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7/27/2019 Two Port Networks_upload

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Two Port Networks

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Circuit Theory 2Asst. Prof. Kalyana Veluvolu

Email: [email protected]

ELEC 244-09

Dr. Kalyana Veluvolu

Why two-port network ?

When circuit designer confronted with acomplex and challenging problem, asensible approach to the problem is to

break the circuit up into a set ofmanageable subproblems, solve eachseparately and then link the subproblemsolutions together.

Introduction

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Coaxial cable between cities.

Transformers.

Transistors, Operational Amplifiers.Power transmission and distributionsystems.Modeling electronic devices.

Automatic control systems.Parameters completely describe circuitbehavior in terms of V-I at each port.

There are many practical circuits aretwo-port circuit :

Knowing the two-port parameters enables usto treat the 2-port as a black box whenembedded within a large network !

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The modular receiver designed by interconnected

two-ports. Each module task defines a simpletransformation, some desired relationship between

the molules input signal and its output.

Real life examples :Radio Receiver

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Harddisk Drive Design


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Biomedical ApplicationsDr. Kalyana Veluvolu

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Robotic ApplicationsDr. Kalyana Veluvolu

8

Industrial Robotic ApplicationsDr. Kalyana Veluvolu


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A two-port circuit is an electrical network with two

separate ports for input and output.

A one-port circuit contains exactly two terminalsat which connections to external elements.

One-port / Two-port Network

12


In many applications, what is most

importantly is to obtain :

The voltage and current relationships at the external

terminals. Parameters for which completely describe circuitbehavior in terms of at each port.

Modeling electronic devices. Remember your circuittransfer function !!!

Knowing the two-port parameters, enables us to treatthe two-port as a Black box when embedded within alarger network.

IV

)(sH

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General Conditions :

No energy stored within circuit N.

No independent sources inside circuit N.Dependent sources allowed inside N.Assume that and .11 II 22 II

N

Input port Output port

A

B

C

D

14


CB

DA

CA

NOT ALLOWED

however, connections ;

Only terminal variables are of

interest.

Inside circuit N : No interest whatsoever !

2211 and,, VIVI

All external connections to be made at ports ONLY,

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Two-Port Network Parameters

The two-port network may be driven by voltagesources or current sources.

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Two-port networks will be studied in the s-Domain.

So, we drop the s-argument for convenience,

writing

)(1 sV 1V

The basic objective :

To relate to .

Two of these four variables are independent. i.etwo simultaneous equations are sufficient.

We can categorize the combination into six categories.

11 and VI 22 and VI

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(1) z-Parameters (Impedance)),(

),(

212

211

IIfV

IIfV

(2) y-Parameters (Admittance)),(

),(

212

211

VVfI

VVfI

(3) h-Parameters (Hybrid)),(

),(

212

211

VIfI

VIfV

Six Categories

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All the 6 sets of parameters are network functions.

(4) g-Parameters (Inverse-hybrid)),(

),(

212

211

IVfV

IVfI

(6)11112

11112

),(

),(

HIGVIVfI

FIEVIVfV Inverse TransmisionParameters

(5)),(),(

221

221

IVfIIVfV ABCD - (Transmision)

Parameters

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(1) Impedance Parameters

The terminal voltages can be related to the terminalcurrents as :

2221212

2121111

IzIzV

IzIzV

In matrix form :

2

1

2

1

2221

1211

2

1 ][IIz

II

zzzz

VV

where z terms are called the impedance parameters

orz-parameters and have units of[ohms].20


The values of the parameters can be evaluated by

open-circuiting the input or output port. i.e. setting

or .01I 02I

where

01

111

2I

I

Vz

02

112

1I

I

V

z

01

221

2I

I

Vz

02

222

1I

I

Vz

Open circuit input impedance

Open circuit transfer impedance from port 1 to port 2

Open circuit transfer impedance from port 2 to port 1

Open circuit output impedance

11z

12z

21z

22z21

Thus,


11z 21z 12z 22z

Finding and .11z 21z

12z 22zFinding and .

22

Determination of parameters

and


1

111I

Vz

1

221I

Vz

We can obtain and by connecting a voltageto port 1 with port 2 open-circuited as shown in

Figure (a) and obtain and , we then get

11z 21z 1V

1I 2V

2

112

I

Vz

2

222

I

Vz

Similarly, we obtain and by connecting a

voltage to port 2 with port 1 open-circuited as shown

in Figure (b) and obtain and , we then get

2V

2I 1V

12z 22z

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Reciprocal Network

When the two-port network is linear and has

no dependent sources, the transfer impedances

are equal i.e.

