hui transmission lines

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Hon Tat Hui Transmission Lines Basic Theories

NUS/ECE EE2011

1

Transmission Lines Basic Theories

1 Introduction

At high frequencies, the wavelength is much smaller than

the circuit size, resulting in different phases at different

locations in the circuit.

Quasi-static circuit theory cannot be applied. We need to

use transmission line theory.


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A transmission line is a two-port network connecting

a generator circuit at the sending end to a load at the

receiving end.

Unlike in circuit theory, the length of a transmission line

is of utmost importance in transmission line analysis.

z0


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3 AC Steady-State Analysis3.1 Distributed parameter representation


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R = resistance per unit length, (/m)

L = inductance per unit length, (H/m)

G = conductance per unit length, (S/m)

C = capacitance per unit length, (F/m)

z= increment of length, (m)

We use the following distributed parameters to

characterize the circuit properties of a transmission line.

These parameters are related to the physical properties of

the material filling the space between the two wires.

where, , = permittivity, permeability, conductivity

of the surrounding medium.

=''CL

='

'

C

G(See Text Book No.3,

pp. 432-433)


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For the coaxial and two-wire transmission lines, thedistributed parameters are related to the physical

properties and geometrical dimensions as follows:

Surface

resistivity ofthe conductors

(See Text

Book No.3,

pp. 445-447)


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3.2 Equations and solutionsConsider a short section z of a transmission line

(dropping the primes onR,L, G, Chereafter) :

Using KVL and KCL circuit theorems, we canderive the following differential equations for this

section of transmission line.

Generator Load


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( ) ( )( , ), ( , ) , 0

( , )( , ) ( , ) ( , ) 0

i z tv z t R zi z t L z v z z t t

v z z t i z t G zv z z t C z i z z t

t

+ =

+ + + =

By letting z0, these lead to coupled equations:

( , ) ( , )( , )

( , ) ( , )( , )

v z t i z t Ri z t Lz t

i z t v z t Gv z t C

z t

= +

= +

General Transmission Line Equations Coupled!


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For sinusoidal varying voltages and currents, we can usephasor forms.

( ) ( ) tjezVtzv Re, =

( ) ( ) tjezItzi Re, =

V(z) and I(z) are called phasors of v(z,t) and i(z,t). In

terms of phasors, the coupled equations can be written as:

( )( ) ( )

( )( ) ( )

dV zR j L I z

dzdI z

G j C V z dz

= +

= +


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After decoupling,

( )

( )

22

2

22

2

( )

( )

d V zV z

dz

d I zI z

dz

=

=

( )( )j R j L G j C = + = + +

is the complex propagation constant whose real part is

the attenuation constant (Np/m) and whose imaginarypart

is the phase constant (rad/m). Generally, these

quantities are functions of .


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Solutions to transmission line equations:

0 0

0 0

( ) ( ) ( )

( ) ( ) ( )

z z

z

V z V z V z

V e V e

I z I z I z

I e I e

+

+

+

+

= +

= +

= +

= +

Forwardtravelling

wave.

Backwardtravelling

wave.

0 0 0 0, , ,V V I I

+ +

= wave amplitudes in the forward andbackward directions atz= 0. (They

are complex numbers in general.)


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Z0 and are the two most important parameters of

a transmission line. They depend on the

distributed parameters (RLGC) of the line itself

and but not the length of the line.

CjG

LjR

CjG

LjRZ

+

+=

+=

+=0

( )( )CjGLjRj ++=+=


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For lossless transmission lines,R = G = 0.Parameters for lossless transmission lines

==

=

LC

0

11

velocityphase ==== LCup

jkjfjjj =====

=

22

constantnpropagatiocomplex


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LCff

ff

up 121

lineiontransmissthealongwavelength

=====

=

C

L

CjG

LjR

Z

=

++=

=

impedancesticcharacteri0


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Voltage and current along the line:

( )

( ) jkzjkz

jkzjkz

eIeIzI

eVeVzV

+

+

+=

+=

00

00

LjLjk

jk

L

eVV

eVeV

z

z

===

=

==

+

+

0

00

0

0

0

0atvoltageincident

0atvoltagereflected

Define a reflection coefficient atz= 0 as L:


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In terms of the reflection coefficient L, the totalvoltage and current can be written as:

( )

( )jkzLjkz

jkzjkz

jkzjkz

eeV

eV

VeV

eVeVzV

2

0

2

0

00

00

1

1

+=

+=

+=

+

+

+

+

( )

( )jkzLjkz

jkzjkz

jkzjkz

eeI

eV

Ve

Z

V

eZ

Ve

Z

VzI

2

0

2

0

0

0

0

0

0

0

0

1

1

=

=

=

+

+

+

+

In subsequent analyses, we will consider only lossless

transmission lines.


