Prof. David R. Jackson Dept. of ECE Notes 1 ECE 5317-6351 Microwave Engineering Fall 2011 Transmission Line Theory 1

Prof. David R. JacksonDept. of ECE

Notes 1

ECE 5317-6351 Microwave Engineering

Fall 2011

Transmission Line Theory

1

A waveguiding structure is one that carries a signal (or power) from one point to another.

There are three common types: Transmission lines Fiber-optic guides Waveguides

Waveguiding Structures

Note: An alternative to waveguiding structures is wireless transmission using antennas. (antenna are discussed in ECE 5318.)

2

Transmission Line

Has two conductors running parallel Can propagate a signal at any frequency (in theory) Becomes lossy at high frequency Can handle low or moderate amounts of power Does not have signal distortion, unless there is loss May or may not be immune to interference Does not have Ez or Hz components of the fields (TEMz)

Properties

Coaxial cable (coax)Twin lead

(shown connected to a 4:1 impedance-transforming balun)

3

Transmission Line (cont.)

CAT 5 cable(twisted pair)

The two wires of the transmission line are twisted to reduce interference and radiation from discontinuities.

4


Microstrip

h

w

er

er

w

Stripline

h

Transmission lines commonly met on printed-circuit boards

Coplanar strips

her

w w

Coplanar waveguide (CPW)

her

w

5


Transmission lines are commonly met on printed-circuit boards.

A microwave integrated circuit

Microstrip line

6

Fiber-Optic GuideProperties

Uses a dielectric rod Can propagate a signal at any frequency (in theory) Can be made very low loss Has minimal signal distortion Very immune to interference Not suitable for high power Has both Ez and Hz components of the fields

7

Fiber-Optic Guide (cont.)Two types of fiber-optic guides:

1) Single-mode fiber

2) Multi-mode fiber

Carries a single mode, as with the mode on a transmission line or waveguide. Requires the fiber diameter to be small relative to a wavelength.

Has a fiber diameter that is large relative to a wavelength. It operates on the principle of total internal reflection (critical angle effect).

8

Fiber-Optic Guide (cont.)

http://en.wikipedia.org/wiki/Optical_fiber

Higher index core region

9

http://upload.wikimedia.org/wikipedia/commons/4/46/Optical-fibre.svg




Waveguides

Has a single hollow metal pipe Can propagate a signal only at high frequency: > c

The width must be at least one-half of a wavelength

Has signal distortion, even in the lossless case Immune to interference Can handle large amounts of power Has low loss (compared with a transmission line) Has either Ez or Hz component of the fields (TMz or TEz)

Properties

http://en.wikipedia.org/wiki/Waveguide_(electromagnetism) 10

Lumped circuits: resistors, capacitors, inductors

neglect time delays (phase)

account for propagation and time delays (phase change)

Transmission-Line Theory

Distributed circuit elements: transmission lines

We need transmission-line theory whenever the length of a line is significant compared with a wavelength.

11

Transmission Line

2 conductors

4 per-unit-length parameters:

C = capacitance/length [F/m]

L = inductance/length [H/m]

R = resistance/length [/m]

G = conductance/length [ /m or S/m]

Dz

12


z

,i z t

+ + + + + + +- - - - - - - - - - ,v z tx x xB

13

RDz LDz

GDz CDz

z

v(z+z,t)

+

-

v(z,t)

+

-

i(z,t) i(z+z,t)

( , )( , ) ( , ) ( , )

( , )( , ) ( , ) ( , )

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

tv z z t

i z t i z z t v z z t G z C zt


14

RDz LDz

GDz CDz

z

v(z+z,t)

+

-

v(z,t)

+

-

i(z,t) i(z+z,t)

Hence

( , ) ( , ) ( , )( , )

( , ) ( , ) ( , )( , )

v z z t v z t i z tRi z t L

z ti z z t i z t v z z t

Gv z z t Cz t

Now let Dz 0:

v iRi L

z ti v

Gv Cz t

“Telegrapher’sEquations”

TEM Transmission Line (cont.)

