Terminated Two-Port Circuits
Zin
ZTh
 In a typical application, the two-port circuit is driven at port 1 and
loaded at port 2.
 From the two-port equations, and including relationships at the
input and output, a number of characteristics of terminated twoport circuits, such as Thevenin impedance, Thevenin voltage,
input impedance, current gain, voltage gain, etc, can be derived.
69
Terminated Two-Port Circuits
Assuming the z parameters of the network are given, find the input
impedance.
Z in =
V1
I1
V1 = z11 I1 + z12 I 2
Zin
ZTh
V2 = -Z L I 2 = z21 I1 + z22 I 2
V2 = z21 I1 + z22 I 2
z21 I1
 I2 = z22 + Z L
V1 = z11 I1 + z12 I 2 = z11 I1 - z12
z21 I1
z22 + Z L
V1
z12 z21
Z in = = z11 I1
z22 + Z L
Note: Z in = z11 only if Z L  ¥, which represents the open-circuit
condition for z parameters.
70
Terminated Two-Port Equations
71
Terminated Two-Port Equations
72
Terminated Two-Port Equations
73
Example 11
The b parameters of the network are given. Find a) voltage V2 ,
b) the average power delivered to the load.
 20 3k 
b =  2 mS 0.2 


a) From table:
V2
DbZ L
=
Vg
b12 + b11Z g + b22 Z L + b21Z g Z L
(20 ⋅ 0.2 - 3⋅103 ⋅ 2 ⋅10-3 )⋅ 5⋅103
10
=
=
3
3
3
-3
19
-3⋅10 - 20 ⋅ 500 - 0.2 ⋅ 5 ⋅10 - 2 ⋅10 ⋅ 500 ⋅ 5 ⋅10
V2 =
b) Average power:
10
10
Vg = 500 V = 263.16 V
19
19
P2 =
V22
2 RL
2
=
(263.16 V)
2 ⋅ 5 k
= 6.93 W
74
Example 12
Given the a parameters and Vg =50mV, Zg =100, ZL =5k, find
a) the average power delivered to the load,
b) the load resistance for maximum power transfer.
5 104
a = 
 1μS
a) From table:I 2 =
10  

0.03
-Vg
a11Z L + a12 + a21Z g Z L + a22 Z g
=
(5⋅10
-4
-50 ⋅10-3 V
3
-6
⋅ 5 ⋅10 + 10 + 10
3
)
⋅100 ⋅ 5 ⋅10 - 0.03⋅100 
= -5 mA
1 2
1
2
PL = I 2 Z L = (-5 mA) 5 k = 62.5 mW
2
2
75
Example 12 cont’d
b) Load resistance for maximum power transfer:
Vg =50mV, Zg =100, ZL =5k
5 104
a = 
 1μS
From table: Z = a12 + a22 Z g = 10  - 0.03⋅100 
Th
-4
-6
a11 + a21Z g
5 ⋅10
+10
10  

0.03
S ⋅100 
= 11.67 k
*
Z L = ZTh
= 11.67 k
76
Example 13
Two identical amplifiers are cascaded; their h parameters are given.
Find the voltage gain V2 /Vg .
1k 0.0015
 h =  100 100μS 


Convert to a parameters:
h (0.05)
a A11 

 5 103 ,
100
h21
a A21 
 h22 100μS

 1μS,
100
h21
5 103
 a    aA  aA   
 1μS
10   5 103

0.01   1μS
a A12
 h11 1k


 10 
100
h21
a A22 
1
1

 0.01
h21 100
10   10.25 106 0.095  


0.01   9.5 109 S 1.1104 
77
Example 13 cont’d
10.25 106 0.095  
 a    aA1  aA2   
9
4 


9.5
10
S
1.1
10


V2
ZL
=
a) From table:
Vg (a11 + a21Z g ) Z L + a12 + a22 Z g
=
104
(10.25⋅10-6 + 9.5⋅10-9 ⋅ 500)⋅104 + 0.095 +1.1⋅10-4 ⋅ 500
104
=
= 33,333.33
0.15 + 0.095 + 0.055
78
Example 14
Evaluate V2/Vs in the following circuit.
Series connection: add z parameters.
z parameters of 10  resistor:
z11  z12
z22  z12
z12
z12  10 
z11  z12  0  z11  z12  10 
z22  z12  0  z22  z12  10 
Add z matrices
12 8 
10 10 


 z  


 8 20
10 10 
 22 18 



18 30
79
Example 14 cont’d
Find V2/Vs from table
V2
z21Z L

( z11  Z g )( z22  Z L )  z12 z21
Vs
 z 
 22 18 
18 30 


Z g  5 , Z L  20
V2
18  20

Vs
(22  5)(30  20)  182
 0.351
80
Ladder Network Synthesis
 LC ladder networks for low-pass filters of
(a) odd order, (b) even order
81
Ladder Network Synthesis
 To synthesize the transfer function of the LC ladder
 To make the design less complicated, assume Zs=0.
82
Ladder Network Synthesis
83
Example 15
Design the LC ladder network terminated with a 1  resistor that has
the following normalized transfer function
84
Example 15 cont’d
85
Example 15 cont’d
Note that any realization of y22 will automatically realize y21,
since y22 is the output driving-point admittance, that is, the
output admittance of the network with the input port shortcircuited.
86
Example 16
Realize the following transfer function using an LC ladder network
terminated in a 1 resistor:
87
Example 16 cont’d
88
Example 16 cont’d
89