EKT 441 MICROWAVE COMMUNICATIONS CHAPTER 3: MICROWAVE NETWORK ANALYSIS (PART 1)

EKT 441MICROWAVE COMMUNICATIONS

CHAPTER 3:

MICROWAVE NETWORK ANALYSIS (PART 1)

NETWORK ANALYSIS

Many times we are only interested in the voltage (V) and current (I) relationship at the terminals/ports of a complex circuit.

If mathematical relations can be derived for V and I, the circuit can be considered as a black box.

For a linear circuit, the I-V relationship is linear and can be written in the form of matrix equations.

A simple example of linear 2-port circuit is shown below. Each port is associated with 2 parameters, the V and I.

Port 1 Port 2

R

CV1

I1 I2

V2

Convention for positivepolarity current and voltage

+

-

NETWORK ANALYSIS

For this 2 port circuit we can easily derive the I-V relations.

We can choose V1 and V2 as the independent variables, the I-V

relation can be expressed in matrix equations.

21

11

2

221

VCjVI

CVjII

RR

C

I1I2

V2jCV2

R

V1

I1

V

2

211

1 VVIR

2 - Ports

I2

V2V1

I1

Port 1 Port 2

R

CV1

I1 I2

V2

2

1

2221

1211

2

1V

V

yy

yy

I

I

2

111

11

2

1V

V

CjI

I

RR

RR

Network parameters(Y-parameters)

NETWORK ANALYSIS

To determine the network parameters, the following relations can be used:

For example to measure y11, the following setup can be used:

0211

11

VV

Iy

0121

12

VV

Iy

0212

21

VV

Iy

0122

22

VV

Iy

This means we short circuit the port

2

1

2221

1211

2

1V

V

yy

yy

I

I

VYI or

2 - Ports

I2

V2 = 0V1

I1

Short circuit

NETWORK ANALYSIS

By choosing different combination of independent variables, different network parameters can be defined. This applies to all linear circuits no matter how complex.

Furthermore this concept can be generalized to more than 2 ports, called N - port networks.

2 - Ports

I2

V2V1

I1

2

1

2221

1211

2

1I

I

zz

zz

V

VV1 V2

I1 I2

2

1

2221

1211

2

1V

I

hh

hh

I

VLinear circuit, because allelements have linear I-V relation

ABCD MATRIX

Of particular interest in RF and microwave systems is ABCD parameters. ABCD parameters are the most useful for representing Tline and other linear microwave components in general.

221

221

2

2

1

1

DICVI

BIAVV

I

V

DC

BA

I

V

02

1

2

IV

VA

02

1

2

VI

VB

02

1

2

VI

ID

02

1

2

IV

IC

(4.1a)

(4.1b)

2 -Ports

I2

V2V1

I1

Take note of the direction of positive current!

Short circuit Port 2Open circuit Port 2

ABCD MATRIX

The ABCD matrix is useful for characterizing the overall response of 2-port networks that are cascaded to each other.

3

3

33

33

1

1

3

3

22

22

11

11

1

1

I

V

DC

BA

I

V

I

V

DC

BA

DC

BA

I

VI2’

V2V1

I1I2

V3

I3

11

11

DC

BA

22

22

DC

BA

Overall ABCD matrix

THE SCATTERING MATRIX

Usually we use Y, Z, H or ABCD parameters to describe a linear two port network.

These parameters require us to open or short a network to find the parameters.

At radio frequencies it is difficult to have a proper short or open circuit, there are parasitic inductance and capacitance in most instances.

Open/short condition leads to standing wave, can cause oscillation and destruction of device.

For non-TEM propagation mode, it is not possible to measure voltage and current. We can only measure power from E and H fields.


Hence a new set of parameters (S) is needed which Do not need open/short condition. Do not cause standing wave. Relates to incident and reflected power waves, instead of

voltage and current.

• As oppose to V and I, S-parameters relate the reflected and incident voltage waves.• S-parameters have the following advantages:1. Relates to familiar measurement such as reflection coefficient, gain, loss etc.2. Can cascade S-parameters of multiple devices to predict system performance (similar to ABCD parameters).3. Can compute Z, Y or H parameters from S-parameters if needed.

• As oppose to V and I, S-parameters relate the reflected and incident voltage waves.• S-parameters have the following advantages:1. Relates to familiar measurement such as reflection coefficient, gain, loss etc.2. Can cascade S-parameters of multiple devices to predict system performance (similar to ABCD parameters).3. Can compute Z, Y or H parameters from S-parameters if needed.

THE SCATTERING MATRIX Consider an n – port network:

Each port is considered to beconnected to a Tline withspecific Zc.

