Fundamentals of Electric CircuitsChapter 19

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Overview

• The concept of a two-port network.

• The relationship between input and output current and voltages.

• Combinations of networks in series, parallel, and cascaded.

2

Figure 19.1

Two Port Network

• A two port network is an electrical network with two separate ports for input and output.

• The two port network has terminal pairs acting as access points.

• This means that the current entering one terminal of a pair leaves the other terminal in the pair.

• Three terminal devices, such as transistors can be configured as two port devices.

4

Parameters• To characterize a two-port network requires

that we relate the terminal quantities V1, V2, I1, and I2.

• Out of these only two are independent.• The terms that relate to these voltages and

currents are called parameters.• Impedance and admittance parameters are

commonly used in the synthesis of filters.• They are also important in the design and

analysis of impedance-matching networks and power distribution networks.

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Impedance Parameters

• A two-port network may be either voltage driven or current driven

• The terminal voltages can be related to the terminal currents as:

6

1 11 1 12 2

2 21 1 22 2

V z I z I

V z I z I

Impedance Network II• The values of the parameters can be

evaluated by setting the input or output port open circuits (i.e. set the current to zero).

• These are referred to

asthe open-circuit

impedance parameters.

7

2 1

2 1

1 111 12

1 20 0

2 221 22

1 20 0

I I

I I

V Vz z

I I

V Vz z

I I

Open Circuit Parameters

• These parameters are as follows:

• z11 Open circuit input impedance

• z12 Open circuit transfer impedance from port 1 to port 2

• z21 Open circuit transfer impedance from port 2 to port 1

• z22 Open circuit output impedance

• When z11=z22, the network is said to be symmetrical.

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z Parameters

• When the network is linear and has no dependent sources, the transfer impedances are equal (z12=z21), the network is said to be reciprocal.

• This means that if the input and output are switched, the transfer impedances remain the same.

• Any two-port network that is composed entirely of resistors, capacitors, and inductors must be reciprocal.

9

Figure 19.4

z Parameters II

• It should be noted that an ideal transformer has no z parameters.

• The equivalent circuit for two port networks is shown below:

11

Example

12

Example

13

1 2Find and I I

o1 1 2 1

2 1 2 2 2

40 20 , =100 0

30 50 , = -10

j

j

V I I V

V I I V I

Example

14

1 2Find and I I

Admittance Parameters

• If impedance parameters do not always exist, then an alternative is needed for these cases.

• This need can be met by expressing the terminal currents in terms of terminal currents:

• The y terms are known as admittance parameters.

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1 11 1 12 2

2 21 1 22 2

I y V y V

I y V y V

y Parameters.

• The y parameters can be determined by short circuiting either the input or output ports (thus setting their voltages to zero).

• Because of this, the y parameters are also called the short circuit admittance parameters.

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2 1

2 1

1 111 12

1 20 0

2 221 22

1 20 0

V V

V V

I Iy y

V V

I Iy y

V V

Figure 19.12

Short Circuit Parameters

• These parameters are as follows:

• y11 Short circuit input admittance

• y12 Short circuit transfer admittance from port 1 to port 2

• y21 Short circuit transfer admittance from port 2 to port 1

• y22 Short circuit output admittance

• The impedance and admittance parameters are collectively called the immitance parameters.

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Equivalent Circuit

• For a network that is linear and has no dependent sources, the transfer admittances are equal.

• A reciprocal network (y12=y21) can be modeled with a -equivalent circuit.

• Otherwise the more general equivalent network (right) is used.

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Example

20

Obtain the y parameters

Example

21

Obtain the y parameters

Example

22

Obtain the y parameters

Hybrid Parameters

• Sometimes the z and y parameters do not always exist.

• There is thus a need for developing another set of parameters.

• If we make V1 and I2 the dependent variables:

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1 11 1 12 2

2 21 1 22 2

V h I h V

I h I h V

Hybrid Parameters II

• The h terms are known as the hybrid parameters, or simply h-parameters.

• The name comes from the fact that they are a hybrid combination of ratios.

• These parameters tend to be much easier to measure than the z or y parameters.

• They are particularly useful for characterizing transistors.

• Transformers too can be characterized by the h parameters.

