microwave network analysis

8/10/2019 Microwave Network Analysis

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Chapter 4. Microwave Network Analysis

It is much easier to apply the simple and intuitive idea

of circuit analysis to a microwave problem than it is

to solve Maxwells equations for the same problem. Maxwells equations for a given problem is complete,

it gives the E & H fields at all points in space.

Usually we are interested in only the V & I at a set of

terminals, the power flow through a device, or some

other type of global quantity.

A field analysis using Maxwells equations forproblems would be hopelessly difficult.

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4.1 Impedance and Equivalent Voltages andCurrentsEquivalent Voltages and Currents

The voltage of the + conductor relative to theconductor

After having defined and determined a voltage,current, and characteristic impedance, we can proceedto apply the circuit theory for transmission lines to

characterize this line as a circuit element.

0

C

V E dl

I H dl

VZ

I

3

Figure 4.1 (p. 163)Electric and magnetic field lines for an arbitrary two-conductor

TEM line.

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Figure 4.2 (p. 163)Electric field lines for the TE10mode of a rectangular waveguide.

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( , , ) sin ( , )

( , , ) sin ( , )

j z j z

y y

j z j z

x x

TE

j a xE x y z A e Ae x y e

a

j a xH x y z A e Ah x y e

a


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The Concept of Impedance

Various types of impedance

Intrinsic impedance ( ) of the medium: depends

on the material parameters of the medium, and is equal to

the wave impedance for plane waves.

Wave impedance ( ): a characteristic of

the particular type of wave. TEM, TM and TE waves each

have different wave impedances which may depend on the

type of the line or guide, the material, and the operating

frequency.

Characteristic impedance ( ): the ratio ofV/I for a traveling wave on a transmission line. Z0for TEM

wave is unique. TE and TM waves are not unique.

/

/ 1/w t t wZ E H Y

0 01/ /Z Y L C

10

Geometry of a partially filled waveguide

Geometry of a partially filled waveguide and itstransmission line equivalent.

Reflection coefficient

11

An arbitrary one-port network.

The complex power delivered to this network is:

wherePlis real and represents the average power dissipated by the

network

12 ( )

2 l m e

SP E H ds P j W W

12

If we define real transverse modal fields, eand h,

over the terminal plane of the network such that

with a normalization

The input impedance is

If the network is lossless, thenPl= 0 andR= 0. Then

Zinis purely imaginary, with a reactance

( , , ) ( ) ( , )

( , , ) ( ) ( , )

j z

t

j z

t

E x y z V z e x y e

H x y z I z h x y e

, ,

1S

e h ds

e1 1

2 2SP VI e h ds VI

e

2 2 21 12 2

2 ( )l m ein

P j W WV VI P Z R jX

I I I I

2

4 ( )m eW WXI


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Even and Odd Properties of Z() and ()

Consider the driving point impedance, Z(), at the

input port of an electrical network.V() = I()

Z().

Since v(t) must be real v(t) = v*(t),

Re{V()} is even in , Im{V()} is odd in . I()

holds the same as V().

1( ) ( )2

j tv t V e d

( ) ( ) ( )

( ) ( )

j t j t j tV e d V e d V e d

V V

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )V Z I Z I V Z I

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The reflection coefficient at the input port

0 0

0 0

0 0

0 0

2 2

( ) ( ) ( )( )

( ) ( ) ( )

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

( ) ( ) ( ) ( )

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

Z Z R Z jX

Z Z Z Z jX

R Z jX R Z jX

Z Z jX Z Z jX

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Impedance and Admittance Matrices

At the nthterminal plane, the total voltage and current

is whenz= 0.

The impedance matrix

Similarly,

where

,n n n n n nV V V I I I

V Z I

I Y V

11 12 1

121

1

N

N NN

Y Y Y

YY Z

Y Y

16

An arbitrary N-port

microwave network.


