TWO-PORT NETWORKS

In many situations one is not interested in the internal organization of anetwork. A description relating input and output variables may be sufficient

A two-port model is a description of a network that relates voltages and currentsat two pairs of terminals

Admittance parametersImpedance parametersHybrid parametersTransmission parameters

LEARNING GOALS

Study the basic types of two-port models

Understand how to convert one model into another

ADMITTANCE PARAMETERS

The admittance parameters describe the currents in terms of the voltages

2221212

2121111

VyVyIVyVyI

The first subindex identifies

the output port. The secondthe input port.

1 port to applied is voltage a and circuited-short is port

the when 2 port into flowingcurrent the determines 21y

The computation of the parameters follows directly from the definition

02

112

01

111

12

VV V

IyVIy

02

222

01

221

12

VV V

IyVIy

The network contains NO independent sources

LEARNING EXAMPLEFind the admittance parameters for the network

2221212

2121111

VyVyIVyVyI

][23)

211( 1111 SyVI

2111, yy determine to used Circuit

2I

1212 21

211 VIII ][

21

21 Sy

2212 , yy determine to used Circuit][

65

31

21

2222 SyVI

][21

6553

323

12221 SyVII

Next we show one use of this model

An application of the admittance parameters

2221212

2121111

VyVyIVyVyI

212

211

65

21

21

23

VVI

VVI

221 4,2 IVAI

The model plus the conditions at theports are sufficient to determine theother variables.

21

21

41

65

210

21

232

VV

VV

22 41VI

][112

][1186

13

2

2

21

AI

VV

VV

Determine the current through the4 Ohm resistor

IMPEDANCE PARAMETERS

The network contains NO independent sources

2221212

2121111

IzIzVIzIzV

The ‘z parameters’ can be derived in a manner similar to the Y parameters

01

221

01

111

22

II I

VzIVz

02

222

02

112

11

II I

VzIVz

LEARNING EXAMPLE Find the Z parameters

Write the loop equations

)(42)(42

1222

2111

IIjIjVIIjIV

24442

2221

1211

jzjzjzjz

212

211

244)42(

IjIjVIjIjV

rearranging

01

221

01

111

22

II I

VzIVz

02

222

02

112

11

II I

VzIVz

2221212

2121111

IzIzVIzIzV

LEARNING EXAMPLEUse the Z parameters to find the current through the 4 Ohmresistor

2221212

2121111

IzIzVIzIzV

Output port constraint

22 4IV

Input port constraint

11 )1(012 IV

212

211

244)42(

IjIjVIjIjV

21 )24(40 IjIj

21 4)43(12 IjIj

)43( j4j

2))43)(24(16(48 Ijjj 73.13761.12I

HYBRID PARAMETERS

The network contains NO independent sources

2221212

2121111

VhIhIVhIhV

admittance output circuit-opengain current forward circuit-short

gain voltage reverse circuit-openimpedance input circuit-short

22

21

12

11

hhhh

These parameters are very common in modeling transistors

01

221

01

111

22

VV I

IhIVh

02

222

02

112

11

II V

IhVVh

LEARNING EXAMPLE Find the hybrid parameters for the network

1I

1V

2I

2V2221212

2121111

VhIhIVhIhV

1I

1V

2I

02 V

14))3||6(12( 1111 hIV

32

636

2112

hII

1V

2I

2V01 I

32

636

1221

hVV

][91

9 222

2 ShVI

TRANSMISSION PARAMETERS

The network contains NO independent sources

221

221

DICVIBIAVV

02

1

02

1

22

II V

ICVVA

02

1

02

1

22

VV I

IDIVB

ratio current circuit-short negative Dadmittance transfer circuit-open C

impedance transfer circuit-short negative Bratio voltage circuit openA

ABCD parameters

LEARNING EXAMPLEDetermine the transmission parameters

221

221

DICVIBIAVV

02

1

02

1

22

II V

ICVVA

02

1

02

1

22

VV I

IDIVB

jAV

j

jV

111

1

12

jVII

jV

2

112

1

02 I when02 V when

112 11

11

1

Ij

I

j

jI

jD 1

211 )1(12)1||1(1 IjjjI

jV

jB 2

PARAMETER CONVERSIONS

If all parameters exist, they can be related by conventional algebraic manipulations.As an example consider the relationship between Z and Y parameters

