EKT 441MICROWAVE COMMUNICATIONS

CHAPTER 3:MICROWAVE NETWORK ANALYSIS (PART 1)

NETWORK ANALYSIS Most electrical circuits can be modeled as a “black box” that

contains a linear network comprising of R, L, C and dependant sources.

Has four terminals, 2-input ports and 2-output ports Hence, large class of electronics can be modeled as two-port

networks, which completely describes behavior in terms of voltage (V) and currents (I) (illustrated in Fig 1 below)

Figure 1

NETWORK ANALYSIS Several ways to characterize this network, such as

1. Impedance parameters2. Admittance parameters3. Hybrid parameters4. Transmission parameters

Scattering parameters (S-parameters) is introduced later as a technique to characterize high-frequency and microwave circuits

NETWORK ANALYSISImpedance Parameters Considering Figure 1, considering network is linear, principle of

superposition can be applied. Voltage, V1 at port 1 can be expressed in terms of 2 currents as follow;

Since V1 is in Volts, I1 and I2 are in Amperes, Z11 and Z12 must be in Ohms. These are called impedance parameters

Similarly, for V2, we can write V2 in terms of I1 and I2 as follow;

2121111 IZIZV

2221212 IZIZV

NETWORK ANALYSISImpedance Parameters (cont) Using the matrix representation, we can write;

Or

Where [Z] is called the impedance matrix of the two-port network

2

1

2221

1211

2

1

II

ZZZZ

VV

IZV

NETWORK ANALYSISImpedance Parameters (cont) If port 2 of the network is left open, then I2 will be zero. In this

condition;

Similarly, when port 1 of the network is left open, then I1 will be zero. In this condition;

01

111

2

I

IVZ

01

221

2

I

IVZand

02

112

1

I

IVZ

02

222

1

I

IVZand

NETWORK ANALYSISExample 1 Find the impedance parameters of the 2-port network shown

here

NETWORK ANALYSISExample 1: Solution If I2 is zero, then V1 and V2 can be found from Ohm’s Law as 6I1.

Hence from the equations

Similarly, when the source is connected at port 2 and port 1 has an open circuit, we find that;

66

1

1

01

111

2II

IVZ

I

66

1

1

01

221

2II

IVZ

I

212 6IVV

NETWORK ANALYSISExample 1: Solution Hence, from

Therefore,

66

2

2

02

112

1II

IVZ

I

66

2

2

02

222

1II

IVZ

I

6666

2221

1211

ZZZZ

NETWORK ANALYSISExample 2 Find the impedance parameters of the 2-port network shown

here

NETWORK ANALYSISExample 2: Solution As before, assume that the source is connected at port-1 while

port 2 is open. In this condition, V1 = 12I1 and V2 = 0. Therefore,

Similarly, with a source connected at port-2 while port-1 has an open circuit, we find that,

1212

1

1

01

111

2II

IVZ

I

001

221

2IIVZ

22 3IV 01 Vand

and

NETWORK ANALYSISExample 2: Solution Hence,

Therefore,

002

112

1

I

IVZ

33

2

2

02

222

1II

IVZ

I

30012

2221

1211

ZZZZ

and

NETWORK ANALYSISAdmittance Parameters Consider again Figure 1. Assuming the network is linear,

principle of superposition can be applied. Current, I1 at port 1 can be expressed in terms of 2 voltages as follow;

Since I1 is in Amperes, V1 and V2 are in Volts, Y11 and Y12 must be in Siemens. These are called admittance parameters

Similarly, we can write I2 in terms of V1 and V2 as follow;

2121111 VYVYI

2221212 VYVYI

NETWORK ANALYSISAdmittance Parameters (cont) Using the matrix representation, we can write;

Or

Where [Y] is called the admittance matrix of the two-port network

2

1

2221

1211

2

1

VV

YYYY

II

VYI

NETWORK ANALYSISAdmittance Parameters (cont) If port 2 of the network has a short circuit, then V2 will be zero. In

this condition;

Similarly, with a source connected at port 2, and a short circuit at port 1, then V1 will be zero. In this condition;

01

111

2

V

VIY

01

221

2

V

VIYand

02

112

1

V

VIY

02

222

1

V

VIYand

NETWORK ANALYSISExample 3 Find the admittance parameters of the 2-port network shown

here

NETWORK ANALYSISExample 3: Solution If V2 is zero, then I1 is equal to 0.05V1, I2 is equal to -0.05V1.

