Impedance parameters

I P
M A
P
E I1 I2
A
z11 z12 M
D + Input Output
+
V1
_ Port z 21 z 22 Port
V2
_ E
A T
E
C
E S
Copyright © 2012 Ahmad Nauman. All Rights Reserved

A two-port network may be voltage-driven as

I1 I2

V1 Linear V2
Network

or current-driven as

+ +
I1 V1 Linear I2
_ V2
Network _


From previous Figures

V1 z11I1 z12 I 2
(a)
V2 z 21I1 z 22 I 2

Only two of the four variables (V1, V2, I1 and I2) are independent/known.
The other two can be found using equation (a)

And in matrix form as

V1 z11 z12 I1
V
V2 z 21 z 22 I2


or
V1 I1
z
V2 I2

Where z terms are called impedance parameters or
simply z-parameters

z11 z12
z
z 21 z 22


I1 I2=0
+
V1 Linear V2
Network _

The values of the parameters can be evaluated by setting I2 = 0
(output port open-circuited).

V1 V2
z11 z 21
I1 I2 0
I1 I2 0


I1=0 I2
+
V1 Linear V2
_ Network

The values of the parameters can be evaluated by setting I1 = 0
(output port open-circuited).

V1 V2
z12 z 22
I2 I1 0
I2 I1 0


z11 Open - circuit input impedance.

z12 Open Circuit Transfer impedance from port 1 to port 2.
z 21 Open Circuit Transfer impedance from port 2 to port 1.

z 22 Open - circuit output impedance.


• Since the z-parameters are obtained by open-
circuiting the input or output port, they are
also called the open-circuit impedance
parameters.
• Since they are obtained as a ratio of voltage
and current.
• And the parameters are obtained by open-
circuiting port 2 ( I2 = 0) or port1 ( I1 = 0).
• Units of z-parameter are in ohms’ (Ω)


Impedance parameters are commonly used;
• In the synthesis of filter.
• They are also useful in the design and analysis
of impedance-matching networks.
• And power distribution networks.


I1 z1 z2 I2
+ +
z3
V1 V2
_ _

General form is

V1 z11 z12 I1
V2 z 21 z 22 I2


z1 z2

+ +
I1 z3 I2 V2
V1
_ _

Find the z-parameters.


z1

+

V1 I1 z3 I2

_

Applying KVL in loop 1
V1 z1I1 z 3 (I1 I2 )
(z 1 z 3 )I1 z 3I 2

But we also know that

V1 z11I1 z12 I 2


z2

+
I1 z3 I2 V2

_

V2 z2I2 z 3 (I1 I2 )
(z 3 )I1 (z 2 z 3 )I 2

But we also know that

V2 z 21I1 z 22 I 2

I1 z1 z2 I2
+ +
z3
V1 V2
_ _

V1 z1 z 3 z3 I1
V2 z3 z 2 z3 I2

And the z-parameters are

z1 z 3 z3
z
z3 z 2 z3


20 50

+ +

V1 25 V2

_ _

Figure 17.19 (a)

Find the z-parameters.


20 50

+ +

V1 25 V2

_ _

Solution:

As we know that z-parameters in T Network are

z1 z 3 z3
z
z3 z 2 z3

So from above figure

20 25 25
z
25 50 25


20 50

+ +

V1 25 V2

_ _

Solution:

45 25
z
25 75


If each two-port has common reference node
for its input and output, and if the references
are connected together as indicated in Figure.

I1=I1A I2=I2A
+ + +
V1A _ Network A _ V2A +

V1 I1 I2 V2

_ V1B + Network B + V2B _
_ _
I1=I1B I2=I2B


I1 flows through the input ports of the two
networks in series. A similar statement holds for
I2. Thus, ports remain ports after the
interconnection. It follows that I = IA = IB.
And
V VA VB zA IA z BIB

So
(z A z B )I zI
Where
z zA zB

Zg Z out
I2

+

V1 V2

_

From this circuit

Vs V1 I1Z g (a)


z in
As we know that
V1 z11 I1 z12 I 2

From eq. (a)

Vs z11I1 z12 I 2 I1Z g

Vs (z 11 Z g )I1 z12 I 2

Vs z12 I 2
I1 (b)
(z 11 Z g ) (z 11 Z g )

zout
We also know that

V2 z 21I1 z 22 I 2 (c)


z 21Vs z12 z 21I 2
V2 z 22 I 2
(z 11 Z g ) (z 11 Z g )

z 21 z12 z 21
V2 Vs z 22 I2
(z11 Z g ) (z11 Z g )

Thus output impedance in terms of z-parameters

z12 z 21
Z out z 22 -
(z 11 Z g )


Z out
I2 z21
zth
+ z11 z g
Vs V2
z12z 21
_ Zout z22
z11 z g

If the generator impedance z 0 is zero, the simpler expression
g

z12z 21
Zout z22
z11
z11z 22 z12z 21 z
z11 11


20 50

+ +
25
I1 I2
V1 V2
0.5V2

_ _

Figure 17.19 (c)


20

+
25
I1 I2
V1
0.5V2

_

Solution:


V1 20I1 25 (I1 I 2 ) 0.5V2
V1 45I1 25I2 0.5V2 (a)


50

+
25
I1 I2
V2
0.5V2

_

Solution:


V2 50I2 25 (I1 I 2 ) 0.5V2
V2 0.5V2 25I1 75I2
V2 50I1 150I2 (b)


Solution:
Substituting equation (b) into (a), we get;

V1 45I1 25I2 0.5(50I1 150I 2 )
V1 (45 25)I1 (25 75)I 2
V1 70I1 100I 2 (c)

As we know that

V1 z11 z12 I1
V2 z 21 z 22 I2

From equation (c) and (b), we get;

V1 70 100 I1
V2 50 150 I 2

Solution:

We know z-parameters are

z11 z12
z
z 21 z 22

So [z] will be

70 100
z
50 150


Impedance parameters

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