Two-port networks







Review of one ports
Various two-port descriptions
Terminated nonlinear two-ports
Impedance and admittance matrices of two-ports
Other two-port parameter matrices
The hybrid matrices
The transmission matrices
1-port
2-port
2-port
2-port
1-port
Thevenin’s Equivalent Circuit
Norton’s Equivalent Circuit

i

N
v
i
NO
v


 
eO C
v ( t )  eO C ( t ) 

t
h ( t ,  ) i ( ) d 
t0
0
For LTI network
v ( t )  eO C ( t ) 

t
h ( t   ) i ( ) d 
0
In frequency domain
V ( s )  EOC ( s )  Z ( s ) I ( s )
0
No independent sources
t0
iD(t)
+
vd(t)
vD(t)
+
VD
Nonlinear
one port
-
-
vA=VA+va
VA
|va|p
0
va
t
0
t
iD(mA)
3.5
3
2.5
id
2
ID
1.5
Bias point
t
Q
1
0.5
VD
0
0.5
0.55
0.6
0.65
0.75
0.7
vd
t
vD(V)
iD (mA)
1.52
1.50
1.48
1.46
id
ID
t
1.44
1.42
1.40
VD
1.38
0.699
0.6995
0.7005
0.7
0.701
vD (V)
For small v d
vd
t
ID  ISe
For DC bias
V D / nV T
For DC bias + small signal
iD  I S e
 ISe
v D / nV T
 IS e
(V D  v d ) / nV T
V D / nV T v d / nV T
e
 IDe
v d / nV T
From Taylor’s series expansion
e
x
1 x 
x
2
2!

x
3
3!
 ...
2
3


x
x
iD  I D  1  x 

 ... 


2!
3!


Where x  v d / nVT
For x
1 or  v d
nVT
i D  I D 1  x   I D 
 ID
id  
 nV T
ID
nV T

vd
vd 

rd

vd
id
 n VT 


I
 D 
+
iD = ID+ id
vD = VD+ vd
vd
-
 nV T 
rd  

 ID 
id
+
rd
iD(mA)
3.5
3
2.5
Slope at Q point = g d 
2
Bias point
ID
1.5
t
Q
1
0.5
VD
0
0.5
0.55
0.6
0.65
0.75
0.7
t
vD(V)
1
rd
Example If V  10 V  10 m V sin(100 t ) find v D
V
V D  0 .7 V
ID 
10 kW
iD
rd 
vD
vd 
10 V  0 . 7 V
10 k W
nV T
ID
rd
rd  10 k W

 0 . 93 mA
2  25 mV
 53 . 8 W
0 . 93 mA
1 0 mV sin( 100 t )  53 . 5 μV sin( 100 t )
v D  0 . 7 V  53 . 5 μ V sin( 100 t )
Two-port networks
LTI one ports
I1
+
V1
-
One port
network
Fig. 1
Zin Yin
Input impedance
Z in 
Input admittance
V1
I1
Yin 
I1
V1
Two-port networks
Example 1
Determine the input impedance of the circuit in Fig. 2
I1
 I1
Z2
Fig. 2
Z3
Zin
I in  I 1    I 1 
V in
Z2
Vin  (1   ) Z 2 I in
Z in  (1   ) Z 2
Example 2
Determine the output impedance of the circuit in Fig. 3
I1
I out
+
Z1
 I1
Vout
Fig. 3
Z3
Z out
-
I out   I 1   I 1  (1   )
V out
Z1
Z out 
V out
I out

Z1
1 
Two-port networks




Circuits can be considered by theirs terminal variables
Voltages and currents are terminal’s variables
Complex circuit can be analyzed more easily.
There are many kinds of two port parameters.
I1
+
V1
-
I2
Two port
network
Fig. 4 A two port network
+
V2
-
Common-Emitter (CE) Fixed-Bias Configuration
Removing DC effects of VCC and Capacitors
Small signal equivalent circuit
Hybrid equivalent model
re equivalent
model
Various two-port descriptions
i  g (v)
or
i1  g 1 ( v1 , v 2 )
i2  g 2 ( v1 , v 2 )
Port current
Port voltage
v  r (i)
or
v1  r1 ( i1 , i2 )
v 2  r2 ( i1 , i2 )
Or hybrid
v1  h1 ( i1 , v 2 )
i2  h2 ( i1 , v 2 )
Two-port networks
The Y parameter
The admittance or Y parameter of a two port network is defined by
 I 1   y11
  