The two-port network is said to be Reciprocal.

2112 zz

24

Interchanging a voltage source at one port with an

ideal ammeter at the other port produces the same

reading in a reciprocal two-port network.


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Symmetrical Network

When the two-port network input and outputimpedances are equal i.e.

The two-port network is said to be Symmetrical.

2211zz

This implies that the network has mirrorlike symmetry

about some center line; that is a line can be found that

divides the network into two similar halves.


26

Symmetrical NetworkExamples


For a reciprocal network, the T-equivalent circuit in Figure

(a) can be used. If the network is not reciprocal, a moregeneral equivalent network is shown in Figure (b).

Equivalent Circuit

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2112zz

1222zz z

1211zz

12z

11z

22z

For reciprocity :

symmetric : 2211 zz



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Example 1

Determine the z-parameters for the circuit.

1V11z 21zTo determine and : apply a voltage source to the

input port and leave the output port open.

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60)4020(

1

1

1

111

I

I

I

VzThen

4040

1

1

1

221

I

I

I

Vzand

12z 22z 2VTo determine and : Apply a voltage source tothe output port and leave the input port open .

Then

4040

2

2

2

112

I

I

I

Vz

and 70)4030(

2

2

2

222

I

I

I

Vz

7040

4060z

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The terminal voltages can be related to the terminal

currents as :

In matrix form :

2221212

2121111

VyVyI

VyVyI

2

1

2

1

2221

1211

2

1][V

Vy

V

V

yy

yy

I

I

where yterms are called the admittance parameters

orY- parameters and have units of [siemens].

(2)Admittance Parameters

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where

01

111

2VV

Iy

02

112

1VV

Iy

01

221

2V

V

Iy

02

222

1V

V

Iy

The values of the parameters can be determined

by setting or .01V 02V

Short circuit input admittance11y

Short circuit transfer admittance from port 2 to port 112yShort circuit transfer admittance from port 1 to port 221y

22y Short circuit output admittance31

Thus,



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and11y 21y 12y 22y

11y 21y(a) Finding and 12y 22y(b) Finding and

32


We can obtain and by connecting a current

to port 1 and short circuiting port 2 as shown in

Figure (a) and obtain and , we then get

11y 21y

1I

1V 2I

1

1

11 V

Iy

1

2

21 V

I

y

12y 22y

2I1I2V

Similarly, we obtain and by connecting a

voltage to port 2 and short circuiting port 1 as

shown in Figure (b) and obtain and .

we then get

2

112

VIy

2

222

VIy

Note : The impedance and admittance parameters are

collectively referred to as immittance parameters.33


For a reciprocal network, the -equivalent circuit in Figure(a) can be used. If the network is not reciprocal, a moregeneral equivalent network is shown in Figure (b).

Equivalent Circuit

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2112yyFor reciprocity :

symmetric : 2211 yy

1211yy

1222yy

12y

11y

22y


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Application : Synthesis of filters

LCladder networks for lowpass filters.

(a) Odd order

(b) Even order



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Example 2

Determine the y-parameters for the network.

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11y 21y

1I

To determine and : short circuit the output port

and connect a current source to the input port.


111 3

4

)2//4( IIV S75.0

3

41

1

1

111

I

I

V

I

y

Since the 8- is short circuited, the 2- resistor is inparallel with the 4- resistor.

37

Hence

By current division

1123

2

24

4III S5.0

3

43

2

1

1

1

221

I

I

V

Iy

2I12y 22yTo get and , short circuit the input port and

connect a current source to the output port. The 4-is short circuited, so that the 2- and 8- resistors are inparallel.


2215

4

28

8III S5.0

5

854

2

2

2

112

I

I

V

Iy

S

625.05.0

5.075.0y

By current division

Thus

38

Hence

2225

8)2//8( IIV S625.0

5

82

2

2

222

I

I

V

Iy


39

yz Relationships

2

1

2

1

2221

1211

2

1][V

Vy

V

V

yy

yy

I

I

If the matrix [y] is non-singular i.e. invertible,

then

2

11

2

1][

I

Iy

V

V

2

1

1121

1222

2

1

I

I

y

y

y

yy

y

y

y

V

V

where

y-parameters

(1)

12212211yyyyy



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40

z-parameters

2

1

2221

1211

2

1I

I

zz

zz

V

V(2)

Comparing eqns (1) and (2)

1

1121

1222

2221

1211][ y

y

y

y

yy

y

y

y

zz

zz

Likewise

1

1121

1222

2221

1211][ 1z

z

z

z

zz

z

z

z

yy

yy

where12212211zzzzz


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(3) Hybrid Parameters

The zand y parameters of a two-port network do not alwaysexist.

