Two-port networks
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Transcript Two-port networks
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
t0
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
t0
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
ZY
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