ece 451 – jose schutt-aine 1 ece 451 advanced microwave measurements transmission lines jose e....

ECE 451 – Jose Schutt-Aine 1

ECE 451Advanced Microwave Measurements

Transmission Lines

Jose E. Schutt-AineElectrical & Computer Engineering

University of [email protected]


Faraday’s Law of Induction

Ampère’s Law

Gauss’ Law for electric field

Gauss’ Law for magnetic field

Constitutive Relations

BE

t

H J D

t

D

0B

B H D E

Maxwell’s Equations


l

l

z

Wavelength : l

l = propagation velocity

frequency

Why Transmission Line?


In Free Space

At 10 KHz : l = 30 km

At 10 GHz : l = 3 cm

Transmission line behavior is prevalent when the structural dimensions of the circuits are comparable to the wavelength.

Why Transmission Line?


Let d be the largest dimension of a circuit

If d << l, a lumped model for the circuit can be used

circuit

z

l

Justification for Transmission Line


circuit

z

l

If d ≈ l, or d > l then use transmission line model

Justification for Transmission Line


L: Inductance per unit length.

C: Capacitance per unit

length.

L

z

C

I

V

+

-

V IL

z t

I VC

z t

Vj LI

z

Ij CV

z

Assumetime-harmonicdependence

, ~ j tV I e

Telegraphers’ Equations


L

z

C

I

V

+

-

22

2

VLCV

z

22

2

ICLI

z

2V V Ij L j Lj CV

z z z z

2 I I V

j C j Lj CIz z z z

TL Solutions


( )

Forward Wave Backward Wave

j z j zV z V e V e

z

Zo

forward wave

backward wave

LC

o

LZ

C

(Frequency Domain)

( ) j z j z

o o


V VI z e e

Z Z

TL Solutions


( , ) cos cos


V z t V t z V t z

LC

o

LZ

C

( , ) cos coso o


V VI z t t z t z

Z Z

1v

LC

v

f

Propagation constant

Characteristic impedanceWavelength

Propagation velocity

TL Solutions


At z=0, we have V(0)=ZRI(0)

But from the TL equations:

(0)V V V

(0) o o

V VI

Z Z

Which gives

R

o

ZV V V V

Z

RV V

R oR

R o

Z Z

Z Z

is the load reflection coefficientwhere

Reflection Coefficient


- If ZR = Zo, GR=0, no reflection, the line is matched

- If ZR = 0, short circuit at the load, GR=-1

- If ZR inf, open circuit at the load, GR=+1

( ) j z j zRV z V e e

( ) j z j zR

o

VI z e e

Z

2( ) 1j z j zRV z V e e

2( ) 1j z

j zR

o

V eI z e

Z

Reflection Coefficient

V and I can be written in terms of R


( )( )

( )

j z j zR

o j z j zR

e eV zZ z Z

I z e e

tan

tanR o

oo R

Z jZ lZ l Z

Z jZ l

Generalized Impedance

Impedance transformatio

n equation


( ) tanoZ l jZ l

( )tan

oZZ lj l

- Short circuit ZR=0, line appears inductive for 0 < l < l/2

- Open circuit ZR inf, line appears capacitive for 0 < l < l/2

2

( ) o

R

ZZ l

Z

- If l = l/4, the line is a quarter-wave transformer

Generalized Impedance


( )Backward traveling wave at( )

Forward traveling wave at ( ) b

f

V zzz

z V z

2 2( )

j z

j z j zRj z

V e Vz e e

V e V

2( ) j lRl e Reflection coefficient

transformation equation

1 ( )( )

1 ( )

o

zZ z Z

z( )

( )( )

o

o

Z z Zz

Z z Z

Generalized Reflection Coefficient


2( ) 1j z j zRV z V e e

2 2( ) 1 1 j z j z j z

R RV z V e e V e

max 1 RV

min 1 RV

Maximum and minimummagnitudes given by

Voltage Standing Wave Ratio (VSWR)

We follow the magnitude of thevoltage along the TL


max

min

1

1

R

R

VVSWR

V

Voltage Standing Wave Ratio (VSWR)

Define Voltage Standing Wave Ratio as:

It is a measure of the interaction between forward and backward waves


VSWR – Arbitrary Load

Shows variation of amplitude along line


VSWR – For Short Circuit Load

Voltage minimum is reached at load


Voltage maximum is reached at load

VSWR – For Open Circuit Load


VSWR – For Open Matched Load

No variation in amplitude along line


Application: Slotted-Line Measurement

- Measure VSWR = Vmax/Vmin

- Measure location of first minimum


Application: Slotted-Line Measurement

At minimum, ( ) pure real Rz

Therefore, min2min( ) j d

R Rd e

min2 j dR R e So,

1

1

R

VSWR

VSWRSince min21

1

j dR

VSWRe

VSWRthen

1

1

R

R oR

Z Zand


Summary of TL Equations

tan

tanR o

oo R

Z jZ lZ l Z

Z jZ l

2( ) j lRl e

1 ( )( )

1 ( )

o

zZ z Z

z

( )( )

( )

o

o

Z z Zz

Z z Z

Impedance Transformation

Reflection Coefficient Transformation

Reflection Coefficient – to Impedance

Impedance to Reflection Coefficient

2( ) 1 j z j z

Ro

VI z e e

Z 2( ) 1j z j z

RV z V e e

Voltage Current


+- Vg =100 V

Zo = 50

Zg =(10+j10)

d=l d=0

ZR=(30+j40) Zin

Ig

Consider the transmission line system shown in the figure above.(a) Find the input impedance Zin.

(30 40) 500.5 90

(30 40) 50R o

RR o

Z Z j

Z Z j

4.0.7252( ) 0.5 90 0.5 72

jj lRd l e e

Example 1


1 ( ) 1 0.5 7250

1 ( ) 1 0.5 72in o

lZ Z

l

64.361 51.738 39.86 50.54inZ j

(b) Find the current drawn from the generator

100 01.552 39.11

(10 10)(39.86 50.54)g

gg in

VI A

Z Z j j

1.207 0.981gI j A

Example 1 – Cont’


c) Find the time-average power delivered to the load

1( ) 64.361 51.73 1.5562 39.11 100.159 12.62

2in gV l Z I

*1 1Real ( ) Real 100.159 12.62 1.5562 39.11

2 2gP V l I

*1Real ( ) 48.26

2 gP V l I W



+- Vg

Zo = 50

d=l d=0

L=1 H

C=100 pF

R=50

In the system shown above, a line of characteristic impedance 50 W is terminated by a series R, L, C circuit having the values R = 50 W, L = 1 mH, and C = 100 pF.

(d) Find the source frequency fo for which there are no standing waves on the line.



No standing wave on the line if ZR = R = Zo which occurs at the resonant frequency of the RLC circuit. Thus,

8

6 10

1 1 1015.92

22 2 10 10of Hz MHz

LC


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