TWO-PORT
CIRCUITS
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5.1 Definition of Two-Port Circuits
5.2 Classification of Two-Port Parameters
5.3 Finding Two-Port Parameters
5.4 Analysis of the Terminated Two-Port Circuit
5.5 Interconnected Two-Port Circuits
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5.1 Definition of Two-Port Circuits
Consider a linear two-terminal circuit N consisting
of no independent sources as follows :
For a, b two terminals,
if Iin = Iout , then it
constitutes a port.
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5.1 Definition of Two-Port Circuits
Now consider the following linear four-terminal
circuit containing no independent sources.
+
V1
−
I1
I2
I1 '
I2 '
+
V2
−
with I1=I1'
I2=I2'
Then terminals a , b constitute the input port
and terminals c , d constitute the output port.
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5.1 Definition of Two-Port Circuits
+
V1
−
I1
I2
I1 '
I2 '
+
V2
−
No external connections exist between the input
and output ports.
The two-port model is used to describe the
performance of a circuit in terms of the voltage
and current at its input and output ports.
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5.1 Definition of Two-Port Circuits
Two-port circuits are useful in communications,
control systems, power systems, and electronic
systems.
They are also useful for facilitating cascaded
design of more complex systems.
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5.2 Classification of Two-Port Parameters
There are four terminal variables , namely V1 ,V2 ,
I1 , I2 , only two of them are independent.
Hence , there are only six possible sets of two-port
parameters.
4 C2 =
4×3
=6
2 ×1
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5.2 Classification of Two-Port Parameters
(1) The impedance , or Z , parameters
I1
I2
+
V1
−
V1   z11
V  =  z
 2   21
+
N
V2
−
z12   I1 
, Z ij : in Ω
z22   I 2 
For two-port networks , four parameters are generally
required to represent the circuit.
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5.2 Classification of Two-Port Parameters
(2) The admittance , or Y , parameters
I1
I2
+
+
N
V1
−
 I1   y11
I  =  y
 2   21
V2
−
y12  V1 
, Yij : in S



y22  V2 
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5.2 Classification of Two-Port Parameters
(3) The hybrid , or h , parameters
I1
+
V1
−
V1   h11
 I  = h
 2   21
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I2
N
h12   I1 
,
h22  V2 
+
V2
−
h11 : in Ω
h22 : in S
h12 & h22 scalars
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5.2 Classification of Two-Port Parameters
(4) The inverse hybrid , or g , parameters
I1
+
I2
+
N
V1
−
 I1   g11
V  =  g
 2   21
g12  V1 
,
g 22   I 2 
V2
−
g11 : in S
g 22 : in Ω
g12 & g 21 scalars
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5.2 Classification of Two-Port Parameters
(5) The transmission , or a , parameters
I1
+
V1
−
 V1   a11
 I  = a
 1   21
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I2
N
+
V2
−
a12 : in Ω
a12   V2 
,
a21 : in S
a22   − I 2 
a11 & a22 scalars
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5.2 Classification of Two-Port Parameters
(6) The inverse transmission , or b , parameters
I1
+
V1
−
I2
N
+
V2
−
b12 : in Ω
 V2   b11 b12   V1 
b21 : in S
 I  = b
  ,
 2   21 b22   − I1  b & b scalars
11
22
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5.3 Finding Two-Port Parameters
Method 1 : Calculate or measure by invoking
appropriate short-circuit and opencircuit conditions at the input and
output ports.
Method 2 : Derive the parameters from another set
of two-port parameters.
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5.3 Finding Two-Port Parameters
Method 1 : Choose Z parameters as an illustration.
V1 = z11 I1 + z12 I 2
V2 = z21 I 1 + z22 I 2
when I2 = 0 , output port is open
V1 = z11 I1 , ∴ z11 =
V2 = z21 I1 , ∴ z21 =
V1
I1
, input impedance
I 2 =0
V2
I1
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, transfer impedance
I 2 =0
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5.3 Finding Two-Port Parameters
When I1 = 0 , input port is open
V1 = z12 I 2 , ∴ z12 =
V2 = z 22 I 2 , ∴ z 22 =
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V1
I2
V2
I2
, transfer impedance
I 1 =0
, output impedance
I 1 =0
When the two-port does not contain any
dependent source, then z12=z21.
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5.3 Finding Two-Port Parameters
z11 & z22 are called driving-point impedances.
z12 & z21 are called transfer impedances.
When z11=z22, the two-port circuit is said to be
symmetrical.
When z12=z21, the two-port circuit is called a
reciprocal circuit.
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5.3 Finding Two-Port Parameters
Example 1 : Find the Z parameters of the T-network
Assign mesh currents
as shown :
R2   I1  V1 
 R1 + R2
=
 R
R2 + R3   I 2  V2 
2

