ckt.)

Electromagnetic Fields :

Electomagnetic fields around a.coQxial line

E - electic field

H-magnetic field

/l

r /lillt..'V

\I

I

Note:



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 & the signal is propagating into the page. For

cpen-wire & twin-lead transmission lines, the TEM wave propagates in the space

between & 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 & less expensive than the coaxial

structure, the ribbon results a less,structural integrity & more radio frequency interference

since .the TEM fields are not a fined & shielded the balanced coaxial line is use to

maintain radiation of the TEM flreld.

TWO WIRE, LIIIE:

i. Two Wire line:

'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 separated only by xc & are

held apart by spacers every fcw inches.

l.



3. fwo-wire Parallel (ribbon or hvi' lead) - the same as the two,wireopen line, except that uniform spacing is assured by embedding the twowires in a low-loss dielectric, usualy polyethyrene. Eq. TV read in

DtELEc'relcSUEATH

4. Oval-Two-WireParallel

5. Two-Wire-shielded - consist of parallel conductors separated from each

other' & 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 frommoisture or mechanical damage.

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

,*rfloo



from the outer concluctor

iutervals. The spacers are

rnaterial possessing good

frequencies.

by insulating spacers, or beads, at regular

marJe uf Pyrex, polystyrene, or some other

i,rsulating characteristics & low loss at high

,)

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 highsuch 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

ftsryf

rUELyEaE



.i.

hrtrc& Line Fundamentr. ts:

Eniralent ckt. of Transmispion line

le-I t uttlr LEN 1 uurT

-l

LrNerH I

fir tt, ingl,l

"il1Primar Constants:

R - series resistance (O/unit length) l A cc'o r,tvTs

L - series inductance (p iunit length) J

G - shunt conductance (u/unit length)

/ 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

surse imnedance)

the characteristic.' impedance 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',

,o =, * -*-- (r)

E = IIZ *!ttZo1 =

4'zo

zn =EIz=zo=#

Z=R,- jwl,

y = G.r jwC

:

Itz + /r(h)t = Itz +,/r,

l'Y, + zo' ' Uz'v/;



1,;qat,: r: = *

7.o+ZozY =Z+ZZoY +Zo

zozY=ZZoYtz

ZozY-hYZ=z ,

ZoY(Zo-Z)=Z:rlrI

..:i, : r. 11:ti a.: ;

Zo 772 '

ZoY(Zo\= Z

ZozY = Ztfr

Zo = -la\lr

But:

Z;'R+ jwlY =G+ Jwe.

4_@h =lG'* i*c 1

General Equation of Zo(O)

At Low IrooBep0iesRIV iu'r,

,r7 j*c

I-

zo=^lL t.fuYq

'

&A 0 t B, "P tl 6&tl 6lt tl€;! :lZ 11 JUL :''...,....6 dtjwl

:

*a:{T;.,u

Characteristic impe{ance of lossless or dissip6ionless lines



Propagation Coeffi cient, (6)Determines the variation of current or vottage with distance

alongatransmissionline.

e---

Like Zo, the 6 arso depends on the primary constants and theangular velocity of the signal.

S = per unit length

-Ihis

is a complex 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 ' '- 1 1 (phase shift of 2n rad occurs over a distanceof l wavelength)

l, = wavelength ; C= speed of lightf = frequency, Hz

I =-9f

f,=3x108

im



USEFULL EQUIVALENTS:

'1. 1 NEPER = 8.585 dB

z. o= - o'toz

tr'rz3t"E6'; Lncn

Where:

O = dian:lter of AWG wiren = AWG no.

3. 1 inch = 1000 mils

4. Area (cir. mits) = (dialrirt)2

5. l\=Efwhere:

Vp = vetocity of propagationf = 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 actas an antenna if the separation of the conductors is an

a ppreciable fraction of wavelength.

Z' 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

l.

(at H.F)

where: d = wirediameter

D = center to center spacing

Characteristic lmped ance, Zo

Zo=

or

. Zo -r2o h?D '

Er ' rr)

where: Er = dielectric constant

= relative permittivity

Er = I forair

rcr,Tt

l-Dd = wire diameter

276

"ler



-,for high impedance applications

- typhical : 3000

R; Series Resistance

1s0o - 600C,

3. L, $erieg Inductancc

7=t-6fl);s1tmr'd

F = dielectric permeability

[r = Fo for a transmission line ; p-po

h= 4rcx l0'7 tr/n for free space

!