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5 Infinitely Long Transmission LineFor an infinitely long transmission line, there can be no

reflected wave (backward travelling wave). So for an

infinite long transmission line, there is only a forwardtravelling wave.

( )

( )

( )

( ) 000)( Z

zI

zV

zI

zVzZ ===

+

+

0=L

( ) ( )

( ) ( ) jkz

jkz

eIzIzI

eVzVzV

++

++

==

==

0

0


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6 Terminated Transmission Line

Note the two coordinate systems and their relation:z= measuring from the left to the right

= measuring from the right to the left

loadsource

= -z

i

L

(

)Z()

z = -d

= dz = 0

= 0

z


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In thezcoordinate system,

( )

VeVeV jkjk =+ +00

( )zIeIeI jkzjkz =+ + 00

In the (= -z) coordinate system,

( ) IeIeI jkjk =+ + 00

( )zVeVeV jkzjkz =+ + 00

We will use the coordinate system in subsequentanalyses.


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( ) Ljkjk

eV

eV ===+

0

0

000

0

0

0 ZI

V=

+

+

The characteristic impedance in the coordinate system is:

The reflection coefficient at = 0 in the coordinate

system is:

As L is obtained at = 0 (the load position), it iscalled the reflection coefficient at the load.


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Putting the expressions for V0+

and V0-

into the equationsfor the voltage and current, we have:

( ) ( ) ( )[ ]( ) ( )[ ]

kjZkZI

eeZeeZIV

LL

jkjkjkjkLL

sincos

2

1

0

0

+=

++=

( ) ( ) ( )[ ]

( ) ( )[ ]

kjZkZZ

I

eeZeeZZ

II

LL

jkjkjkjk

LL

sincos

2

1

0

0

0

0

+=

++=


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Using V() andI(

), we can obtain the impedanceZ(

) at

an arbitrary point on the transmission line as:

( )( ) ( )( )

kjZZkjZZZ

IVZ

L

L

tantan)(

0

00

++==

The reflection coefficient at the loadL can be expressedas:

( )

( ) 00

0

0

0

0

2

12

1

ZZZZ

ZZI

ZZI

VV

L

L

LL

LLL

+=

+

== +


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In fact, we can further define a reflection coefficient ()at any point on the transmission line by:

( )

kj

L

kj

jk

jk

eeV

V

eV

eV 22

0

0

0

0

pointatvoltageincident

pointatvoltagereflected

+

+

===

=

( ) ( )( )

( ) ( )( )00

00

2

12

1

ZZIeV

ZZIeV

jk

jk

=

+=

+

As we know (by solving the two equations on page 22

with 0):


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( )( ) ( )[ ]

( ) ( )[ ]( )( )

0

0

0

0

2

12

1

ZZZZ

ZZI

ZZI

+=

+

=

Therefore, alternatively we can write,

( ) ( )( )

+=

11

0ZZ

Then,


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At the position of the generator (= d),

( )( )kdjZZkdjZZ

ZdZZL

Li

tan

tan)(

0

00

+

+===

( ) kdjLi

ii e

ZZ

ZZd 2

0

0 =+

===

i

Vg


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Example 1A 100- transmission line is connected to a load consisted

of a 50- resistor in series with a 10-pF capacitor.

(a) Find the reflection coefficient L at the load for a 100-MHz signal.

(b) Find the impedance Zin at the input end of the

transmission line if its length is 0.125.


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(b) d=0.125

( )

( )

( )

( )

0

00

00

0

( 0.125 )

tan 4

tan 4

14 3717 25 5544

29.32 60.65

in

L

L

L

L

Z Z

Z jZ

Z Z jZ

Z jZZ

Z jZ. - j .

= =

+

= +

+=

+

=

=

See animation Transmission Line Impedance Calculation

Normalizedzin = 0.1437-j 0.2555


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6.1 Voltage/current maxima and minima

( )

( )

kj

L

jk

kjjk

jkjk

eeV

eV

VeV

eVeVV

2

0

2

0

00

00

1

1

+

+

+

+

+=

+=

+=

( )

( )

2

0

2

0

0

1

1

1

L

j k

L

j k

L

V V e

V e

V

+

+

+

= +

= +

= +

|L|1

( )2

a complex number

Lj k

Le

=

=

Lj

LL e

=


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=L-2k

1 +

1

0

Complex plane of (1+)

Re

Im

( )2 'L kzLe

=1 L

1 L+

0=m

M

( )V

See animation Transmission Line Voltage Maxima and Minima


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( ) ( )( )

[ )

max

is maximum when 1 1 2 2

, 0,1,2,4 2

Note: has to be specified in the range , .

