EE 41139 Microwave Techniques 1

Lecture 3

Impedance and Equivalent Voltages and Currents for Non-TEM Lines

Impedance Properties of One-Port Networks

Impedance, Admittance and Scattering Matrices

Signal Flow Graphs

Impedance and Equivalent Voltages and Currents for Non-

TEM Lines

show how circuit and network concepts can be extended to handle many microwave analysis and design problems of practical interest

TEM Lines

equivalent voltage and current can be defined uniquely for TEM-type lines (require two conductors) but not so for non-TEM lines

TEM Lines

for non-TEM lines, voltage and current are only defined for a particular waveguide mode, V is related to Et and I to Ht where t denotes the transverse component

TEM Lines

the product of the equivalent V and I should yield the power flow of the mode

V/I for a single traveling wave should be equal to the characteristic impedance of the line or can be normalized to 1

TEM Lines

for an arbitrary waveguide mode with a +ve and -ve (in z) traveling waves

E x y z e x y A e A ee x y

CV e V e

H x y z h x y A e A eh x y

CI e I e

tj z j z j z j z

( , , ) ( , )( )( , )

TEM Lines

we can write the voltage and current of an equivalent transmission line as

V z V e V ej z j z( )

I z I e I ej z j z( )

TEM Lines

i.e., we are only interested in certain quantities and these quantities can be derived using circuit and network theory

TEM Lines

the incident power is given by

C Ce h zds V Iinc

C C e h zdsS

TEM Lines

the characteristic impedance of the equivalent transmission line is

TEM Lines

the wave impedance is given by

TEM Lines

if we choose the characteristic impedance of the line equal to that of the wave impedance, i.e.,

which can be either TE or TM modes

CZo w w 1

TEM Lines

therefore, if one can measure the voltage and current for each mode, the field in the waveguide can be determined as sum of the field for each mode

TEM Lines

E x y zV

Ce e x y

H x y zI

Ce h x y

( , , ) ( ) ( , )

TEM Lines

we reiterate that there are various types of impedance

intrinsic impedance

depends on material parameters but is equal to the wave impedance of a plane wave in a homogeneous medium

TEM Lineswave impedance of a particular type of wave, namely, TEM, TE and TM

depends on frequency, materials and boundary conditions

TEM Lines

characteristic impedance

unique for TEM waves, non-unique for TE and TM

TEM Lines

we will calculate the wave impedance of the TE10 mode waveguide

now consider the field equations for the TE10 rectangular waveguide

mode, the field components are

TEM Lines

ak,eAeA

akjH,0H

xsineAeAeAeA

akjE,0E

zjzjzj10

TEM Lines

xsineAeA

1H zjzj

TEM Lines

compare with the transmission line equations and the incident power

eVeV)z(V

TEM Lines

*xy CCA

AabdsHE

10TE21

TEM Lines

therefore, we can relate the field and circuit parameters for the TE10 waveguide mode

we can also the transverse resonance technique to look at the wavenumber of the TE10 mode in the y direction (height of the waveguide)

TEM Linesthe waveguide can be regarded as a transmission line with certain characteristic impedance and is shorted at both ends

y = y’

TEM Linesusing the transmission line equation to transfer the short circuit to y =y'

according to the transverse resonance technique, we have impedance looking downward +

impedance looking upward = 0

0)'yb(ktanjZ'yktanjZ yy

TEM Lines

for the above equation to be true for any value of y', ky must be zero

this is correct as ky = n/b where n = 0 for TE10 mode

0)'yb(ktanjZ'yktanjZ yy

consider the arbitrary one-port network shown here

Zinone-portnetwork

P E H ds P j W Ws

l m e 1

22* ( )

assume that

if then

E x y z V z e x y etj z( , , ) ( ) ( , )

H x y z I z h x y etj z( , , ) ( ) ( , )

e h dss

1 P VI e h ds VIs

the input impedance

for a lossless network, R = 0, therefore

P j W W

IR jXin

2 2 42 2

the input impedance is purely imaginary

the reactance is positive for an inductive load (Wm > We) and is negative for a capacitive load

Foster s reactance theorem: the rate of change of the reactance for the lossless one-port network with frequency is

from Maxwells equations

and the vector identity

E j H xH j E ,

( ) ( ) ( )A B xA B xB A

( ) ( | | | | )

( ) ( ) ( )

H j E H

H dv j E H dv

H n ds j W W

( ) ( )* *EH E

H ds j W Wes

( ) ( )* * * *VI

e h V Ieh V

I e h VIe

h ds j W Ws

( ) ( )* * * *VI

e h V Ieh V

I e h VIe

h ds j W Ws

Note that

and therefore,

E x y z V z e x y e

H x y z I z h x y e

( , , ) ( ) ( , )

( )* *

eh ds real

V= j XI for lossless line

( ) ( )

| | ( )

* * * *

I e h ds VI V

I j W W

Isimilarly

e m e m

poles and zeros must alternate in position as the slope is always positive

zero poleX

Even and Odd Properties of Z(w) and G(w)

define the Fourier transform as

note that v(t) must be a real quantity, i.e., v(t) = v*(t), therefore,

or V(-) = V*()

v t V e dj t( ) ( ) 1

V e d V e d V e dj t j t j t( ) * ( ) * ( )

note that we can only measure V(), we need its complex conjugate to obtain v(t), similar arguments hold for I()

V*(-) = V()=Z()I()=Z*(-)I*(-)=Z*(- Z() =Z*(-

therefore, the real part of Z, i.e, R is an even function of while the imaginary part X is an odd function of

the reflection coefficient also has an even real part and an odd imaginary part

( )( )

( ) ( )

( )( )

( ) ( )

( ) ( ), . . , ( ) * ( )