Linear n - portnetwork

T-line orwaveguide

Port 2

Port 1

Port n

Reference planefor local z-axis(z = 0)

Zc2

Zc1

Zcn

THE SCATTERING MATRIX There is a voltage and current on each port. This voltage (or current) can be decomposed into the incident (+) and reflected component (-).

V1+ V1

-

Linear n - portNetwork

Port 2

Port 1

Port n

z = 0

V1

I1

+z

Port 1

111 VVV

V1

V1+

+

- V1-

+

1111

111

VV

III

cZ

222

22

0 VVVV

eVeVzV zjzj

222

22

0 IIII

eIeIzI zjzj

THE SCATTERING MATRIX The port voltage and current can be normalized with respect to the impedance

connected to it. It is customary to define normalized voltage waves at each port as:

ciii

ci

ii

ZIa

Z

Va

(4.3a)

Normalizedincident waves

Normalizedreflected waves

ciii

ci

ii

ZIb

Z

Vb

(4.3b)

i = 1, 2, 3 … n

THE SCATTERING MATRIX Thus in general:

Vi+ and Vi

- are propagatingvoltage waves, which canbe the actual voltage for TEMmodes or the equivalent voltages for non-TEM modes.(for non-TEM, V is defined proportional to transverse Efield while I is defined propor-tional to transverse H field, see[1] for details).

Vi+ and Vi

- are propagatingvoltage waves, which canbe the actual voltage for TEMmodes or the equivalent voltages for non-TEM modes.(for non-TEM, V is defined proportional to transverse Efield while I is defined propor-tional to transverse H field, see[1] for details).

V2+

V2-

V1+ V1

-

Vn+

Vn-


T-line orwaveguide

Port 2

Port 1

Port nZc1

Zc2

Zcn

THE SCATTERING MATRIX If the n – port network is linear (make sure you know what this means!), there is a linear relationship between the

normalized waves. For instance if we energize port 2:

V2+

V1-

Vn-

Port 2

Port 1

Port nZc1

Zc2

Zcn

V2-


Constant thatdepends on the network construction

2222

VsV

22

VsVnn

2121

VsV

THE SCATTERING MATRIX Considering that we can send energy into all ports, this can be

generalized to:

Or written in Matrix equation:

Where sij is known as the generalized Scattering (S) parameter, or just S-parameters for short. From (4.3), each port i can have different characteristic impedance Zci

nnnnnnn

VsVsVsVsV 332211

(4.4a)

(4.4b)or

nn

VsVsVsVsV13132121111

nnVsVsVsVsV

23232221212

VSV

nnnnn

n

n

nV

V

V

sss

sss

sss

V

V

V

:

...

:::

...

...

:2

1

21

22221

11211

2

1


Consider the N-port network shown in figure 4.1.

Figure 4.1: An arbitrary N-port microwave network


Vn+ is the amplitude of the voltage wave incident on port n.

Vn- is the amplitude of the voltage wave reflected from port n.

The scattering matrix or [S] matrix, is defined in relation to these incident and reflected voltage wave as:

nNNN

N

n V

V

V

SS

S

SSS

V

V

V

.

.

.

....

......

......

......

.....

...

.

.

.2

1

1

21

11211

2

1

[4.1a]


VSVor [4.1b]

jkforVj

iij

k

V

VS

,0

A specific element of the [S] matrix can be determined as:

[4.2]

Sij is found by driving port j with an incident wave Vj+, and measuring

the reflected wave amplitude, Vi-, coming out of port i.

The incident waves on all ports except j-th port are set to zero (which means that all ports should be terminated in matched load to avoid reflections).Thus, Sii is the reflection coefficient seen looking into port i when all other ports are terminated in matched loads, and Sij is the transmission coefficient from port j to port i when all other ports are terminated in matched loads.

THE SCATTERING MATRIX For 2-port networks, (4.4) reduces to:

Note that Vi+ = 0 implies that we terminate i th port with its

characteristic impedance. Thus zero reflection eliminates standing wave.

2

1

2

1

2221

1211

2

1

V

VS

V

V

ss

ss

V

V(4.5a)

(4.5b)01010202

2

1

12

2

2

22

1

2

21

1

1

11

VVVV

V

Vs

V

Vs

V

Vs

V

Vs


2 – Port Zc2

Zc2Zc1

Zc1Vs

V1+

02021

2

21

1

1

11

VV

V

Vs

V

Vs

V1-

V2-

V1-

2 – PortZc1

Zc2Zc1

Zc2Vs

V2-

V2+

01012

1

12

2

2

22

VV

V

Vs

V

Vs

Measurement of s11 and s21:

Measurement of s22 and s12:

THE SCATTERING MATRIX Input-output behavior of network is defined in terms of normalized

power waves S-parameters are measured based on properly terminated

transmission lines (and not open/short circuit conditions)