15

To combine these, take the derivative of the first one with

respect to z:

2

2

2

2

v i iR L

z z z t

i iR L

z t z

vR Gv C

t

v vL G C

t t

Switch the order of the derivatives.


16

2 2

2 2( ) 0

v v vRG v RC LG LC

z t t

The same equation also holds for i.

Hence, we have:

2 2

2 2

v v v vR Gv C L G C

z t t t


17

2

2

2( ) ( ) 0

d VRG V RC LG j V LC V

dz

2 2

2 2( ) 0

v v vRG v RC LG LC

z t t


Time-Harmonic Waves:

18

Note that

= series impedance/length

2

2

2( )

d VRG V j RC LG V LC V

dz

2( ) ( )( )RG j RC LG LC R j L G j C

Z R j L

Y G j C

= parallel admittance/length

Then we can write:2

2( )

d VZY V

dz


19

Let

Convention:

Solution:

2 ZY

( ) z zV z Ae Be

1/2

( )( )R j L G j C

principal square root

2

2

2( )

d VV

dzThen


is called the "propagation constant."

/2jz z e

j

0, 0

attenuationcontant

phaseconstant

20


0 0( ) z z j zV z V e V e e

Forward travelling wave (a wave traveling in the positive z direction):

0

0

0

( , ) Re

Re

cos

z j z j t

j z j z j t

z

v z t V e e e

V e e e e

V e t z

g0t

z0

zV e

2

g

2g

The wave “repeats” when:

Hence:

21

Phase Velocity

Track the velocity of a fixed point on the wave (a point of constant phase), e.g., the crest.

0( , ) cos( )zv z t V e t z

z

vp (phase velocity)

22

Phase Velocity (cont.)

0

constant

t z

dz

dtdz

dt

Set

Hence pv

1/2

Im ( )( )p

vR j L G j C

In expanded form:

23

Characteristic Impedance Z0

0

( )

( )

V zZ

I z

0

0

( )

( )

z

z

V z V e

I z I e

so 00

0

VZ

I

+ V+(z)-

I+ (z)

z

A wave is traveling in the positive z direction.

(Z0 is a number, not a function of z.)

24

Use Telegrapher’s Equation:

v iRi L

z t

sodV

RI j LIdz

ZI

Hence0 0

z zV e ZI e

Characteristic Impedance Z0 (cont.)

25

From this we have:

Using

We have

1/2

00

0

V Z ZZ

I Y

Y G j C

1/2

0

R j LZ

G j C

Characteristic Impedance Z0 (cont.)

Z R j L

Note: The principal branch of the square root is chosen, so that Re (Z0) > 0. 26

00

0 0

j z j j z

z z

z j zV e e

V z V e V

V e e e

e

e

0

0 cos

c

, R

os

e j t

z

z

V e t

v z t V z

z

V z

e

e t

Note:

wave in +z direction wave in -z

direction

General Case (Waves in Both Directions)

27

Backward-Traveling Wave

0

( )

( )

V zZ

I z

0

( )

( )

V zZ

I z

so

+ V -(z)-

I - (z)

z

A wave is traveling in the negative z direction.

Note: The reference directions for voltage and current are the same as for the forward wave.

28

General Case

0 0

0 00

( )

1( )

z z

z z

V z V e V e

I z V e V eZ

A general superposition of forward and backward traveling waves:

Most general case:

Note: The reference directions for voltage and current are the same for forward and backward waves.

29

+ V (z)-

I (z)

z

1

2

12

0

0 0

0 0

0 0

z z

z z

V z V e V e

V VI z e e

Z

j R j L G j C

R j LZ

G j

Z

C

I(z)

V(z)+- z

2mg

[m/s]pv

guided wavelength g

phase velocity vp

Summary of Basic TL formulas

30

Lossless Case

0, 0R G

1/ 2

( )( )j R j L G j C

j LC

so 0

LC

1/2

0

R j LZ

G j C

0

LZ

C

1pv

LC

pv

(indep. of freq.)(real and indep. of freq.)31

Lossless Case (cont.)1

pvLC

In the medium between the two conductors is homogeneous (uniform) and is characterized by (, ), then we have that

LC

The speed of light in a dielectric medium is1

dc

Hence, we have that p dv c

The phase velocity does not depend on the frequency, and it is always the speed of light (in the material).

