Download 2-port matrix description

CIRCUITS and
SYSTEMS – part II
Prof. dr hab. Stanisław Osowski
Electrical Engineering (B.Sc.)
Projekt współfinansowany przez Unię Europejską w ramach Europejskiego Funduszu Społecznego. Publikacja dystrybuowana jest bezpłatnie
Lecture 13
Two-port networks
Definition of 2-port
2-port network is a 4-terminal circuit structure of two terminals
forming input and 2 terminals forming output port
at following conditions satisfied
I1  I1' ,
I 2  I 2'
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2-port matrix description
Admittance form - (I1, I2) expressed as function of (U1, U2)
Impedance form - (U1, U2) expressed as function of (I1, I2)
Hybrid form - (U1, I2) expressed as function of (I1, U2)
Inverse hybrid form - (I1, U2) expressed as function of (U1, I2)
Chain form - (U1, I1) expressed as function of (U2, I2)
Inverse chain form - (U2, I2) expressed as function of (U1, I1).
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2-port matrix description (cont.)
1) Admittance desription
 I1  Y11 Y12  U1 
U1 
 I   Y Y  U   Y U 
 2   21 22   2 
 2
2) Impedance desription
U1   Z11 Z12   I1 
 I1 
 Z 
U   Z



 2   21 Z 22   I 2 
I 2 
Y  Z 1
3) Hybrid desription
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U1   H11
 I   H
 2   21
H12   I1 
 I1 
 H 



H 22  U 2 
U 2 
2-port matrix description (cont.)
4) Inverse hybrid desription
 I1  G11 G12  U1 
U1 
 G 
U   G



 2   21 G22   I 2 
 I2 
G  H 1
5) Chain desription
U1   A11
 I   A
 1   21
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A12   U 2 
 U2 
 A




A22   I 2 

I
 2
Example
Determine the matrix description of 2-port
Solution:
I1  I  I 2  YU 2  1  Z 2Y  I 2 

U1  U 2  Z1I1  Z 2 I 2
U1  1  Z1Y U 2  Z1  Z2  Z1Z2Y  I 2 
Chain description
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U1  1  Z1Y
I   Y
 1 
Z1  Z 2  Z1Z 2Y   U 2 
  I 
1  Z 2Y
 2 
Transfer function and 2-port
description
Any transfer function can be expressed through the 2-port
parameters. For example voltage transfer function
• chain matrix description
H u (s) 
U 2 (s)
1

U 1 ( s ) A11
• admittance matrix description
H u (s) 
8
U 2 (s)
Y
  21
U1 ( s)
Y22
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Input impedance
• Impedance in general form
U1 ( s) A11  A12Yo
Z we ( s) 

I1 ( s) A21  A22Yo
• No load of the 2-port (Yo=0)
A11
Z we ( s ) 
A21
• Short circuit of output port (Zo=0)
Z we ( s ) 
A12
A22
Example
Determine the input impedance of the 2-port
Chain matrix
Voltage transfer function
1  Z1Y
A
 Y
Z1  Z 2  Z1Z 2Y 

1  Z 2Y

U 2 (s)
1
1
Z
Tu ( s) 



U1 ( s ) A11 1  Z1Y Z  Z1
Input impedance
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U1 ( s) A11  A12Yo (1  Z1Y )  ( Z1  Z 2  Z1Z 2Y )Yo
Z we ( s) 


I1 ( s) A21  A22Yo
Y  (1  Z 2Y )Yo
Connections of 2-ports
Chain connection
A  A1 A 2 ,
A  A1 A 2    A n ,
A1 A 2  A 2 A1
Series connection
Z  Z1  Z 2 ,
n
Z   Zi
i 1
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Connections of 2-ports (cont.)
Parallel connection
Y  Y1  Y2 ,
n
Y   Yi
i 1
Series-parallel connection
H  H1  H 2 ,
n
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H   Hi
i 1
Connections of 2-ports (cont.)
Parallel-series connection
G  G1  G 2 ,
n
G   Gi
i 1
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Gyrator
U1   0
 I   G
 1  z
Rz   U 2 
0   I 2 
Loading gyrator by the impedance Zo the input impedance of the
connection is inversely proportional to Zo.
Z we
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A11  A12Yo Rz2


A21  A22Yo Z o
Gyrator loaded by capacitor
Let us assume that Zo = 1/sC . In such case the input impedance
of this connection represents the inductor.
Z we  sRz2C
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L  Rz2C
Negative impedance converters
(NIC)
Negative impedance converter (NIC) converts either the current or
voltage with the negative sign.
•NIC with inversion of current (INIC)
0  U2 
U1  1
 I   0  K   I 
i 
2
 1 
•NIC with inversion of voltage (VNIC)
U1   Ku
I   0
 1 
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0  U 2 
1  I 2 
NIC loaded by an impedance
Input impedance of the connection
Z we 
Z
U1
U2

 o
I1  K i (  I 2 )
Ki
Loading NIC by impedance Zo we implement the negative Zo.
Observe that applying NIC we may introduce the instability to the circuit!
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