Download Chapter 14 Frequency Response

Chapter 18
Two-port Networks
the four terminals have
four-terminal been paired into ports
network
KCL
two-port
network
At all times, the instantaneous current flowing
into one terminal is equal to the instantaneous
current flowing out the other.
i1
i2
i4
i3
i1+i2+i3+i4=0(KCL)
i1
i1=-i2 ; i3=-i4
i4
V1  z11I1  z12 I 2
V1  AV2  BI 2
V1  h11I1  h12V2
I1  CV2  DI 2
I 2  h21I1  h22V2
I1  y11V1  y12V2
V2  aV1  bI1
I1  g11V1  g12 I 2
I 2  y21V1  y22V2
I 2  cV1  dI1
V2  g 21V1  g 22 I 2
V2  z 21I1  z 22 I 2
The network is linear(without independent sources).
Impedance Parameters
Impedance or z parameters are defined by
V1  z11I1  z12 I 2
V2  z21I1  z22 I 2
impedance
matrix Z
z11 
V1
I1 I  0
2
V1  z11I1  z12 I 2  Voc1
V2  z21I1  z22 I 2  Voc2
Open-circuit input impedance.
z12  VI 21
I1  0
Open-circuit transfer impedance from port 1 to port 2
z 21  VI12
I 2 0
Open-circuit transfer impedance from port 2 to port 1
z 22  VI 22
I1  0
Open-circuit output impedance.
Determining of the z parameters: (a) finding z11 and z21, (b)
finding z12 and z22
Examples
(a) T equivalent circuit (for reciprocal case only), (b) general
equivalent circuit
Admittance Parameters
Admittance or y parameters are defined by
 I1   y11
   
 I 2   y21
y12  V1 
 
y22 V2 
admittance
matrix Y
I1  y11V1  y12V2
I 2  y21V1  y22V2
I1  y11V1  y12V2  I sc1
I 2  y21V1  y22V2  I sc2
y11  VI11
V2  0
Short-circuit input admittance.
y12  VI12
V1  0
Short-circuit transfer admittance from port 1 to port 2
y21  VI21
V2  0
Short-circuit transfer admittance from port 2 to port 1
y22  VI22
V1  0
Short-circuit output admittance.
Determination of the y parameters: (a) finding y11 and y21,
(b) finding y12 and y22.
(a) -equivalent circuit (for reciprocal case only), (b) general
equivalent circuit.
Hybrid Parameters
Hybrid or h parameters are defined by
V1  h11I1  h12V2
I 2  h21I1  h22V2
hybrid matrix
Z
h11  VI11
V2  0
h12  VV12
I1  0
Open-circuit reverse voltage gain
I2
I1 V  0
2
Short-circuit forward current gain
h21 
h22  VI22
I1  0
Short-circuit input impedance.
Open-circuit output admittance.
The h-parameter equivalent network of a two-port network
Inverse hybrid parameters (g parameters)
I1  g11V1  g12 I 2
V2  g 21V1  g 22 I 2
The g-parameter model of a two-port network
Transmission Parameters
Transmission or T parameters are defined by
V1   A B  V2 
   


 I1   C D   I 2 
V1  a11V2  a12 I 2
I1  a21V2  a22 I 2
Transmission
matrix T
A  VV12
I 2 0
B   VI 21
C  VI12
V2  0
I 2 0
D   II12
V2  0
Open-circuit voltage ratio
Negative short-circuit transfer impedance
Open-circuit transfer admittance
Negative short-circuit current ratio
Inverse transmission parameters
V2  aV1  bI1
I 2  cV1  dI1
Reciprocal Two-Port Circuits
------------- linear and has no dependent source
If a two-port circuit is reciprocal, the following
relationships exist among the port parameters:
z12  z21
y12  y21
h12  h21
g12   g 21
T  AD  BC  1
T   ad  bc  1
Symmetric Two-Port Circuit
A reciprocal two-port circuit is symmetric if its ports can
be interchanged without disturbing the values of the
terminal currents and voltages.
If a two-port circuit is symmetric, the following
relationships exist among the port parameters: (besides
those exist in reciprocal)
z11  z 22
y11  y22
h  h11h22  h12h21  1
g  g11 g 22  g12 g 21  1
A D
ad
Question: How many calculations or measurements are
needed to determine a set of parameters of a two-port
circuit?
For a general two-port with sources:
For a general linear two-port:
6
4
For a reciprocal two-port:
3
For a symmetric two-port:
2
Relationships between
parameters
Example:
z parameters
y parameters
V1  z11I1  z12 I 2
I1  y11V1  y12V2
V2  z21I1  z22 I 2
I 2  y21V1  y22V2
z 22
z12
 y11 
, y12  
,
z
z
z 21
z11
y 21  
, y 22 
z
z
where
z  z11z22  z12 z21
Interconnection of networks
Series connection of
two two-port networks
Z   Z a   Zb 
Parallel connection of two
two-port networks
Y   Ya   Yb 
Cascade connection of
two two-port networks
T   Ta Tb 
Transistor amplifier with source and load resistance