International Journal of Advanced Engineering Research and Technology (IJAERT) 334
Volume 3 Issue 10, October 2015, ISSN No.: 2348 – 8190
A-B-C-D Parameter model for Two-port Networks
Emerole Kelechi C1, Matthew Gabriel Chinedum2
1,2
Department of Electrical and Electronics Engineering, Federal Polytechnic Nekede, Owerri, Imo State, Nigeria
ABSTRACT
III.
Over the years, measurement of transmission system has
lingered with focus on how best to establish a network or
chain of cascaded network in order to implement or
measure the credibility and efficiency of a transmission
system. The two-port network analysis is aligned in a
well orderly and rigorously treated format in which it is
possible to ascertain the measurement of a two-port
network through its input and output terminals.
Keywords - Transmission, network, input, output,
terminals, two-port
I.
A-B-C-D PARAMETER
For any two port network there is a linear relationship
between input voltage and current and output voltage
and current for such a network we have
I2
I1
A
B
V2
V1
C
1
D
INTRODUCTION
Fig 2: Two Port Network with ABCD parameter
A two port network system is any network with four
terminals that is having two accessible input terminals
and two accessible output terminals. A two port network
includes attenuators, filters, transformers, amplifiers and
also transmission system. A two-port network is mostly
comprised of an input voltage and output voltage, an
input current and output current each are interwoven and
aid the easy measurement in a transmission system. The
transmission system is measured in this context using
series impedance and also a shunt impedance and further
merging or cascading of this two categories given room
or arises to other complex cascaded connection of a
transmission network.
II.
=
V1 =A V2 + B I2 …………………….. (1)
I1 = C V2 + D I2 ……………………... (2)
Output current open circuited
A=
TWO PORT NETWORK
B=
V2 = 0
C=
I2 = 0
I2
………(Z)
(4)
impedance (Ω) ohms
(5)
(Y) Simmers
…………Admittance
Output voltage short circuited
V2
V1
11
11
……….. no unit
Output voltage short circuited
A two port network is a network which has two input
and two output terminals. It can also be referred to as
four terminal network or quadruple network, such
network are as in important as in transmission element
thus making it necessary to analyze the input and output
terminals in relation to its voltage and current
importance.
I1
(3)
I2=0
D=
Fig 1 Two port network
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V2 = 0
…………no unit
(6)
International Journal of Advanced Engineering Research and Technology (IJAERT) 335
Volume 3 Issue 10, October 2015, ISSN No.: 2348 – 8190
IV.
ABCD parameters for determining
simple transmission network/lines
Where
Y= shunt admittance refer to fig 3 for V1, V2, I1, I2
Arranging the equation in matrix from we have:
For any series impedance in a simple transmission lines
such the equation is given as follows:
V1 = V2 + Z I2 ……………….
(7)
I1 = 0 V2 + I2 …………………
(8)
Where: Z – series impedance
V1 – sending voltage
V2 – receiving voltage
I1 – input current
I2 – output current
This series impedance in a simple transmission line can
further be expressed in matrix form as follows:
=
the following values are derived.
A =
1
B =
O
C =
Y
D =
1
VI.
HALF T-NETWORK
The circuit configuration of a half T – network can be
shown as follows:
Here the matrix formation is used to match the
equivalence in the ABCD formation and the value
deduced as given below:
A
=
1
B
=
Z
C
=
0
D
=
1
V.
Shunt admittance
parameters
in
V
I2
γ
V1
V2
Fig 5 Half T-Network
I2
Y
Z
ABCD
In shunt admittance network the circuit can be
represented as follows:
I1
I1
V
A Half T – network is a combination of a series
impedance and shunt admittance. it can be
considered as a cascaded connection that exist
between a single series impedance and a single
shunt admittance, with the series impedance coming
first then followed by the shunt admittance.
Putting the transmission parameter in matrix
2
1
Multiplying out the matrix values we have
Fig 4 Shunt admittance network
Here the admittance (Y) is place in parallel in the circuit
and the equation is written as follows:
V1 = V2 + O I2
(9)
I1 = YV2 + I2
(10)
=
Representing the values according to its ABCD
alignment
A
=
1 +ZY
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International Journal of Advanced Engineering Research and Technology (IJAERT) 336
Volume 3 Issue 10, October 2015, ISSN No.: 2348 – 8190
B
C
D
VII.
=
=
=
A full T – network has impedance Z1 and Z2 with
shunt admittance (Y) running in parallel and in
between Z1 and Z2 to for the T – transmission
network
Placing a full T – network in matrix we have:
Z
Y
1
HALF π - NETWORK
I1
Z
1
1
1
V1
I2
Y
V2
For unsymmetrical network we have
Fig 6 Half π-Network
A half π- network is referred to as a simple
transmission network having single series
impedance thereby forming a cascaded network of a
series or half π – network
Representing the parameter in matrix
The linear parameters are:
A
=
1 + Z1Y
B
=
Z2 + Z1 Z2 Y + Z1
C
=
Y
D
=
YZ2 + 1
For Symmetrical Network
Z1 = Z2 = Z, A = D
The linear parameters for symmetrical network are:
A
=
1 + ZY
B
=
2Z + Z2Y
C
=
Y
D
=
YZ + 1
=
Therefore the parameters linearly are put as follows:
A
=
1
B
=
Z
C
=
Y
D
=
1 +ZY
VIII.
IX.
