RF Circuit Design - [Ch1-2] Transmission Line Theory

Chapter 1-2

Transmission Line Theory

Chien-Jung Li

Department of Electronics Engineering

National Taipei University of Technology

Department of Electronic Engineering, NTUT

Common Types of Transmission Lines

Two-wire line

Coaxial

Microstrip

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Transmission Line Theory

• At high frequencies, especially when the wavelength is not longer than the dimension of circuitry, conventional circuit theory no longer holds.

• Circuitry perspective on electromagnetic waves.

• Transmission-line effects: Standing waves generated (related to position)

Load impedance changes

Departs from max power transmission

• Transmission-line effect is obvious as frequency or line length increases.

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Theory

Conventional

Circuit Theory

Microwave

Engineering Optics

4/40


Electrical Model for Transmission Line

Source

Source

impedance

Load

impedance

• In conventional circuit theory, you can easily find the voltage

appears at load by: L

s

s L

Zv v

Z Z

Transmission line

sv

sZLZ

l

• If the transmission line is not just an ideal interconnection?

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Distributed Circuit Model

• Assume that the transmission line is uniform and the length l is divided into many identical

sections Δx.

• R: Ω/m, L: H/m, C: F/m, G: S/m

G x

L xR x

C xsZ

svLZ

l

dx dx dx

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A Section of the Transmission Line

• The voltages and currents along the trans-mission line are functions of position and time.

• Input: ,

• Output: ,

• R: finite conductivity G: dielectric loss

,v x t

,i x tR x L x

G x C x ,v x x t

,i x x t

,v x t ,i x t

,v x x t ,i x x t

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Transmission Line Equations (I)

• Apply Kirchhoff’s voltage law (KVL)

• Apply Kirchhoff’s current law (KCL)

,, , ,

i x tv x t v x x t R x i x t L x

t

,, , ,

v x x ti x t i x x t G x v x x t C x

t

,v x t

,i x tR x L x

G x C x ,v x x t

,i x x t

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Transmission Line Equations (II)

, ,,

v x t i x tRi x t L

x t

, ,,

i x t v x tGv x t C

x t

, , ,,

v x t v x x t i x tRi x t L

x t

, , ,,

i x t i x x t v x x tGv x x t C

x t

• Rearrange Δx then we have:

• Assume that Δx is very small

The partial differential equations describe the voltages and currents along

the transmissions, called transmission line, or telegrapher equation.

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Lossless Transmission Line

, ,,

v x t i x tRi x t L

x t

, ,,

i x t v x tGv x t C

x t

, ,v x t i x tL

x t

, ,i x t v x tC

x t

,v x t

,i x t

,v x x t

,i x x tL x

C x

• Of particular interest in microwave electronics is the lossless

transmission line, i.e., R=G=0.

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Sinusoidal Steady-state Analysis

f(x) and g(x) are the real functions of position, and , describe

the positional dependence of the phase.

, cos Re Rej t x j x j tv x t f x t x f x e f x e e

, cos Re Rej t x j x j ti x t g x t x g x e g x e e

j xI x g x e

j xV x f x e , Re j tv x t V x e

, Re j ti x t I x e

• Time-domain representation

x x

• Phasor-domain representation

time-domain

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Solve for the Voltage

V x I xL j LI x

x t

I x V xC j CV x

x t

dV xj LI x

dx

dI xj CV x

dx

V x dV x

x dx

2

2

2

d V x dI xj L LCV x

dx dx

• Laplace equation

2 2

2 2

2 2

d V x d V xLCV x V x

dx dx

LC• Propagation constant

• General solution

j x j xV x Ae Be ,where A, B are complex constant

• Lossless transmission line equations

(Think that if approaches zero?)

( is also called the “phase constant” that represents the phase change per

meter for the wave traveling along the path)

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Solve for the Current

1 1 j x j xdv x

I x A j e B j ej L dt j L

• Solve for the current

LC

• Define

0

L L LZ

CLC

j x j xA e B eL L

j x j xV x Ae Be

0 0

j x j xA BI x e e

Z Z

dV xj LI x

dxWith and

,where

Z0 is the characteristics impedance of the transmission line, for the lossless

condition Z0 is real.

