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Fundamentals of
Electric Circuits
Chapter 19
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
Overview
• In this chapter, the concept of a two-port
network.
• The relationship between input and output
current and voltages will be cataloged and
described.
• Combinations of networks in series, parallel,
and cascaded will be discussed.
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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.
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One port or two
terminal circuit
Two port or four
terminal circuit
• It is an electrical network
with two separate ports for
input and output.
• No independent
sources.
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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:
V1  z11 I1  z12 I 2
V2  z21 I1  z22 I 2
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z11 
V1
I1
and
z 21 
I 2 0
V2
I1
I 2 0
z11 = Open-circuit input
impedance
z21 = Open-circuit transfer
impedance from port 1 to port 2
z12 
V1
I2
and
I1  0
z 22 
V2
I2
I1  0
z12 = Open-circuit transfer
impedance from port 2 to port 1
z22 = Open-circuit output
impedance
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z11 
z12 
V1
I1
V1
I2
and
z 21 
I 2 0
and
I1  0
z 22 
V2
I1
I 2 0
V2
I2
I1  0
•When z11 = z22, the two-port network is
said to be symmetrical.
•When the two-port network is linear and
has no dependent sources, the transfer
impedances are equal (z12 = z21), and the
two-port is said to be reciprocal.
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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).
z11 
z21 
V1
I1
V2
I1
z12 
I 2 0
z22 
I2 0
V1
I2
I1  0
V2
I2
I1  0
• These are referred to as the open-circuit
impedance parameters.
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Example 1
Determine the Z-parameters of the following circuit.
I1
I2
V2
V1
Answer:
z11 
z12 
60 40
z


40 70
V1
I1
V1
I2
and
z 21 
I 2 0
and
I1  0
z 22 
V2
I1
I 2 0
V2
I2
I1  0
 z11 z12 
z


z 21 z 22 
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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.
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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:
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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:
I1  y11V1  y12V2
I 2  y21V1  y22V2
• The y terms are known as admittance
parameters.
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y Parameters.
• The y parameters can be determined by short
circuiting either the input or output ports
(thus setting their voltages to zero).
y11 
I1
V1 V 0
y12 
I2
V1
y22 
2
y21 
V2  0
I1
V2
V1  0
I2
V2
V1  0
• Because of this, the y parameters are also
called the short circuit admittance
parameters.
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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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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:
V1  h11 I1  h12V2
I 2  h21I1  h22V2
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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:
h11 
V1
I1 V 0
h12 
I2
I1
h22 
2
h21 
V2  0
V1
V2
I1  0
I2
V2
I1  0
• The parameters h11, h12, h21, and h22 represent
an impedance, a voltage gain, a current gain,
and an admittance respectively.
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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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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:
I1  g11V1  g12 I 2
V2  g 21V1  g 22 I 2
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g Parameters II
• The values of the g parameters are
determined as:
g11 
g 21 
I1
V1
V2
V1
g12 
I 2 0
g 22 
I2 0
I1
I2
V1  0
V2
I2
V1  0
• The equivalent model is shown below:
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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
V1  AV2  BI 2
I1  CV2  DI 2
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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:
A
C
V1
V2
I1
V2
B
I 2 0
D
I 2 0
V1
I2
V2  0
I1
I2
V2  0
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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.
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Inverse Transmission
Parameters
• We can also derive parameters based on the
relationship of the input to the output
variables.
V2  aV1  bI1
I 2  cV1  dI1
• These inverse transmission parameters are:
a
c
V2
V1
I2
V1
b
I1  0
d 
I1  0
V2
I1
V1  0
I2
I1
V1  0
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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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