Rl 16.8J7

where: Rl = pCUm

f = fi:q.rHz

d ,= wire diamerer, cm

R2 = CY100ft.

f = freq., MHz

d = wire diameter, inch.

4. C, Shunt Capacitance



u=ft;F/md.:..

where:

5. G, Shunt Conductance

G=O'atRF

Er= relative permittivity, ( dielectric constant)

6.

d=..W/unitlength

@RF:

a;J@: 6=iw,{E=j0

7. Attenuation constani a

Id,L =

07

4.35^RdZ=Zo

wnere:



a? =. clB lunitlanght

rrx. R=Qlm

a*d*'lm' ' ' ,'.

B. Coaxial Lines : at HF

l. Charaeteristic Impedaauguh

D= lrqrgB pliilrgrH oriTtr,hxrFF coNtuCTDR

d, Cn 14, pthl,lElER trTtFTHNEF CDitDtr6,Sr

Zo

or

:ffir'{1o

zn=ftry$,st , :

'.,t.'

where: Er* dielecuic constsst

,']r i:i'ioi air ", . ."'

For low impeciaace appiicatioari',.t'

4m _ lJgg,:r,.. i:' -: :,,i;,,

Typical: 75Oi:. ,. , ''.,.''i:.,''.i..'1.,- ,i;., ,t . ,..,. -.

,

i,.t

I



: =

-)

)

fil = 41.tiJ f t7+ o)or

R2 *0,1f,4,* j) ,,

Rl= nCt l cm

f = freq,,Hz :

:'

:t

' ! r, -'R2 = CrnAAfr. .

3. 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 L.m



GeneralEquations for Zoz

Two Wire Lines:

Coorial Lines:

h=J-Eb#)

^=*E^rrf

Dieler.tric e po+tap$ of ,MnF,thl.t,

Material ffietectric Constant ( APPror)

1,0Air

Glass ( electrical) 3.8 - 14.5

, I r I !l

- . - l-Mica(deceiPal) 4.0 - 9.0



Paper (dry) 1.5 - 3.0

Flexiglass 2.6 -3.5

polyethylene

Polystyrene 2.4 -3.0

Quartz 5.0

Styrofoam 1.03

Teflon

Line Perfurmance Calculation:

Consider a section of an infinite line with fundamental constant ( & L, G & C)

concentates in very small units of length d/.

If the current flows through an impedance the voltage drop is IZ & therefore the voltage

drop along the element of line dl will be:

2.t

dRL

dE =*IZdlor

#=-IZ +(t)

{

l+



':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 dI = -,Eydl :

dI

-=-Ey+(2)I

Differentiating (1) & Q)tc, eliminate E & I with respect to l:

#=-z#-+i:r

d2I dE_= -+(4)dlz 'dl '\/

Eliminating undesired variables by r;ubstituting (3) e $) with (l) e-nd (2) respectively:

d2E

_=_Z(_Ey)=Zy+(5)dl2

;

d2I

-E-y(-IZ)=.ZyI -+(6)

: dlz

(5) & (6) are sirnpte diffl equati,: ; of the 2nd ord,.:r whose general solution.for these

fomrs are:

E*Ae-t@ +Be'ffi +(7)

1 =1Ae-'@ - ur'* lh-+ (B)



Nois U) A.(8) are General Solutions/ Equations:

rr-.'i:.';'..';::l:i.- . I l, ',r,

:,::..-:-: li.-" I l

AITALYSIS OF TRANSMISSION LINE:

Reference.Transmission Line :

egSworrc h

s)uRcE

€LR

Wglurz,- NL0ao

ZS =Ez Js,

+ X..0

+,- r' "rr':

7s ,ET

Zg = internal irapedanco oftho genersior

Eg= generator voltage

Zs = sending end / souree lapq{agcc

Is = sending end./ source cun€-nl ,: j j:..i.: ,l

Es = sending end / source voltage

Zr = receiving end llodimpedance'

E = voltage



"'1

','!""!' \ ,l a ?/- x fto'* tl-'SottrcoI=current I or d'l*ru' tl'w /oa./

Z: impedancf I

S: total length of the transmission line

General Equations:

E=Ae-e+Bee1

J =L1Ae-* - Be*lLO

Where:fzy =JZY

Note: A & B are arbitrary constant whose values depend on the Load

Case I : AC SteaCy State - Liues with No Reflections

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\

'

'ZoSolving For A:

tr. in terms of gending e-nd:

set: x=0

in (l)':

I =!!e-*'solution in terms of senfir9 errd

-Zo

L in terms of receiving end:

set x=s.:

fhen: "-."1

E=Er=Aea

\

.Er,-ofu'',,''A=-= uR-

e-6

Subttituting A ia (1) & (2): ::, '

fii,= Ereee& *bufi d=s-r.AISOI l. 1,.:,. ,.tr,' ,' ::i..' ,

| = +.rnr'oZo

J =E* ,*fut



, . ::

Ei6.l e seen at any point along the line is always,equal: tor h:for,transrnission

Elineswith no reflections. This can !s prov€n by dedvir.rg,Z =4:,t

.:. 1. Using Sending end Solution: .

E'= Es.e*d

I =4 ,-*Zo

Es.eayE

I 4,r*Zo

:.2 = Zo

2. Using Receiving End Solution:

E =,E^edD'

I - "R-oilZO

D p-du un€,=-=.+I L"*

;.2 = Zo

If atransmission line has a finite length & if the line is not terminated at its own

cbamcteristic impedancc (Z a *Jo) thcn reflections wlll occur, Reflections are signalsi

tlutgobacktothosource(i.e.thesaaresignalsnotabeorbedbytheload), :,, : ',",' i,



2,, * Zo

TheGeneral Solution for lines with reflections can be derived in the following manner.

l. Determine the value of A & B frorn the General Equations

2. Substitute A & B in the General Equarions

General Equations:

E=Aea+Bza'

I =Ll.e"-e - Betul2o,"".

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 =2AIs(Zs + Zo) =Ztl

fiz _g&

(t' t z;1

Sub: (1) & (2)

Es-IsZo=)B i Es=lsk

:. IsZs - IsZo =28 i

Is(Zs - Zo) =28 .t

i

'n=!<-zo)2'

Sub: A & B in the General Equations:



li

' Add: ($) +'(4)

n = lyZs + Zo)e-e + (Zs - Zo)ea' f2":'-. '..,.. -, . : :

1 = Lylzs + Zo)e-e - (h - Zo)ee I2Zo-'-_:

2. in terms of Receiving En.1:

Set: x=s : :

Eo=Ae&+Bee+(3)^

tI*=L1Ae-e -8etu1

ZOI^,Zo=Ae-tu,*Beb i&)

solutions in the terms ofsending end

,r , , l=$<r-+zo1etu'

Sub. (3) - (4)

ER-I RZn =ZB.ee

I*Zr-I*h=ZBe-e' i En * lxZn

I n(Z * - Zo) =ZBc-*

E^+I*Zo=ZAca

! r;Zr + I *Zo *ZAe*&

I *(Z* + Zo) *2Ac'e

- Zo)ea =!;l!:r

, : ,',,i::s{tbstitufingA & B ilths Genenrl Equations:

a =$uzr+za)etue'* +(z^ . zopeatul "' r'

,dL.t



... :

. Case III. Dissipatiion Less /Loss Less Lines

dffilTZo= I ' = -l-\G + jwc llc

d=J(X + jwl)\i+ jwC) = jwlLC = jBj

tia ;Genoral Equations:

;'E=Ae-iB +Berfi

I1 =!1tle'rr - Beiftl

7n-

: . . .i .. : ,

At tte $endlng End: ( x* 0)



',',.i-,?i-ffiF'

Es = Ar * B, = /rZ, -+ (l)

It, = )(A- B) + (2)

ZO

Add (1) a Q)z

-Za)(wsft -sinpr)]'-,,

Sinplifvinei

T

g= f-122 r

cos

ft -jltusia

Bxl2' o '

E = Escos p*- i*zosinfu

.:

::i

''I

T



\

E =,-Er[cos Px- iffsinpxl

I =Islcostx-

iffsapxlt,

In torms of Receiving End : ( x=s ) .,'