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Time-domain Results

, Re Re

j x t j x tj tv x t V x e Ae Be

0 0

, Re Rej x t j x tj t A B

i x t I x e e eZ Z

cos cosA x t B x t

0 0

cos cosA B

x t x tZ Z

j x j xV x Ae Be

0 0

j x j xA BI x e e

Z Z

• Time-domain results can be easily drawn from the phasor.

Voltage Wave

Current Wave

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Wave Wavelength

j xV x Ae

• Consider

Take a look on the term , the phase means how many radians

change for the wave to travel through the distance of x. If distance x is

equal to a wavelength long, i.e., :

j xe x

2

xx

2

0x x

0t t T

distance

time

phase

0xx

2

xx

1 , Re cosj x j tv x t Ae e A t x

x

For simplification, assume the

wave starts from x=0 and t=0.

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Wave Velocity

In the vacuum, 7

0 4 10 Wb/A-mL

0 8.85419 F/mC

8

,

0 0

1 light speed 3 10 /p vacuumv c m s

0

0

0

377 L

ZC

,

0

p vaccumv

fis the wavelength in vacuum

2 1

2 2pv f

T LC

• Wave velocity can be found: ( the wave goes one wavelength long in a period of T)

is the intrinsic/characteristic

impedance of vacuum (or free-space)

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Wave Propagation in Material

0 0r

8

0 0

1 3 10/p

r r r

cv m s

Propagation in material with relative dielectric constant r

(non-magnetic material)

0p r

g

r

cv

f f

Take water for an example:

8

7

,

3 103.32 10 /

81.5p waterv m s

0

, 00.1181.5

g water

The propagation speed is slower than that in vacuum, and the wavelength

is also shorter than that in vacuum.

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Wave at a Certain Point

2

t Tt

2T

1 , cosv x t A x t

0x

1 0, cosv t A t

2t

00

tt

A

A

1 0, cosv t A t

0x x l

0x

x l• Consider

At position

We only pay attention to this point

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Wave at a Certain Time

2

xx

2

1 , cosv x t A x t

0t

1 ,0 cosv x A x

2 x

00

xx

A

A

1 ,0 cosv x A x

0x x l

0x

x l

• Consider

At time

We now pay attention to the whole line

at any time instant (here, t=0)

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Wave Propagation versus x and t

2 x

A

A

t

t T

2t T

x

x

t

20/40


Wave at Point x=

2 x

A

A

t

t T

2t T

t

0x x l

We only pay attention to this point

x

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Terminated Transmission Line

LZ LZ0Z 0Z

j xAe

j xBe j xBe

j xAe

0x x l 0dd l

j x j xV x Ae Be

0 0

j x j xA BI x e e

Z Z

IN d

1 1

j d j dV d A e B e

1 1

0 0

j d j dA BI d e e

Z Z

1

jA Ae l 1

jB Be lwhere and

d xl

incident wave

reflected wave

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Reflection Coefficient

1 1

j d j dV d A e B e 1

jA Ae l 1

jB Be l

2 21 1

0

1 1

j dj d j d

IN j d

B e Bd e e

A e A

10

1

0IN

B

A

where and

• Moves from the load (at d=0) toward the source (at d=l)

where is the load reflection coefficient, which is the value

of at d=0:

0

IN d

incident wave reflected wave

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Reflection Coefficient at Load

2

1 0 1 01j d j d j d j dV d A e e A e e

21 10 0

0 0

1j d j d j d j dA AI d e e e e

Z Z

00

0

j d j d

IN j d j d

V d e eZ d Z

I d e e

00

0

10

1IN LZ Z Z

00

0

L

L

Z Z

Z Z

1 1

j d j dV d Ae B e

1 1

0 0

j d j dA BI d e e

Z Z

Input impedance of the transmission line at any position d

is defined as

Use boundary condition at load (d=0)