 I 2   y 21
y12   V1 
 
y 22  V 2 
or in scalar form
I 1  y11V1  y12V 2
I 2  y 21V1  y 22V 2
The Y parameter
The Y parameters can found from
y11 
y 21 
I1
V1
y12 
V2  0
I2
V1
y 22 
V2  0
I1
V2
V1  0
I2
V2
V1  0
These parameters are call short-circuited admittance parameters
The Y parameter
Example 3
Determine the admittance parameters from the circuit in Fig 5.
I1
Y2
+
V1
I2
Y1
Y3
0.5V1
-
+
V2
Fig 5.
-
I1  Y1V1  Y2 (V1  V 2 )  (Y1  Y2 )V1  Y2V 2
I 2  0.5V1  Y3V 2  Y2 (V 2  V1 )  (0.5  Y2 )V1  (Y2  Y3 )V 2
 I 1   Y1  Y2
  
 I 2   0.5  Y2
 Y2   V1 
 
Y2  Y3  V 2 
y11  Y1  Y 2 , y12   Y 2
y 21  0.5  Y 2 , y 22  Y 2  Y3
The Y parameter
Example 4
Compute the y-parameter of the circuit in Fig.6
1W
I1
I2
Iˆ1
1:a
+
V1
-
1W
+
ˆ
1W V1
-
+
V2
-
Fig.6
1
ˆ
ˆ
I 1  V1  (V1  V1 )  2V1  V1  2V1  V 2
a
1 ˆ
1
1
2
I 2   I 1    Vˆ1  (V1  Vˆ1 )    V1 
V2
2


a
a
a
a

2

 I1 
   
I2   1
 a
1
 
a  V1 
 
2  V 2 
2
a 
y11  2 , y12   1
y 21   1
a
a
, y 22  2
a
2
Y parameter analysis of terminated two-port
I1
+
V1
-
I2
+
Two port
network
V2 YL
-
Fig. 9 Terminated two-port
Y-parameter equations
 I 1   y11
  
 I 2   y 21
 I 1   y11
  
 0   y 21
y12   V1 
 
y 22  V 2 
  V1 
 
 Y L  V 2 
y12
y 22
I 2  Y LV 2
Y parameter analysis of terminated two-port
From Crammer’s rules
V1 
The input admittance Yin
I1
y 12
0
y 22  Y L

y 11
y 12
y 21
y 22  Y L
Yin  y11 
and
y 21
y 22  YL
y 11 ( y 22  Y L )  y 12 y 21
y12 y 21
y11 ( y 22  Y L )
y 2 1V1   ( y 2 2  Y L )V 2
V2  
( y 22  Y L ) I 1
V1
Y parameter analysis of terminated two-port

y y
I1  y11V1  y12V 2   y11  12 21
y 22  Y L

V2
Gain:

V1
Rs
+
y 21
y 22  Y L
I1
+
vs

 V1

V1
y11
I2
y 12V2
y 21V1
+
y22
V2
-
-
-
Yin
Fig 10 Terminated two-port Y-parameter model
YL
Two-port networks
The Z parameter
The impedance or Z parameter of a two port network is defined by
 V1   z11
  
V 2   z 21
z12   I 1 
 
z 22   I 2 
or in scalar form
V1  z11 I 1  z12 I 2
V 2  z 21 I 1  z 22 I 2
The Z parameter
The Z parameters can be found from
z11 
z 21 
V1
I1
z12 
I2 0
V2
I1
z 22 
I2 0
V1
I2
I1  0
V2
I2
I1  0
These parameters are call open circuit impedance parameters
The Z parameter
Example 6
Determine the impedance parameters from the circuit in Fig 11
3W
I1
I2
+ 4I 2 +
+
V1
0.1F
V2
-
-
Fig 11.
In frequency domain
V1  4 I 2 
V2  3 I 2 
 z1 1
Z  
 z 21
10
s
( I1  I 2 ) 
10
s
10
( I1  I 2 ) 
10
z1 2   s
  
z 22   1 0
 s
s
I 1  (4 
10
10
s
I 1  (3 
s
4 s  10 

s

3s  10 

s
)I2
10
s
)I2
The Y parameter
Example 7
Compute the z-parameter of the circuit in Fig.12
I1
R2
I2
+
V1
R1
-
R3
I3
+
V2
Fig.12
-
V1  R1 I1  R1 I 3
V 2  R3 I 2  R3 I 3
0   R1 I1  R3 I 2  ( R1  R 2  R3 ) I 3
I3 
R1
R1  R 2  R 3
I1 
R3
R1  R 2  R 3
I2
The Z parameter
2
V1  ( R1 

