Lecture 10 Introduction to Microwave Network Analysis Analysis. Microwave Networks: Voltages and Currents • the theory of microwave networks was developed to enable circuit- like

Lecture 10

Introduction to Microwave Network Analysis

Microwave Networks: Voltages and Currents

• the theory of microwave networks was developed to enable circuit-like analysis methods which are simpler than field methods

• it also enables the integrated analysis of microwave structures with conventional lumped components (ICs, chip transistors, resistors, etc.)

• usual low-frequency definitions of voltage and current are not valid

V d

E L

CI d H L

if E is not conservative, voltage integral depends on the chosen integration path and is thus ambiguous

if there are no metallic leads (dielectric guides), the current integral is meaningless and the voltage one is ambiguous

ElecEng4FJ4 LECTURE 10: MICROWAVE NETWORK ANALYSIS 2

• network parameters (Z, Y, ABCD) based on voltages and currents are often inadequate

Equivalent Voltages and Currents

considerations in defining equivalent voltages and currents• equivalent voltage and current describe a traveling wave mode• voltage must be proportional to the transverse E field of a mode• current must be proportional to the transverse H field of a mode• the product of the rms voltage and rms current must produce the

power carried by the mode • in the case of a TEM line, the ratio of the voltage and the current

(V/I) must equal the characteristic impedance Z0

• in the case of a waveguide of uniform cross-section, the V/I ratio must equal to the wave impedance Zw if known

• the V/I ratio is taken as 1 if characteristic and wave impedances are not available (general waveguides, numerical solutions)

• the choice of the V/I ratio does not affect the scattering parametersElecEng4FJ4 LECTURE 10: MICROWAVE NETWORK ANALYSIS 3

Equivalent Voltages and Currents (2)

consider the transverse components of a field traveling along +z

1 1ˆ( ) 2 2

e ee eSV I

V IS ds V IC C

e h z

complex power and normalization:

ˆ ( ) V ISds C C e h z

0E 0H

set normalization condition as ˆ( ) 1S

ds e h z 1 /I VC C


physical meaning of e and h: the rms field phasors of the traveling wave in the waveguide such that it carries 1 W power

00 ( , )( , )

( , , ) ( , ) ( , , ) ( , )e ej z j z

V I

x yx y

V Ix y z x y e x y z x y eC C

HE

E e H h

ˆ0.5 ( )S

S ds E H z

(proof on next slide)

Equivalent Voltages and Currents (3) – optional

2

Let ,

ˆ( ) ( )

j jr r

jV I V I r rS S

e e

ds C C C C e d

e e h h

e h z e h s

2( ) 1 ( ) 1 jr r V IS S

d d C C e e h s e h s

real

2

Let | | ( absorbs the angle of entirely)1

||1

||

jV V e

j

j jVV

IV

C

C C e VeCe CC e

E

Loss-free case: φ = 0

(now absorbs the angle of ) eI H

ˆif ( ) 1 then 1/I VSds C C e h zProof:

( , ) is realV

x yC

e

( , ) is realI

x yC

h

PROOF


Equivalent Voltages and Currents – Summary

general definitions of voltage and current following from the normalization

0 0

0 0

/ ( , ) ( , )

( , ) ( , ) /

e

V

V

S

e S

Vx y x yC

x y C I x y

d

d

E e Eh h s

eH h He s

0

0

ˆ( )1 ˆ( )

e V S

e SV

V C ds

I dsC

E h z

e H z

⇒


wave power ( ) does not depend on the choice of CV

modal vectors e and h do not depend on the choice of CV either

apply normalization

voltage-to-current ratio however does depend on the choice of CV

Re{ }e eV I

Voltage-to-Current Ratio

2 20

0

1

ˆ( )

ˆ( )e

VS

S

Ve

dsVI ds

C C

E h z

e H z

integrals in numerator and denominator cancel; follows from Maxwell’s equations in a source-free medium:

/ /

jj

E H hH Ε e

/ /

jj

e h Hh e E

( )( )

jj

h E h He H e Ε

( )( )

jj

H e H hE h E e

( )( ) ( )( ) ) (( )

j jj j

h E E h EH h E e hH e e H e HH h E e

( ) ( ) (*) E h e H ElecEng4FJ4 LECTURE 10: MICROWAVE NETWORK ANALYSIS 7

PROOF – OPTIONAL

voltage/current DEF 1: using a known characteristic impedance Z0(suitable for TEM TLs)

20 0 0 , 1 /e

V IVe

V C Z C Z C ZI

0 ˆ( )e SV Z ds E h z

0

1 ˆ( )e SI ds

Z e H z

00

( , , ) ( , ) ( , , ) ( , )e j z j ze

Vx y z x y e x y z Z I x y eZ

E e H h

1/2 1/2

1 1/2 1 1/2UNITS: [ ] , [ ] , [ ] , [ ] , [ ] m , [ ] m

V I e eC C V V I A

e h


Voltage-to-Current Ratio: Definition 1

ˆ ˆ( , ) ( , )( , ) and ( , )w w

x y x yx y x yZ Z

z e z Eh H

voltage/current DEF 2: using a known wave impedance Zw (suitable for waveguides of homogeneous cross-section)

2 ew V wV

e

VZ C C ZI

ˆ( )e w SV Z ds E h z 1 ˆ( )e S

wI ds

Z e H z

( , , ) ( , ) ( , , ) ( , )e j z j zw e

w

Vx y z x y e x y z Z I x y eZ

E e H h

units are the same as with DEF 1



• the meaning of Zw:

• in TEM TLs: wZ

voltage/current DEF 3: impedance set to unity (general, used for waveguides of heterogeneous cross-section that are analyzed numerically)

2 1 1eV IV

e

V C C CI

ˆ( )e SV ds E h z ˆ( )e S

I ds e H z

( , , ) ( , ) ( , , ) ( , )e

je e

V

z j zx y z x y e x y z x y eV I

E e H h

1/2 1/2

1 1/2 1 1/2UNITS: [ ] none, [ ] none, [ ] , [ ] , [ ] m , [ ] m

V I e eC C V W I W

e h

root-power wave

e eV I



Repeat the example for slide 8 this time setting voltage-to-current ratio to unity. Do you obtain the same modal vectors? (homework)

Microwave Network: Voltage/Current Formulation

• at each port incident and reflected voltage/current waves can be defined

N-port network

• at the nth port:

( 1, , )

n n n

n n n

n N

V V VI I I

(omit ~ for simpler notation)


[Pozar]

Impedance and Admittance Matrices – Review

1 11 12 1 1

2 21 22 2

2

N

N N NN N

V Z Z Z IV Z Z I

V Z Z I

• relate the total voltages to the total currents at the ports

0 for all k

iij

j I k j

VZI

all ports except port j are open-circuited

1 11 12 1 1

2 21 22 2

2

N

N N NN N

I Y Y Y VI Y Y V

I Y Y V

impedance or Z matrix

0 for all k

iij

j V k j

IYV

all ports except port j are short-circuited

admittance or Y matrix

1Y ZElecEng4FJ4 LECTURE 10: MICROWAVE NETWORK ANALYSIS 12

Impedance Matrix: Example

2

1

1

2

111

1 0

222

2 0

112

2 0?

221 12

1 0

?

?

?

I

I

I

I

VZIVZIVZIVZ ZI

A CZ Z

B CZ Z

CZ

1I 2I


Transmission (ABCD) Matrix (aka Cascade Parameters)

• defined for a 2-port network (see Fig. a)

• particularly useful when cascading 2-port networks (see Fig. b)

1 2 2

1 2 2

V AV BII CV DI

1 2

1 2

V A B VI C D I

1 1 1 2 2 3

1 1 1 2 2 3

V A B A B VI C D C D I


Reciprocal Networks consider any two ports (e.g., ports 1 and 2) in a network containing

only linear media (no plasma, ferrites or active devices)

short-circuit all other ports – the network is now a 2-port network

apply Reciprocity Theorem in the volume of the network for two possible sources a and b residing outside the network