0 0 when 0LZ ZProperly terminated (matched line):

LZ V d

0dd l

IN d

INZ d

I d

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Input Impedance of a Terminated Line

0 0

0

0 0

j d j d

L L

IN j d j d

L L

Z Z e Z Z eZ d Z

Z Z e Z Z e

00

0

cos sin

cos sinL

L

Z d jZ dZ

Z d jZ d

00

0

tan

tanL

L

Z jZ dZ

Z jZ d

00

0

L

L

Z Z

Z Z

00

0

j d j d

IN j d j d

V d e eZ d Z

I d e e

It gives the value of the input impedance at any position d along the

transmission line

At d=0 0IN LZ Z

At d=l

00

0

tan

tanL

IN

L

Z jZZ Z

Z jZ

ll

l

A very important property of a transmission line is the ability to change

a load impedance to another value of impedance as its input.

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Voltage Standing-wave Ratio (VSWR)

2

1 01 j dV d A e

1 0max1V d A 1 0min

1V d A

0max

0min

1

1

V dVSWR

V d

The addition of the two waves traveling in opposite directions in a

transmission line produces a standing-wave pattern – that is,

a sinusoidal function of time whose amplitude is a function of position.

2

1 0 1 01j d j d j d j dV d A e e Ae e

Magnitude of the voltage along the line:

The voltage standing-wave ratio (VSWR):

Next we will discuss 4 important cases of the terminated line, which are

matched line, short-circuited line, open-circuited line, and quarter-wave line.

and

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Matched Line (Properly Terminated)

• Matched line: the characteristic impedance is equal to the

load impedance, i.e., IN LZ d Z

0LZ Z

0dd l

l 0INZ Z

0INZ d Z

0Z

There is no reflection wave, the input impedance is at any location d,

VSWR has its minimum value of 1, i.e., 0 0 1VSWRand 0Z

No matter how long the line is, there is no transmission line effects .

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Short-Circuited Line

• For , it follows that , , and the

input impedance at a distance d from the load, , is

given by 0 tanscZ d jZ d

The amplitude of the incident and reflected waves are

the same since (total reflection from the load), VSWR attains

its largest value of infinity.

0LZ

0dd l

l l 0 tanINZ jZ

0 tanscZ d jZ d

0Z

0LZ 0 1 VSWR

scZ d

When

Short-circuited load is transformed to

open-circuited.

0 1

l

4

l scZ

When

Short-circuited load is still short-

circuited seen from the input.

l

2

l 0scZ

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Open-Circuited Line

• For , it follows that , , and the

input impedance at a distance d from the load, , is

given by

The amplitude of the incident and reflected waves are

the same since (total reflection from the load), VSWR attains

its largest value of infinity.

LZ 0 1 VSWR

ocZ d

When

Open-circuited load is transformed to

short-circuited.

0 1

l

4

l 0ocZ

When

Open-circuited load is still open-

circuited seen from the input.

l

2

l ocZ

0 cotocZ d jZ d

LZ

0dd l

l l 0 cotINZ jZ

0 cotocZ d jZ d

0Z

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Quarter-wave Line

• Quarter-wave transformer:

2

04IN

L

ZZ

Z

0 4IN LZ Z Z

In order to transform a real impedance to another real impedance

given by , a quarter-wave line with real characteristic impedance

of value

can be used.

LZ

0d

4d

2

0

4IN

L

ZZ

Z

0Z

l

4

d

LZ 4INZ

Example: , how to trans-

form it to at certain frequency?

75 LZ50

0 4 50 75 61.2 IN LZ Z Z

You can simply use a transmission line

with 61.2 Ohm characteristic impedance! 4

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Half-wave Line

• Half-wave line:

2IN LZ Z

No matter what the line impedance is, when a half-wave line is used

and it does not affect the impedance seen from the input.

