EGR 271 – Circuit Theory I
Reading Assignment: Chapter 18 in Electric Circuits, 9th Ed. by Nilsson
One-port networks
Earlier in the course we saw that a network N consisting or resistors and dependent
sources can be represented by an equivalent resistance, Req . This type of network is
a one-port network since it has one pair of terminals.
I
I
+
Network N
(dependent sources,
resistors)
V
_
+
R
eq
V
_
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EGR 271 – Circuit Theory I
Two-port networks
Suppose that a network N has two ports as shown below. How could it be
represented or modeled? A common way to represent such a network is to use one of
6 possible two-port networks. These networks are circuits that are based on one of 6
possible sets of two-port equations. These equations are simply different
combinations of two equations that relate the variables V1 , V2 , I1 , and I2 to one
another. The coefficients in these equations are referred to as two-port parameters.
I
I
1
2
+
Input
Port
V
1
_
+
Network N
(dependent sources,
resistors)
V
_2
Output
Port
(N is a 2-port network)
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EGR 271 – Circuit Theory I
Note that I1 , I2 , V1 , and V2 are labeled as shown by convention. Often there is a
common negative terminal between the input and the output so the figure above
could be redrawn as:
I
I
1
2
+
Input
Port
+
V
1
Network N
(dependent sources,
resistors)
(N is a 2-port network)
_
V
2
Output
Port
_
3
EGR 271 – Circuit Theory I
Two-port equations
Two-port equations are sets of two equations relating the four variables labeled on
the diagram of the two-port network above. There are 6 possible ways to form sets
of two equations which express two of the variables in terms of the other two
variables. The 6 possible sets of equations are shown below.
z-parameters
g-parameters
z-parameter equations:
g-parameter equations:
V1  z11  I1  z12  I 2
I1  g11  V1  g12  I2
V2  z 21  I1  z 22  I 2
V2  g 21  V1  g 22  I2
y-parameters
Chain (or T or ABCD) parameters
y-parameter equations:
Chain parameter equations:
I1  y11  V1  y12  V2
V1  A  V2  B  (I 2 )
I 2  y 21  V1  y 22  V2
I1  C  V2  D  (I 2 )
h-parameters
Inverse chain (or T-1 or A'B'C'D' parameters
h-parameter equations:
Inverse-chain parameter equations:
V1  h 11  I1  h 12  V2
V2  A'V1  B'(I1 )
I 2  h 21  I1  h 22  V2
I2  C'V1  D'(I1 )
Applications:
z- and y-parameters: circuit modeling
y-parameters: modeling transistor capacitance at high frequencies
h- parameters: electronics (modeling transistors)
Chain- and inverse-chain parameters: system representation (relates input and output)
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EGR 271 – Circuit Theory I
Calculation of z-parameters
Show that
V
z11  1
I1
I2  0
z - parameterequations:
V1  z11  I1  z12  I 2
V2  z 21  I1  z 22  I 2
Similarly show expressions for the other z-parameters.
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EGR 271 – Circuit Theory I
Example: Determine the z-parameters using the
z-parameter definitions for the network shown
I
I
below.
1
2
+
V
10
30
20
1
_
z - parameterequations:
V1  z11  I1  z12  I 2
+
V
V2  z 21  I1  z 22  I 2
2
_
Circuit 2
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EGR 271 – Circuit Theory I
Calculation of y-parameters
Show that
I
y11  1
V1
V2  0
y - paramet erequations:
I1  y11  V1  y12  V2
I 2  y 21  V1  y 22  V2
Similarly show expressions for the other y-parameters.
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EGR 271 – Circuit Theory I
Example: Determine the y-parameters using the
y-parameter definitions for the network shown
below.
I
1
+
10
1
0.3 I
X
_
I
I1  y11  V1  y12  V2
2
I 2  y 21  V1  y 22  V2
+
I
V
y - paramet erequations:
6
X
V
2
_
Circuit 3
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EGR 271 – Circuit Theory I
Reading Assignment: Chapter 18 in Electric Circuits, 8th Ed. by Nilsson
Example: Determine the h-parameters for the
network shown below.
