Ch 2_Transmission Line Theory

7/29/2019 Ch 2_Transmission Line Theory

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ELEC 4750UELEC 4750U

Microwave and RF CircuitsMicrowave and RF Circuits

Faculty of EngineeringFaculty of Engineering

and Applied Scienceand Applied Science

ChapterChapter TwoTwo

Transmission LineTransmission Line TheoryTheory


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Transmission Line TheoryTransmission Line Theory

ELEC 4750UDr. A. Saleh2


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THE LUMPEDTHE LUMPED--ELEMENT CIRCUIT MODELELEMENT CIRCUIT MODEL

FOR A TRANSMISSION LINEFOR A TRANSMISSION LINE

The piece of line of infinitesimal lengthThe piece of line of infinitesimal length z can bez can be

modeled as a lumpedmodeled as a lumped--element circuitelement circuit



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Kirchhoffs voltage lawKirchhoffs voltage law

Kirchho s current lawKirchho s current law


DividingDividing z and taking the limit asz and taking the limit as z 0z 0


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For the sinusoidal steadyFor the sinusoidal steady--state condition, with cosinestate condition, with cosine--

based phasors, the above equations simplified tobased phasors, the above equations simplified to


This resultant equations are similar to MaxwellsThis resultant equations are similar to Maxwellscurl equationscurl equations


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Wave Propagation on a Transmission LineWave Propagation on a Transmission Line


The complex propagation constantThe complex propagation constant


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The traveling voltage and current areThe traveling voltage and current are

To express the characteristic impedance, ZTo express the characteristic impedance, Z00, in terms of lines, in terms of lines

parameters:parameters:



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In time domain, the voltage waveform be expressed asIn time domain, the voltage waveform be expressed as

The wavelength isThe wavelength is


The phase velocity isThe phase velocity is


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The Lossless LineThe Lossless Line

In many practical cases the loss of the line is very smallIn many practical cases the loss of the line is very small

and can be neglected.and can be neglected.

In such case , R = G = 0 , which results toIn such case , R = G = 0 , which results to



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FIELD ANALYSIS OF TRANSMISSION LINESFIELD ANALYSIS OF TRANSMISSION LINES

The transmission line parameters (R, L, G, C) can beThe transmission line parameters (R, L, G, C) can be

derived in terms of the electric and magnetic fields ofderived in terms of the electric and magnetic fields of

the transmission linethe transmission line



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ConsiderConsider thethe specificspecific casecase ofof aa coaxialcoaxial lineline withwith fieldsfields EE

andand HH ,, asas shownshown inin thethe figurefigure


Let the voltage and the current between the conductors beLet the voltage and the current between the conductors be


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The timeThe time--average stored magnetic energy per unit length ofaverage stored magnetic energy per unit length of

line isline is


The timeThe time--average stored electric energy per unit length isaverage stored electric energy per unit length is


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The power loss per unit length due to the finite conductivityThe power loss per unit length due to the finite conductivity

of the metallic conductors isof the metallic conductors is

Where RWhere Rss is the surface resistance of the conductorsis the surface resistance of the conductors


The timeThe time--average power dissipated per unit length in a lossyaverage power dissipated per unit length in a lossy

dielectric isdielectric is


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EXAMPLE 2.1 TRANSMISSION LINE PARAMETERS OF A COAXIAL LINEEXAMPLE 2.1 TRANSMISSION LINE PARAMETERS OF A COAXIAL LINE

The fields of a traveling TEMThe fields of a traveling TEM

wave inside the coaxial line canwave inside the coaxial line can

be expressed asbe expressed as


The parameters of the coaxial line can be calculated fromThe parameters of the coaxial line can be calculated from

the given fields as followsthe given fields as follows


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The power flow (in the z direction) on the coaxial line may beThe power flow (in the z direction) on the coaxial line may be

computed from the Poynting vector ascomputed from the Poynting vector as



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THE TERMINATED LOSSLESSTHE TERMINATED LOSSLESSTRANSMISSION LINETRANSMISSION LINE

When the TL is terminated with load we expect that someWhen the TL is terminated with load we expect that someof the incident wave will be reflected on the line and someof the incident wave will be reflected on the line and some

other will be transmitted to the load.other will be transmitted to the load.