(proof given later)

32

0 0z zV z V e V e

Where do we assign z = 0?

The usual choice is at the load.

I(z)

V(z)+-

zZL

z = 0

Terminating impedance (load)

Ampl. of voltage wave propagating in negative z direction at z = 0.

Ampl. of voltage wave propagating in positive z direction at z = 0.

Terminated Transmission Line

Note: The length l measures distance from the load: z33

What if we know

@V V z and

0 0V V V e

z zV z V e V e

0V V e

0 0V V V e

Terminated Transmission Line (cont.)

0 0z zV z V e V e

Hence

Can we use z = - l as a reference plane?

I(z)

V(z)+-

zZL

z = 0


34

( ) ( )z zV z V e V e


0 0z zV z V e V e

Compare:

Note: This is simply a change of reference plane, from z = 0 to z = -l.

I(z)

V(z)+-

zZL

z = 0


35

0 0z zV z V e V e

What is V(-l )?

0 0V V e V e

0 0

0 0

V VI e e

Z Z

propagating forwards

propagating backwards


l distance away from load

The current at z = - l is then

I(z)

V(z)+-

zZL

z = 0


36

20

0

1 L

VI e e

Z

200

00 0 1

VV eV V e ee

VV

Total volt. at distance l from the load

Ampl. of volt. wave prop. towards load, at the load position (z = 0).

Similarly,

Ampl. of volt. wave prop. away from load, at the load position (z = 0).

0

21 LV e e

L Load reflection coefficient

Terminated Transmission Line (cont.)I(-l )

V(-l )+

l

ZL-

0,Z

l Reflection coefficient at z = - l

37

20

2

2

0

0

2

0

1

1

1

1

L

L

L

L

V V e e

VI e e

Z

V eZ Z

I e

Input impedance seen “looking” towards load at z = -l .


I(-l )

V(-l )+

l

ZL-

0,Z

Z

38

At the load (l = 0):

0

10

1L

LL

Z Z Z

Thus,

20

00

20

0

1

1

L

L

L

L

Z Ze

Z ZZ Z

Z Ze

Z Z


0

0

LL

L

Z Z

Z Z

2

0 2

1

1L

L

eZ Z

e

Recall

39

Simplifying, we have

00

0

tanh

tanhL

L

Z ZZ Z

Z Z


202

0 0 00 0 2

2 0 00

0

0 00

0 0

00

0

1

1

cosh sinh

cosh sinh

L

L L L

L LL

L

L L

L L

L

L

Z Ze

Z Z Z Z Z Z eZ Z Z

Z Z Z Z eZ Ze

Z Z

Z Z e Z Z eZ

Z Z e Z Z e

Z ZZ

Z Z

Hence, we have

40

20

20

0

2

0 2

1

1

1

1

j jL

j jL

jL

jL

V V e e

VI e e

Z

eZ Z

e

Impedance is periodic with period g/2

2

/ 2

g

g

Terminated Lossless Transmission Line

j j

Note: tanh tanh tanj j

tan repeats when

00

0

tan

tanL

L

Z jZZ Z

Z jZ

41

For the remainder of our transmission line discussion we will assume that the transmission line is lossless.

20

20

0

2

0 2

00

0

1

1

1

1

tan

tan

j jL

j jL

jL

jL

L

L

V V e e

VI e e

Z

V eZ Z

I e

Z jZZ

Z jZ

0

0

2

LL

L

g

p

Z Z

Z Z

v

Terminated Lossless Transmission Line

I(-l )

V(-l )+

l

ZL-

0 ,Z

Z

42

Matched load: (ZL=Z0)

0

0

0LL

L

Z Z

Z Z

For any l

No reflection from the load

A

Matched LoadI(-l )

V(-l )+

l

ZL-

0 ,Z

Z

0Z Z

0

0

0

j

j

V V e

VI e

Z

43

Short circuit load: (ZL = 0)

0

0

0

01

0

tan

L

Z

Z

Z jZ

Always imaginary!Note:

B

2g

scZ jX

S.C. can become an O.C. with a g/4 trans. line

0 1/4 1/2 3/4g/

XSC

inductive

capacitive

Short-Circuit Load

l

0 ,Z

0 tanscX Z

44

Using Transmission Lines to Synthesize Loads

A microwave filter constructed from microstrip.

