8/12/2019 TLW CH6

1/49

Transmission Lines

Transmission lines and waveguides may be defined as devices used

to guide energy from one point to another (from a source to a load).

Transmission lines can consist of a set of conductors, dielectrics orcombination thereof. As we have shown using Maxwells equations, we

can transmit energy in the form of an unguided wave (plane wave) through

space. In a similar manner, Maxwells equations show that we can

transmit energy in the form of a guided wave on a transmission line.

Plane wave propagating in air

unguided wave propagation

Transmission lines / waveguides guided wave propagation

Transmission line examples

8/12/2019 TLW CH6

2/49

8/12/2019 TLW CH6

3/49

Transmission Line Definitions

Almost all transmission lines have a cross-sectional geometry which

is constant in the direction of wave propagation along the line. This type

of transmission line is called a uniform transmission line.

Uniform transmission line- conductors and dielectrics maintain the

same cross-sectional geometry along the transmission line in

the direction of wave propagation.

Given a particular conductor geometry for a transmission line, only

certain patterns of electric and magnetic fields (modes) can exist for

propagating waves. These modes must be solutions to the governingdifferential equation (wave equation) while satisfying the appropriate

boundary conditions for the fields.

Transmission line mode- a distinct pattern of electric and magnetic

field induced on a transmission line under source excitation.

The propagating modes along the transmission line or waveguide may be

classified according to which field components are present or not presentin the wave. The field components in the direction of wave propagation

are defined as longitudinalcomponents while those perpendicular to the

direction of propagation are defined as transversecomponents.

Transmission Line Mode Classifications

Assuming the transmission line is oriented with its axis along thez-

axis (direction of wave propagation), the modes may be classified as

1. Transverse electromagnetic(TEM) modes- the electric and

magnetic fields are transverse to the direction of wave

propagation with no longitudinal components [Ez=Hz= 0].

TEM modes cannot exist on single conductor guiding

8/12/2019 TLW CH6

4/49

structures. TEM modes are sometimes called transmission line

modes since they are the dominant modes on transmission

lines. Plane waves can also be classified as TEM modes.

2. Quasi-TEM modes - modes which approximate true TEM

modes for sufficiently low frequencies.

3. Waveguide modes - either Ez, Hz or both are non-zero.

Waveguide modes propagate only above certain cutoff

frequencies. Waveguide modes are generally undesirable on

transmission lines such that we normally operate transmission

lines at frequencies below the cutoff frequency of the lowest

waveguide mode. In contrast to waveguide modes, TEMmodes have a cutoff frequency of zero.

8/12/2019 TLW CH6

5/49

Transmission Line Equations

Consider the electric and magnetic fields associated with the TEM

mode on an arbitrary two-conductor transmission line (assume perfect

conductors). According to the definition of the TEM mode, there are no

longitudinal fields associated with the guided wave traveling down the

transmission line in thezdirection (Ez=Hz= 0).

From the integral form of Maxwells equations, performing the line integral

of the electric and magnetic fields around any transverse contour C1gives

where ds= dsaz. The surface integrals of Eand Hare zero-valued since

there is noEzorHzinside or outside the conductors (PECs, TEM mode).

0

0

8/12/2019 TLW CH6

6/49

The line integrals of Eand Hfor the TEM mode on the transmission line

reduce to

These equations show that the transverse field distributions of the TEM

mode on a transmission line are identical to the corresponding static

distributions of fields. That is, the electric field of a TEM at any frequency

has the same distribution as the electrostatic field of the capacitor formed

by the two conductors charged to a DC voltage Vand the TEM magneticfield at any frequency has the same distribution as the magnetostatic field

of the two conductors carrying a DC currentI. If we change the contour

C1 in the electric field line integral to a new path C2 from the

conductor to the + conductor, then we find

These equations show that we may define a unique voltage and current at

any point on a transmission line operating in the TEM mode. If a unique

voltage and current can be defined at any point on the transmission line,

then we may use circuit equations to describe its operation (as opposed to

writing field equations).