For example : An ideal transformer has no z -parameters.

The defining equations for the

two-port network are :

So, there is a need for developing another set of parameters.


2221212

2121111

VhIhIVhIhV

2

1

2

1

2221

1211

2

1][V

Ih

V

I

hh

hh

I

V

we obtain

In matrix form :

where h terms are known as the hybrid parameters

orh-parameters

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The third set of parameters is based on making

and the dependent variables. Thus,1

V

2I


01

111

20V

I

Vh

02

112

10I

V

Vh

01

221

2

VI

Ih

02

2

22

10I

V

Ih

11h 12h 21h 22hThe parameters , , and represent animpedance, a voltage gain, a current gain and admittance

resprctively hybridparameters.

The values of the parameters can be determined as

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where 11h

12h

21h

22h Short circuit output admittance

Short circuit input admittance

Short circuit transfer admittance from port 2 to port1

Short circuit transfer admittance from port 1 to port 2

The procedure for calculating the h parameters is similar to that used for

the zand yparameters.


D K l V l l D K l V l l


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Equivalent Circuit

The h parameters equivalent network is shown.

44

Application : Transistor circuits - Common emitter amplifier.

For reciprocity :

symmetric : .2112hh

121122211hhhhh


Example 3

Determine the hybrid parameters for the two port

network.

45

11h 21h

1ITo find and : short circuit the output port and

connect a current source to the input port.


1123

2

36

6III

3

2

1

221

I

Ih

To obtain and , open circuit the input port and

connect a voltage source to the output port.12h 22h

2V

By current division

Hence

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41

111

I

Vh

111 4)6//32( IIVHenceDr. Kalyana Veluvolu

3

2

2

112

V

Vh

222 9)63( IIV

S9

1

2

222

V

Ih

9

1

3

23

24

h

Hence

Also

Thus

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By voltage division

2213

2

36

6VVV


Dr Kalyana Veluvolu

Dr Kalyana Veluvolu


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(5) ABCD - (Transmision) Parameters

221

221

DICVI

BIAVV

The transmission parameters model provides a

measure of how a circuit transmits voltage and currentfrom source to a load.

In matrix form :

2

2

2

2

1

1][

I

VT

I

V

DC

BA

I

V


49


A B CandD

The transmission parameters are determined as

02

1

2 0IV

VA

02

1

2

0V

I

VB

02

1

2IV

IC

02

1

2

V

I

ID

A and D are dimensionless, B is in ohms and Cis in

siemens.

For reciprocity :

symmetric :

1BCAD

DA


50

Since the transmission parameters provide a direct

relationship between input and output variables, theyare very useful in ;

cascaded networks

transmission line

telephone systems

microwave networks radar systems

Application ofABCD - (Transmision) Parameters


51

Example 4Dr. Kalyana Veluvolu

Dr Kalyana Veluvolu Dr Kalyana Veluvolu


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53

(1)

(2)

Subt. Eqn (2) into (1) gives


54


Interconnection of Networks

Two-port networks can be used asbuilding blocks to design morecomplicated circuits.

A large complex network may be dividedinto sub-networks for the purposes ofanalysis and design before being

interconnected to form the complexnetwork.

55


Dr. Kalyana Veluvolu Dr. Kalyana Veluvolu


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Cascade connection

Parallel connection

Series connection

Series-Parallel connection

Parallel-Series connection

There are five waysof interconnection

56

y

Cascade Connection

57

We notice that

y

58

Or


Parallel Connection

59




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60

We notice that

Or

Series Connection

61

We notice that

62

and that

Or


63

Example 5

bbbb zzzz 22112112 10




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(1)

(2)

(3)

(4)

Subst. eqns (3) & (4) into (1) gives

Subst. eqn (4) into (2) yields

(5)

(6)

Subst. eqn (6) into (5) we get

Therefore, all the six sets of networkparameters can be used tocharacterize a wide range of two-

port networks. depending on theway two-ports are interconnected to

form a large network.

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END


Thanks for attending my lectures :-)

Best of luck with your Exams!

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