∴ z11 = R1 + R2
z12 = z21 = R2
z22 = R2 + R3
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5.3 Finding Two-Port Parameters
A two-port can be replaced by the following
equivalent circuit.
V1 = z11 I 1 + z12 I 2
V2 = z21 I 1 + z22 I 2
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5.3 Finding Two-Port Parameters
In case the two-port is reciprocal, z12=z21,
then it can also be represented by the
T-equivalent circuit.
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5.3 Finding Two-Port Parameters
Example 2 : Find the Z parameters
Assign mesh currents as follows and write
down the mesh equation.
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5.3 Finding Two-Port Parameters
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0
−4   I1  V1 
2 + 4
 0
10 + 8
8   I 2  = V2 

 −4
8
4 + 6 + 8  I 3   0 
 I1  V1 
A B    
C D   I 2  = V2 

 I   0 
 3  
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5.3 Finding Two-Port Parameters
I 
V 
A  1  + BI 3 =  1  ...... (1)
I2 
V2 
I 
C  1  + DI 3 = 0
.......... (2)
 I2 
I 
From (2) , I 3 = -D -1C  1  ...... (3)
 I2 
Substitute (3) into (1)
I 
I 
 I  V 
A  1  - BD -1C  1  =  A - BD -1C   1  =  1 
 I2 
 I2 
 I 2  V2 
−1
∴ [ Z ] = A − BD C
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5.3 Finding Two-Port Parameters
Example 3: Find the Z parameters of the following
circuit by definition of Z parameters.
I1
8Ω
I2
+
V1
−
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4Ω
8Ω
+
V2
−
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5.3 Finding Two-Port Parameters
Step1: Let I2 = 0 and apply V1
V1
I2 = 0
8Ω
I1
I1 = V1 / (4Ω / /(8 + 8)Ω) =
8Ω
4Ω
+
V2
−
20
V1
64
8
1
V1 = V1
8+8
2
V 64 16
∴ z11 = 1 =
= Ω
I1 20 5
V2 =
1
V
V2 2 1 1
8
= z11 = Ω
z21 = =
2
5
I1
I1
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5.3 Finding Two-Port Parameters
Step2: Let I1 = 0 and apply V2
I 2 = V2 / (8Ω / /(8 + 4)Ω ) =
I1 = 0
+
V1
4Ω
−
8Ω
I2
8Ω
5
V2
24
4
1
V2 = V2
4+8
3
V
24
V2 ∴ z22 = I 2 = 5 Ω
2
V1 =
1
V2
V
1
8
z12 = 1 = 3 = z 22 = Ω
I2
I2
3
5
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16 / 5 8 / 5 
∴[Z ] = 
Ω
 8 / 5 24 / 5 
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5.3 Finding Two-Port Parameters
Example 4: Containing dependent source case
R1
I1
R2
I2
+
+
V1
α I1
−
−
Step1: Let I2 = 0 and apply V1
I1
V2
R2 I 2 = 0
+
R1
V1
α I1
V2
−
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5.3 Finding Two-Port Parameters
I1 =
I1
V1
R1
R2
α I1
I2 = 0
+
V2
−
⇒ I1 =
V1 − α I1
R1
V1
R1 + α
V2 = α I1
∴ z11 =
z21 =
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V1
= R1 + α
I1
V2
=α
I1
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5.3 Finding Two-Port Parameters
Step2: Let I1 = 0 and apply V2
I1 = 0 R1
R2
I1 = 0
I2
+
V2
α I1
V1
∴ I2 =
V2
R2
V1 = 0
−
∴ z12 =
V1
=0
I2
z22 =
V2
= R2
I2
R + α
, ∴[ Z ] =  1
 α
0
R2 
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5.3 Finding Two-Port Parameters
Example 5: Find the y parameters of the following
circuit.
Gb
I1
+
V1
I2
+
Ga
Gc
−
V2
−
Use nodal analysis
I1
I1
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Ga
V1
Gb
I2
V2
Gc
I2
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5.3 Finding Two-Port Parameters
I1
I1
V1
Gb
I2
V2
Ga
Gc
I2
Nodal equation
−Gb  V1   I1 
Ga + Gb
=
 −G
Gb + Gc  V2   I 2 
b