Transmission lines are typically electrically long (severalwavelengths) such that we cannot accurately describe the voltages and

currents along the transmission line using a simple lumped-element

equivalent circuit. We must use a distributed-elementequivalent circuit

which describes each short segment of the transmission line by a lumped-

element equivalent circuit.

8/12/2019 TLW CH6

7/49

Consider a simple uniform two-wire transmission line with its

conductors parallel to thez-axis as shown below.

Uniform transmission line - conductors and insulating medium

maintain the same cross-sectional geometry along the entiretransmission line.

The equivalent circuit of a short segment zof the two-wire transmission

line may be represented by simple lumped-element equivalent circuit.

8/12/2019 TLW CH6

8/49

R = series resistance per unit length (/m) of the transmission line

conductors.

L= series inductance per unit length (H/m) of the transmission line

conductors (internal plus external inductance).

G = shunt conductance per unit length (S/m) of the media betweenthe transmission line conductors (insulator leakage current).

C= shunt capacitance per unit length (F/m) of the transmission line

conductors.

We may relate the values of voltage and current at zandz+ zby

writing KVL and KCL equations for the equivalent circuit.

KVL

KCL

Grouping the voltage and current terms and dividing by zgives

Taking the limit as z 0, the terms on the right hand side of the

equations above become partial derivatives with respect tozwhich gives

Time-domain

transmission line

equations

(coupled PDEs)

8/12/2019 TLW CH6

9/49

For time-harmonic signals, the instantaneous voltage and current may be

defined in terms of phasors such that

The derivatives of the voltage and current with respect to time yield j

times the respective phasor which gives

Frequency-domain

(phasor) transmission

line equations

(coupled DEs)

Note the similarity in the functional form of the time-domain and the

frequency-domain transmission line equations to the respective source-free

Maxwells equations (curl equations). Even though these equations were

derived without any consideration of the electromagnetic fields associated

with the transmission line, remember that circuit theory is based on

Maxwells equations.

Just as we manipulated the two Maxwell curl equations to derive thewave equations describing Eand Hassociated with an unguided wave

(plane wave), we can do the same for a guided (transmission line TEM)

wave. Beginning with the phasor transmission line equations, we take

derivatives of both sides with respect toz.

We then insert the first derivatives of the voltage and current found in the

original phasor transmission line equations.

8/12/2019 TLW CH6

10/49

The phasor voltage and current wave equations may be written as

Voltage and

current wave

equations

where

is the complex propagation constant of the wave on the

transmission line given by

Just as with unguided waves, the real part of the propagation constant (

)

is the attenuation constant while the imaginary part (

) is the phase

constant. The general equations for and in terms of the per-unit-lengthtransmission line parameters are

