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ENE 428Microwave Engineering

Lecture 10 Signal Flow Graphs and Excitation of Waveguides

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Review (1)• Two-port network

- At low frequencies, the z, y, h, or ABCD parameters are basic network input-output parameter relations. The parameters are readily measured using short- and open-circuit tests at the terminals.

- At RF or microwave frequency, these parameter are difficult to measure

- At high frequencies (in microwave range), scattering parameters (S parameters) are defined in terms of traveling waves and completely characterize the behavior of two-port networks.

- S parameters can be easily measured using matched loads which ensure the stability of the network.

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Signal flow graphs and applications• A convenient technique to represent and analyzed the

transmission and reflection of waves in a microwave amplifier.

• Relations between the variables can be obtained using Mason’s rule.

• The flow graph technique permits expression, such as power gains and voltage gains of complex microwave amplifiers, to be derived easily.

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Rules of signal flow graph constructions

1. Each variable is designated as a node. 2. The S parameters and reflection coefficients are

represented by branches. 3. Branches enter dependent variable nodes and emanate

from independent variable nodes. The independent variable nodes are the incident waves, and the reflected waves are dependent variable nodes.

4. A node is equal to the sum of the branches entering it.

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Signal flow graph of the S parameters of a two-port network

• Observe that b1 and b2 are the dependent variable nodes and a1 and a2 are the independent variable nodes.

S11 S22

S21

S12

a1

b1

b2

a2

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Signal flow graph of a signal generator

+

-

+

-

Ig

Vg

ZS

ES

ag

bg

GS

1bS bg

ag

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Signal flow graph of a load impedance

+

-

IL

VLZL

aL

bL

GL

aL

bL

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Signal flow graph of a microwave amplifier

• Observe that the nodes bg, ag, bL, and aL are identical to a1, b1, a2, and b2, respectively.

S11 S22

S21

S12

a1

b1

b2

a2

GLGS

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Mason’s rule

• Mason’s rule is used to determine the ratio of transfer function T of a dependent to an independent variable.

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

( ) ( ) ( ) ...

1 1 21 2P 1 L 1 L 2 P 1 L 1

T1 L 1 L 2 L 3

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Variables’descriptions (1)

P1, P2, (and so on) = paths connecting the dependent and independent variables whose transfer function T is to be determined. A path is defined as a set of consecutive, codirectional branches along which no node is encountered more than once as we move in the graph from the independent to the dependent node.

L(1)= the sum of all first-order loops. A first-order loop is defined as the product of the branches encountered in a round trip as we move from a node in the direction of the arrows back to that original node.

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Variables’descriptions (2)

L(2)= the sum of all second-order loops. A second-order loop is defined as the product of any two nontouching first-order loops.

L(3)= the sum of all third-order loops. A third-order loop is defined as the product of any three nontouching first-order loops.

L(4), L(5), and so on represent fourth-, fifth-, and higher order loops.

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Variables’descriptions (3)

L(1)(P)

= the sum of all first-order loops that do not touch the path P between the independent and dependent variables.

L(2)(P) = the sum of all second-order loops that do not touch the path P between the independent and dependent variables.

L(3)(P), L(4)(P) and so on represent third-, fourth-, and higher order loops that do not touch the path P.

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Ex1 Use Mason’s rule to obtain b1/bS as shown in a microwave amplifier’s signal flow graph

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Applications of Signal flow graphs (1)•The calculation of the input reflection coefficient, IN

Observing that P1 = S11, P2 = S21L S12, L(1) = S22 L , and L(1)(1) = S22 L, we can use Mason’s rule to obtain

( ).1 11 22 L 21 L 12 12 21 L

IN 111 22 L 22 L

b S 1 S S S S SS

a 1 S 1 S

G G GG

G G

S11 S22

S21

S12

a1

b1

b2

a2

GL

GIN

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Applications of Signal flow graphs (2)•The calculation of the output reflection coefficient, OUT

Observing that P1 = S22, P2 = S21S S12, L(1) = S11 S , and L(1)(1) = S11 S, we can use Mason’s rule to obtain

22 11 21 12 12 21222

2 11 11

(1 ).

1 1S S S

OUTS S

S S S S S SbS

a S S

G G GGG G

S11 S22

S21

S12

a1

b1

b2

a2

GS

GOUT

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Excitation of Waveguides

• Several propagating modes can be excited in the waveguide along with evanescent modes.

