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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 www.ijaert.org

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 www.ijaert.org

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) www.ijaert.org