, E* = ls-itu + pblfi = IaZn+ G)

4=)Udin - Bein)+ (4)

Solve (3) & (4) Simultaueourly:

,e=,, $tz rl zo)eik

':T

B=t@R-Zo1e-$

Sending:End Solutions

.:,

fE=+lG*+zo)eiu +(zn-zo'1e-tu1 ; d=s-x

f

4= *Ilz.u + Zo)(cospd+ jsii fd) + (Z o - Zo\(cos Bd - j sn Bd)ji:--!-. :

Slilq/tFl/tl6 "r : : :r

'r:i::': '!i:1; i.:iir: i ::

n =lVzrcosfr r ih{nfilE * Encos B{ + grZosinfd : ;,,

: i,'. r.

,'': tl .

i



E=Enlcos N*i{"i"pd))

I = I nlcosN + i $sinprJ Recei.rirrg End Solutions

,

, -lrr,,rZ*+ jzoTanPd-,

,ar-

-lJvt- 'Zo+ jZ*Tanpd'

Note:

Wave Length (1,) = The distance between poinrs that have corresponding phase in two

consecutive cycles in a periodic wave.

^ 3xlos m/7-e - Is

ft--

f f

=2trRAD, =360'

Special Ca,$es: trmpedance of Loss Lcss Llnes

l, When; Zr=Zo

zsn=h

(matched line & frequency impedance)



s=L2

2. When : (or,rlven rnultiples)

z* + jzoffon(41*l* = zoE-----------4'Y I

Zo+ 1Z*Tan(+X*)AI

7Zs = Zo("R,.)

ZO

Zs*Za

z.uRi

(

4. When i Z * u shurt ciresit *0 ,, ,

J.1. i:-

'3. When : S =L (orodd multiples)4

z^+jzoran<4><$ t(R' k=zol ! ?-1.#zo+ iz.ran(+xi; Yk

:l

Let: *=ranffn*r= Tatt))o

=q' r ' a , '' ri l

, zn/ + i?n' k=Zol#1 &ot.i7

/k ' !'R

h=-

;r :.r:arll

:

..

7a+ JZ.xTwNo.,,,,,,,! -, :,, .,.,,,,,.,,:ri..... .t: :.:

7.s=+jZoTanff (inductive),:

.l

5.rVhen:Zp=OpenCircuit=o I,.,



; ;

t., ,, I

h'Zo+ jZ*TanftS' I

a'LR

)T@fi

-l

Tanfit

7apa

-- jTanfi

/6 = -JtuCotg ( eapacirive)

b*)* i

,:;.

.:... '

,lt.

ii;

It'ra

6.zo- Jm

.: , Zsc=InFs impedfficc ( seudiqgcndimpedarrce)

Withtherreceiving erd *o{t eircEited (21= 0)

With the receiving end open circuircd (Zn d)

, r,:';::;6fidsenfre ffite&atTyhves (toss Lcat Llaffi) '

,:'' ,'E=EI+E-

I=1"+I-Whc: t :, :. , " .r ,..,

. '.t:qf = Incident Waves,

' '= uaves tavsliflg from the sending end to ttlc lpad

: ,, . , "'i -.,:: .i . . E, I'= R€fltcted waves i'

= rvaves fraveling from the load to the source



,, u.:.':l-,,

l,'' .: ,..i,:.Ii,

\i,

.j.

GeneralEquations in Terms of Receiving End

..

t = * (zn * zo)e+i\a * | {z* - za)e' ita

E = En* e*iFd + Ex'gj9a

| = *(Za+ z,) e tpd*V.a-z-le

.jpd

| = In* ei+iFd + lx'g-jFd '

REFLECTION COEFFICIENT (r) i

i

- The negative indicated a dlrection r€versal took place

,=5{=-I*-

ForVoltage:

Tv=Tt

I

4@*-2,)T=!e-'

$tza+zn)

, 'a,



. :,',,,,,=jFai:Eo

t,.,,,,, ' t,Zp*Zg

. For Current

i

.a

..'.. t:

filQTe r is a complex guanttty, lt csri be.ffiae:r*lrl &MNGE: 0 TO 1

!F: r =E perfect match

r i 0; mhmatch

r=lPeifegtmatch

transline_handout1

Documents