I
I
1
+
V
_
200
1
100
h - paramet erequations:
2
V1  h11  I1  h12  V2
+
300
V
I 2  h 21  I1  h 22  V2
2
_
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EGR 271 – Circuit Theory I
Calculation of 2-port parameters using network equations
Consider the z-parameter equations shown below.
V1  z11  I1  z12  I 2
V2  z 21  I1  z 22  I 2
Note that V1 and V2 are functions of I1 and I2 . If general sources, I1 and I2 are added
to a network and the voltages V1 and V2 are calculated, the result will be expressions
for V1 and V2 that are functions of I1 and I2 . So the z-parameter equations are
naturally generated.
Similarly, y-parameters can be found by adding two general voltage sources V1 and
V2 and solving for the currents I1 and I2 .
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EGR 271 – Circuit Theory I
Example: Determine the y-parameters using network equations for the network
shown below.
1
I
+
V
_
I
1
1
1
1
1
2
+
V
2
_
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EGR 271 – Circuit Theory I
Modeling Two-Port Networks
We have seen how two-port parameters can be determined for a given network.
Additionally, two-port parameters might be specified for a certain device by the
manufacturer (such as h-parameter values for a transistor). How are these parameters
used? They are used to form a circuit model for the device or circuit. A circuit model
is developed using the two-port parameter equations.
z-parameter model
The z-parameter equations are:
V1  z11  I1  z12  I 2
V2  z 21  I1  z 22  I 2
What sort of circuit model could be drawn that would satisfy these equations?
I1
I2
+
V1
_
+
?
V2
_
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EGR 271 – Circuit Theory I
Development of the z-parameter model:
One possible circuit model could be developed by
treating each of the two-port parameter equations as
KVL equations (illustrate). This results in the
following circuit.
z - parameterequations:
V1  z11  I1  z12  I 2
V2  z 21  I1  z 22  I 2
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EGR 271 – Circuit Theory I
Development of the y-parameter model:
One possible circuit model could be developed by
treating each of the two-port parameter equations as KCL
equations (illustrate). This results in the following
circuit.
y - paramet erequations:
I1  y11  V1  y12  V2
I 2  y 21  V1  y 22  V2
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EGR 271 – Circuit Theory I
Development of the h-parameter model:
One possible circuit model could be developed by
treating one of the two-port parameter equations as a
KVL equation and the other as a KCL equation
(illustrate). This results in the following circuit.
h - paramet erequations:
V1  h11  I1  h12  V2
I 2  h 21  I1  h 22  V2
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EGR 271 – Circuit Theory I
Summary:
Note: This page will be provided on the final exam.
I2
I1
+
z11
V1
+
z - parameterequations:
V2
V1  z11  I1  z12  I 2
_
V2  z 21  I1  z 22  I 2
+
y - paramet erequations:
V2
I1  y11  V1  y12  V2
_
I 2  y 21  V1  y 22  V2
+
h - paramet erequations:
V2
V1  h11  I1  h12  V2
_
I 2  h 21  I1  h 22  V2
z22
z12 I 2
+
_
+
_
z21 I 1
_
z-parameter model
I2
I1
+
V1
y 11
y12 V2
y21 V1
y 22
_
y-parameter model
I2
I1
+
V1
h11
h12V 2
+
_
h21I 1
_
h-parameter model
h 22
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EGR 271 – Circuit Theory I
Example: Consider Network N shown below.
I
2
10 V
+
_
I
1
+
2
+
+
Network N
V
_
1
V
_
2
8
V
L
_
Two tests were run on the circuit above in order to determine the z-parameters. The
results of the tests are shown below.
Test #1 (output open-circuited)
Test #2 (input open-circuited)
V1 = 2V
V2 = 4V
V1 = 2V
V2 = 4V
I1 = 1A
I2 = 0
I1 = 0
I2 = 2A
A) Use the test results to determine the z-parameters.
B) Redraw the circuit shown above using the z-parameter model and the
z-parameter values determined in part A.
C) Determine VL.
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EGR 271 – Circuit Theory I
Example: Consider Network N shown below.
I
2k
50 V
+
_
I
1
+
2
+
+
Network N
V
_
1
V
2
_
10k
V
L
_
Determine VL if the h-parameters for Network N are:
h11 = 2.5k , h12 0.001, h21 = 125, h22 = 25 mho
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