LZ

0d

2d

2IN LZ Z

0Z

2IN LZ Z

l

2

d

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Voltage on the Shorted-circuited Line

1 12 sinj d j dV d A e e j A d

2

1, Re Re 2 sinj t

j tv d t V d e A d e

When

1 14 2 sin 2 2V j A j A

12 2 sin 0V j A

4d

When

2

d

• Voltage on the short-circuited line:

incident wave reflected wave

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Standing-wave Pattern

1, 2 sin cos2

v d t A d t

2

3

2

2

max

min

V dVSWR

V d

V d

1 max2A V d

min

0 V dd

d

2

4

3

4

In order to proceed we need to know the value of the complex constant A1.

For simplicity let us assume that A1 is real. Hence we obtain

2

3

2

2

,v d t

12A

min

0 V dd

d

2

4

3

4

12A

32

t 54

t

3,4 4

t 2

t

0,t

33/40


Example(I)

100 50sZ j 50 50LZ j

10 0sv

50 5010 0 3.92 11.31

50 50 100 50L

L s

L s

jZV V

Z Z j j

Consider the circuit shown below that there is no transmission line

between the source and load, find the voltage at the terminal of the

load.

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Example(II)

Find the load reflection coefficient, the input impedance, and the

VSWR in the transmission line shown, the length of the

transmission line is and its characteristic impedance is 50

Ohm.

10 0sv

100 50sZ j 50 50LZ j

l

00

0

50 50 500.447 63.44

50 50 50L

L

jZ Z

Z Z j

50 50 50 tan458 50 100 50

50 50 50 tan45IN

j jZ j

j j

0

0

1 1 0.4472.62

1 1 0.447VSWR

8

0 50Z

8

8INZ

LV

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Example(III)

8 100 508 10 0 5.59 26.57

8 100 50 100 50

IN

s

IN s

Z jV V

Z Z j j

2

1 01j d j dV d A e e

4 218 5.59 26.57 1 0.447 63.44

j j

V A e e 1 3.95 63.44A

0 3.95 63.44 1.77 5 45LV V

100 50sZ j

10 0sv

8 100 50INZ j

Equivalent circuit

Transmission-line effect makes things different! It is not always bad, since it

cab be used for matching and makes maximum power transfer possible.

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V x

I xR x j L x

G x j C x

V x x

I x x

Lossy Transmission Line

dV xR j L I x

dx

dI xG j C V x

dx

2

2

2

d V xV x

dx

j R j L G j C

Complex propagation constant

is the attenuation constant in

nepers/m and propagation constant

is in rads/m

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Reflection Coefficient and Input Impedance

, Re Rej x t j x tj t x xv x t V x e Ae e Be e

0 0

, Re Rej x t j x tj t x xA B

i x t I x e e e e eZ Z

With x=l - d

1 1

d dV d A e B e

1 1

0 0

d dA BI d e e

Z Z

l1A Ae l1B Be

The reflection coefficient of the terminated lossy line

21

0

1

dd

IN d

B ed e

A ewhere

010

1 0

0 LIN

L

Z ZB

A Z Z

00

0

tanh

tanhL

IN

L

Z Z dZ d Z

Z Z d

where and

The impedance of the terminated lossy line

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Summary (I)

00

0

tanh

tanhL

IN

L

Z Z dZ d Z

Z Z d

2

0

d

IN d e

00

0

L

L

Z Z

Z Z

0

0

1

1VSWR

00

0

tan

tanL

IN

L

Z jZ dZ d Z

Z jZ d

2

0

j d

IN d e

pv

2

LC• Propagation constant

• Reflection coefficient at load

• Reflection coefficient at any position lossless

• Input impedance

lossless

j where

• Phase velocity

• Wavelength

• Voltage standing wave ratio

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Summary (II)

• Transmission line affects the input impedance, and you

may not get the voltage you want at the load. On the other

hand, this property is useful when you need impedance

matching to get maximum power transfer.

• Both the wave velocity and wavelength decreases when the

wave travels from vacuum into the material that has a

relative dielectric constant greater than 1.

• Generally speaking, when the circuit dimension is

under , the transmission effects can be considered

negligible.

20

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