Hence from the equations above;

Similarly, with a source connected at port 2 and port 1 having a short circuit, we find that;

SVV

VIY

V

05.005.0

1

1

01

111

2

SVV

VIY

V

05.005.0

1

1

01

221

2

212 05.0 VII

NETWORK ANALYSISExample 3: Solution (cont) Hence, from

Therefore,

SVV

VIY

V

05.005.0

2

2

02

112

1

SVV

VIY

V

05.005.0

2

2

02

222

1

05.005.005.005.0

2221

1211

YYYY

NETWORK ANALYSISExample 4 Find the admittance parameters of the 2-port network shown

here

NETWORK ANALYSISExample 4: Solution Assuming that a source is connected to at port-1 while keeping

port 2 as a short circuit, we find that;

And if voltage across 0.2S is VN, then;

Therefore;

AVVI 111 325.00225.0

025.02.01.0025.02.01.0

VVVIVN 25.3325.0225.00225.0

025.02.01

11

AVVI N 12 25.32.02.0

NETWORK ANALYSISExample 4: Solution (cont) Therefore;

Similarly, with a source at port-2 and port-1 having a short circuit;

SVIY

V

0692.0325.00225.0

01

111

2

SVIY

V

0615.025.32.0

01

221

2

AVVI 212 325.0025.0

025.02.01.0025.01.02.0

NETWORK ANALYSISExample 4: Solution (cont) And if voltage across 0.1S is VM, then,

Therefore,

Hence;

VVVIVM 25.32

325.0125.0025.0

025.01.02

22

SVIY

V

0615.025.32.0

02

112

1

SVIY

V

0769.0325.0025.0

02

222

1

AVVI M 21 25.32.01.0

NETWORK ANALYSISExample 4: Solution (cont) Therefore,

0769.00615.00615.00692.0

2221

1211

YYYY

NETWORK ANALYSISHybrid Parameters Consider again Figure 1. Assuming the network is linear, principle of

superposition can be applied. Voltage, V1 at port-1 can be expressed in terms of current I1 at port-2 and voltage V2 at port-2, as follow;

Similarly, we can write I2 in terms of I1 and V2 as follow;

Since V1 and V2 are in volts, while I1 and I2 are in amperes, parameter h11 must be in ohms, h12 and h21 must be dimensionless, and h22 must be in siemens.

These are called hybrid parameters.

2121111 VhIhV

2221212 VhIhI

NETWORK ANALYSISHybrid Parameters (cont) Using the matrix representation, we can write;

Hybrid parameters are especially important in transistor circuit analysis. The parameters are defined as follow; If port-2 has a short circuit, then V2 will be zero.

This condition gives;

2

1

2221

1211

2

1

VI

hhhh

IV

01

111

2

V

IVh

01

221

2

V

IIhand

NETWORK ANALYSISHybrid Parameters (cont) Similarly, with a source connected to port-2 while port-1 is open;

Thus, parameters h11 and h21 represent the input impedance and the forward current gain, respectively, when a short circuit is at port-2.

Similarly, h12 and h22 represent reverse voltage gain and the output admittance, respectively, when port-1 has an open circuit.

In circuit analysis, these are generally denoted as hi, hf, hr and ho, respectively.