This is very useful is microwave engineering.

45

00

0

tan

tanL

inL

Z jZ dZ Z d Z

Z jZ d

inTH

in TH

ZV d V

Z Z

I(-l)

V(-l)+

l

ZL

-0Z

ZTH

VTH

d

Zin

+

-

ZTH

VTH

+ZinV(-d)

+

-

Example

Find the voltage at any point on the line.

46

Note: 021 j

LjV V e e

0

0

LL

L

Z Z

Z Z

20 1j d j d in

THin TH

LV dZ

Ze V

ZV e

2

2

1

1

jj din L

TH j dm TH L

Z eV V e

Z Z e

At l = d :

Hence

Example (cont.)

0 2

1

1j din

TH j din TH L

ZV V e

Z Z e

47

Some algebra: 2

0 2

1

1

j dL

in j dL

eZ Z d Z

e

2

20 20

2 220

0 2

20

20 0

2

0

20 0

0

111

1 11

1

1

1

1

j dL

j dj dLL

j d j dj dL TH LL

THj dL

j dL

j dTH L TH

j dL

j dTH THL

TH

in

in TH

eZ

Z ee

Z e Z eeZ Z

e

Z e

Z Z e Z Z

eZ

Z

Z

Z Z

Z Z Ze

Z Z

Z

2

0

20 0

0

1

1

j dL

j dTH THL

TH

e

Z Z Z Ze

Z Z

Example (cont.)

48

2

02

0

1

1

jj d L

TH j dTH S L

Z eV V e

Z Z e

20

20

1

1

j din L

j din TH TH S L

Z Z e

Z Z Z Z e

where 0

0

THS

TH

Z Z

Z Z

Example (cont.)

Therefore, we have the following alternative form for the result:

Hence, we have

49

2

02

0

1

1

jj d L

TH j dTH S L

Z eV V e

Z Z e

Example (cont.)

I(-l)

V(-l)+

l

ZL

-0Z

ZTH

VTH

d

Zin

+

-

Voltage wave that would exist if there were no reflections from the load (a semi-infinite transmission line or a matched load).

50

2 2

2 2 2 20

0

1 j d j dL L S

j d j d j d j dTH L S L L S L S

TH

e eZ

V d V e e e eZ Z

Example (cont.)

ZL0Z

ZTH

VTH

d

+

-

Wave-bounce method (illustrated for l = d ):

51

Example (cont.)

22 2

22 2 20

0

1

1

j d j dL S L S

j d j d j dTH L L S L S

TH

e e

ZV d V e e e

Z Z

Geometric series:

2

0

11 , 1

1n

n

z z z zz

2 2

2 2 2 20

0

1 j d j dL L S

j d j d j d j dTH L S L L S L S

TH

e eZ

V d V e e e eZ Z

2j dL Sz e

52

Example (cont.)

or

2

0

202

1

1

1

1

j dL s

THj dTH

L j dL s

eZV d V

Z Ze

e

2

02

0

1

1

j dL

TH j dTH L s

Z eV d V

Z Z e

This agrees with the previous result (setting l = d ).

Note: This is a very tedious method – not recommended.

