DIT, Kevin St.
Electric Circuit Theory
DT287/2
Two-Port network parameters
Summary
We have seen previously that a two-port network has a pair of input terminals and a
pair of output terminals figure 1. These circuits were analysed using several circuit
techniques such as mesh, nodal, superposition, Thevenin and Norton to predict the
behavior of the network. A further technique is to use the input and output current and
voltages II, Io, Vi, Vo to generate sets of parameters. By considering a combination of
currents and voltages and taking the input current or the input voltage or a combination
of current and voltage as the independent variable we generate six different types of
parameters. For example if we consider the input and output current as the
independent variables, we generate the Z parameters. If we take the input and output
voltages as the independent variables, we generate the Y parameters. Another set of
parameters, the hybrid (mixed Z and Y) or h-parameters uses the input current and
output voltage as the independent variables. The remaining three combinations have
specialized use and will not be discussed here.
The Z - Parameters
These parameters are often referred to as the open-circuit parameters and have uses in
analyzing passive networks and some active networks. Here I1 and I2 are the
independent variables. The equations are:
E1 = I 1 Z11 + I 2 Z12
E 2 = I 1 Z 21 + I 2 Z 22
(1)
And in matrix form
E1
E2
=
I 2  Z 11
I 2  Z 21
Z 12 
Z 22 
(2)
Figure 43: A two - port network.
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Electric Circuit Theory
DT287/2
The four Z parameters have the dimensions of impedance and for the general case can
be considered complex (R + jX). We can define each of these parameters by letting
certain currents to be zero. Open circuiting the input or the output terminals produces
the name open circuit parameters. We define the Z – parameters as:
Z 11 =
E1
I1
i.e. open-circuit at the o/p = Input or driving point impedance.
I 2 =0
Similarly
Z 12 =
E1
I2
i.e. open circuit at the L/P = Reverse transfer impedance.
I1 = 0
Also
Z 22 =
Z 21 =
E2
I2
E2
I1
= output impedance.
I1 = 0
= forward transfer impedance.
I 2 =0
Figure 44: Obtaining an expression for the output impedance.
The Z-parameters can be calculated by using the definition for each parameter and
solve for a particular current or voltage.
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DIT, Kevin St.
Electric Circuit Theory
DT287/2
Z-equivalent circuit
Figure 45: The Z- equivalent circuit.
The next step in this technique is to define of obtain expressions for the current gain,
voltage gain and power gain and also expressions for input and output impedance. We
can write the circuit parameters in matrix form as:
E1
E2
I 2  Z 11
I 2  Z 21
=
Z 12 
Z 22 
(3)
We can say that E 2 = − I 2 Z L . The negative sign takes care of the direction of current
and polarity of E2 Substituting for E2 in the above matrix using
E2 = − I 2 Z L
E1
− I2ZL
=
I 2  Z 11
I 2  Z 21
(4)
Z 12 
Z 22 
(5)
Also the input voltage is expressed as:
E1 = − I 1 Z S
− I1 Z S
E2
=
I 2  Z 11
I 2  Z 21
(6)
Z 12 
Z 22 
(7)
− I 1 Z S = I 1 Z 11 + I 2 Z 12
0 = I 1 ( Z 11 + Z S ) + I 2 Z 12
− I 2 Z L = I 1 Z 21 + I 2 Z 22
0 = I 1 Z 21 + I 2 ( Z 22 + Z L )
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Electric Circuit Theory
DT287/2
Obtain an expression for the voltage gain
Av =
Vout E 2
I2Z L
=
=
Vin
E1 Z 22 + Z L
 0   I 1   Z 11 + Z s
E  = I   Z
21
 2   2 
(8)
Z 12 
Z 22 
(9)
 Z11 + Z S 0 
 Z
E2 
21
I2 = 
 Z11 + Z S Z12 
 Z
Z 22 
21

I2 =
(10)
E2 ( Z11 + Z s )
( Z11 + Z S ) Z 22 − Z12 Z 21
(11)
I2
Z11 + Z S
=
E 2 Z 22 ( Z11 + Z S ) − Z12 Z 21
Z out =
(12)
E2
Z Z
= Z 22 − 12 21
I2
Z11 + Z S
(13)
Current Gain
I2
so in order to find an
I1
expression, for the current gain in terms of the Z-parameters we must solve for I1 and
I2 separately.
The current gain for a two-port network is defined as:
E1
E2
=
I 2  Z 11
I 2  Z 21
Z 12 
Z 22 
(14)
The matrix equation is changed if a load ZL is present then E 2 = − I 2 Z L
 E1   I 1   Z 11
 0  =  I  Z
   2   21
Z 12 
Z 22 + Z L 
(15)
To solve for I1
Z 12 
 E1
0 Z +Z 
22
L
I1 = 
=
Z 12 
 Z 11
Z