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A matched 3B attenuator with a 50 Characteristic impedance

Evaluation of Scattering Parameters

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Show how [S][Z] or [Y]. AssumeZ0nare all

identical, for convenienceZ0n= 1.

where

Therefore,

For a one-port network,

,n n n n n n n nV V V I I I V V

[ ][ ] [ ][ ] [ ][ ] [ ] [ ] [ ]

([ ] [ ])[ ] ([ ] [ ])[ ]

Z I Z V Z V V V V

Z U V Z U V

1 0 0

0 1[ ]

0 1

U

1 1[ ] [ ][ ] ([ ] [ ]) ([ ] [ ])S V V Z U Z U

1111

11

1

1

zS

z

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To find [Z],

Reciprocal Networks and Lossless Networks

As in Sec. 4.2, the [Z] and [Y] are symmetric forreciprocal networks, and purely imaginary for

lossless networks.

From

1

[ ][ ] [ ][ ] [ ] [ ]

[ ] ([ ] [ ])([ ] [ ])

Z S U S Z U

Z U S U S

1

2

1[ ] ([ ] [ ])[ ]

2

n n nV V I

V Z U I

1

2

1[ ] ([ ] [ ])[ ]

2

n n nV V I

V Z U I

1

1

1

[ ] ([ ] [ ])([ ] [ ]) [ ]

[ ] ([ ] [ ])([ ] [ ])

[ ] ([ ] [ ]) ([ ] [ ])t

t t

V Z U Z U V

S Z U Z U

S Z U Z U

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If the network is reciprocal, [Z]t= [Z].

If the network is lossless, no real power delivers to

the network.

1[ ] ([ ] [ ]) ([ ] [ ])

[ ] [ ]

t

t

S Z U Z U

S S

1 1Re{[ ] [ ] } Re{([ ] [ ] )([ ] [ ] )}2 2

1Re{([ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] )}

2

1 1[ ] [ ] [ ] [ ] 0

2 2

t t t

av

t t t t

t t

P V I V V V V

V V V V V V V V

V V V V

[ ] [ ] [ ] [ ]([ ][ ]) ([ ][ ])

[ ] [ ] [ ] [ ]

t t

t

t t

V V V V S V S V

V S S V


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Generalized Scattering Parameters

Figure 4.10 (p. 181)AnN-port network with different characteristic impedances.

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0 0/ , /n n n n n na V Z b V Z

0

0

0

( )

1( )

n n n n n n

n n n n n n

n

V V V Z a b

I V V Z a bZ

2 2

2 2

1 1Re Re2 2

1 1

2 2

n n n n n n n n n

n n

P V I a b b a b a

a b

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The generalized scattering matrix can be used to

relate the incident and reflected waves,

b S a

0 fork

iij

j a k j

bS

a

_

0 fork

iij

j V k j

VS

V

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Figure on page 183


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4.4 The Transmission (ABCD) Matrix

The ABCD matrix of the cascade connection of 2 or

more 2-port networks can be easily found by

multiplying the ABCD matrices of the individual 2-

ports.

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1 2 2

1 2 2

V AV BI

I CV DI

1 2

1 2

V VA B

I C D I

1 21 1

1 11 2

V VA B

C DI I

32 2 2

2 22 3

VV A B

C DI I

31 1 1 2 2

1 1 2 21 3

VV A B A B

C D C DI I

Ex. 4.6 Evaluation of ABCD Parameters

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Relation to Impedance Matrix

From the Z parameters with -I2,

1 1 11 2 12

1 1 21 2 22

V I Z I Z

I I Z I Z

2

2 2 2

2

2

1 1 1111 21

2 1 210

1 1 11 2 12 1 1 22 11 22 12 2111 12 11 12

2 2 2 1 21 210 0 0

1 121

2 1 210

1 2 2222 21

2 20

/

1/

/

I

V V V

I

V

V I ZA Z Z

V I Z

V I Z I Z I I Z Z Z Z Z B Z Z Z Z

I I I I Z Z

I IC ZV I Z

I I ZD Z Z

I I

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If the network is reciprocal,Z12=Z21, andAD-BC=1.

Equivalent Circuits for 2-port Networks

Table 4-2

A transition between a coaxial line and a microstrip

line. Because of the physical discontinuity in thetransition from a coaxial line to a microstrip line,

electric and/or magnetic energy can be stored in the

vicinity of the junction, leading to reactive effects.


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Figure 4.12 (p.

188)A coax-to-microstrip

transition and equivalent

circuit representations.