The general solutions to the voltage and current wave equations are

~~~~~ ~~~~~

+z-directed waves

z-directed waves

8/12/2019 TLW CH6

11/49

The coefficients in the solutions for the transmission line voltage and

current are complex constants (phasors) which can be defined as

The instantaneous voltage and current as a function of position along the

transmission line are

Given the transmission line propagation constant, the wavelength and

velocity of propagation are found using the same equations as for

unbounded waves.

The region through which a plane wave (unguided wave) travels ischaracterized by the intrinsic impedance(

) of the medium defined by the

ratio of the electric field to the magnetic field. The guiding structure over

which the transmission line wave (guided wave) travels is characterized the

characteristic impedance(Zo) of the transmission line defined by the ratio

of voltage to current.

8/12/2019 TLW CH6

12/49

If the voltage and current wave equations defined by

are inserted into the phasor transmission line equations given by

the following equations are obtained.

Equating the coefficients on e zand e zgives

The ratio of voltage to current for the forward and reverse traveling waves

defines the characteristic impedance of the transmission line.

8/12/2019 TLW CH6

13/49

The transmission line characteristic impedance is, in general, complex and

can be defined by

The voltage and current wave equations can be written in terms of the

voltage coefficients and the characteristic impedance (rather than the

voltage and current coefficients) using the relationships

The voltage and current equations become

These equations have unknown coefficients for the forward and reverse

voltage waves only since the characteristic impedance of the transmission

line is typically known.

8/12/2019 TLW CH6

14/49

Special Case #1 Lossless Transmission Line

A lossless transmission line is defined by perfect conductors and a

perfect insulator between the conductors. Thus, the ideal transmission line

conductors have zero resistance ( = ,R=0) while the ideal transmissionline insulating medium has infinite resistance ( =0, G=0). The

equivalent circuit for a segment of lossless transmission line reduces to

The propagation constant on the lossless transmission line reduces to

Given the purely imaginary propagation constant, the transmission line

equations for the lossless line are

8/12/2019 TLW CH6

15/49

The characteristic impedance of the lossless transmission line is purely real

and given by

The velocity of propagation and wavelength on the lossless line are

Transmission lines are designed with conductors of high conductivity and

insulators of low conductivity in order to minimize losses. The lossless

transmission line model is an accurate representation of an actual

transmission line under most conditions.

8/12/2019 TLW CH6

16/49

Special Case #2 Distortionless Transmission Line

On a lossless transmission line, the propagation constant is purely

imaginary and given by

The phase velocity on the lossless line is

Note that the phase velocity is a constant (independent of frequency) so

that all frequencies propagate along the lossless transmission line at the

same velocity. Many applications involving transmission lines require thata band of frequencies be transmitted (modulation, digital signals, etc.) as

opposed to a single frequency. From Fourier theory, we know that any

time-domain signal may be represented as a weighted sum of sinusoids.

A single rectangular pulse contains energy over a band of frequencies. For

the pulse to be transmitted down the transmission line without distortion,

all of the frequency components must propagate at the same velocity. This

is the case on a lossless transmission line since the velocity of propagation

is a constant. The velocity of propagation on the typical non-idealtransmission line is a function of frequency so that signals are distorted as

different components of the signal arrive at the load at different times.

This effect is called dispersion. Dispersion is also encountered when an

unguided wave propagates in a non-ideal medium. A plane wave pulse

propagating in a dispersivemedium will suffer distortion. A dispersive

medium is characterized by a phase velocity which is a function of

frequency.

For a low-loss transmission line, on which the velocity of propagationis near constant, dispersion may or may not be a problem, depending on

the length of the line. The small variations in the velocity of propagation

on a low-loss line may produce significant distortion if the line is very

long. There is a special case of lossy line with the linear phase constant