−Gb 
G + Gb
∴[ y ] =  a
in S unit
Gb + Gc 
 −Gb
Note that y12 = y21 = −Gb
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5.3 Finding Two-Port Parameters
A linear reciprocal two-port can be represented by the
following equivalent Π circuit .
I1
I2
− y12
+
V1
−
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y11 + y12
y22 + y12
+
V2
−
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5.3 Finding Two-Port Parameters
Similarly , a linear two-port can also be represented
by the following equivalent circuit with dependent
sources .
I1
I2
+
+
y11
V1
y12V2
y21V1
V2
y22
−
−
I1 = y11V1 + y12V2
I 2 = y 21V1 + y 22V2
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5.3 Finding Two-Port Parameters
A linear two-port can be represented by the following
equivalent circuit .
I1
+
V1
−
h11
I2
h12V2
h21 I1
h22
+
V2
−
V1 = h11 I 1 + h12V 2
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I 2 = h 2 1 I 1 + h 22V 2
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5.3 Finding Two-Port Parameters
Example 6 : Find the transmission parameters
of the following circuits
，
(b)
(a)
V1   a11
 I  = a
 1   21
a1 2   V 2 
a 2 2   − I 2 
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5.3 Finding Two-Port Parameters
For circuit (a) and when I2=0
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I1 = − I 2 = 0 ∴ a21 =
I1
=0
V2
V2 = V1
V1
=1
V2
∴ a11 =
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5.3 Finding Two-Port Parameters
For circuit (a) and when V2=0
 a11
∴
 a21
V1 = RI1 ∴a12 =
V1 V1
= =R
−I2 I1
I1 =−I2 ∴a22 =
I1
=1
−I2
a12  1 R 
=

a22  0 1 
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5.3 Finding Two-Port Parameters
Similarly, for circuit (b)
a11 a12   1 0
a a  = G 1

 21 22  
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5.3 Finding Two-Port Parameters
Method (2)
The 6 sets of parameters relate the same input
and output terminal variables, hence they are
interrelated.
A systematical procedure for obtaining a set of
parameters from another one is given as
follows for reference.
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5.3 Finding Two-Port Parameters
Step1: Arrange the given two port parameters
in the following standard form:
k11V1+k12I1+k13V2+k14I2=0
k21V1+k22I1+k23V2+k24I2=0
Step2: Separate the independent variables and
the dependent variables of the desired
parameter set.
Step3: Find the solution of the dependent
variable vector.
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5.3 Finding Two-Port Parameters
Example 7: Given Z parameters, find h parameters.
Step1 From V1=z11I1+z12I2
V2=z21I1+z22I2
In the standard form
1V1-z11I1+0V2-z12I2=0
0V1+z21I1-1V2+z22I2=0
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5.3 Finding Two-Port Parameters
Step2 1