V2 

 z 11

 z 21
R1
R1  R 2  R 3
R1 ( R 2  R 3 )
R1  R 2  R 3
R1 R 3
R1  R 2  R 3
R1 R 3
R1  R 2  R 3
I1 
) I1 
R1 R 3
R1  R 2  R 3
R1 R 3
I2
I2
R1  R 2  R 3
2
I1  ( R3 
I1 
R3
R1  R 2  R 3
R 3 ( R1  R 2 )
R1  R 2  R 3
 R1 ( R 2  R 3 )
z 12   R1  R 2  R 3
  
R1 R 3 )
z 21  
R  R  R
2
3
 1
)I2
I2


R1  R 2  R 3

R 3 ( R1  R 2 ) 
R 1  R 2  R 3 
R1 R 3 )
Z parameter analysis of terminated two-port
I1
+
V1
-
I2
+
Two port
network
V2 Z L
-
Fig. 14 Terminated two-port
Z-parameter equations
 V1   z11
  
V 2   z 21
V1   z11
  
 0   z 21
z12   I 1 
 
z 22   I 2 
  I1 
 
 Z L  I2 
z12
z 22
V2   Z L I 2
Z parameter analysis of terminated two-port
From Crammer’s rules
I1 
The input impedance Zin
V1
z12
0
z 22  Z L
z11
z12
z 21
z 22  Z L
Z in  z11 
z12 z 21
z 22  Z L
and
z 21 I1   ( z 22  Z L ) I 2
I2  
z 21
z 22  Z L

I1
( z 22  Z L )V1
z11 ( z 22  Z L )  z12 z 21
Z parameter analysis of terminated two-port

z z
V1  z11 I 1  z12 I 2   z11  12 21
z 22  Z L

Z in
V2
V V
ZL
z 21
 1 2 
Vs
V s V1
Z in  Z s z 22  Z L Z in
Gain:
Rs
+

ZL
V1
-
z 21
z 22  Z L Z in  Z s
I1
+
vs

 I1

I2
z11
z 12 I 2
+
+
-
-
z22
z 21 I1
-
Z in
Fig 15 Terminated two-port Z-parameter model
+
V2
-
ZL
Z parameter analysis of terminated two-port
Example 9
The circuit in Fig 16 is a two-stage transistor amplifier. The Z-parameters
for each stage are
I1
Vs
+
V1
-
Z in
Z1
1.0262  10 6
Z2  
6
1.0258  10
I2
k
0.5W
+
-
2.667 

6, 667 
2 W
 350
Z1  
6
  10
Stage 1
+
V2
-
Z in 2
Determine a) The input impedance Z in 2 and Z in
b) The overall voltage gain
6, 790.8 

6, 793.5 
Z2
Stage 2
I out
+
Vout
16W
-
Fig 16
c) Check the matching of the load and output impedance
Z parameter analysis of terminated two-port
Solution
Z in 2  z11 
z12 z 21
z 22  Z L
6
 1.0262  10 
6790.8  1.0258  10
6793.5  16
 3,159 W
V out

V2
ZL
z 21
z 22  Z L Z in 2
6

16(1.0258  10 )
(16  6793.5)3,159
 0.7629
6
Z parameter analysis of terminated two-port
Z L1  2 k // Z in 2  2000 // 3159  1224 . 7 W
Z in  z11 
z12 z 21
z 22  Z L 1
 350 
2.667  10
6
6667  1224.7
 687.9 W
V2
Vs

Z L1
z 21
Z L 1  z 22 Z s  Z in
1224.7


 1224.7  6667
 10

 
  75  687.9
6




  203.4
0.902
V2
Vs

V1 V 2
V s V1

Z in
225.6
ZL
z 21
Z in  Z s z 22  Z L Z in

ZL
z 21
z 22  Z L Z in  Z s
Z parameter analysis of terminated two-port
The overall voltage gain
AVS 
V out