that produces a distortionless line.

8/12/2019 TLW CH6

17/49

A transmission line can be made distortionless (linear phase constant) by

designing the line such that the per-unit-length parameters satisfy

Inserting the per-unit-length parameter relationship into the generalequation for the propagation constant on a lossy line gives

Although the shape of the signal is not distorted, the signal will suffer

attenuation as the wave propagates along the line since the distortionless

line is a lossytransmission line. Note that the attenuation constant for a

distortionless transmission line is independent of frequency. If this were

not true, the signal would suffer distortion due to different frequencies

being attenuated by different amounts.

In the previous derivation, we have assumed that the per-unit-length

parameters of the transmission line are independent of frequency. This isalso an approximation that depends on the spectral content of the

propagating signal. For very wideband signals, the attenuation and phase

constants will, in general, both be functions of frequency.

For most practical transmission lines, we find thatRC> GL. In order

to satisfy the distortionless line requirement, series loading coils are

typically placed periodically along the line to increaseL.

8/12/2019 TLW CH6

18/49

Transmission Line Circuit

(Generator/Transmission line/Load)

The most commonly encountered transmission line configuration is

the simple connection of a source (or generator) to a load (ZL) through thetransmission line. The generator is characterized by a voltage Vg and

impedanceZgwhile the transmission line is characterized by a propagation

constant

and characteristic impedanceZo.

The general transmission line equations for the voltage and current as a

function of position along the line are

The voltage and current at the load (z= l ) is

We may solve the two equations above for the voltage coefficients in terms

of the current and voltage at the load.

8/12/2019 TLW CH6

19/49

Just as a plane wave is partially reflected at a media interface when the

intrinsic impedances on either side of the interface are dissimilar ( 1 2),

the guided wave on a transmission line is partially reflected at the load

when the load impedance is different from the characteristic impedance of

the transmission line (ZL Zo). The transmission line reflection coefficient

as a function of position along the line [

(z)] is defined as the ratio of the

reflected wave voltage to the transmitted wave voltage.

Inserting the expressions for the voltage coefficients in terms of the load

voltage and current gives

The reflection coefficient at the load (z=l) is

and the reflection coefficient as a function of position can be written as

8/12/2019 TLW CH6

20/49

Note that the ideal case is to have L= 0 (no reflections from the load

means that all of the energy associated with the forward traveling wave is

delivered to the load). The reflection coefficient at the load is zero when

ZL=Zo. IfZL=Zo, the transmission line is said to be matchedto the load

and no reflected waves are present. If ZL Zo, a mismatchexists andreflected waves are present on the transmission line. Just as plane waves

reflected from a dielectric interface produce standing waves in the region

containing the incident and reflected waves, guided waves on a

transmission line reflected from the load produce standing waves on the

transmission line (the sum of forward and reverse traveling waves).

The transmission line equations can be expressed in terms of the

reflection coefficients as

The last term on the right hand side of the above equations is the reflectioncoefficient

(z). The transmission line equations become

Thus, the transmission line equations can be written in terms of voltagecoefficients (Vso

+,Vso ) or in terms of one voltage coefficient and the

reflection coefficient (Vso+, ).

The input impedance at any point on the transmission line is given

by the ratio of voltage to current at that point. Inserting the expressions for

the phasor voltage and current [Vs(z) andIs(z)] from the original form of

8/12/2019 TLW CH6

21/49

the transmission line equations gives

The voltage coefficients have been determined in terms of the load voltage

and current as

Inserting these equations for the voltage coefficients into the impedance

equation gives

We may use the following hyperbolic function identities to simplify this

equation:

8/12/2019 TLW CH6

22/49

Dividing the numerator and denominator by cosh [

(lz)] gives

Input impedance at any

point along a general

transmission line

For a lossless line,

=j

and Zois purely real. The hyperbolic tangent

function reduces to

which yields

