ELECTRICAL TECHNOLOGY B.L THERAJA A.K THERAJA 21 rights reserved. Pvt.Ltd. Ltd.AllAll 2016 bybyS.S.Chand Copyright reserved. rights Chand & & Company Company Pvt.

2016 Copyright A TEXTBOOK OF 22 Copyright 2016 by S. Chand & Company Pvt. Ltd. All rights reserved. 2 DC Network Theorems R E T P A

H C C-2

Electric Circuits and Network Theorems Kirchhoffs Laws Determination of Voltage Sign Assumed Direction of Current Solving Simultaneous Equations Determinants Solving Equations with Two Unknowns Solving Equations With Three Unknowns Independent and Dependent Sources Maxwells Loop Current Method Mesh Analysis Using Matrix Form Nodal Analysis with Voltage Sources Nodal Analysis with Current Sources Source Conversion Ideal Constant-Voltage Source Ideal Constant-Current Source

Copyright 2016 by S. Chand & Company Pvt. Ltd. All rights reserved. 23 DC Network Theorems Contd. C-2 DC Network Theorems 24 Superposition Theorem Thevenin Theorem How to Thevenize a Given Circuit ? General Instructions for Finding Thevenin Equivalent Circuit Reciprocity Theorem

Delta/Star Transformation Star/Delta Transformation Compensation Theorem Nortons Theorem How to Nortanize a Given Circuit ? General Instructions for Finding Norton Equivalent Circuit Millmans Theorem Generalised Form of Millmans Theorem Maximum Power Transfer Theorem Power Transfer Efficiency Copyright 2016 by S. Chand & Company Pvt. Ltd. All rights reserved.

Electric Circuits and Network Theorems Copyright 2016 by S. Chand & Company Pvt. Ltd. All rights reserved. There are certain theorems, which when applied to the solutions of electric networks, wither simplify the network itself or render their analytical solution very easy. Different electric circuits (according to their properties) are defined below : Circuit

Passive Network Parameters Active Network Liner Circuit Node

Non-linear Circuit Branch Bilateral Circuit Loop

Unilateral Circuit Mesh Electric Network Contd. 25 Electric Circuits and Network Theorems Copyright 2016 by S. Chand & Company Pvt. Ltd. All rights reserved. 26

Kirchhoffs Laws* Ohms law and are used for solving electrical networks which may not be readily solved by the latter. Kirchhoffs laws, two in number, are particularly useful (a) in determining the equivalent resistance of a complicated network of conductors and (b) for calculating the currents flowing in the various conductors. 27 Copyright 2016 by S. Chand & Company Pvt. Ltd. All rights reserved. These laws are more comprehensive than Determination of Voltage Sign

should be paid to the algebraic signs of voltage drops and e.m.fs., otherwise results will come out to be wrong. Following sign conventions is suggested : Sign of Battery E.M.F. Sign of IR Drop 28 Copyright 2016 by S. Chand & Company Pvt. Ltd. All rights reserved. In applying Kirchhoffs laws to specific problems, particular attention Assumed Direction of Current assuming proper direction of current usually arises. The direction of current flow may be assumed either clockwise or anticlockwise. If the assumed direction of current is not the actual direction, then on solving the question, this current will be found to have a minus sign. If the answer is positive, then assumed

direction is the same as actual direction (Example 2.10). However, the important point is that once a particular direction has been assumed, the same should be used throughout the solution of the question. 29 Copyright 2016 by S. Chand & Company Pvt. Ltd. All rights reserved. In applying Kirchhoffs laws to electrical networks, the question of Solving Simultaneous Equations Electric circuit analysis with the help of Kirchhoffs laws usually involves solution of two or three simultaneous equations. These equations can be solved by a systematic elimination of the variables but the procedure is often lengthy and laborious and hence more liable to error.