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The voltage and current ofThe voltage and current of

the incident wavethe incident wave


At the load (z = 0), the load impedance must beAt the load (z = 0), the load impedance must be

This ratio is defined as the voltage reflection coefficient, This ratio is defined as the voltage reflection coefficient,


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The total voltage and current waves onThe total voltage and current waves on

the line can then be written asthe line can then be written as


e t mee t me--average power ow a ong t e ne at t e po nt zaverage power ow a ong t e ne at t e po nt z

is given asis given as


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The presence of a reflected wave leads to standing waves, andThe presence of a reflected wave leads to standing waves, and

the magnitude of the voltage on the line is not constantthe magnitude of the voltage on the line is not constant


The standing wave ratioThe standing wave ratio


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Reflection coefficient and inputReflection coefficient and input

impedance at any distance z =impedance at any distance z = --ll


The transmission line impedance equationThe transmission line impedance equation


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Special Cases of Lossless Terminated LinesSpecial Cases of Lossless Terminated Lines

oo The shortThe short--circuited linecircuited line

ZZLL

= 0 , then == 0 , then = --1 , SWR1 , SWR

The voltage and current on the line areThe voltage and current on the line are


The input impedance isThe input impedance is


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oo

The openThe open--circuited linecircuited lineZZLL 0 , then = 1 , SWR0 , then = 1 , SWR

The voltage and current on the line areThe voltage and current on the line are


The input impedance isThe input impedance is


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oo The transmission lines withThe transmission lines with

some special lengthssome special lengths

It means thatIt means that


Then, we haveThen, we have

This means that a halfThis means that a half--wavelength line (or any multiplewavelength line (or any multipleof /2) does not alter or transform the load impedance,of /2) does not alter or transform the load impedance,

regardless of its characteristic impedanceregardless of its characteristic impedance


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It means thatIt means that

Then, we haveThen, we have



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A Transmission Line (ZA Transmission Line (Z00) feeding a different) feeding a different

Transmission Line (ZTransmission Line (Z11

))

Assume no reflectionsAssume no reflections

from its far end so that thefrom its far end so that the

reflection coefficient isreflection coefficient is


Then the voltage for z < 0 isThen the voltage for z < 0 is

The voltage for z > 0 isThe voltage for z > 0 is

The transmission coefficient, TThe transmission coefficient, T


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THE SMITH CHARTTHE SMITH CHART

What Smith chart?What Smith chart?

Why Smith chart?Why Smith chart?



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The reflection coefficient at the load can beThe reflection coefficient at the load can beexpressed in term of the normalized Z as:expressed in term of the normalized Z as:


This complex equation can be reduced to two real equationsThis complex equation can be reduced to two real equations

&&


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The real and imaginary parts can beThe real and imaginary parts can be

separated and rearranging to giveseparated and rearranging to give


Which are two families of (resistance andWhich are two families of (resistance and

reactance) circles in thereactance) circles in the rr,, iiplane. These circlesplane. These circles

are orthogonal to each otherare orthogonal to each other


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The Smith chart can also be used toThe Smith chart can also be used tographically solve the transmissiongraphically solve the transmission

line impedance equationline impedance equation



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The normalized impedance lineThe normalized impedance line

ELEC 4750UDr. A. Saleh

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This line is called the pure resistance line, and formsThis line is called the pure resistance line, and forms

the reference for measurements made on the chartthe reference for measurements made on the chart


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The constant resistance circlesThe constant resistance circles

For example, circle A passesFor example, circle A passes

through the center of the chart, sothrough the center of the chart, soit represents all points on the chartit represents all points on the chart

with a normalized resistance of 1.0.with a normalized resistance of 1.0.


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This particular circle isThis particular circle is

sometimes called thesometimes called the unityunity

resistance circleresistance circle


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The constant reactance circlesThe constant reactance circles


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Impedance , Admittance and SWRImpedance , Admittance and SWR


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The constructed circle isThe constructed circle iscalled the VSWRcalled the VSWR circlecircle


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Outer circle parametersOuter circle parameters

(A) the pure reactance circle(A) the pure reactance circle

(B) the wavelength distance(B) the wavelength distance


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generator end of the TLgenerator end of the TL

(C) either the transmission(C) either the transmission

or reflection coefficient angleor reflection coefficient anglein degreesin degrees