Vs
V out V 2
V2 Vs
 0 . 7629  (  203 . 4 )
  155 . 2 V / V
Out put impedance
Z out 
V2
I2
Vs 0
The detail is left to the student to show that
Z out  z 22 
z12 z 21
R s  z11
Z parameter analysis of terminated two-port
Z out 1  z 22 
z12 z 21
R s  z11
 6667 
2.667  10
6
0.5  350
 14.276 k W
R s 2  Z out 1 // 2 k  1.7542 k W
Z out  6793.5 
6790.8  1.0258  10
6
1754.24  1.0262  10
6
 16.93 W
Therefore the load is closely matched to the output impedance
The h-parameter (Hybrid parameter)
H-parameter is the combination of Z and Y parameter
defined by
 V1   h11
  
 I 2   h21
h12   I 1 
 
h22  V 2 
or in scalar form
V1  h11 I 1  h12V 2
I 2  h 21 I 1  h 22V 2
H-parameter is commonly used in transistor modeling.
The h-parameter
The h parameters can found from
h1 1 
h2 1 
h2 2 
h1 2 
V1
I1


y 21
 y 22 
 
 
I1  0
y1 2
y1 1
z1 2 z 2 1
z 22
z 21
z 22
y1 2 y 2 1
y1 1
I1  0
V1
V2
 z1 1 
y1 1
V2  0
I2
V2
y1 1
V2  0
I2
I1
1

z1 2
z 22

1
z 22
The h-parameter
Rs
I1
h11
I2
+
vs
+
V1
-
-
h 12V2
+
h 21 I1
+
-
V2
h22
Z in
Fig 17 Hybrid parameter model
-
ZL
The h-parameter
Example 10
Determine the h-parameter of the two-port circuit shown in Fig. 18
+
+
Fig. 18
R
1:a
I1
V1
V1 
2
I1 
1
V2
-
-
V2
a
a
V 2  Vˆ2
Vˆ2 V 2
I2 
 

R
R
R
 
1
a
I 1  0V 2
+
Vˆ2
-
R
I2
1 ˆ
V1  V 2
a
I1   aI 2
R
ˆ
V2  V2  R I 2 
I1  V 2
a
 R
V1   a 2
    1
 I 2  
 a
1

a  I1 
 
V
0  2 

The h-parameter
Example 10
Find the h-parameter of the circuit in Fig. 19 assuming L1=L2=M=1H
I1
Iˆ1
V1
-
1W
I2
+
+
Fig. 19
M
1W
L1
Vˆ2
L2 -
In frequency domain
V1  sL 1 Iˆ1  sMI
+
V2
2
Iˆ1  I 1  V 1
(1  sL1 )V1  sM I 2  sL1 I 1
-
The h-parameter
V 2  Vˆ2  I 2
Vˆ2  sL 2 I 2  sM Iˆ1  sL 2 I 2  sM ( I 1  V1 )
V 2  (1  sL2 ) I 2  sM ( I1  V1 )
sM V1  (1  sL 2 ) I 2  sM I 1  V 2
In matrix form
1  sL 1

 sM
 sM
 V1   sL 1
   
 (1  sL 2 )   I 2   sM
 V1  1  sL1
   
 I 2   sM
 sM


 (1  sL 2 ) 
1
 sL1

 sM
0   I1 
 
 1 V 2 
0   I1 
 
 1  V 2 
The h-parameter
With L1=L2=M=1 H
 V1  1  s
   
I2   s
s


 (1  s ) 
 s


2 s  1  s
1
1
s

s
0   I1 
 
 1  V 2 
s   I1 
 
s  1 V 2 
The inverse hybrid parameter
(g- parameter)
g-parameter is defined by
 I 1   g 11
  
V 2   g 21
g 12   V1 
 
g 22   I 2 
or in scalar form
I 1  g 11V1  g 12 I 2
V 2  g 21V1  g 22 I 2
g-parameter is an alternative form of hybrid representation.
The g parameters can found from
g11 
g 21 
g 22 
g12 
I1
V1

V1

I2 0
V2
I2
z 21
 
z1 1
y 21
 z 22 
z1 2 z 2 1
z1 2
y1 2
 
V1  0
y1 2 y 2 1
z1 1
z1 1

y 22
w h ere  h  h1 1 h 2 2  h1 2 h 2 1

y 22
 

y 22
 
h
h
1

h2 2
h2 1
y 22
V1  0
I1
I2
 y1 1 
z1 1
I2 0
V2
1
h1 2
h
h1 1
h
Inverse hybrid parameter model
Conversion of Two-port parameters
Two port parameters can be converted to any form as follows
From
And
 I 1   y11
  
