# ELECTRICAL TECHNOLOGY B.L THERAJA A.K THERAJA 15 1 ELECTRICAL TECHNOLOGY B.L THERAJA A.K THERAJA 15 1 rights reserved. Pvt.Ltd. Ltd.AllAll 2016 bybyS.S.Chand Copyright reserved. rights Chand & & Company

Company Pvt. 2016 Copyright A TEXTBOOK OF Copyright 2016 by S. Chand & Company Pvt. Ltd. All rights reserved. 15 2 A.C. Network Analysis 15 R E T

P A H C C-15 A.C. Network Analysis Kirchhoff's Laws Mesh Analysis Nodal Analysis Superposition Theorem Thevenins Theorem Reciprocity Theorem Nortons Theorem Maximum Power Transfer Theorem-General Case Maximum Power Transfer Theorem Millimans Theorem

15 3 Copyright 2016 by S. Chand & Company Pvt. Ltd. All rights reserved. Introduction Introduction network theorems in Chapter 2 of this book. The same laws are applicable to a.c. networks except that instead of resistances, we have impedances and instead of taking algebraic sum of voltages and currents we have to take sum.

15 4 the phasor Copyright 2016 by S. Chand & Company Pvt. Ltd. All rights reserved. We have already discussed various d.c. Kirchhoffs Laws for d.c. networks except that instead of algebraic sum of currents and voltages, we take phasor or vector sums for a.c. networks. Kirchhoffs Current Law. According to this law, in any electrical network, the phasor sum of the currents meeting at a junction is zero. In other words, I = 0 ...at a junction

Kirchhoffs Voltage Law. According to this law, the phasor sum of the voltage drops across each of the conductors in any closed path (or mesh) in a network plus the phasor sum of the e.m.fs. connected in that path is zero. In other words, IR + e.m.f. = 0 15 5 ...round a mesh Copyright 2016 by S. Chand & Company Pvt. Ltd. All rights reserved. The statements of Kirchhoffs laws are similar to those given in Art. 2.2 Mesh Analysis It has already been discussed in Art. 2.3. Sign convention regarding the voltage drops across various impedances and the

e.m.f.s is the same as explained in Art. 2.3. The circuits may be solved with the help of KVL or by use of determinants and Cramers rule or with the help of impedance matrix [Zm]. 15 6 Copyright 2016 by S. Chand & Company Pvt. Ltd. All rights reserved. Nodal Analysis This technique is the same although we have to deal with circuit impedances rather than resistances and take phasor sum of

voltages and currents rather than algebraic sum. 15 7 Copyright 2016 by S. Chand & Company Pvt. Ltd. All rights reserved. This method has already been discussed in details Chapter 2. Superposition Theorem In any network made up of linear impedances and containing more than one source of e.m.f., the current flowing in any branch is the phasor sum of the currents that would flow in that branch if each source were considered separately, all other e.m.f. sources being replaced for the time being, by their respective internal impedances (if any). 15 8

Copyright 2016 by S. Chand & Company Pvt. Ltd. All rights reserved. As applicable to a.c. networks, it states as follows : Thevenins Theorem The current through a load impedance ZL connected across any two terminals A and B of a linear network is given by Vth/(Zth + ZL) wher Vth is the open-circuit voltage across A and B and Zth is the total impedance of the network as viewed from the open-circuited terminals A and B with all voltage sources replaced by their internal impedances (if any) and current sources by infinite impedance. 15 9

Copyright 2016 by S. Chand & Company Pvt. Ltd. All rights reserved. As applicable to a.c. networks, this theorem may be stated as follows : Reciprocity Theorem and a single voltage source or a single current source. This theorem may be stated as follows : If a voltage source in branch A of a network causes a current of 1 in branch B, then shifting the voltage source (but not its impedance) to branch B will cause the same current I in branch A. It may be noted that currents in other branches will generally not remain the same. A simple way of stating the above theorem is that if an ideal voltage source and an ideal ammeter are interchanged, the ammeter reading would remain the same. The ratio of the input voltage in branch A to the output current in branch B called the transfer impedance. 15 10

is Copyright 2016 by S. Chand & Company Pvt. Ltd. All rights reserved. This theorem applies to networks containing linear bilateral elements Nortons Theorem Any two terminal active linear network containing voltage sources and impedances when viewed from its output terminals is equivalent to a constant current source and a parallel impedance. The constant current is equal to the current which would flow in a short-circuit placed across the terminals and the parallel impedance is the total impedance of the network when viewed from open-circuited terminals after voltage sources have been replaced by their internal impedances (if any) and current sources by infinite impedance.

15 11 Copyright 2016 by S. Chand & Company Pvt. Ltd. All rights reserved. As applied to a.c. networks, this theorem can be stated as under : Maximum Power Transfer Theorem As explained earlier in Art. this theorem is particularly useful for analysing communication networks where the goals is transfer of maximum power between two circuits and not highest efficiency. 15 12 Copyright 2016 by S. Chand & Company Pvt. Ltd. All rights reserved. Maximum Power Transfer Theorems General Case

when the source has a fixed complex impedance and delivers power to a load consisting of a variable resistance or a variable complex impedance. 15 13 Copyright 2016 by S. Chand & Company Pvt. Ltd. All rights reserved. We will consider the following maximum power transfer theorems Millmans Theorem It permits any number of parallel branches consisting of voltage sources and impedances to be reduced to a single equivalent voltage source and equivalent impedance. Such multi-branch circuits are frequently encountered in both electronics and power applications.