Determinants and Cramers rule provide a simple and straight method for solving network equations through manipulation of their coefficients. 2 10 Copyright 2016 by S. Chand & Company Pvt. Ltd. All rights reserved. Determinants Copyright 2016 by S. Chand & Company Pvt. Ltd. All rights reserved. 2 11 Solving Equations with Two Unknowns Copyright 2016 by S. Chand & Company Pvt. Ltd. All rights reserved.

2 12 Solving Equations with Three Unknowns Copyright 2016 by S. Chand & Company Pvt. Ltd. All rights reserved. 2 13 Independent and Dependent Sources Copyright 2016 by S. Chand & Company Pvt. Ltd. All rights reserved. Those voltage or current sources, which do not depend on any other quantity in the circuit, are called independent sources. An independent d.c. voltage source is shown in Fig. 2.20 (a) whereas a time-varying voltage source is shown in Fig. 2.20 (b). The

positive sign shows that terminal A is positive with respect to terminal B. In other words, potential of terminal A is v volts higher than that of terminal B. Similarly, Fig. 2.20 (c) shows an ideal constant current source whereas Fig. 2.20 (d) depicts a time-varying current source. The arrow shows the direction of flow of the current at any moment under consideration. Contd. 2 14 Independent and Dependent Sources Copyright 2016 by S. Chand & Company Pvt. Ltd. All rights reserved. 2 15

Maxwells Loop Current Method This method which is particularly well-suited to coupled circuit solutions employs a system of loop or mesh currents instead of branch currents (as in Kirchhoffs laws). Here, the currents in different meshes are assigned continuous paths so that they do not split at a junction into branch currents. This method eliminates a great deal of tedious work involved in the branch-current method and is best suited when energy sources are voltage sources rather than current sources. Basically, this method consists of writing loop voltage equations

by Kirchhoffs voltage law in terms of unknown loop currents. 2 16 Copyright 2016 by S. Chand & Company Pvt. Ltd. All rights reserved. Mesh Analysis Using Matrix Form Copyright 2016 by S. Chand & Company Pvt. Ltd. All rights reserved. 2 17 Nodal Analysis With Sources unlike loop-current method which is based on Kirchhoffs voltage law. However, like loop current method, nodal method also has the advantage that a minimum number of equations need be written to

determine the unknown quantities. Moreover, it is particularly suited for networks having many parallel circuits with common ground connected such as electronic circuits. 2 18 Copyright 2016 by S. Chand & Company Pvt. Ltd. All rights reserved. The node-equation method is based directly on Kirchhoffs current law Nodal Analysis with Current Sources Copyright 2016 by S. Chand & Company Pvt. Ltd. All rights reserved. Consider the network of Fig. 2.68 (a) which has two current sources and three nodes out of which 1 and 2 are independent ones whereas No. 3 is the reference node. The given circuit has been redrawn for ease of understanding and is shown in Fig. 2.68 (b). The current directions have been taken on the assumption that

1. both V1 and V2 are positive with respect to the reference node. That is why their respective currents flow from nodes 1 and 2 to node 3. 2. V1 is positive with respect to V2 because current has been shown flowing from node 1 to node 2. Contd. 2 19 Nodal Analysis with Current Sources Copyright 2016 by S. Chand & Company Pvt. Ltd. All rights reserved. 2 20

Source Conversion Copyright 2016 by S. Chand & Company Pvt. Ltd. All rights reserved. A given voltage source with a series resistance can be converted into (or replaced by) an equivalent current source with a parallel resistance. Conversely, a current source with a parallel resistance can be converted into a vaoltage source with a series resistance. Suppose, we want to convert the voltage source of Fig. 2.75 (a) into an

equivalent current source. First, we will find the value of current supplied by the source when a short is put across in termials A and B as shown in Fig. 2.75 (b). This current is I = V/R. A current source supplying this current I and having the same resistance R connected in parallel with it represents the 2 21 equivalent source. It is shown in Fig. 2.75 (c).