L

L

L

M

L

VV k n

n

n

+ = +

= =

= + =

( ) ( )( ) ( )

( )

[ )

min

is minimum when 1 1

2 2 1

2 1 , 0,1, 2,4 4

Note: has to be specified in the range , .

L

L

Lm

L

V

V k n

n n

= = = +

+ = + =


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( ) 20

0

0

1

1

j kLI I e

V

Z

+

+

=

=

As current is

Current is maximum when voltage is minimum and

minimum when voltage is maximum.

( ) ( )

=

++= L

LM n

nI with,,2,1,0,

4

12

4at

max

( )

=+= L

Lm n

nI with,,2,1,0,

24at

min


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( )

( )

( )( )

0max

0min

voltage standing wave ratio (VSWR)

1 1

(dimensionless)11

L L

LL

S

V V

V V

+

+

=

+ +

= = =

1

1

+

=

S

S

L

|V(z)|

|V|max

|V|min

|I|max

|I|min

|I(z)|

load load

lmax lmax

Define a voltage standing wave ratio (VSWR) as:


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6.2 Power flow in a transmission line

Power flow at any pointzon a transmission line

is given by:

( ) ( ) ( ){ }zIzVzPav *Re21=

Power delivered by the source:

{ }*Re2

1igs IVP =

Power dissipated in the source impedanceZg:

{ } { } { }giiigZZZ ZIIIZIVP ggg Re21

Re2

1Re

2

1 2** ===


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Power input to the transmission line:

( ) ( ) ( ){ }

{ } { } { }

=

=

===

==

*

2

*

*

2**

*

1Re

2

1Re

2

1

Re

2

1Re

2

1Re

2

1

Re21

i

i

i

ii

iiiiiii

avi

ZV

Z

VV

ZIIIZIV

dIdVdPP

( ) ( ) ( ){ }

{ } { } { }

=

=

===

==

*

2

*

*

2**

*

1Re

2

1Re

2

1

Re2

1

Re2

1

Re2

1

00Re2

10

L

L

L

LL

LLLLLLL

avL

ZV

Z

VV

ZIIIZIV

IVPPPower dissipated in the terminal impedance:


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Solutions

The following information is given:

m310

Hz10MHz100

,5050,50,V60

m,1.5,50

8

8

0

===

==+===

==

ccu

f

jZZV

dZ

p

Lgg

1.110

0

50 50 500.2 0.4 0.45

50 50 50

jLL

L

Z Z j j e

Z Z j

+ = = = + =+ + +

The reflection coefficient at the load is:


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rad11.1,45.0Therefore, ==

LL

load)(from them27.009.04

11.1

0when,24

Then,

===

=+=

n

nLM

( )( )kdjZZkdjZZZZ

L

Li

tantan

0

00

++=

The input impedance Zi looking at the input to thetransmission line is:

/


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A24.048.0505050

60

:islineiontransmisstheinput toat thecurrentThe

jjZZ

VI

ig

g

i =++

=+

=

{ } W2.750288.02

1Re

2

1 2==== iiiL ZIPP

As the transmission line is lossless, power delivered to the

loadPL is equal to the power input to the transmission line

Pi. Hence,

( )+=

++

++

= 5050

5.13

2tan505050

5.13

2

tan50505050 j

jj

jjZi


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giig ZIVV +=

Vi andIi are related to the source voltage Vgas:

From the expressions ofVi, Ii, and Vg, we can find V0+

.

( )( )kdjLgg

jkd

g

eZZ

eZVV

2

0

0

01

+

+=

kdj

L

jkd

i eeVV2

0 1 + += ( )kdjLjkdi ee

Z

VI 2

0

0 1 +

=

At = d, V(d) = Vi andI(d) =Ii.

tcoefficienreflectionsource0

0 =+

=

ZZ

ZZ

g

g

g

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( )

( )( ) ( )

( )( )( )

( )

kj

L

jk

kdj

Lgg

jkd

g

kj

L

jk

kdj

Lgg

jkd

g

eeeZZ

eVI

ee

eZZ

eZVV

2

2

0

2

2

0

0

11

1

1

+

=

+

+

=

Putting V0+ into the expressions of V() andI(), we have:

Now the voltage and current on the transmission line are

expressed in terms of the known parameters of thetransmission line.

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Example 3

A 1.05-GHz generator circuit with a series impedanceZg= 10and voltage source given by:

is connected to a load ZL = (100 + j50) through a 50-, 67-cm-long lossless transmission line. The phase velocity of the line is

0.7c, where c is the velocity of light in a vacuum. Find the

instantaneous voltage and current v(,t) and i(

,t) on the line and

the average power delivered to the load.