− z12  V1   z11
=
z 22   I 2   − z 21
0

Step3 solve
V1 
I 
 2
V1   1
 I  = 0
 2 
h
=  11
 h21
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0   I1 
1  V 2 
−1
− z12   z11
z 22   − z 21
h12   I1 
h22  V2 
0   I1 
1  V2 
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5.3 Finding Two-Port Parameters
A d ifferent so lu tio n ap proach
V1 = z11 I 1 + z1 2 I 2 ........( A )
V 2 = z 21 I1 + z 22 I 2
........( B )
Fro m ( B ) on e can o btain
I2 =
− z 21
1
I1 +
V2
z 22
z 22
....( C )
= h 21 I 1 + h 2 2 I 2
Fro m ( A ) an d ( C )
V1 = z1 1 I 1 + z12 ( h 21 I 1 + h 2 2 I 2 )
= ( z11 + z1 2 h 2 1 ) I 1 + z12 h 2 2 I 2
@ h1 1 I 1 + h12V 2
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5.3 Finding Two-Port Parameters
Example 8 :
Two sets of measurements are made on a two-port
resistive circuit. The first set is made with port 2
open, and the second set is made with port 2
short-circuited. The results are as follows :
Port 2 open
Port 2 short - circuited
V1 = 10mV
V1 = 24mV
I 1 = 10μA
I 1 = 20μA
V2 = -40V
I 2 = 1mA
Find h parameters from there measurements.
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5.3 Finding Two-Port Parameters
V1 = h11 I1 + h12 V2 ……… (A)
I2 = h21 I1 + h22 V2 ……… (B)
When port 2 is short circuited, V2=0
V1=24mV, I1=20μA, I2=1mA
Hence, from (A) and (B)
24mV = h11(20μA) + 0
1mA = h21(20μA) + 0
∴ h11 = 1.2 KΩ , h21 = 50 ……(C)
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5.3 Finding Two-Port Parameters
When port 2 is open, I2=0
V1=10mV, I1=10μA, V2=-40V
Hence, from (A), (B) and (C)
10mV = 1.2 KΩ (10μA) + h12 (-40V)
0 = 50 (10μA) + h22 (-40V)
∴ h12 = 5x10-5 , h22 = 12.5 μS
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5.4 Analysis of the Terminated Two-Port
Circuit
Reciprocal Theorem
Version 1 : For a reciprocal circuit, the interchange
of an ideal voltage source at one port
with an ideal ammeter at the other port
produces the same ammeter reading.
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5.4 Analysis of the Terminated Two-Port
Circuit
V1 = z11 I1 + z12 I 2
V2 = z21 I1 + z22 I 2
Q V1 = VS , I 2 = -I A1 , V2 = 0
VS = z11 I1 + z12 (-I A1 )
0 = z12 I1 + z22 (-I A1 )
∴ I A1 =
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z12VS
2
z11 z22 - z12
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5.4 Analysis of the Terminated Two-Port
Circuit
Q V1 = 0 , I A2 = -I 1 , V2 = VS
∴ 0 = z11 (-I A2 ) + z12 I 2
VS = z12 (-I A2 ) + z22 I 2
∴ I A2 =
z12VS
= I A1
2
z11 z22 - z12
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5.4 Analysis of the Terminated Two-Port
Circuit
The effect of reciprocity on the two-port
parameters is given by
z12 = z21 , or y12 = y21
a11a22 - a12 a21 = 1 , or
b11b22 - b12 b21 = 1 , or
h12 = -h21 , or g12 = -g 21
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5.4 Analysis of the Terminated Two-Port
Circuit
Examples of symmetric two ports.
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5.4 Analysis of the Terminated Two-Port
Circuit
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5.4 Analysis of the Terminated Two-Port
Circuit
There are mainly 6 interested characteristics
for a terminated two-port circuit in practical
applications.
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5.4 Analysis of the Terminated Two-Port
Circuit
input impedance Zin @ V1 / I1
output current I2
Thevenin equivalent looking into port 2.
current gain I2 / I1
voltage gain V2/ V1
voltage gain V2/ Vg
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5.4 Analysis of the Terminated Two-Port
Circuit
The derivation of any one of the desired
expressions involves the algebraic manipulation
of the two-port equations along with the two
constraint equations imposed at input and output
terminals.
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5.4 Analysis of the Terminated Two-Port Circuit
Example 9 : Use Z parameters as an illustration
Zg
Vg
+-
I1
+
V1
−
I2
[Z ]
+
V2
ZL
−
two-port equation
V1 = z11 I1 + z12 I 2 LL ( A )
V2 = z 21 I1 + z 22 I 2 LL ( B )
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5.4 Analysis of the Terminated Two-Port Circuit
input port constraint
Zg
Vg = I1 z g + V1 LL (C )
+-
Vg
I1
I2
+
[Z ]
V1
−
+
ZL
V2
−
Output port constraint
V2 = − z L I 2 LL ( D)
(1) Find Zin = V1 / I1
From (D) and (B) ,
I2 =
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− z21 I1
LL ( E )
z22 + z L
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5.4 Analysis of the Terminated Two-Port Circuit
Substitute (E) into (A)
Z in = z11 −
z12 z21
z22 + zL
Zg
Vg
I1
+
V1
−
I2