• Formalism for excitation of the give wg mode due to an arbitrary electric and magnetic current source will be determined.

• Excitation of the wg using aperture coupling will also be briefly discussed.

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Single mode excitation using current sheets

x

y

z

a

b

Js or Ms

( ) cos sin , (1)

( ) sin cos , (2)

( ) sin cos , (3)

( ) cos sin (4)

j zx mn

j zy mn

j zx mn

j zy mn

n m x n yE A e

b a b

m m x n yE A e

a a b

m m x n yH A e

a a b

n m x n yH A e

b a b

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The amplitudes must be equal to satisfy BCs.

ˆ( ) 0 (5)

ˆ ( ) (6)

9999999999999999999999999999

9999999999999999999999999999 z

z s

E E a

a H H J

When (5) is applied to (1) and (2), we get

The discontinuity in the transverse magnetic field in (6) is equalto the electric surface current density thus at z = 0,

.mn mnA A

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The surface current density at z = 0 can be found.

This current will excite only the TEmn mode sinceMaxwell’s equations and all boundary conditions are satisfied.

ˆ ˆ( ) ( )

2 2ˆ ˆcos sin sin cos .

s x x y y

mn mn

J y H H x H H

A n A mm x n y m x n yx y

b a b a a b

99999999999999

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The electric current that excites only the TM mode can be determined analogously.

2 2ˆ ˆcos sin sin cos .

TM mn mns

B m B nm x n y m x n yJ x y

a a b b a b

99999999999999

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Excitation of WG modes by an arbitrary electric or magnetic current source.

z

z1 z2

V

E+, H+E -, H -J or M

2

2

1

1

ˆ( ) , (7)

ˆ( ) , (8)

ˆ( ) , (9)

ˆ( ) , (10)

99999999999999

9999999999999999999999999999

99999999999999

9999999999999999999999999999

j znnn n n zn

n n

j znn nn n znn n

j znnn n n zn

n n

j znn nn n znn n

E A E A e ze e z z

H A H A h zh e z z

E A E A e ze e z z

H A H A h zh e z z

where n represents any possible TE or TM mode.

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The unknown amplitude An+ can be

determined using the Lorentz reciprocity theorem. (1)

Let the volume V be the region between the wg walls,

Let and , And let be the nth wg mode in the –z

direction,

1 2 2 1 2 11 2( ) ( ) (11)s VE H E H d S E J E J dv 999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999

where S is a closed surface enclosing the volume V, and are the fields due to the current source .,i iE H

9999999999999999999999999999iJ

99999999999999

1E E

9999999999999999999999999999

1H H

9999999999999999999999999999

2 2,E H9999999999999999999999999999

2

2

ˆ( ) (12)

ˆ( ) . (13)

9999999999999999999999999999

9999999999999999999999999999

j zn n zn

j zn n zn

n

n

E E e ze e

H H h zh e

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The unknown amplitude An+ can be

determined using the Lorentz reciprocity theorem. (2)Substitution (12) and (13) into (11) with and , gives

The portion of the surface integral over the wg walls vanishes because the tangential electric field is zero. This reduces the integration to the guide cross section, S0, at the planes z1 and z2.

1J J9999999999999999999999999999

2 0J 99999999999999

( ) . (14)

99999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999

n n ns VE H E H d S E Jdv

0 0

0

ˆ ˆ ˆ( ) ( ) (15)

ˆ 0, .

999999999999999999999999999999999999999999m n m nS S zn zn

m nS

E H dS e ze h zh zds

e h zds for m n

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The unknown amplitude An+ can be

determined using the Lorentz reciprocity theorem. (3)Using (7)-(10) and (15), then reduces (14) to

Since the second integral vanishes, this further reduces to

where

2 1( ) ( )

. (16)

99999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999

9999999999999999999999999999

n n n n n n n nz zn n

nV

A E H E H dS A E H E H dS

E Jdv

1 1ˆ( ) ,

999999999999999999999999999999999999999999j z

n nV Vn znn n

nA E Jdv e ze Je dvP P

0 ˆ2 .n nsnP e h zds

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The unknown amplitude An- can be

determined using the same procedure.

By repeating with and , we get

These above results can be applied to any type of wg such as stripline and microstrip, where modal fields can be defined.

1 1ˆ( ) .j znn nV Vn zn

n n

A E Jdv e ze Je dvP P

9999999999999999999999999999999999999999992 nE E

99999999999999999999999999992 nH H

9999999999999999999999999999