1* ][][ tss

THE SCATTERING MATRIXReciprocal and Lossless networks Impedance and admittance matrices are symmetric for reciprocal

networks A symmetric network happens when:

It is also purely imaginary for lossless network (no real power can be delivered to the network)

A matrix that satisfies the condition of (4.6b) is called a unitary matrix

tss ][][

1* ][][ tss

(4.6a)

(4.6b)

THE SCATTERING MATRIX Transpose of a Matrix (taken from Engineering

Maths 4th Ed by KA Stroud)

dc

baS

db

catS

Transpose of [S], written as [S]t

THE SCATTERING MATRIX Symmetrical Matrix (taken from Engineering

Maths 4th Ed by KA Stroud)

2212

2111

aa

aaS

It is symmetrical when aij = aji

When a [S] is symmetric, it is also reciprocal

If

THE SCATTERING MATRIXReciprocal and Lossless networks (cont) The matrix equation in (4.6b) can be re-written in;

OR

(4.7)

0

1

1

*

1

*

N

kkjki

N

kkiki

SS

SS For i = j

For i ≠ j

Used to determine reciprocality for a 2 port

network

1||||2111SS

THE SCATTERING MATRIX (Ex)

Find the S parameters of the 3 dB attenuator circuit shown in Figure 4.2.

Figure 4.2: A matched 3 dB attenuator with a 50 Ω characteristic impedance.


From the following formula, S11 can be found as the reflection coefficient seen at port 1 when port 2 is terminated with a matched load (Z0 =50 Ω);

The equation becomes;

jforkV

j

i

ijkV

VS

0

0220

)1(

0

)1(

0

)1(

0

1

1

11 Z

in

in

VV ZZ

ZZ

V

VS

On port 2


To calculate Zin(1), we can use the following formula;

Thus S11 = 0. Because of the symmetry of the circuit, S22 = 0.

S21 can be found by applying an incident wave at port 1, V1

+, and measuring the outcome at port 2, V2-. This is

equivalent to the transmission coefficient from port 1 to port 2:

50)5056.8(8.141

)5056.8(8.14156.8

)1(

inZ

0

1

2

212

VV

VS


From the fact that S11 = S22 = 0, we know that V1- = 0 when

port 2 is terminated in Z0 = 50 Ω, and that V2+ = 0. In this

case we have V1+ = V1 and V2

- = V2.

Where 41.44 = (141.8//58.56) is the combined resistance of 50 Ω and 8.56 Ω paralled with the 141.8 Ω resistor. Thus, S21 = S12 = 0.707

11227071.0

56.850

50

56.844.41

44.41VVVV


A two port network is known to have the following scattering matrix:

a) Determine if the network is reciprocal and lossless.

b) If port 2 is terminated with a matched load, what is the return loss seen at port 1?

c) If port 2 is terminated with a short circuit, what is the return loss seen at port 1?

02.04585.0

4585.0015.0S


Q: Determine if the network is reciprocal and lossless Since [S] is not symmetric, the network is not reciprocal. Taking the 1st

column, (i = 1) gives;

So the network is not lossless. Q: If port two is terminated with a matched load, what is the return loss

seen at port 1? When port 2 is terminated with a matched load, the reflection coefficient

seen at port 1 is Γ = S11 = 0.15. So the return loss is;

dBRL 5.16)15.0log(20||log20

1745.0)85.0()15.0(|||| 222

21

2

11 SS

Used to determine reciprocality for a 2 port

network

1||||2111SS


Q: If port two is terminated with a short circuit, what is the return loss seen at port 1?

When port 2 is terminated with a short circuit, the reflection coefficient seen at port 1 can be found as follow

From the definition of the scattering matrix and the fact that V2+ = - V2

- (for a short circuit at port 2), we can write:

2121112121111

VsVsVsVsV

2221212221212VsVsVsVsV


The second equation gives;

Dividing the first equation by V1+ and using the above result gives the

reflection coefficient seen as port 1 as;

1

22

21

2 1V

S

SV

22

2112

11

1

2

1211

1

1

1 S

SSS

V

VSS

V

V

452.02.01

)4585.0)(4585.0(15.0

00


The return loss is;

Important points to note: Reflection coefficient looking into port n is not equal to Snn, unless all

other ports are matched Transmission coefficient from port m to port n is not equal to Snm,

unless all other ports are matched S parameters of a network are properties only of the network itself

(assuming the network is linear) It is defined under the condition that all ports are matched Changing the termination or excitation of a network does not change its

S parameters, but may change the reflection coefficient seen at a given port, or transmission coefficient between two ports

dBRL 9.6)452.0log(20||log20

EKT 441 MICROWAVE COMMUNICATIONS CHAPTER 3: MICROWAVE NETWORK ANALYSIS (PART 1)

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