Ch. 2. Transmission Line Analysis - Universitetet i oslo · IFI5481: RF kretser, teori og design Institutt for Informatikk Svein-Erik Hamran Ch. 2. Transmission Line Analysis Phase

Institutt for InformatikkIFI5481: RF kretser, teori og design

Svein-Erik Hamran

Ch. 2. Transmission Line Analysis

Phase velocity

rp cv 1

Traveling voltage wave

0, sinxEV z t t z

• Voltage has a time and space variation

• Space is neglected for low frequency applications

• For RF there can be a large spatial variation

Institutt for Informatikk

Consequences of spatial voltage variations

• For low frequency (1MHz) Kirchhoff’s laws apply

• For high frequency (1GHz) Kirchhoff’s laws do not applyanymore

• Solution: Consider elements of infinitesimal length

IFI5481: RF kretser, teori og design

Svein-Erik Hamran


Svein-Erik Hamran

Kirchhoff’s laws on a microscopic level

• Over a differential section we can again use basic circuit theory

• Model takes into account line losses and dielectric losses

• Ideal line (lossless) involves only L and CDistributed parameters R, G, L and C


Svein-Erik Hamran

Two-wire transmission line

• Alternating electric field between conductors

• Alternating magnetic field surrounding conductors

• dielectric medium tends to confine field inside material


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Coaxial line

• Electric field contained between conductors

• Perfect shielding of magnetic field

• TEM mode up to a certain cutoff frequency

Always used for externally

connected RF systems or

measuring equipment.

Also LAN.


Svein-Erik Hamran

Microstrip lines

Low dielectric medium High dielectric medium

Printed circuit board

(PCB) section with

ground plane to

prevent excessive field

leakage, interference,

and radiation loss


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Other TEM configurations

Triple-layer lineReduced radiation losses

Parallel plate lineLow impedance, high power


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Transmission line representation

• Detailed analysis is based on differential section

• Analysis applies to many types of transmission lines

such as coax cables, two-wire, microstrip, etc.


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Pros and cons of electric circuit

representation

• Clear intuitive physical

picture

• Yields a standardized

two-port network

representation

• Serves as building

blocks to go from

microscopic to

macroscopic forms

• Basically a one-

dimensional representation

(cannot take into account

interferences)

• Material nonlinearities,

hysteresis, and

temperature effects are not

accounted for


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Basic electromagnetism

• Ampère’s law

JH

SJlH

dd

• Faraday’s law

dt

d

dt

d

BE

SBlE

dd

r

IH

222 a

IrH

tHB cos00

tHadt

dV sind 00

2 SB


Svein-Erik Hamran

Line parameters for specific cases

Generic electric

equivalent circuit

representation

Check out

example 2.3

in text book

to get feeling

of numbers!


Svein-Erik Hamran

General transmission line equation

KVL:

KCL:

Coupled first-order differntial equations

zILjR

dz

zdV

zzVzzILjRzV

z

0

zVCjG

dz

zdI

zzIzCjGzzVzI

z

0


Svein-Erik Hamran

Traveling voltage and current waves

2

2

20

d V zV z

dz

2

2

20

d I zI z

dz

Complex propagation constant:

j R j L G j C

Solutions:

z zV z V e V e z zI z I e I e


Svein-Erik Hamran

General line impedance definition

Characteristic line impedance:

z z

dV zR j L I z I z V e V e

dz R j L

0

R j L R j L V VZ

G j C I I

0

1 z zI z V e V eZ

Note: Z0 is not a conventional impedance, but is a

characteristic of the positive and negative traveling waves


Svein-Erik Hamran

Lossless transmission line

Lossless implies:

R = 0 and G = 0

Consider lossless parallell

plate transmission line with:

d

wC

w

dL and

w

d

w

d

C

LZ

r

3770

Characteristic impedance:

37700 fZ is the

wave impedance of free space


Svein-Erik Hamran

Microstrip transmission line

hw

ff

f

hwff

f

h

w

h

w

Zor

h

w

w

hZZ

444.1ln3

2393.1

4

8ln

20

hw

rr

hw

rreff

w

hor

h

w

w

h

2

12

2

1

121

2

1

2

1104.0

121

2

1

2

1


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Teminated lines - Voltage reflection coefficient

