Lectures Microwave Engineering

8/3/2019 Lectures Microwave Engineering

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Microwave Circuits andSystems



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Objectives

Develop understanding of fundamentals of RF/microwave circuits

Analyze and design microwave circuits

Learn transmission line, scattering parameters, the Smith chart,

passive and active devices Simulate microwave circuits using popular EDA tools:

Advanced Design System (ADS), CST (Computer SimulationTechnology)

Use Vector Network Analyzer for S parameter measurement

Study and Analyze Microwave communication systems



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Aim

After this part the participants should be able to:

• Apply electromagnetic theory to calculations regarding waveguides andtransmission lines• Describe, analyze and design simple microwave circuits and devices eg

matching circuits, couplers, antennas and amplifiers.• Describe and coarsely design common systems such as radar andmicrowave transmission links.• Describe common devices such as microwave vacuum tubes,high-speedtransistors.

• Handle microwave equipment and be able to make measurements(VNA).



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Textbook

Pozar

David M Pozar, Microwave Engineering- 2nd Ed.,John Wiley , 1998

ReferencesReference materials such as component datasheet, FCCregulations, wireless standards, papers, useful links, etc.:

in class

http://www.amazon.com/gp/product/images/0471170968/ref=dp_image_0?ie=UTF8&n=283155&s=books

http://www.amazon.com/gp/product/images/0471170968/ref=dp_image_0?ie=UTF8&n=283155&s=books



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Analysis

L j R ),( t z I

dz

dI z I

),( t zV C jG dz

dV zV

zdt

dI L z RI z

dz

dV

dt

dI L RI

dz

dV

zdt

dV C zGV z

dz

dI

dt

dV C GV

dz

dI

From Kirchoff Voltage Law Kirchoff current law

zdt

dV C zGV z

dz

dI I I

z

dt

dI L z RI z

dz

dV V V

(a) (b)



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Analysis

Let’s V=Voe jt , I = Ioe jt

Therefore

V jdt

dV

I jdt

dI

then

I L j R

dz

dV V C jG

dz

dI

1

2

Differentiate with respect to z

dz

dI L j R

dz

V d

2

2

V C jG L j Rdz

V d

2

2

V dz

V d 2

2

2

dz

dV C jG

dz

I d

2

2

I C jG L j Rdz

I d

2

2

I dz

I d 2

2

2



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Analysis

The solution of V and I can be written in the form of

where

C jG

L j R Z o

Let say at z=0 , V=VL , I=IL and Z=ZL

Therefore

and L

L

L Z

I

V

5 6

3 4

C jG L j R j and

z z Be AeV

B AV L 0 Z

B A I L

0 Z

Be Ae I

z z



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Analysis

Solve simultaneous equations ( 5 ) and (6)

Inserting in equations ( 3) and (4) we have

22)(

z z

o L

z z

L

ee Z I

eeV zV

22)(

z z

o

L z z

L

ee

Z

V ee I z I

2

o L LZ I V

B

2

o L L Z I V A



1012/4/2011 1012/4/2011

Analysis

2)cosh(

z z ee z

2)sinh(

z z ee z

We have

)sinh()cosh()( z Z I zV zV o L L

)sinh()cosh()( z Z

V z I z I

o

L L

)sinh()cosh(

)sinh()cosh(

)(

)()(

z Z

V z I

z Z I zV

z I

zV z Z

o

L L

o L L

and



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Analysis

)sinh()cosh(

)sinh()cosh()(

z Z z Z

z Z z Z Z z Z

Lo

o Lo

)tanh(

)tanh()(

z Z Z

z Z Z Z z Z

Lo

o Lo

Or further reduce

or

For lossless transmission line , = j since 0

)tan(

)tan()(

z jZ Z

z jZ Z Z z Z

Lo

o Lo

)cos()cosh( z z j

)sin()sinh( z j z j

C

L Z o

C jG

L j R Z o

Lossy line

Lossless line

C jG L j R

LC



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II-Loaded Transmission Line

Example: Antenna is a load (Feeder to antenna)

Load impedance: the input impedance of the antenna



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Standing wave

)()( jBd d j eeV d V

)2

cos()sin(2),( t d V t d v



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Voltage:

Current

Impedance

)sin(2)( d jV d V

)cos(2

)(0

d Z

V d I

)tan()( 0 d jZ d Z in

Special terminal conditions

Input impedance of short circuit transmission line



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What is about open circuit transmission line?