(a) Geometry of the

transition. (b)

Representation of the

transition by a black

box.

(c) A possible equivalentcircuit for the transition

[6].

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Figure 4.13 (p. 188)Equivalent circuits for a reciprocal two-port network. (a) T equivalent.

(b) equivalent.

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4.5 Signal Flow Graphs

Very useful for the features and the construction of

the flow transmitted and reflected waves.

Nodes: Each port, i, of a microwave network has 2nodes, aiand bi. Node aiis identified with a wave

entering port i, while node biis identified with a wave

reflected from port i. The voltage at a node is equal to

the sum of all signals entering that node.

Branches: A branch is directed path between 2 nodes,

representing signal flow from one node to another.Every branch has an associated Sparameter or

reflection coefficient.

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Figure 4.14 (p. 189)The signal flow graph representation of a two-port network. (a)

Definition of incident and reflected waves. (b) Signal flow graph.


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Figure 4.18 (p. 192)Signal flow path for the two-port network with general source and

load impedances of Figure 4.17.

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Figure 4.19 (p. 192)

Decompositions of the flow graph of Figure 4.18 to find in=b1/a1and out= b2/a2. (a) Using Rule 4 on node a2. (b) Using

Rule 3 for the self-loop at node b2. (c) Using Rule 4 on node b1. (d)

Using Rule 3 for the self-loop at node a1.

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Figure 4.20 (p. 193)Block diagram of a network analyzer measurement of a two-port

device.

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Figure 4.21a (p. 194)Block diagram and signal flow graph for the Thruconnection.


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Figure 4.21b (p. 194)Block diagram and signal flow graph for theReflectconnection.

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Figure 4.21c (p. 194)Block diagram and signal flow graph for theLineconnection.

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Figure 4.22 (p. 198)Rectangular waveguide

discontinuities.

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Some common microstripdiscontinuities. (a) Open-

ended microstrip. (b) Gap

in microstrip. (c) Change in

width.

(d) T-junction. (e) Coax-to-

microstrip junction.


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Figure 4.24 (p. 200)Geometry of anH-plane step (change in width) in rectangular

waveguide.

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Figure 4.25 (p. 203)Equivalent inductance of an H-plane asymmetric step.

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Figure on page 204Reference: T.C. Edwards, Foundations for Microwave Circuit Design, Wiley, 1981.

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Figure 4.26 (p. 205)An infinitely long rectangular waveguide with surface current

densities atz= 0.


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Figure 4.27 (p. 206)An arbitrary electric or magnetic current source in an infinitely

long waveguide.

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Figure 4.28 (p. 208)A uniform current probe in a rectangular waveguide.

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Figure 4.29 (p. 210)

Various waveguide and other transmission line configurations usingaperture coupling. (a) Coupling between two waveguides wit an

aperture in the common broad wall. (b) Coupling to a waveguide

cavity via an aperture in a transverse wall. (c) Coupling between

two microstrip lines via an aperture in the common ground plane. (d)

Coupling from a waveguide to a stripline via an aperture.

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Figure 4.30 (p. 210)Illustrating the development of equivalent electric and magnetic

polarization currents at an aperture in a conducting wall (a) Normal

electric field at a conducting wall. (b) Electric field lines around an

aperture in a conducting wall. (c) Electric field lines around electricpolarization currents normal to a conducting wall. (d) Magnetic field

lines near a conducting wall. (e) Magnetic field lines near an

aperture in a conducting wall. (f)Magnetic field lines near magnetic

image theory to the


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image theory to the

problem of an aperture in

the transverse wall of a

waveguide. (a) Geometry

of a circular aperture in

the transverse wall of a

waveguide. (b) Fieldswith aperture closed. (c)

Fields with aperture open.

(d) Fields with aperture

closed and replaced with

equivalent dipoles.

(e) Fields radiated by

equivalent dipoles forx< 0; wall removed by

image theory.

(f) Fields radiated by

equivalent dipoles forz>

0; all removed by image

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Figure 4.32 (p. 214)Equivalent circuit of the aperture in a transverse waveguide wall.

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Figure 4.33 (p. 214)Two parallel waveguides coupled through an aperture in a

common broad wall.

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