Open line: ZL , Γ0 = 1 Wave fully reflected with same polarity as incident wave

Short circuit: ZL = 0, Γ0 = -1 Wave fully reflected with opposite polarity of incident wave

Load match: ZL = Z0, Γ0 = 0 No reflection when load matches line impedance

0

00

ZZ

ZZ

V

V

L

L

Load impedance:

0

z z

z z

V z V e V e

I z V e V e Z

Reflection coefficient:

0

000

1

1

0

00

ZVV

VVZ

I

VZZ L


Svein-Erik Hamran

Lossless transmission line

j j LC For lossless line (R = G = 0):

Voltage and current waves:

Phase velocity:

0 LC

zjzj eeZ

VzI

0

0

zjzj eeVzV 0

LCfvp

1


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Standing waves

Short circuit: ZL = 0, Γ0 = -1

Wave fully reflected with opposite

polarity of incident wave

2cossin2Re),(

sin2

tdVVetdv

djVeeVdV

tj

djdj

Standing wave pattern:

d = λ

Note: d = -z


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Standing wave ratio (SWR)

Generally:

dZ

dAdI

ddAeeVdV djdj

1

11

0

20

djed 20

Reflection coefficient:

SWR is a measure of mismatch of the load

to the line

SWR=1 (matched)

SWR (total mismatch)

0

0

min

max

min

max

1

1

I

I

V

VSWR

match

Note: SRW applies to lossless lines,

but also works well in low-loss cases


Svein-Erik Hamran

Transformation of load impedance

• Terminated lossless transmission line ( )

- Input impedance:

- Used:

CLZ 0

lj

L

lj

L

lj

L

lj

L

ljlj

ljlj

ineZZeZZ

eZZeZZZ

ee

eeZ

lI

lVldZ

00

000

0

00

0

00

ZZ

ZZ

V

V

L

L

djZZ

djZZZdZ

L

Lin

tan

tan

0

00


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Short circuit transmission line

djZdZin tan0

dZ

VdI

djVdV

cos2

sin2

0


Svein-Erik Hamran

Open circuit transmission line

djZdZin cot0

dZ

jVdI

dVdV

sin2

cos2

0


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Quarter-wave transmission line

0 /4 L inZ Z Z

For d = λ/4 we have:

20 /4

0 /40 /4

0 /4

2tan

4

24tan

4

L

in

LL

Z jZZ

Z d ZZ

Z jZ

Lamda-quarter transformer

matches given input and output

impedances by choosing a line

with characteristic impedance

(narrowband matching, Zin =Z0):

0 /4 0line LZ Z Z Z

500MHz 1.5GHz


Svein-Erik Hamran

Transmission line with source and load

Also have to consider the

impedance matching at the

source!

Gin

inGininininin

ZZ

ZVVVVV

1

lj

in

inin e

ZZ

ZZld 2

0

0

0

Reflection coefficients:

Input voltage (d = l):

ljSout

G

GS e

ZZ

ZZ 2

0

0

Transmission coefficients:

0

21

ZZ

ZT

in

ininin

0

00

21

ZZ

ZT

L

L


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Input Power

Gin

in

in

G

in

inin

ZZ

ZVVV

11

2

0

2

* 12

1Re

2

1in

in

inininZ

VIVP

in

inin ZZ

1

10

S

SG ZZ

1

10

2

2

2

0

2

11

1

8

1in

inS

SGin

Z

VP

Lossless TL:

2 2

2

022

00

111

8 1

G S

inj l

S

VP

Z e


Svein-Erik Hamran

Lossless TL - special cases

Load and source matched to line:

0

2

8

1

Z

VP

Gin 00 S

Maximum available power

provided by the source

Match at load and mismatch at source:

2

0

2

18

1s

Gin

Z

VP 00

Power usually measured in dBm:

mW

WPdBmP

1log10


Svein-Erik Hamran

Return and insertion losses

inin

i

r

P

PRL

log20log10log10

2

21log10log10log10 in

i

ri

i

t

P

PP

P

PIL

IL dB0 dB

Documents

Ch. 2. Transmission Line Analysis - Universitetet i oslo · IFI5481: RF kretser, teori og design Institutt for Informatikk Svein-Erik Hamran Ch. 2. Transmission Line Analysis Phase