Input impedance at any point along a lossless

transmission line

Special Case #1 Open-circuited lossless line ( ZL , L= 1)

Special Case #2 Short-circuited lossless line (ZL 0, L= 1)

8/12/2019 TLW CH6

23/49

8/12/2019 TLW CH6

24/49

The impedance characteristics of a short-circuited or open-circuited

transmission line are related to the positions of the voltage and current

nulls along the transmission line. On a lossless transmission line, the

magnitude of the voltage and current are given by

The equations for the voltage and current magnitude follow the crank

diagramform that was encountered for the plane wave reflection exampleand the minimum and maximum values for voltage and current are

For a lossless line, the magnitude of the reflection coefficient is constant

along the entire line and thus equal to the magnitude of

at the load.

The standing wave ratio (s) on the lossless line is defined as the ratio of

maximum to minimum voltage magnitudes (or maximum to minimum

current magnitudes).

The standing wave ratio on a lossless transmission line ranges between 1

8/12/2019 TLW CH6

25/49

8/12/2019 TLW CH6

26/49

8/12/2019 TLW CH6

27/49

From the equations for the maximum and minimum transmission line

voltage and current, we also find

It will be shown that the voltage maximum occurs at the same location as

the current minimum on a lossless transmission line and vice versa. Using

the definition of the standing wave ratio, the maximum and minimum

impedance values along the lossless transmission line may be written as

Thus, the impedance along the lossless transmission line must lie within

the range ofZo/sto sZo.

8/12/2019 TLW CH6

28/49

Example

A source [Vsg=100 0oV,Zg=Rg= 50 ,f= 100 MHz] is connected

to a lossless transmission line [L= 0.25

H/m, C=100pF/m, l=10m]. For

loads ofZL=RL= 0, 25, 50, 100 and , determine (a.) the reflectioncoefficient at the load (b.) the standing wave ratio (c.) the input impedance

at the transmission line input terminals (d.) voltage and current plots along

the transmission line.

8/12/2019 TLW CH6

29/49

RL( ) (a.)Zin( ) (b.) L (c.) s

0 0

1

25 25

0.333 2

50 50 0 1

100 100 +0.333 2

+1

8/12/2019 TLW CH6

30/49

RL( ) Vs(z) max(V) Vs(z) min(V) Vs(l ) (V)

0 100 0 0

25 66.7 33.3 33.3

50 50 50 50

100 66.7 33.3 66.7

100 0 100

8/12/2019 TLW CH6

31/49

8/12/2019 TLW CH6

32/49

Smith Chart

The Smith chart is a useful graphical tool used to calculate the reflectioncoefficient and impedance at various points on a (lossless) transmission line

system. The Smith chart is actually a polar plot of the complex reflection

coefficient (z) [ratio of the reflected wave voltage to the forward wave

voltage] overlaid with the corresponding impedance Z(z) [ratio of overall

voltage to overall current].

8/12/2019 TLW CH6

33/49

The corresponding standing wave

ratio sis

The magnitude of Lis constant

on any circle in the complex

plane so that the standing wave

ratio (VSWR) is also constant on

the same circle.

Smith chart center L = 0 Constant VSWR

(no reflection - matched, s= 1) circle

Outer circle

L = 1

(total reflection, s= )

Once the position of Lis located on the Smith chart, the location of the

reflection coefficient as a function of position [

(z)] is determined using

the reflection coefficient formula.

This equation shows that to locate (z), we start at Land rotate through

an angle of z=2 (z l ) on the constant VSWR circle. With the load

located atz= l, moving from the load toward the generator (z < l) defines

a negative angle z (counterclockwise rotation on the constant VSWR

circle). Note that if z= 2 , we rotate back to the same point. Thedistance traveled along the transmission line is then

Thus, one complete rotation around the Smith chart (360o) is equal to one

half wavelength.

8/12/2019 TLW CH6

34/49

On the Smith chart ...

CW rotation toward the generator

CCW rotation toward the load

/2=360o =720o

and sare constant on a lossless transmission line. Moving

from point to point on a lossless transmission line is equivalent

to rotation along the constant VSWR circle.

All impedances on the Smith chart are normalized to thecharacteristic impedance of the transmission line (when using

a normalized Smith chart).

The points along the constant VSWR circle represent the complex

reflection coefficient at points along the transmission line. The reflection

coefficient at any given point on the transmission line corresponds directly

to the impedance at that point. To determine this relationship between L

andZL, we first solve (1) forzL.

where rL and xL are the normalized load resistance and reactance,

respectively. Solving (2) for the resistance and reactance gives equations