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L L

Lin

Z Z

l jZ Z l jZ Z Z

2

0

0

0

)4tan()4tan()4 / (

Quarter-wave transmission line



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)1( ininin V V )1(0

inin

in Z V I

}Re{2

1 *in

inin

I V P

)1(2

1 2

0

2

in

in

in Z

V P

)1(1

1

8

1 2

2

2

0

2

in

inS

SG

in Z

V P

Power considerations



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Two special cases:

Load and sourceMatched line

Mismatch at source,But match at load

How to measure power?

00 S

00

0

2

8

1

Z

V P

G

in

2

0

2

18

1S

G

in Z

V P

mW

mW PdBmP

1

][log10][



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Return and insertion losses

Return loss:

Insertion loss:

][log20log10)log(102

dBP

P RL inin

t

r

][1log10)log(10)log(102

dB

P

PP

P

P IL in

i

r t

i

t



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III- The Smith chart

What for ?When ?

How to understand ?How to determine microwave transmission parameters ????

Note: Please print the complete Smith chart .



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Normalized impedance

ir

ir inin

j

j

d

d jxr z Z d Z

1

1

)(1

)(1 / )( 0

ir

d j j

jeed L

2

0)(

Real part of normalizedimpedance

Imaginary part of normalizedimpedance

22

22

)1(

1

ir

ir r

22)1(

2

ir

i x



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Inversion of complex reflection coefficient(constant normalized resistance)

222

)1

1()1( r r

r ir



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Inversion of complex reflection coefficient(constant normalized resistance)

222

)

1

()

1

()1( x xir



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Generic Smith Chart computation

• Normalize load impedance

• Find reflection coefficient

• Rotate reflection coefficient

• Record normalized input impedance

• De-normalize input impedance

L L z Z

0 L z

)()( d Z d z inin

)(d zin

)(0

d



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Admittance Transformation(Smith Chart)

• Impedance represetation in Smith Chart

• Admittance representation in Smith Chart

)(1

)(1

d

d

jxr zin

)(1

)(1

)(1

)(11

0 d e

d e

d

d

zY

Y y

j

j

in

inin

180 degreePhase shift



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Transformation2

1

2

111 j y j z inin



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Alternative: re- interpretation

Instead of rotating the reflection coefficient about 180 degree, we keep the

location fixed and rotate the entire Smith Chart by 180 degree.



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Re-interpretation leads to ZY-Smith Chart

The Smith Chart inits original form iskept for impedance

display,but a second SmithChart is rotated by180 degree forAdmittance display.



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IV- Some microwave transmission lines

Two wirecable Coaxial

cable

Microstripeline

Rectangularwaveguide

Circularwaveguide

Stripline



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Parallel wire cable

d a for ad or ad L / ln2 / cosh 1

d a for

ad or

ad C

/ ln2 / cosh 1

ad Z o 2 / cosh1 1

Where a = radius of conductord = separation between conductors



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Coaxial cable

abC

/ ln2

ab L / ln2

ab Z o / ln

2

1

Where a = radius of inner conductorb = radius of outer conductorc = 3 x 108 m/s

r

cc

ck f 2

ba

k c

2

a

b



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Micro strip

whe r

t

t=thickness of conductor

Substrate

Conducted strip

Ground



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Characteristic impedance of Micro-strip line