Hence

53

I(-l)

V(-l)+

l

ZL-

0 ,Z

At a distance l from the load:

*

*

2

0 2 2 * 2*0

1Re 1 1

1R

2

e2

L L

Ve e

Z

V I

e

P

2

20 2 4

0

11

2 L

VP e e

Z

If Z0 real (low-loss transmission line)

Time- Average Power Flow

20

20

0

1

1

L

L

V V e e

VI e e

Z

j

*2 * 2

*2 2

L L

L L

e e

e e

pure imaginary

Note:

54

Low-loss line

2

20 2 4

0

2 2

20 02 2* *0 0

11

2

1 1

2 2

L

L

VP d e e

Z

V Ve e

Z Z

power in forward wave power in backward wave

2

20

0

11

2 L

VP d

Z

Lossless line ( = 0)

Time- Average Power Flow

I(-l)

V(-l)+

l

ZL-

0 ,Z

55

00

0

tan

tanL T

in TT L

Z jZZ Z

Z jZ

2

4 4 2g g

g

00

Tin T

L

jZZ Z

jZ

0

20

0

0in in

T

L

Z Z

ZZ

Z

Quarter-Wave Transformer

20T

inL

ZZ

Z

so

1/2

0 0T LZ Z Z

Hence

This requires ZL to be real.

ZLZ0 Z0T

Zin

56

20 1 Lj j

LV V e e

20

20

1

1 L

j jL

jj jL

V V e e

V e e e

max 0

min 0

1

1

L

L

V V

V V

max

min

V

VVoltage Standing Wave Ratio VSWR

Voltage Standing Wave Ratio

I(-l )

V(-l )+

l

ZL-

0 ,Z

1

1L

L

VSWR

z

1+ L

1

1- L

0

( )V z

V

/ 2z 0z

57

Coaxial Cable

Here we present a “case study” of one particular transmission line, the coaxial cable.

a

b ,r

Find C, L, G, R

We will assume no variation in the z direction, and take a length of one meter in the z direction in order top calculate the per-unit-length parameters.

58

For a TEMz mode, the shape of the fields is independent of frequency, and hence we can perform the calculation using electrostatics and magnetostatics.

Coaxial Cable (cont.)

-l0

l0

a

b

r0 0

0

ˆ ˆ2 2 r

E

Find C (capacitance / length)

Coaxial cable

h = 1 [m]

r

From Gauss’s law:

0

0

ln2

B

AB

A

b

ra

V V E dr

bE d

a

59

-l0

l0

a

b

r

Coaxial cable

h = 1 [m]

r

0

0

0

1

ln2 r

QC

V ba

Hence

We then have

0 F/m2

[ ]ln

rCba


60

ˆ2

IH

Find L (inductance / length)

From Ampere’s law:

Coaxial cable

h = 1 [m]

r

I

0ˆ

2 r

IB

(1)b

a

B d S

h

I

I z

center conductorMagnetic flux:


61

Note: We ignore “internal inductance” here, and only look at the magnetic field between the two conductors (accurate for high frequency.

Coaxial cable

h = 1 [m]

r

I

0

0

0

1

2

ln2

b

r

a

b

r

a

r

H d

Id

I b

a

0

1ln

2r

bL

I a

0 H/mln [ ]2

r bL

a

Hence


62

0 H/mln [ ]2

r bL

a

Observation:

0 F/m2

[ ]ln

rCba

0 0 r rLC

This result actually holds for any transmission line.


63

0 H/mln [ ]2

r bL

a

For a lossless cable:

0 F/m2

[ ]ln

rCba

0

LZ

C

0 0

1ln [ ]

2r

r

bZ

a

00

0

376.7303 [ ]


64

-l0

l0

a

b

0 0

0

ˆ ˆ2 2 r

E

Find G (conductance / length)

Coaxial cable

h = 1 [m]

From Gauss’s law:

0

0

ln2

B

AB

A

b

ra

V V E dr

bE d

a


65

-l0

l0

a

b

J E

We then have leakIG

V

0

0

(1) 2

2

22

leak a

a

r

I J a

a E

aa

0

0

0

0

22

ln2

r

r

aa

Gba

2[S/m]

lnG

ba

or


66

Observation:

F/m2

[ ]ln

Cba

G C

This result actually holds for any transmission line.

2[S/m]

lnG

ba

0 r


67

G C

To be more general:

tanG

C

tanG

C

Note: It is the loss tangent that is usually (approximately) constant for a material, over a wide range of frequencies.