R jX Z

R jX Zi e

( )( )

( ) ( )

( )( )

( ) ( )

( ) ( ), . . , ( ) * ( )

R jX Z

R jX Zi e

V , I n n

N-port microwave network, each port has a reference plane tn

at the reference plane of port N, we have

if we are only interested in knowing the relationship among the voltages and currents at the ports, we can define a impedance matrix Z so that

[ ] [ ][ ]V Z I

the element Zij of the impedance matrix

is given by

similar equations can be written for the admittance matrix

jI k jk

for reciprocal network, the impedance (admittance) matrix is symmetric

for lossless network, all matrix elements are purely imaginary

the scattering matrix relate the voltage waves incident on the ports to those reflected from the ports

the scattering parameter is written as

[ ] [ ][ ]V S V

each element is given by

Sij is found by driving port j with an incident wave of voltage Vj

+, and measuring the reflected amplitude coming Vi

-, out of port i

jV k jk

the incident waves on all ports except the jth port are set to zero which implies that all these ports are terminated with match loads

for a reciprocal network [S] = [S]t, i.e., the matrix is symmetric

for a lossless network

for all I,j

the scattering parameters can be readily measured by a Network Analyzer

S Skik

Nkj ij

A Shift in Reference Planes

the S parameters relate the amplitude of traveling wave incident on and reflected from a microwave network, phase reference planes must be specified for each port of the network

we need to know how the S parameters change when the reference planes are moved

z lz l =0=l1

V1‘-

port NlN

let the original reference at zl=0, the

incident and reflected port voltages are related by

at the new reference planes at zn=ln

[ ] [ ][ ]V S V

[ ' ] [ ' ][ ' ]V S V

for a lossless transmission line

V V e l

In matrix form, we have

[ ] [ ] [ ]' '

[ ] [ ] [ ]' 'V

note that each diagonal term is shifted by twice the electrical length of the shift in the reference plane, i.e., the shift is a round trip shift

[ ' ] [ ]S

Generalized Scattering Parameters

note that not all the ports are of the same characteristic impedance, let the nth port has a characteristic impedance of Zon

we define a new set of wave amplitude as

an represents an incident wave at the

nth port and bn represents a reflected

wave from that port

n n on

at the reference plane, we have

V V V Z a b

n n n on n n

( ) ( )1 1

the average power delivered to the nth port is

the average power delivered through port n is the incident power minus the reflected power

P V I a b b a b a a bn n n n n n n n n n n 1

22 2 2 2Re{ } Re{| | | | ( )} (| | | | )* * *

P V I a b b a b a a bn n n n n n n n n n n 1

22 2 2 2Re{ } Re{| | | | ( )} (| | | | )* * *

a generalized scattering matrix can be defined with the matrix element given by

ja k jk

i oiV k jk

note that it only depends on the ratio of the characteristic impedances, not the characteristic impedance themselves

Signal Flow Graph

the primary components of a signal flow graph are nodes and branches

Signal Flow Graph

for a two-port network, we have

[S]port 1 port 2

Signal Flow Graph

each port has two nodes, node a is identify with a wave entering the port while node b is identify with a wave reflected from the port

nodes a and b are connected by a branch and each branch is associated with a scattering parameter

Simplification of Signal Flow Graphs

series rule: two branches, whose common node has only one incoming and one outgoing wave many be combined to form a single branch

V2 V3V1

S21 S32

V1 V3S21S32

132213 VSSV

parallel rule: two branches that are parallel may be combined as

V 1 V 2

Sa Sb+

1ba2 V)SS(V

self-loop rule: when a loop has a self-loop, it can be eliminated

V2 V3V1

S21 S32

S21/(1-S22)

splitting rule:a node may be split into two separate nodes

V2 V3V1

S21 S32

V4 S21

S42 V4

142212424 VSSVSV

Mason’s rule

independent variable node is the node of an incident wave

dependent variable node is the node of a reflected wave

Mason’s rule

path is a series of codirectional branches from an independent node to a dependent node, along which no node is crossed more than once, the value of a path is the product of all the branch coefficients along the path

Mason’s rule

first-order loop is the product of branch coefficient encountered in a round trip from a node back to that same node, without crossing the node twice

second-order loop is the product of any two nontouching first-order loop

third-order loop is the product of three nontouching first-order loop

Mason’s rule

the Mason’s rule for the ratio T of the wave amplitude of a dependent variable to the wave amplitude of an independent variable is given as

,are the coefficients of the possible paths connecting the independent and dependent variables

TP L L P L

221 1 2 1 1

1 1 2 3

[ ( ) ( ) ] [ ( ) ]

( ) ( ) ( )

P P1 2, ,

Mason’s rule

are the sums of all the first-order, second-order, … loops

are the sums of all first-order, second-order, … loops that do not touch the first path between the variables

L L( ), ( )1 2

L L( ) , ( )1 21 1

Mason’s rule

are the sums of all first-order, second-order, … loops that do not touch the second path between the variables and so on, for all the path between the independent and dependent variables

L L( ) , ( )1 22 2

flow graph simplification

find in

Splittling rule

self-loop rule

1-S22l

series rule follows by the parallel rule yields

Mason’s rule

Two paths

TP L L P L

221 1 2 1 1

1 1 2 3

[ ( ) ( ) ] [ ( ) ]

( ) ( ) ( )

a2P S Sl2 21 12 P S1 11

first-order loop

there is no second-order loopthe sum of all the first-order loop not touching P1

the sum of all the first-order loop not touching P2 is zero

L S l( )1 22

L S l( )1 122

Therefore,

this is the same result obtained by the other method

S S S S

11 22 21 12

EE 41139 Microwave Techniques 1

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