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THE QUARTERTHE QUARTER--WAVE TRANSFORMERWAVE TRANSFORMER

The quarterThe quarter--wave transformer uses for impedancewave transformer uses for impedance


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ma c ng an a so us ra es e proper es oma c ng an a so us ra es e proper es o

standing waves on a mismatched linestanding waves on a mismatched line


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The Impedance ViewpointThe Impedance Viewpoint

RRLL andand ZZoo are both real and knownare both real and known

lossless quarter wavelength TL oflossless quarter wavelength TL ofunknown characteristic impedance Zunknown characteristic impedance Z11


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The input impedanceThe input impedance ZZinin is given byis given by

for = 0, we must havefor = 0, we must have ZZinin == ZZoo, which yields, which yields


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EXAMPLE 2.5EXAMPLE 2.5

ZZoo = 50= 50 , Z, ZLL = 100= 100 . Find the characteristic impedance of. Find the characteristic impedance ofthe matching section and plot the magnitude of the reflectionthe matching section and plot the magnitude of the reflection

coefficient versus normalized frequency, f /coefficient versus normalized frequency, f / ffoo, where, where ffoo is theis the

frequency at which the line is /4 long.frequency at which the line is /4 long.


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The input impedanceThe input impedance ZZinin is a function of frequency whichis a function of frequency whichcan be written in terms of f /can be written in terms of f / ffoo asas


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The MultipleThe Multiple--Reflection ViewpointReflection Viewpoint



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GENERATOR AND LOAD MISMATCHESGENERATOR AND LOAD MISMATCHES

In general, both generator and load may presentIn general, both generator and load may present

mismatched impedances to the transmission line,mismatched impedances to the transmission line,and thus multiple reflections can occur on the lineand thus multiple reflections can occur on the line



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Assuming lossless TL, the power delivered to the load isAssuming lossless TL, the power delivered to the load is


WhereWhere RRinin andand XXgg are the resistance and reactanceare the resistance and reactance

parts ofparts of ZZinin andand ZZgg (impedance of generator.)(impedance of generator.)

We will discuss three cases of impedanceWe will discuss three cases of impedance


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Load Matched to LineLoad Matched to Line

In this case we haveIn this case we have ZZll== ZZoo, so, so ll= 0, and SWR = 1= 0, and SWR = 1

The power delivered to the load isThe power delivered to the load is


Generator Matched to Loaded LineGenerator Matched to Loaded Line

In this case we haveIn this case we have ZZinin == ZZgg

The power delivered to the load isThe power delivered to the load is


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Conjugate MatchingConjugate Matching

To maximize P, we differentiate with respect to the realTo maximize P, we differentiate with respect to the real

and imaginary parts ofand imaginary parts of ZZinin..


This condition is known as conjugate matching, and itThis condition is known as conjugate matching, and it

results in maximum power transfer to the loadresults in maximum power transfer to the load


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LOSSY TRANSMISSION LINESLOSSY TRANSMISSION LINES

The LowThe Low--Loss LineLoss Line

The transmission line parameters are = +The transmission line parameters are = + jj andand ZZoo


For small lossFor small loss


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Also, the characteristic impedanceAlso, the characteristic impedance ZZoo can be approximatedcan be approximated

as a real quantity:as a real quantity:


This method for the calculation of attenuation requiresThis method for the calculation of attenuation requiresthat the line parameters L, C, R, and G be knownthat the line parameters L, C, R, and G be known


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The Distortion less LineThe Distortion less Line

There is a special case, however, of a lossy line that hasThere is a special case, however, of a lossy line that has

a linear phase factor as a function of frequencya linear phase factor as a function of frequency


is now a linear function of frequency is now a linear function of frequency nadnad the phasethe phasevelocityvelocity vvpp = / will be frequency= / will be frequency--independentindependent

Also note that the attenuation constant, does notAlso note that the attenuation constant, does notdepend on frequencydepend on frequency.


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The Terminated Lossy LineThe Terminated Lossy Line

Assume the loss is small, so thatAssume the loss is small, so that ZZ is a roximatel real,is a roximatel real,


the voltage and current wave on the line arethe voltage and current wave on the line are


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The power delivered to the input of the terminated lineThe power delivered to the input of the terminated line


The power delivered to the loadThe power delivered to the load

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Ch 2_Transmission Line Theory