InputPort
Output Port
+
_ _
+V1 V2
I1 I2
2221
1211
zz
zz
ZPARAMETERS
IMPEDANCE
Copyright © 2012 Ahmad Nauman. All Rights Reserved
LinearNetwork
V1 V2
I1 I2
INTRODUCTIONA two-port network may be voltage-driven as
or current-driven as
LinearNetwork
I1 I2
+
_V1_
+V2
Copyright © 2012 Ahmad Nauman. All Rights Reserved
From previous Figures
Only two of the four variables (V1, V2, I1 and I2) are independent/known.The other two can be found using equation (a)
2121111 Iz Iz V
2221212 Iz Iz V (a)
And in matrix form as
2
1
2221
1211
2
1
I
I
zz
zz
V
V V
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2
1
2
1
I
I z
V
V
or
Where z terms are called impedance parameters or simply z-parameters
zz
zz z
2221
1211
Copyright © 2012 Ahmad Nauman. All Rights Reserved
LinearNetwork
V1
I1 I2=0
_
+V2
The values of the parameters can be evaluated by setting I2 = 0(output port open-circuited).
0I1
111
2I
V z
0I1
221
2I
V z
Copyright © 2012 Ahmad Nauman. All Rights Reserved
LinearNetwork
V1
I1=0
The values of the parameters can be evaluated by setting I1 = 0 (output port open-circuited).
0I2
112
1I
V z
0I2
222
1I
V z
V2
I2
_
+
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impedance.input circuit -Openz11
impedance.output circuit -Openz22
12z
21z
Open Circuit Transfer impedance from port 1 to port 2.
Open Circuit Transfer impedance from port 2 to port 1.
Copyright © 2012 Ahmad Nauman. All Rights Reserved
• 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)
Copyright © 2012 Ahmad Nauman. All Rights Reserved
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.
USAGE:
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T-NETWORK
+
_ _
+V1 V2
I1 I21z
2z
3z
2
1
2221
1211
2
1
I
I
zz
zz
V
V
General form is
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T-NETWORK
+
_ _
+
V1V2
1z
2z
3zI1 I2
Find the z-parameters.
Copyright © 2012 Ahmad Nauman. All Rights Reserved
+
_
V1I1
1z
3z
Applying KVL in loop 1
)I (I z Iz V 213111
I2
Iz )Iz (z 23131
But we also know that
2121111 Iz Iz V
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_
+
V2
2z
3zI1 I2
Applying KVL in loop 2
)I (I z Iz V 213222
)Iz (z )I(z 23213
But we also know that
2221212 Iz Iz V
2
1
323
331
2
1
I
I
zzz
zzz
V
V
+
_ _
+V1 V2
I1 I21z
2z
3z
And the z-parameters are
zzz
zzz z
323
331
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D-PROBLEM 17.8(A) -HAYT
+
_ _
+
V1V2
20 50
25
Figure 17.19 (a)
Find the z-parameters.
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+
_ _
+
V1V2
20 50
25
Solution:
As we know that z-parameters in T Network are
zzz
zzz z
323
331
255025
252520 z
So from above figure
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+
_ _
+
V1V2
20 50
25
Solution:
7525
2545 z
AnsweR
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SERIES CONNECTION
Network A+
_ _
+
V1 V2
I1=I1A
Network B
I1=I1B
I1
+
+
_
_
I2=I2A
I2=I2B
+
+
_
_
I2
V1A
V1B
V2A
V2B
If each two-port has common reference node for its input and output, and if the references are connected together as indicated in Figure.
Copyright © 2012 Ahmad Nauman. All Rights Reserved
SERIES CONNECTIONI1 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
So
BBAABA Iz Iz V V V
I z )Iz (z BA
WhereBA z z z
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_
+
V21V
I2
g11s ZI V V
THÉVENIN EQUIVALENTgZ
outZ
From this circuit
(a)
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As we know that
From eq. (a)
g1212111 I Iz Iz ZVs
2121g11 Iz I)(z Z Vs
)(z
Iz
)(z I
g11
212
g111 ZZ
V s
We also know that
(b)
(c)2221212 I z Iz V
outz
2121111 Iz I z V
inz
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222g11
22112
g11
21 Iz)(z
Izz
)(z
z V
ZZ
Vs2
)(z
zz - z
g11
211222 Z
Z out
Thus output impedance in terms of z-parameters
2g11
211222
g11
21 I )(z
zzz
)(z
z V
ZV
Z s2
Copyright © 2012 Ahmad Nauman. All Rights Reserved
_
+
V2
g11
211222out zz
zzzZ
sV
I2
g
th zz
zz
11
21
outZ
If the generator impedance is zero, the simpler expression
0zg
11
211222out z
zzzZ
11
z
11
21122211
z
zzzz
Copyright © 2012 Ahmad Nauman. All Rights Reserved
D-PROBLEM 17.9 - HAYT
+
_ _
+
V1 V2
20 50
25
Figure 17.19 (c)
2V5.0
I1I2
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Applying KVL in loop 1
+
_
V1
20
25
2V5.0
I1I2
2211
22111
0.5V 25I 45I V
0.5V )I (I 25 20I V
(a)
Solution:
Copyright © 2012 Ahmad Nauman. All Rights Reserved
_
+
V2
50
25
2V5.0
I1I2
Applying KVL in loop 2
212
2122
22122
150I I50V
75I 25I V5.0V
0.5V )I (I 25 50I V
(b)
Solution:
Copyright © 2012 Ahmad Nauman. All Rights Reserved
Substituting equation (b) into (a), we get;
211
211
21211
I100 70I V
I)75(25 25)I(45 V
)I1500.5(50I 25I 45I V
From equation (c) and (b), we get;
(c)
2
1
2
1
I
I
15050
10070
V
V
As we know that
2
1
2221
1211
2
1
I
I
zz
zz
V
V
Solution:
Copyright © 2012 Ahmad Nauman. All Rights Reserved
We know z-parameters are
zz
zz z
2221
1211
So [z] will be
15050
10070 z
AnsweR
Solution:
Copyright © 2012 Ahmad Nauman. All Rights Reserved
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