for the resistance and reactance circles.

(2)

8/12/2019 TLW CH6

35/49

In a similar fashion, the reflection coefficient as a function of

position

(z) along the transmission line can be related to the impedance

as a function of positionZ(z). The general impedance at any point along

the length of the transmission line is defined by the ratio of the phasor

voltage to the phasor current.

The normalized value of the impedancezn(z) is

(3)

8/12/2019 TLW CH6

36/49

Note that Equation (2) is simply Equation (3) evaluated atz = l.

Thus, as we move from point to point along the transmission line plotting

the complex reflection coefficient (rotating around the constant VSWR

circle), we are also plotting the corresponding impedance.

Once a normalized impedance is located on the Smith chart for a

particular point on the transmission line, the normalized admittance at thatpoint is found by rotating 180ofrom the impedance point on the constant

reflection coefficient circle.

(3)

(2)

8/12/2019 TLW CH6

37/49

The locations of maxima and minima for voltages and currents along

the transmission line can be located using the Smith chart given that these

values correspond to specific impedance characteristics.

Voltage maximum, Current minimum Impedance maximum

Voltage minimum, Current maximum Impedance minimum

8/12/2019 TLW CH6

38/49

8/12/2019 TLW CH6

39/49

8/12/2019 TLW CH6

40/49

8/12/2019 TLW CH6

41/49

Example (Smith chart)

A 60 lossless line has a maximum impedanceZin= (180 + j0) at

a distance of /24 from the load. If the line is 0.3 , determine (a.) s(b.)

ZLand (c.) the transmission line input impedance.

(a.)

(b.) [Zin]maxoccurs at the rightmost point on the s=3 circle. From thispoint, move /24 toward the load (CCW) to findZL.

(c.) Insert series transmission line section and fromZL, move 0.3 toward

the generator (CW) to findZin.

8/12/2019 TLW CH6

42/49

DP#1 (117.2 + j78.1)

DP#2 (20.0 + j2.8)

8/12/2019 TLW CH6

43/49

Quarter Wave Transformer

When mismatches between the transmission line and load cannot be

avoided, there are matching techniques that we may use to eliminate

reflections on the feeder transmission line. One such technique is thequarter wave transformer.

IfZo ZL, > 0

(mismatch)

Insert a /4 length section of different transmission line (characteristicimpedance =Zo ) between the original transmission line and the load.

The input impedance seen looking into the quarter wave transformer is

Solving for the required characteristic impedance of the quarter wave

transformer yields

8/12/2019 TLW CH6

44/49

Example (Quarter wave transformer)

Given a 300 transmission line and a 75 load, determine the

characteristic impedance of the quarter wave transformer necessary to

match the transmission line. Verify the input impedance using the Smith

chart.

DP#1 (75.0 + j0.0)

DP#2 (300.0 - j0.0)

8/12/2019 TLW CH6

45/49

8/12/2019 TLW CH6

46/49

with the transmission line segment

8/12/2019 TLW CH6

47/49

Ytl= Yo+jB [Admittance of the terminated t-line section]

Ys= jB [Admittance of the stub (short or open circuit)]

Yin= Ytl+ Ys= Yo [Overall input admittance]

[Transmission line characteristic admittance]

In terms of normalized admittances (divide by Yo), we have

ytl= 1 +jb ys= jb yin= 1

Note that the normalized conductance of the transmission line segment

admittance (ytl= g +jb) is unity (g= 1).

Single Stub Tuner Design Using the Smith Chart

1. Locate the normalized load impedancezL(rotate 180oto find

yL). Draw the constant VSWR circle [Note that all points on

the Smith chart now represent admittances].

2. From yL, rotate toward the generator (CW) on the constant

VSWR circle until it intersects the g= 1. The rotation distance

is the distance dwhile the admittance at this intersection pointisytl= 1 +jb.

3. Beginning at the stub end (short circuit admittance is the

rightmost point on the Smith chart, open circuit admittance is

the leftmost point on the Smith chart), rotate toward the

generator (CW) until the point at ys= jb is reached. This

rotation distance is the stub length l.

8/12/2019 TLW CH6

48/49

Short circuited stub tuners are most commonly used because a

shorted segment of transmission line radiates less than an open-circuited

section. The stub tuner matching technique also works for tuners in series

with the transmission line. However, series tuners are more difficult to

connect since the transmission line conductors must be physicallyseparated in order to make the series connection.

Example

Design a short-circuited shunt stub tuner to match a load of

ZL = (60 j40) to a 50 transmission line.

Short circuit admittance - rightmost point on Smith chart (0o)

Shunt stub reactance

8/12/2019 TLW CH6

49/49