( ) ( ) ( )V30sin10g += ttv

Z0 = 50 ZL

dZg

vg Zi

0

IiA

A

Vi

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Solutions

35.32.0

67.0cm67

m2.01005.11037.0 9

8

p

===

===

d

fu

32

50105010

tcoefficienreflectionsource

0

0=+=+

=

ZZZZ

g

g

g

46.0

0

0 45.05050100

5050100

tcoefficienreflectionload

j

L

L

L

ej

j

ZZ

ZZ=

++

+=

+

=


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Hon Tat Hui Transmission Lines Basic Theories50

( ) ( )( ) ( )( )[ ]46.077.2

22

0

45.020.0

11

=

+=

kjjkj

kjL

jkkdj

Lgg

jkd

g

eee

eeeZZ

eVI

Therefore instantaneous forms are:

( ) ( ){ }( )[ ]{ }

( ) ( )23.3cos58.477.2cos18.1045.018.10Re

Re,

46.077.2

++++=+=

=

ktkt

eeee

eVtv

tjkjjkj

tj

( ) ( ){ } ( )[ ]{ }( ) ( )23.3cos09.077.2cos20.0

45.020.0Re

Re,

46.077.2

+++=

=

=

ktkt

eeee

eIti

tjkjjkj

tj

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( )( )

( )

( )

4.179.21

35.32

tan5010050

35.32

tan5050100

50

tan

tan

0

00

j

jj

jj

kdjZZ

kdjZZZZ

L

Li

+=

++

++

=

+

+=

55.13/

28.04.179.2110

10 jj

ig

g

i ej

e

ZZ

V

I

=++=+=

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Power delivered to the load

= power input to the transmission line at AA

{ }

{ }

{ }

{ }

Watt86.0

4.179.21Re28.0

2

1

Re21

Re2

1

Re

2

1

2

2

*

*

=

+=

=

=

=

j

ZI

IZI

IV

ii

iii

ii

NUS/ECE EE2011


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7 Special Cases of Terminations in a Transmission Line

For a matched line,ZL = Z0. Then,

7.1 Matched line

( ) ( )

( )

( ) ( )( )

linetheoflengthanyfor

0

tan

tan

0

0

0

00

000

=+=

=+

+=

ZZZZ

ZkjZZ

kjZZZZ

Thus, there is no reflection on a matched line. There isonly an incident voltage. It is same as the case of an

infinitely long line.

Note =-z

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Z0

Zin

1

-1

1

-1

Normalized current magnitude

1

-1

Normalized impedance (Zin/Z0)

Z0

z

z

z

z0

0

Note:

Normalized voltage = voltage/max. |voltage|

Normalized current = current/max. |current|

Normalized voltage magnitude

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For a short circuit,ZL = 0. Then

7.2 Short-circuited line

( ) ( )kzjZkjZZ tantan 00sc

in ==

Normalized impedance (=-tan(kz))



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7.3 Open-circuited line

For an open circuit,ZL = . Then

( ) ( )kzjZkjZZ cotcot 00oc

in ==


Normalized impedance (=cot(kz))

Note that:

( )[ ] ( )[ ]20

00

oc

in

sc

in

cottan

Z

kjZkjZZZ

=

=

( )[ ] ( )[ ]

( )

k

kjZkjZZZ

2

00

oc

in

sc

in

tan

cottan

=

=

k.andcompute,and,,Given 0oc

in

sc

in ZZZ


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57

7.4 /4 transmission line terminated inZL

7.5 /2 transmission line terminated inZL

( )( ) LL

Lin

Z

Z

jZZ

jZZZZZ

2

0

0

00

2tan

2tan)4( =

+

+===

( )( ) LL

Lin Z

jZZ

jZZZZZ =

+

+===

tan

tan)2(

0

00

Zin

Z0 ZL

Zin

Z0 ZL

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58

Example 4

The open-circuit and short-circuit impedances measured atthe input terminals of a lossless transmission line of length

1.5 m (which is less than a quarter wavelength) are j54.6

andj103 , respectively.

(a) FindZ0 and kof the line.

(b) Without changing the operating frequency, find theinput impedance of a short-circuited line that is twice

the given length.

(c) How long should the short-circuited line be in orderfor it to appear as an open circuit at the input

terminals?

NUS/ECE EE2011

S l ti


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59

Solution

The given quantities are

(a)

(b) For a line twice as long, = 3 m and k =1.884 rad,

m5.1=

mrad628.0tan1 oc

in

sc

in

1

==

ZZk

232tan0sc

in jkjZZ ==

m102

==k

103scin jZ =

6.54ocin jZ =

75scinoc

in0 == ZZZ


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hui transmission lines

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