[Z ]
+
ZL
V2
−
(2) Find I2 : From (A) and (C)
I1 =
Vg − z12 I 2
z11 + z g
LL ( F )
Substitute (F) into (E)
I2 =
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− z21Vg
( z11 + z g )( z22 + z L ) − z12 z21
LL (G )
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5.4 Analysis of the Terminated Two-Port Circuit
(3) Find VTH and ZTH at port 2 :
With I2=0 , from (A) , (B) and (F)
V1 = z11 I1
V2 = z 21 I1 ⇒ V2 =
I1 =
Vg
z11 + z g
z 21
V1 LL ( H )
z11
LLLLLLL ( I )
From (C) , (H) and (I)
∴ VTH = V 2 =
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z 21
Vg
z g + z11
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5.4 Analysis of the Terminated Two-Port Circuit
With Vg =0 , then V1 = − z g I1 LL ( J )
From (A) and (J) , I 1 =
∴ ZTH =
− z12 I 2
z11 + z g
V2
z z
= z22 − 12 21
I2
z11 + z g
(4) Find current gain I2/I1
From (E) ,
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I2
z21
=−
I1
z L + z22
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5.4 Analysis of the Terminated Two-Port Circuit
(5) Find voltage gain V2/V1 :
From (B) and (D)
V2 = z21 I1 + z22 (−
∴V2 =
V2
)
zL
z21 z L
I1 LLLLLLL (K)
z L + z22
From (A) , (D) , and (K)
( z11 z L + z11 z22 − z122 )
V1 =
I1
z L + z22
∴
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V2
z21 z L
=
LLLLL ( L)
V1 z11 z L + z11 z22 − z122
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5.4 Analysis of the Terminated Two-Port Circuit
(6) Find V2/Vg :
Q
Z in
V2 V2 V1 V2
=
=
Vg V1 Vg V1 ( z g + Z in )
=
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z21 z L
( z11 + z g )( z22 + z L ) − z12 z21
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5.4 Analysis of the Terminated Two-Port Circuit
Example 10 : Given the following circuit, find V2
when RL=5KΩ
b11 = −20
b12 = −3000Ω
b21 = −2ms
b22 = −0.2
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5.4 Analysis of the Terminated Two-Port Circuit
Example 10: (cont.)
V 2 = b 1 1V 1 − b 1 2 I 1 .. . .. . .. . .. . .( A )
I 2 = b 2 1V 1 − b 2 2 I 1 .. . .. . . .. . .. . ( B )
V 1 = 5 0 0 − 5 0 0 I 1 . .. . .. . .. . . .. ( C )
I2 = −
V2
. . . .. . .. . .. . .. . .. . . .. . ( D )
RL
Substitute (C) and (D) into (A) and (B) to eliminate V1
and I2 V 2 − 1 3 × 1 0 3 I 1 = − 1 0 4 .....( E )
V 2 + 6 × 1 0 3 I 1 = 5 × 1 0 3 .....( F )
From (E) and (F) , V 2 =
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5000
= 2 6 3 .1 6 V
19
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5.5 Interconnected Two-Port Circuits
Two-port circuits may be interconnected in five
ways :
(1) in series
(2) in parallel
(3) in series-parallel
(4) in parallel-series
(5) in cascade
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5.5 Interconnected Two-Port Circuits
(1) Series connection
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I1 = I1a = I1b
V1 = V1a + V1b
I2 = I2a = I2b
V2 = V2a + V2b
∴
[ Z] =
 Za  +  Zb 
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5.5 Interconnected Two-Port Circuits
(2) Parallel connection
I1 = I1a + I1b
V1 = V1a = V1b
I2 = I2a + I2b
V2 = V2a = V2b
∴
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[Y] =
 Ya  +  Yb 
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5.5 Interconnected Two-Port Circuits
(3) Series-parallel connection
I1
+
I1a
V1
-
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I2a
V1a +-
Na
I1b
V2a +-
I2
I2b
V1b +-
Nb
V1 = V1a + V1b
I2 = I2a + I2b
V2 = V2a = V2b
[h] =
+
-
V2b +-
I1 = I1a = I1b
∴
V2
 h a  +  h b 
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5.5 Interconnected Two-Port Circuits
(4) Parallel-series connection
I1 = I1a + I1b
V1 = V1a = V1b
I2 = I2a = I2b
V2 = V2a + V2b
∴ g  =  g  +  g 
 a  b
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5.5 Interconnected Two-Port Circuits
(5) Cascade connection
+
V1
I1a
I1
+
V1a
−
−
I 2a
Na
+
V1b
−
−
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'
a12 
 a11
=  '
a22 
 a21
I 2b
+
V2 a
V1 = V1a
V2a= V1b
V2 = V2b
 a11
a
 21
I1b
Nb
+
V2b
I2
−
+
V2
−
I1 = I1a
I1b = -I2a
I2 = I2b
a12 '   a11"
+
a22 '   a21"
a12" 
a22" 
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5.5 Interconnected Two-Port Circuits
Example 11 : Find the [Z] and [Y] parameters of the
following two port network.
R4
I1
R1
I2
R3
+
V1
+
V1
R2
-
-
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5.5 Interconnected Two-Port Circuits
Example 11 : (cont.)
For [y] parameters :
I1a
R1
R3
I1
R2
+
-
I2a
I2
+
V2
-
V1
R4
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5.5 Interconnected Two-Port Circuits
Example 11 : (cont.)
 I1a   y11 y12   V1a 
I  =  y
 