 I 2   y 21
y12   V1 
 
y 22  V 2 
I  YV
 V1   z11
  
V 2   z 21
z12   I 1 
 
z 22   I 2 
V  ZI
 V  ZYV
ZY
1
and
Y  Z
1
 y11

 y 21
 z 22
y12    Z
 
y 22    z 21
  Z
 z12 
Z 

z11 
 Z 
 z11

 z 21
 y 22
z12    Y
 
z 22    y 21
  Y
 y12 
Y 

y11 
 Y 
where
 Z  z11 z 22  z12 z 21
 Y  y11 y 22  y12 y 21
Conversion of Two-port parameters
From y to h
 I 1   y11
  
 I 2   y 21
y12   V1 
 
y 22  V 2 
 y11V1   I 1  y12V 2
 y 21V 1  I 2  y 22V 2
  y11

  y 21
0   V1    1
   
1 I2   0
 V1    y11
   
 I 2    y 21
y12   I1 
 
y 22  V 2 
0

1
1
1

 0
y12   I 1 
 
y 22  V 2 
Conversion of Two-port parameters
 y12
 V1 
1  1
 

y11  y 21
I2 
Hence
 h11

 h 21


h12 

 
h22  


y11 y 22
1
  I1 
 
 y12 y 21  V 2 

y12
y11
y11
y 21
y12 y 21
y11
y 22 
y11






Conversion of Two-port parameters
It can be shown that for the terminated two-port with h-parameter the
following equations can be derived
V2  
Z in 
I1

V1
and
AVS 
V2
Vs

I1
h 22  Y L
V1
Z out 
V2
h 21
V2
I2
 h11 
h12 h 21
h 22  Y L
 h22 
h12 h21
h11  Z s
h21
( h22  Y L ) Z in
V1 V 2
V s V1

h21
1
( h22  Y L ) Z in  Z s
Transmission parameter
The t-parameter or transmission parameters are used in power system
and it is called ABCD parameter. The transmission parameter is defined by
V1   t11
  
 I 1   t 21
t12   V 2 


t 22    I 2 
or
V1   A
  
 I1   C
B   V2 


D   I2 
This means that the power flows into the input port and flow out to the
load from the output port.
t-parameter can be calculated from
t11 
t 21 
V1
V2
t12  
I2 0
I1
V2
t 22  
I2 0
V1
I2
V2  0
I1
I2
V2  0
Open or short circuit at
the output port
Transmission parameter
Example 11
Determine the t-parameter of the circuit shown in Fig 20.
I1
+
V1
-
1 ˆ
1
V1  V 2  (V 2  RI 2 )
a
a
I 1   aI 2
R
1:a
+
I2
+
Vˆ2
V2
-
-
V1   1
    a
 I1   0
Fig 20
  V2 



I
a  2 
R
a
Transmission parameter
One of the most importance characteristics of the two-port circuit with
t-parameter is to determine the overall cascade parameter.
I2
+
I1
+
V1
-
V1 
 V2 
   T1 

I

I
2
 1

Therefore
T1
V2
-
+
V3
-
V 3 
 V4 
   T2 

I

I
 4
 3
V1 
   T1 T2
 I1 
I4
I3
 V4 



I
 4
+
T2
V4
-
V 2  V3 ,
 I2  I3
Inverse Transmission parameter
V 2   A 
  
 I 2  C 
A 
V2
V1
C 
B  
D  
I1  0
V2
I1
I1  0
I2
V1
B    V1 


D     I1 
V1  0
I2
I1
V1  0
Interconnection of two-port network


Two port networks can be connected in series parallel or
cascaded
Series and parallel of two-port have 4 configurations





Series input-series output (Z-parameter)
Series input-parallel output (h-parameter)
Parallel in put-series output (g or h-1-parameter)
Parallel input-parallel output (Y-parameter)
With proper choice of parameters the combined
parameters can be added together.
Interconnection of two-port network
+
+
+
V11
Z1
-
+
V21
+
V11
H1
-
-
+
V2
V1
V1
+
V12
-
-
+
+
Z2
V22
-
-
+
G1
-
H2
-
H=H1+H2
Z=Z1+Z2
+
V12
+
V21
+
V2
+
Y1
V2
V1
+
G2
G=G1+G2
V21
-
-
-
Y2
Y=Y1+Y2
-
Example Bridge-T network
N1 // N2
For network N2
For network N1
1
T   
0
Z4 

1 
Y-parameters of the bridge-t network are