hwwh Z hwFor eeeff

o / 25.0 / 8ln601 /

444.1 / ln667.0393.1 /

3771 /

hwhw Z hwFor

eeeff o

25.0 / 104.0 / 121

2

1

2

1hwwh ee

r r eff

5.0

/ 1212

1

2

1

e

r r

eff wh

1

2ln

t

ht ww e

e

t hhe 2Where

w=width of striph=height andt=thickness



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Micro-strip width

r r

r r o Z A

11.023.011

21

60

2 / 1

r o Z

B

260

2)2exp(

)exp(8 /

A

AhW

r r

r Blb B BhW

61.039.01

2

112ln1

2 /

For A>1.52

For A<1.52



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Simple Calculation

2

377

h

w Z

r

o

2377

/ or Z

hw

Approximation only

Using ADS (Advanced Design System): LinCal



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Micro-strip components

Capacitance

Inductance

Short/Open stub

Open stub

Transformer

Resonator



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Capacitance

Zo Zo Zoc

1c Z C

oc

12sin

2

c Z C

oc

smc

r

/ 103 8

1

For8

For

8



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Inductance

Zo Zo ZoL

1c

Z L oL

1

sinc

Z L oL

smc

r

/ 103

8

1

For8

For

8



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Short Stub

Zo

Z

Zo

Zo ZL

o L Z X / tan 1 eff

o

360

tano Lsc jZ X Z



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Open stub

Zo

Z

Zo

Zo ZL

oc Z X / cot 1 eff

o

360

cotococ jZ X Z



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Short-circuited /2 lossy line

n /2

Zin Zo

o Z R

o

o Z L

2

LC

o2

1

22

R

LQ o

2where

= series RLC resonant cct



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Open-circuited /2 lossy line

n /2

Zin Zo

22

RC Q o

= parallel RLC resonant cct

o Z R

oo Z C

2

C L

o2

1

2where



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Short-circuited /4 lossy line

/4

Zin Zo

o Z R

= parallel RLC resonant cct

oo Z C

4

C L

o2

1

24

RC Q o

2where



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Smith Chart: Homework+ Exercises



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Chapter III- MicrowaveNetwork Analysis



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Network representations

• For a linear system should be a set of (possibly

frequency dependent) parameters that relate “inputs” to“outputs” • n-port representations

- Impedance matrix (Z parameters) • “input”:currents • “output”:voltage

- Admittance matrix(Y parameters) • input:currents • output:voltage

- h-parameters matrix - Scattering matrix (S parameters) • input: incident traveling (voltage) wave • output: “reflected” traveling wave

- ABCD matrix



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What “parameters” should you use?

• it depends on the type of network…

•Example: view voltages as “inputs”, currents as “outputs” - units for Y parameters: (ohm)-1

If it’s a linear network then we have

N

j

NN Nj N N

jN jj j j

j

N j

N

j

V

V

V

V

Y Y Y Y

Y Y Y Y

Y Y Y

Y Y Y Y

I

I

I

I

2

1

21

21

22221

111211

2

1



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0

0

0

1

21

21

22221

111211

2

1

NN Nj N N

jN jj j j

j

N j

N

j

Y Y Y Y

Y Y Y Y

Y Y Y

Y Y Y Y

I

I

I

I

Y parameter representation

• how would you measure these?

- SHORT circuit ALL ports but port 1, apply “unit” voltage toport 1, measure all currents- Repeat for other ports

+ source : second index+response : first index

11212111 ,...,,...,, N N j j Y I Y I Y I Y I



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Y parameter applications

• “shunt” connected two-port networks

total

total

total

total

V

V

Y Y

Y Y

Y Y

Y Y

I

I

2

1

2221

1211

2221

1211

2

1



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N

j

NN Nj N N

jN jj j j

j

N j

N

j

I

I

I I

Z Z Z Z

Z Z Z Z

Z Z Z Z Z Z Z

V

V

V V

2

1

21

21

22221

111211

2

1

Z parameters

• it depends on the type of network…

•Example: view current as “input”, voltages as outputs -Units for Z parameters: ohmsIf it’s a linear network then we have



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0

0

0

1

21

21

22221

111211

2

1

NN Nj N N

jN jj j j

j

N j

N

j

Z Z Z Z

Z Z Z Z

Z Z Z

Z Z Z Z

V

V

V

V

11212111 ,...,,...,, N N j j Z V Z V Z V Z V

Z parameter representation• how would you measure these?-OPEN circuit ALL ports but port 1, apply “unit” current port 1, measure

all voltages-Repeat for other ports• source : second index• response : first index



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total

total

total

total

I

I

Z Z

Z Z

Z Z

Z Z

V

V

2

1

2221

1211

2221

1211

2

1

Z parameter applications

• “series” connected two-port networks



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Hybrid (H) parameters

• input : i1 , v2

• onput : v1 , i2 • units:- h1 : ohms- hi , hf : dimensionless- ho : mhos

• measurement:

- short “output” (port 2) + hi : input impedance at port 1+ hf : current gain from 1 to 2

