KAVOSHCOM RF Communication Circuits Lecture 1: Transmission Lines.

KAVOSHCOM

RF Communication Circuits

Dr. Fotowat-AhmadySharif University of TechnologyFall-1391Prepared by: Siavash Kananian & Alireza Zabetian

Lecture 1: Transmission Lines

A wave guiding 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

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

Transmission Line (cont.)

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 Line (cont.)

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

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

Transmission Line (cont.)

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

Transmission Line (cont.)

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.

TEM Transmission Line (cont.)

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

TEM Transmission Line (cont.)

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

TEM Transmission Line (cont.)

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

TEM Transmission Line (cont.)

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

TEM Transmission Line (cont.)

is called the "propagation constant."

/2jz z e

j

0, 0

attenuationcontant

phaseconstant

20

TEM Transmission Line (cont.)

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 VelocityTrack 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 Case0, 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

Terminating impedance (load)

34

( ) ( )z zV z V e V e

Terminated Transmission Line (cont.)

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

Terminating impedance (load)

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

Terminated Transmission Line (cont.)

l distance away from load

The current at z = - l is then

I(z)

V(z)+-

zZL

z = 0

Terminating impedance (load)

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 .

Terminated Transmission Line (cont.)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

Terminated Transmission Line (cont.)

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

Terminated Transmission Line (cont.)

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

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

54

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 RatioI(-l )

V(-l )+

l

ZL-

0 ,Z

1

1L

L

VSWR

z

1+ L

1

1- L

0

( )V z

V

/ 2z 0z

55