( ) ( )a b b aS Sd d E H s E H s

Reciprocity Theorem• consider two separate sets of sources (set a and set b) and their fields

in a linear medium so that superposition applies (linear media)/

/a a a b

a a a b

jj

E H M HH Ε J E

/ /

b b b a

b b b a

jj

E H M HH E J E


PROOF – OPTIONAL

Reciprocity Theorem in Electromagnetics – Optional • combine all 4 equations properly by using the vector identity

( )

a b b a

b a a b a b b a

E H E HH E Ε H H E E H

( )a b b a a b a b b a b a E H E H E J H M E J H M • reciprocity theorem – differential form

( )

( )s

a b b aS

a b a b b a b aV

d

dv

E H E H s

E J H M E J H M

• reciprocity theorem – integral form

• in our case, there are no sources in the volume of the network( ) ( )a b b aS S

d d E H s E H s ElecEng4FJ4 LECTURE 10: MICROWAVE NETWORK ANALYSIS 16

Reciprocal Networks (2)

a

b• field at port 1

1 1 1

1 1 1

a a

b b

VV

E eE e

1 1 1

1 1 1

a a

b b

II

H hH h

• field at port 22 2 2

2 2 2

a a

b b

VV

E eE e

2 2 2

2 2 2

a a

b b

II

H hH h

1 21 1 1 1 1 1 2 2 2 2 2 2( ) ( ) ( ) ( ) 0a b b a a b b aS S

V I V I d V I V I d e h s e h s apply Reciprocity Theorem: only over port cross-sections

unity normalization holds for both ports, e.g.,

1 11 1 2 2( ) ( ) 1

S Sd d e h s e h s

1 1 1 1 2 2 2 2 0a b b a a b b aV I V I V I V I

short

short

short

assume CV = 1 (result doesn’t depend on choice of CV)

( ) 0S


Z and Y Matrices of Reciprocal Networks

substitute Y-matrix equations1 11 1 12 2

2 21 1 22 2

a a a

a a a

I Y V Y VI Y V Y V

1 11 1 12 2

2 21 1 22 2

b b b

b b b

I Y V Y VI Y V Y V

12 21 1 2 2 1( )( ) 0a b a bY Y V V V V 12 21Y Y

• both the Y and Z matrices are symmetric for the reciprocal (linear medium) networks

into1 1 1 1 2 2 2 2 0a b b a a b b aV I V I V I V I

• the ABCD matrix for a reciprocal network fulfills

1AD BC


Loss-free Networks

since the N-port network is loss-free, the net real power delivered to the network must be zero

1 1 1express complex power: ( )2 2 2

T T T TP V I ZI I I Z I

assume the network is reciprocal, ZT = Z

1 1

1Re 02

N N

av m mn nn m

P I Z I

consider the case where all ports are left open-circuited except the nth port

10, all Re 02k av n nn nI k n P I Z I

Re 0, 1, ,nnZ n N


Z and Y Matrices of Loss-free Networks

now let all ports be open-circuited but ports n and m

real

10, all , Re[ ( )] 02k av nm n m n mI k n m P Z I I I I

Re 0, , 1,nmZ n m N

• all elements of the Z matrix for a loss-free network are purely imaginary, i.e., ReZ = 0

• analogous derivation shows that ReY = 0, too


Summary


• voltages and currents for microwave networks depend on the field differently compared to static (or low-frequency) networks

• to calculate equivalent voltages and currents of traveling waves, we need the modal vectors: electric e and magnetic h

• e and h are the rms field phasors of the traveling wave in the TL/waveguide such that it carries 1 W power

• the most general definition sets the voltage-to-current ratio to 1: voltage is the same as current describing the root-power wave

• reciprocal networks feature symmetric Z and Y matrices

• loss-free networks feature purely imaginary Z and Y matrices

1/2ˆ ˆ( ) ( ) , We eS SV ds I ds E h z e H z

Documents

Lecture 10 Introduction to Microwave Network Analysis Analysis. Microwave Networks: Voltages and Currents • the theory of microwave networks was developed to enable circuit- like