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Values• The values of the parameters are:

• The parameters h11, h12, h21, and h22 represent an impedance, a voltage gain, a current gain, and an admittance respectively.

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2 1

2 1

1 111 12

1 20 0

2 221 22

1 20 0

V I

V I

V Vh h

I V

I Ih h

I V

h Parameters• The h-parameters correspond to:

• h11 Short circuit input impedance

• h12 Open circuit reverse voltage gain

• h21 Short circuit forward current gain

• h22 Open circuit output admittance

• In a reciprocal network, h12=-h21.

• The equivalent network is shown below:

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Example

27

Obtain the h parameters

Example

28

Determine the Thevenin equivalent at the output port

g Parameters• A set of related parameters are the g

parameters.• They are also known as the inverse hybrid

parameters.• They are used to describe the terminal

currents and voltages as:

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1 11 1 12 2

2 21 1 22 2

I g V g I

V g V g I

g Parameters II

• The values of the g parameters are determined as:

• The equivalent model is shown below:

30

2 1

2 1

1 111 12

1 20 0

2 221 22

1 20 0

I V

I V

I Ig g

V I

V Vg g

V I

g Parameters• The g parameters correspond to:

• g11 Open circuit input admittance

• g12 Short circuit reverse current gain

• g21 Open circuit forward voltage gain

• g22 Short circuit output impedance

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Transmission Parameters

• Since any combination of two variables may be used as the independent variables, there are many possible sets of parameters that may exist.

• Another set relates the variables at the input and output

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1 2 2

1 2 2

V AV BI

I CV DI

Example

33

Obtain the g parameters

Example

34

Obtain the g parameters

Transmission Parameters II• Note that in computing the transmission

parameters, I2 has a minus sign because it is considered to be leaving the network.

• This is done by convention; when cascading networks it is logical to consider I2 as coming out.

• The transmission parameters are:

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2 2

2 2

1 1

2 20 0

1 1

2 20 0

I V

I V

V VA B

V I

I IC D

V I

Transmission Parameters III

• The transmission parameters correspond to:• A: Open circuit voltage ratio• B: Negative short circuit transfer impedance• C: Open circuit transfer admittance• D: Negative short circuit current ratio• A and D are dimensionless while B is in

ohms and C is in siemens.• These are also known as the ABCD

parameters.

36

Example

37

Obtain the transmission parameters

Example

38

1 2Obtain and . Given T

5 10

0.4 1T

s

I I

Inverse Transmission Parameters

• We can also derive parameters based on the relationship of the input to the output variables.

• These inverse transmission parameters are:

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2 1 1

2 1 1

V aV bI

I cV dI

1 1

1 1

2 2

1 10 0

2 2

1 10 0

I V

I V

V Va b

V I

I Ic dV I

t Parameters

• The inverse transmission parameters, also called t parameters, correspond to:

• a: Open circuit voltage gain• b: Negative short circuit transfer impedance• c: Open circuit transfer admittance• d: Negative short circuit current gain• a and d are dimensionless while b is in ohms

and c is in siemens.

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Table 19.01

Interconnections of Networks

• Often it is worthwhile to break up a complex network into smaller parts.

• The sub-network may be modeled as interconnected two port networks.

• From this perspective, two port networks can be seen as building blocks for constructing a more complex network.

• These connections may be in series, parallel, or cascaded.

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Series Connection• Consider the series

connected network shown here.

• They are considered to be in series because their input currents are the same and their voltages add.

• The z parameter for the whole network is:

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a bz z z

Parallel Connection• Two port networks are in

parallel when their port voltages are equal and the port currents of the larger network are the sums of the individual port currents.

• Consider the network shown.

• Here, the y parameters of the entire network are:

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a by y y

Cascaded Connection

• A cascaded series of networks at first glance appears to be a series connected system.

• But note, that here, the output of one network is directly sent into the input of another network.

• Consider the cascaded network below:

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Cascaded Networks• Understanding the output of a cascaded

system requires considering the flow of a signal through the system.

• A signal enters the first two port network and is changed by the transmission characteristics.

• This altered signal then enters the next two-port network and is again altered by its transmission properties.

• Thus the overall transmission is:

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a bT T T