A full π network is shown below
FULL T – NETWORK
I1
A full transmission network is shown below
I1
Z
Fig 7 Full T-Network
Z
I2
I2
V2
V1
V1
FULL π T – NETWORK
Y
V2
Fig 8 Full π network
From Fig 8 the transmission line is made up of three
cascaded network in the form of Y1 Z, Y2 forming
the full π – network.
Putting is matrix form we have
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International Journal of Advanced Engineering Research and Technology (IJAERT) 337
Volume 3 Issue 10, October 2015, ISSN No.: 2348 – 8190
An ideal transformer has no Ohmic losses and
leakage losses
For unsymmetrical network
ACTUAL TRANSFORMER
I1
The linear parameters for unsymmetrical network
are:
A
= I + Z Y2
B = Z
C = Y1+Y1Y2Z + Y2
D = Y1Z + 1
For Symmetrical network
Y1 = Y2 = Y, A = D
The linear parameter for symmetrical network is
A
=
1 + ZY
B
=
Z
C
=
2Y + Y2Z
D
=
YZ + 1
X.
I1
Z
Y
Fig 10 Actual Transformer
An actual transformer has imperfections and can be
represented as shown in Fig 10 above with the half
T – network cascade at the primary with
conjunction to an ideal transformer
IDEAL TRANSFORMER
A typical circuit network for an ideal transformer in
shown below
There the linear parameters are as follows:
A
=
n + ZYn
B
=
Z 1/n
C
=
Yn
D
=
1/n
XI.
Fig 9 Ideal Transformer
An ideal transformer is a lossless or loss free
transformer, an ideal transformer is perfect and free
from loss and has the transformation ratio of n:I
The equation for an ideal transformer in given as:
V1 = nV2 + 0I2
(11)
I1 = 0V2 + 1/n I2
(12)
Expressing the equation in terms of matrix
IDENTICAL TWO PASSIVE
NETWORK
V1 = AV2 + BI2
(13)
I1 = CV2 + DI2
(14)
CIRCUIT (A)
When a voltage is applied at input terminal and
output terminal is short circuited
then V2 = 0 we have:
V = A (0) + BI2SC
V = BI2SC
V = BI2SC
(15)
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V2
International Journal of Advanced Engineering Research and Technology (IJAERT) 338
Volume 3 Issue 10, October 2015, ISSN No.: 2348 – 8190
CIRCUIT (B)
0 =AV + BI21
BI21 = - AV
I21 = - AV/B
(17)
When the input voltage V1 = 0 the we have
Z2
V2
V1
Where the magnitude of: I11 = - I2SC & I2SC =
-I2SC = CV + DI2
Z1
-I2SC = CV + D (-AV/B)
Fig 13 Lattice Network
-I2SC = CV - DAV/B
-I2SC = (C- DA/B)V
The current I1 divides into Iz1 flowing through Z1 and IZ2
flows through Z2 impedances. The voltage drop in Z1
impedance due to the flow of current Iz1 is given by V21
with the arrow pointing the directions similarly voltage
drop in Z2 impedance due to the flow of current IZ2 in
give by V22
-(V/B) =
Where AD – BC = 1
BC – AD = -1
(18)
(19)
Z1
Z2
I1SC
I1SC
A
V2
B
Z2
Fig 14 Lattice Network
a
I21
I11
C
A
D
B
Z2
V22
V21
3
Fig 15 Voltage triangle vector
1
C
S
D
C
4
b
Fig 11 Passive Network
1
S
LATTICE NETWORK
C
Measurement of lattice network
In the measurement of a Symmetrical lattice
network the circuit is represented below as follows:
V1
Z2
Z2
V2
From the voltage triangle
V2 + V21 = V22
(20)
V2 = V22 – V21
(21)
Applying Kirchhoff’s Voltage Law
Substituting values of V22 & V21 into equation
21 we have
-
Z1
Fig 12 Lattice Network
This circuit or network in Fig. 12 can be redrawn
into several mode of connections including bridge
equivalent network
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V2
International Journal of Advanced Engineering Research and Technology (IJAERT) 339
Volume 3 Issue 10, October 2015, ISSN No.: 2348 – 8190
B=
B=
For an open circuit output, because A = D we
have
(24)
(22)
A=D
For an open circuit output when I2 = 0 we have
XII.
CONCLUSION
A two-port network is a quadruple network that is to say
a two-port network is a four terminal network and this
network is an important element in transmission system
making it possible to easily analyze the input and output
terminal with a relationship to its voltage and current
importance.
REFERENCES
Botton W. (1992). Electrical Circuit Principles;
Longman Group UK Ltd. London.
Boylestad Robert L. (1987). Introductory Circuit
Analysis; Merrill Publishing Co.; Columbus Ohio. 5th
Edition.
Carter Robert C. (1996). Introduction to Electric Circuit
Analysis; Holt, Rinchart and Winston; New York.
Charles Belong and Melvyn, M. Drossman (1976).
“System Circuits for Electrical Engineering” McGrewHill Inc. UK London.
Edward Hudges, (1977). “Electrical Technology”
Longman Group Ltd London.
Engr. Dr. G.C. Ochiagha; Engr. Okoronkwo Charles;
Engr. Igweonu E.I. (2005). Electric Circuit Theory with
Solved Examples: Cheston Nig. Ltd, Enugu.
=
=
(23)
AD–BC=1
A=D
A2 – B C = 1
B C = A2 – 1
B C = (A - 1) (A + 1)
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