- open at input (port 1)+ hr : reverse voltage transfer ratio+ ho : output admittance

2

1

02

1

V

I

hh

hh

I

V

f

r i

0

1

02

1 I

hh

hh

I

V

f

r i

202

1 0

V hh

hh

I

V

f

r i

1211,. I h I I hV f i

2211, V h I V hV

or



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•“Chain” (ABCD) parameters • input: V2 , - I2• output: V

1, - I

1

• units:• - A, D : dimensionless• - B :ohms• - C : mhos

• measurement :- short port 2

+ B : “transfer” impedance + D : reverse current gain

-open port 2

+ A : reverse voltage transfer ratio+ C : ”transfer” admittance

2

2

1

1

I

V

DC

B A

I

V

21

1 0

I DC

B A

I

V

0

2

1

1 V

DC

B A

I

V

2121 , DI I BI V

2121 , CV I AI V



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total

total

total

total

I

V

DC

B A

DC

B A

I

V

2

2

1

1

Application of ABCD parameters

• Cascaded two ports



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Scattering (S) parameters• view travelling waves into the network as “inputs”

• view travelling waves out of the network as “outputs”

• units for S parameters : dimensionless

If it’s a linear network then we have

N

j

NN Nj N N

jN jj j j

j

N j

N

j

a

a

a

a

SSSS

SSSS

SSS

SSSS

b

b

b

b

1

1

21

21

22221

111211

2

1

j j

j Z V a 0 /

j j

j Z V b 0 /



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0

0

0

1

21

21

22221

111211

2

1

NN Nj N N

jN jj j j

j

N j

N

j

SSSS

SSSS

SSS

SSSS

b

b

b

b

S parameter representation

• how would you measure these?

-MATCH ALL ports but port1, apply “unit ” input wave to port 1, measureall reflected waves- repeat for other ports

+source : second index

+response : first index



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2

2

2221

1211

1

1

a

b

T T

T T

b

a

2

2

1

1

abT T

ba

Chain S (transfer T) parameters

•Two port networks only•“mixed” representaion -Port two wavves are “inputs” -Port two waves are “outputs” • units for T parametters : dimensionless



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Scattering Parameters (S-Parameters)

In this course, we focus on S MatrixFor simple, we consider 2-portnetwork.

The behavior of the network can becompletely characterized by itsscattering parameters (S-parameters),or its scattering matrix, [S].

S matrix is used to representmicrowave devices, such as

amplifiers and circulators, and areeasily related to concepts of gain, lossand reflection.

11 12

21 22

S S

S S S

Scattering matrix



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The scattering parameters: ratios ofvoltage waves entering and leaving theports (If the same characteristic

impedance, Zo, at all ports in the networkare the same).

1 11 1 12 2.V S V S V

2 21 1 22 2.V S V S V

11 121 1

21 222 2

,S SV V

S SV V

In matrix form this is written

.V S V

2

1

11

1 0V

V S

V

1

1

12

2 0V

V S

V

1

2

22

2 0V

V S

V

2

2

21

1 0V

V S

V



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Properties:

The two-port network is reciprocal if thetransmission characteristics are thesame in both directions (i.e. S21 = S12).

IA network is reciprocal if it is equal to

its transpose. Stated mathematically,for a reciprocal network

,t

S S

11 12 11 21

21 22 12 22

.

t

S S S SS S S S

12 21S SCondition for Reciprocity:

1) Reciprocity

By inspection :If the network is symmetrical

Reciprocal



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A lossless network does not contain any

resistive elements and there is noattenuation of the signal.

In terms of scattering parameters, anetwork is lossless if

2) Lossless Networks

*

,

t

S S U

1 0

[ ] .0 1U

where [U ] is the unitary matrix

For a 2-port network:

2 2 * *

11 21 11 12 21 22*

2 2* *

12 11 22 21 12 22

1 0

0 1

t S S S S S S

S S S S S S S S

2 2

11 211S S If the network is reciprocal and lossless

* *

11 12 21 220S S S S



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S12 = S21 =0 (Reciprocal)

10

0

0

0

0

011

Z

Z

Z Z

Z Z S

in

inin

0

222 18011 inS

10

01S

Lossless condition check

12

21

2

11 Ss

Example



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