As just derived,

The loss tangent actually arises from both conductivity loss and polarization loss (molecular friction loss), ingeneral.

68

This is the loss tangent that would arise from conductivity effects.

General expression for loss tangent:

c

c c

j

j j

j

tan c

c

Effective permittivity that accounts for conductivity

Loss due to molecular friction Loss due to conductivity


69

Find R (resistance / length)

Coaxial cable

h = 1 [m]


,b rb

a

b

,a ra

a bR R R

1

2a saR Ra

1

2b sbR Rb

1sa

a a

R

1

sbb b

R

0

2a

ra a

0

2b

rb b

Rs = surface resistance of metal

70

General Transmission Line Formulas

tanG

C

0 0 r rLC

0losslessL

ZC

characteristic impedance of line (neglecting loss)(1)

(2)

(3)

Equations (1) and (2) can be used to find L and C if we know the material properties and the characteristic impedance of the lossless line.

Equation (3) can be used to find G if we know the material loss tangent.

a bR R R

tanG

C

(4)

Equation (4) can be used to find R (discussed later).

,iC i a b contour of conductor,

2

2

1( )

i

i s sz

C

R R J l dlI

71

General Transmission Line Formulas (cont.)

tanG C

0losslessL Z

0/ losslessC Z

R R

Al four per-unit-length parameters can be found from 0 ,losslessZ R

72

Common Transmission Lines

0 0

1ln [ ]

2lossless r

r

bZ

a

Coax

Twin-lead

100 cosh [ ]

2lossless r

r

hZ

a

2

1 2

12

s

ha

R Ra h

a

1 1

2 2sa sbR R Ra b

a

b

,r r

h

,r r

a a

73

Common Transmission Lines (cont.)

Microstrip

0 0

1 00

0 1

eff effr reff effr r

fZ f Z

f

0

1200

0 / 1.393 0.667 ln / 1.444effr

Zw h w h

( / 1)w h

21 ln

t hw w

t

h

w

er

t

74

Common Transmission Lines (cont.)

Microstrip ( / 1)w h

h

w

er

t

2

1.5

(0)(0)

1 4

effr reff eff

r rfF

1 1 11 /0

2 2 4.6 /1 12 /

eff r r rr

t h

w hh w

2

0

4 1 0.5 1 0.868ln 1r

h wF

h

75

At high frequency, discontinuity effects can become important.

Limitations of Transmission-Line Theory

Bend

incident

reflected

transmitted

The simple TL model does not account for the bend.

ZTH

ZLZ0

+-

76

At high frequency, radiation effects can become important.

When will radiation occur?

We want energy to travel from the generator to the load, without radiating.

Limitations of Transmission-Line Theory (cont.)

ZTH

ZLZ0

+-

77

r a

bz

The coaxial cable is a perfectly shielded system – there is never any radiation at any frequency, or under any circumstances.

The fields are confined to the region between the two conductors.


78

The twin lead is an open type of transmission line – the fields extend out to infinity.

The extended fields may cause interference with nearby objects. (This may be improved by using “twisted pair.”)

+ -


Having fields that extend to infinity is not the same thing as having radiation, however.

79

The infinite twin lead will not radiate by itself, regardless of how far apart the lines are.

h

incident

reflected

The incident and reflected waves represent an exact solution to Maxwell’s equations on the infinite line, at any frequency.

*1ˆRe E H 0

2t

S

P dS

S

+ -


No attenuation on an infinite lossless line

80

A discontinuity on the twin lead will cause radiation to occur.

Note: Radiation effects increase as the frequency increases.


h

Incident wavepipe

Obstacle

Reflected wave

Bend h

Incident wave

bend

Reflected wave81

To reduce radiation effects of the twin lead at discontinuities:

h

1) Reduce the separation distance h (keep h << ).2) Twist the lines (twisted pair).


CAT 5 cable(twisted pair)

82

Documents

Prof. David R. Jackson Dept. of ECE Notes 1 ECE 5317-6351 Microwave Engineering Fall 2011 Transmission Line Theory 1