Contd. Source Conversion Copyright 2016 by S. Chand & Company Pvt. Ltd. All rights reserved. 2 22 Ideal Constant Voltage Source Copyright 2016 by S. Chand & Company Pvt. Ltd. All rights reserved. It is that voltage source (or generator) whose output voltage remains absolutely constant whatever the change in load current. Such a voltage source must possess zero internal resistance so that

internal voltage drop in the source is zero. In that case, output voltage provided by the source would remain constant irrespective of the amount of current drawn from it. In practice, none such ideal constant-voltage source can be obtained. However, smaller the internal resistance r of a voltage source, closer it comes to the ideal sources described above. Contd. 2 23 Ideal Constant Voltage Source Copyright 2016 by S. Chand & Company Pvt. Ltd. All rights reserved.

2 24 Ideal Constant Current Source It is that voltage source whose internal resistance is infinity. In practice, it is approached by a source which posses very high resistance as compared to that of the external load resistance. As shown in Fig. 2.94 (b), let the 6-V battery or voltage source have an internal resistance of 1 M and let the load resistance vary from 20 K to 200 K. The current supplied by the source varies from 6.1/1.02 = 5.9 A to 6/1.2 = 5 A. As seen, even when load resistance increases 10 times, current decreases by 0.9 A. Hence, the source can be considered, for all practical purposes, to be a constant current source. 2 25

Copyright 2016 by S. Chand & Company Pvt. Ltd. All rights reserved. Superposition Theorem simultaneously in any linear bilateral network, then each e.m.f. acts independently of the others i.e. as if the other e.m.fs. did not exist. The value of current in any conductor is the algebraic sum of the currents due to each e.m.f. 2 26 Copyright 2016 by S. Chand & Company Pvt. Ltd. All rights reserved. According to this theorem, if there are a number of e.m.fs. acting Thevenin Theorem viewed from two output terminals, by a single voltage source with a

series resistance. It makes the solution of complicated networks (particularly, electronic networks) quite quick and easy. 2 27 Copyright 2016 by S. Chand & Company Pvt. Ltd. All rights reserved. It provides a mathematical technique for replacing a given network, as How to Thevenize a Given Circuit? current is required. Find the open-circuit voltage Voc which appears across the two terminals from where resistance has been removed. It is also called Thevenin voltage Vth. Compute the resistance of the whole network as looked into from these two terminals after all voltage sources have been removed leaving behind their internal resistances (if any) and current sources have been replaced by open-circuit i.e. infinite resistance. It is also called Thevenin

resistance Rth or Tt. Replace the entire network by a single Thevenin source, whose voltage is Vth or Voc and whose internal resistance is Rth or Ri. 2 28 Copyright 2016 by S. Chand & Company Pvt. Ltd. All rights reserved. Temporarily remove the resistance (called load resistance R L) whose General Instructions for Finding Thevenin Equivalent Circuit independent current or voltage sources only. However, we often come across circuits which contain both independent and dependent sources or circuits which contain only dependent sources. Procedure for finding the value of Vth and Rth in such cases is detailed below : When Circuit Contains Both Dependent and Independent Sources When Circuit Contains Dependent Sources Only 2 29

Copyright 2016 by S. Chand & Company Pvt. Ltd. All rights reserved. So far, we have considered circuits which consisted of resistors and Reciprocity Theorem In any linear bilateral network, if a source of e.m.f. E in any branch produces a current I in any other branch, then the same e.m.f. E acting in the second branch would produce the same current I in the first branch. In other words, it simply means that E and I are mutually transferrable. The ratio E/I is known as the transfer resistance (or impedance in a.c. systems). Another way of stating the above is the receiving point and the sending point in a network are interchangeable.