2

11

2221

1211

2

1

2

1

2221

1211

2

1

2221212

2121111

VV

zzzz

II

II

zzzz

VV

IzIzVIzIzV

2

1

2221

1211

VV

yyyy

1

2221

1211

2221

1211

zzzz

yyyy

1121

12221zzzz

Z

12212211 zzzzZ withIn the following conversion table, the symbol stands for the determinant of thecorresponding matrix

DCBA

hhhh

yyyy

zzzz

THYZ ,,,2221

1211

2221

1211

2221

1211

INTERCONNECTION OF TWO-PORTSInterconnections permit the description of complex systems in terms of simplercomponents or subsystems

The basic interconnections to be considered are: parallel, series and cascade

PARALLEL: Voltages are the same.Current of interconnectionis the sum of currents

The rules used to derive modelsfor interconnection assume thateach subsystem behaves in thesame manner before and afterthe interconnection

SERIES: Currents are the same.Voltage of interconnection is the sumof voltages

CASCADE:Output of first subsystemacts as input for thesecond

aaaba

aaa

a

aa

a

aa VYI

yyyy

YVV

VII

I

2221

1211

2

1

2

1 ,,bbb VYI manner similar a In

baba

baba

VVVVVVIIIIII

222111

222111

,,

:sconstraint ctionInterconne

ba

ba

VVVIII

ba YYY

VYYVYVYI babbaa )(

Parallel Interconnection: Description Using Y Parameters

YVIVV

yyyy

II

2

1

2221

1211

2

1

ndescriptioctionInterconne

Series interconnection using Z parameters

bbbaaa IZVIZV ,subsystem each of nDescriptio

Interconnection constraints

a b

a b

I I IV V V

IZZIZIZV baba )(

ba ZZZ

SERIES: Currents are the same.Voltage of interconnection is the sumof voltages

Cascade connection using transmission parameters

2 1 2 1

1 1 2 2

1 1 2 2

Interconnection constraints:

a b a b

a b

a b

I I V VV V V VI I I I

a

a

aa

aa

a

a

IV

DCBA

IV

2

2

1

1

b

b

bb

bb

b

b

IV

DCBA

IV

2

2

1

1

2

2

1

1

IV

DCBA

DCBA

IV

bb

bb

aa

aa

Matrix multiplication does not commute.Order of the interconnection is important

CASCADE:Output of first subsystemacts as input for thesecond

2

2

1

1

IV

DCBA

IV

221

221

DICVIBIAVV

LEARNING EXAMPLEFind the Y parameters for the network

11

2

2j1I

1V

2I

2V

1I

1V

2I

2V

12

121 2II

IjVV

11 121 1,2 2

a ay j y j

21 221 1,2 2

a ay j y j

212

211

32

IIVIIV

2113

51

3112 1

bY

][

21

52

21

51

21

51

21

53

Sjj

jjY

LEARNING EXAMPLEFind the Z parameters of the networkNetwork A

Network BUse direct method,or given the Y parameters transform to Z… or decompose the network in a seriesconnection of simpler networks

jj

j

jjj

Za

2342

232

232

2322

1111

bZ

jj

jj

jj

jj

ZZZ ba

2365

2325

2325

2345

LEARNING EXAMPLEFind the transmission parameters

By splitting the 2-Ohm resistor,the network can be viewed as thecascade connection of two identicalnetworks

jjjj

jjjj

DCBA

121

121

2

2

)1()2())(1()1()1)(2()2)(1()2()1(

jjjjjjjjjjjjjj

DCBA

22

22

24122264241

jjjj

DCBA

jjjj

DCBA

121

Two-Ports