 21 Z 22 + Z L 
 E1
0

Z 12 
Z 22 + Z L 
∆Z
(16)
For the input current
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DIT, Kevin St.
Electric Circuit Theory
E1 ( Z 22 + Z L )
∆Z
I1 =
DT287/2
(17)
Similarly
 Z 11
Z
21
I2 = 
∆Z
E1 
0 
(18)
I2 =
− E1 Z 21
∆Z
(19)
Ai =
I2
I1
(20)
Ai =
− E1 Z 21
∆Z
.∆Z E1 ( Z 22 + Z L )
(21)
Ai =
− Z 21
Z 22 + Z L
(22)
Current Gain
To find an expression for the input impedance, we use this matrix and solve for I1. We
write using Cramer's Rule.
I1 =
E1 ( Z 22 + Z L )
Z11 ( Z 22 + Z L ) − Z12 Z 21
I1
Z 22 + Z L
=
E1 Z11 ( Z 22 + Z L ) − Z12 Z 21
(23)
(24)
So
Z in =
E1
Z Z
= Z 11 − 12 21
I1
Z 22 + Z L
(25)
Find an expression for Zout
Two-Port Networks
A two-port network, shown in figure 45, has a pair of input terminals and a pair of
output terminals. Examples of such two-port networks are:
• Transformer
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Electric Circuit Theory
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• T network,
• Series-tuned LCR
We can classify these as
• Symmetrical or asymmetrical networks and
• Balanced or unbalanced networks.
A symmetrical network is defined as a network such that when the i/p and o/p
terminals are interchanged, the electrical properties of the i/p and o/p remain unaltered.
A Tee network with equal series arms is an example of such a symmetrical network. If
the series arms were not equal then the network is asymmetrical. Interchanging the i/p
and o/p would change the electrical properties of the i/p and o/p. Further classification
is a balanced network where the two input arms contain the same elements. A tee
network is an example of an unbalanced network. The characteristic impedance is
defined as the impedance looking into one pair of terminals when the other pair of
terminals is terminated in the characteristic impedance.
Zin = Zo = characteristic impedance
Alternatively it is defined as the input impedance of a network terminated at infinity
Symmetrical and Unbalanced
Total series Arm = Z1
Total shunt Arm = Z2
From the definition of characteristic impedance, the input impedance is the
characteristic impedance when terminated in Zo.
Z in = Z o =
Z1
Z

+ [Z 2 ]  1 + Z o 
2
2

(26)
Z
Zo = 1 +
2
Z1
+ Zo )
2
Z
Z2 + 1 + Zo
2
(27)
Z2 (
Z1
+ Zo )
2
x
Z
(Z 2 + 1 + Z o )
2
(Z 2 +
2
Z Z
ZZ
ZZ
Z
ZZ
2
Zo Z2 + o 1 + Zo = 1 2 + 1 + 1 o + 1 2 + Z2Zo
2
2
4
2
2
(28)
Z o / c = 125Ω
Z s / c = 25 +
(29)
25(100)
= 50Ω
125
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Electric Circuit Theory
DT287/2
Z o = 50(125) ⇒ Z o = 75 Ω
2
Asymmetrical Networks
If a network is asymmetrical we cannot interchange the input and o/p terminals
without affecting the electrical properties of the network. In this situation the
characteristic impedance has a different value when looking at either i/p or o/p. In this
situation we have to introduce the concept of the image impedance
Image Impedance
These are defined for a two-port network as the two impedances, which are such, that
when one of them is connected across the appropriate pair of terminals of the network
the other is seen across the opposite pair of terminals. For example, consider an
asymmetrical T network. To simplify the calculations for different networks we can
show that the characteristic impedance for a two-port network is equal to the square
root of the product of the short - circuit and open circuit impedance. To prove this,
consider the unbalanced symmetrical T network. The input impedance, when the o/p is
open circuited, is
Z1
+ Z2
2
Zo /c =
The input impedance when the o/p is short circuit is
Zs/c
Z1 Z 2
Z
Z
Z
2
= 1 + Z 2 // 1 = 1 +
2
2
2 Z1
+ Z2
2
Z o = {Z o / c Z s / c }
Zo =
2






Z 1 Z1 Z 1 Z 2 Z 1 Z 2
+
+
2 2
2
2
2
Zo =
2
(31)
Z1 Z 2

( Z 1 + Z 2 )  Z 1
2
+
Z
 2
2
1
+ Z2

2

Zo =
Z1
+ Z1 Z 2 ⇒ Z o =
4
(30)
(32)
(33)
2
Z1
+ Z1 Z 2
4
(34)
Thus we can find a value for the characteristic impedance Zo in terms of the elements
of the two-port networks. e.g. Z1 = 50 Ω and Z2 = 100 Ω.
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DIT, Kevin St.
Zo =
Electric Circuit Theory
DT287/2
(50) 2
+ 50(100) = 75Ω
4
Z IN = 25 + 100 // 100 = 75Ω Z o = Z IN
Verify the π network using Zoπ = Z o / c Z s / c
Zo /c =
Z1 (2Z 2 )
2Z1 Z 2
=
Z1 + 2Z 2 Z 1 + 2Z 2
Zo /c =
2Z 2 ( Z1 + 2Z 2 ) 2Z 2 ( Z1 + 2Z 2 )
=
2Z 2 + Z 1 + 2Z 2
Z1 + 4Z 2
 2Z 2 (Z1 + 2Z 2

 Z1 + 4Z 2



Z oπ 2 =
2Z1Z 2
Z1 + 2Z 2
Z oπ 2 =
4Z Z
4Z1Z 2
Z
x 1 = 2 1 2
Z1 + 4Z 2
Z 1 Z1 + 4Z1 Z 2
2
Z oπ =
2
Z1 Z 2
2
2
2
2
Z1 Z 2
Z1
+ Z1Z 2
4
2
⇒ Z oπ =
÷4
÷4
2
2
Z1
+ Z1 Z 2
4
Or in terms of the Tee network characteristic impedance:
Z oπ =
Z1 Z 2
Z oT
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