 2a   21 y 22  V2a 
 1
R R
2 3
 = R1 + (R 2 || R 3 ) = R1 +
R +R
 y11
2
3

 1 = R + (R || R ) = R + R1R 2
3
1
2
3 R +R
y
 22
1
2
I
R2
R2
1
y = y = 1a = - V y ×
=-y ×
12
21
2 22 R1 +R 2 V2
22 R1 +R 2
V2
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5.5 Interconnected Two-Port Circuits
Example 11 : (cont.)
 I1b 
I 
 2b 
=
 y11
y
 21
y  V 
1b
12
y  V 
22   2b 
1

y =
 11 R = y 22
4


 y = y = I1 = - 1
 12 21 V
R

2
4
∴  y  =  y  +  y 
 a  b
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5.5 Interconnected Two-Port Circuits
Example 11 : (cont.)
For [z] parameters：
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5.5 Interconnected Two-Port Circuits
Example 11 : (cont.)
I1a
+
V1a
V2a
−
−
I1b
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I 2a
+
z11 = R 1 || (R 3 + R 4 )
Z22 = R 3 || (R 1 +R 4 )
Z12 = Z21 = Z22 ×
R1
R 1 +R 4
I 2b
+
V1b
+
V2b
Z11 = Z22 = R 2
−
−
Z12 = Z21 = R 2
∴
[ Z] =
 Za  +  Zb 
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5.5 Interconnected Two-Port Circuits
Example 12 : Find the a parameters.
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 1 0
 V1  1 10  
 1 5  V2 
=
 I  0 1   1

 
1  0 1 -I 2 

 1 
2

a12   V2 
a
=  11
 
a 21 a 22   −I 2 
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5.5 Interconnected Two-Port Circuits
Example 13 : A common emitter circuit
 Vb   h ie
 I  = h
 c   fe
h re   I b 
h oe   Vc 
h ie :base input impedence
h re :reverse voltage gain
h fe :forward current gain
h oe :output admittance
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Summary
n
Objective 1 : Understand the definition of 6 sets of
two port parameters.
n
Objective 2 : Be able to find any set of two-port
parameters.
n
Objective 3 : Be able to analyze a terminated
two-port circuit.
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Summary
n
Objective 4 : Understand the reciprocal theorem
for two-port circuits.
n
Objective 5 : Know how to analyze an
interconnected two-port circuits.
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Summary
n Problem : 18.4
18.8
18.11
18.18
18.37
18.38
n Due within one week.
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