02

112

1

I

VVh

02

222

1

I

VIhand

NETWORK ANALYSISExample 5: Hybrid parameters Find hybrid parameters of the 2-port network shown here

NETWORK ANALYSISExample 5: Solution With a short circuit at port-2,

And using the current divider rule, we find that

111 14363612 IIV

112 32

636 III

NETWORK ANALYSISExample 5: Solution (cont) Therefore;

Similarly, with a source at port-2 and port-1 having an open circuit;

And

1401

111

2VIVh 3

2

01

221

2

V

IIh

222 9)63( IIV

21 6IV

NETWORK ANALYSISExample 5: Solution (cont) Because there is no current flowing through the 12Ω resistor,

hence;

Thus,

32

96

2

2

02

112

1

II

VVh

VS

VIh

I91

02

222

1

Shhhh

91

32

3214

2221

1211

NETWORK ANALYSISTransmission Parameters Consider again Figure 1. Since the network is linear, the

superposition principle can be applied. Assuming that it contains no independent sources, Voltage V1 and current at port 1 can be expressed in terms of current I2 and voltage V2 at port-2, as follow;

Similarly, we can write I1 in terms of I2 and V2 as follow;

Since V1 and V2 are in volts, while I1 and I2 are in amperes, parameter A and D must be in dimensionless, B must be in Ohms, and C must be in Siemens.

221 BIAVV

221 DICVI

NETWORK ANALYSISTransmission Parameters (cont) Using the matrix representation, we can write;

Transmission parameters, also known as elements of chain matrix, are especially important for analysis of circuits connected in cascade. These parameters are determined as follow; If port-2 has a short circuit, then V2 will be zero.

This condition gives;

2

2

1

1

IV

DCBA

IV

02

1

2

V

IVB

02

1

2

V

IIDand

NETWORK ANALYSISTransmission Parameters (cont) Similarly, with a source connected at port-1 while port-2 is open,

we find;

02

1

2

I

VVA

02

1

2

I

VICand

NETWORK ANALYSISExample 6: Transmission parameters Find transmission parameters of the 2-port network shown here

NETWORK ANALYSISExample 6: Solution With a source connected to port-1, while port-2 has a short circuit

(so that V2 is zero)

Therefore;

12 II

101

1

2VIVB

11 IV

102

1

2

V

IID

and

and

NETWORK ANALYSISExample 6: Solution (cont) Similarly, with a source connected at port-1, while port-2 is

open (so that I2 is zero)

Hence;

Thus;

12 VV

002

1

2

I

VIC1

02

1

2

I

VVA

01 Iand

and

1011

DCBA

NETWORK ANALYSISExample 7: Transmission parameters Find transmission parameters of the 2-port network shown here

NETWORK ANALYSISExample 7: Solution With a source connected to port-1, while port-2 has a short circuit

(so that V2 is zero), we find that

Therefore;

111 12

111 I

jjI

jV

)2(01

1

2

jIVB

V

112 1

1

11

1

Ij

I

j

jI

jIID

V

102

1

2

and

and

NETWORK ANALYSISExample 7: Solution (cont) Similarly, with a source connected at port-1, while port-2 is

open (so that I2 is zero)

Hence;

Thus;

111111 IjjI

jV

jVIC

I

02

1

2

jVVA

I

102

1

2

121 Ij

V

and

and

jjjj

DCBA

121

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

DICVIBIAVVIV

DCBA

IV

02

1

2

IVVA

02

1

2

VIVB

02

1

2

VIID

02

1

2

IVIC

(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

IV

DCBA

IV

IV

DCBA

DCBA

IVI2’

V2V1

I1I2

V3

I3

11

11DCBA

22

22DCBA

Overall ABCD matrix

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

V2

211

1 VVI R

2 - Ports

I2

V2V1

I1

Port 1 Port 2

R

CV1

I1 I2

V2

2

1

2221

1211

2

1VV

yyyy

II

2

111

11

2

1VV

CjII

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

Iy01

21

12

VV

Iy

0212

21

VV

Iy01

22

22

VV

Iy

This means we short circuit the port

2

1

2221

1211

2

1VV

yyyy

II

VYI or

2 - Ports

I2

V2 = 0V1

I1Short 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

1II

zzzz

VVV1 V2

I1 I2

2

1

2221

1211

2

1VI

hhhh

IVLinear circuit, because all

elements have linear I-V relation

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.

THE SCATTERING MATRIX 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.