2 30 that Copyright 2016 by S. Chand & Company Pvt. Ltd. All rights reserved. It can be stated in the following manner : Delta/Star* Transformation the application of Kirchhoffs Laws, one sometimes experiences great difficulty due to a large number of simultaneous equations that have to be solved. However, such complicated network can be simplified by successively replacing delta meshes by equivalent star system and vice

versa. 2 31 Copyright 2016 by S. Chand & Company Pvt. Ltd. All rights reserved. In solving networks (having considerable number of branches) by Star/Delta Transformation and (iii) given above. Multiplying (i) and (ii), (ii) and (iii), (iii) and (i) and adding them together and then simplifying them, we get 2 32 Copyright 2016 by S. Chand & Company Pvt. Ltd. All rights reserved. This tarnsformation can be easily done by using equations (i), (ii)

Compensation Theorem a) For analysing those networks where the values of the branch elements are varied and for studying the effect of tolerance on such values. b) For calculating the sensitivity of bridge network. 2 33 Copyright 2016 by S. Chand & Company Pvt. Ltd. All rights reserved. This theorem is particularly useful for the following two purposes : Nortons Theorem it is the dual of Thevenins theorem.

Whereas Thevenins theorem reduces a two-terminal active network of linear resistances and generators to an equivalent constant-voltage source and series resistance, Nortons theorem replaces the network by an equivalent constant-current source a parallel resistance. 2 34 and Copyright 2016 by S. Chand & Company Pvt. Ltd. All rights reserved. This theorem is an alternative to the Thevenins theorem. In fact, How To Nortonize a Given Circuit?

above. Remove the resistance (if any) across the two given terminals and put a short-circuit across them. Compute the short-circuit current ISC. Remove all voltage sources but retain their internal resistances, if any. Similarly, remove all current sources and replace them by open-circuits i.e. by infinite resistance. Next, find the resistance R1 (also called RN) of the network as looked into from the given terminals. It is exactly the same as Rth (Art. 2.16). 2 35 The current source (ISC) joined in parallel across Ri between the Copyright 2016 by S. Chand & Company Pvt. Ltd. All rights reserved.

This procedure is based on the first statement of the theorem given General Instructions for Finding Norton Equivalent Circuit already been given in Art. That procedure applies to circuits which contain resistors and independent voltage or current sources. Similar procedures for circuits which contain both dependent and independent sources or only dependent sources are given below : 2 36 Copyright 2016 by S. Chand & Company Pvt. Ltd. All rights reserved. Procedure for finding Norton equivalent circuit of a given network has Millmans Theorem current sources or both.

As Applicable to Voltage Sources This Theorem is a combination of Thevenins and Nortons theorems. It is used for finding the common voltage across any network which contains a number of parallel voltage sources as shown in Fig. As Applicable to Current Sources This theorem is applicable to a mixture of parallel voltage and current sources that are reduced to a single final equivalent source which is either a constant current or a constant voltage source. 2 37 Copyright 2016 by S. Chand & Company Pvt. Ltd. All rights reserved.

This theorem can be stated either in terms of voltage sources or Generalised Form of Millmans Theorem for solving many circuits which are frequently encountered in both electronics and power applications. Consider a number of admittances G1, G2, G3... Gn which terminate at common point 0 (Fig. 2.225). The (Fig. 2.225). The other ends of the admittances are numbered as 1, 2, 3....n. Let O be other point in the network. 2 38 any

Copyright 2016 by S. Chand & Company Pvt. Ltd. All rights reserved. This theorem is particularly useful Maximum Power Transfer Theorem theorem is particularly useful for analysing communication networks. The overall efficiency of a network supplying maximum power to any branch is 50 per cent. For this reason, the application of this theorem to power transmission and distribution networks is limited because, in their case, the goal is high efficiency and not maximum power transfer. However, in the case of electronic and communication networks,

very often, the goal is either to receive or transmit maximum power (through at reduced efficiency) specially when power involved is only a few milliwatts or microwatts. 2 39 Copyright 2016 by S. Chand & Company Pvt. Ltd. All rights reserved. Although applicable to all branches of electrical engineering, this Power Transfer Efficiency Copyright 2016 by S. Chand & Company Pvt. Ltd. All rights reserved. 2 40