Physics
Network Theorems
Network theorems are a set of principles used to analyze electrical circuits. They include the superposition theorem, Thevenin's theorem, Norton's theorem, and others. These theorems provide methods for simplifying complex circuits and determining voltage, current, and power in different parts of the network.
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10 Key excerpts on "Network Theorems"
eBook - PDF- R Britz(Author)
- 2013(Publication Date)
- Future Managers(Publisher)
1 MODULE Network Theorems Introduction The topic of Network Theorems is an essential part of circuit analysis. There are many theorems, but in this chapter we will only focus on the four main theorems. Network Theorems are useful in simplifying analysis of some more complex circuits. But the more useful aspect of Network Theorems is the insight they provide into the properties and behaviour of circuits. L Kirchhoff's Laws i. Kirchhoff ’s first law states that, “If several conductors meet at a point, the total current flowing away from it, i.e. the algebraic sum of the currents, is zero.” I 5 I 1 I 2 I 3 I 4 I 2 + I 3 + I 4 = I 1 + I 5 I = O Figure 1.1 The symbol Σ is “sigma” and means “the algebraic sum of ...” As no current can build up at a junction, the current that flows away from the junction must be equal to the current that flows towards it. ii. Kirchhoff ’s second law states that, “In any closed circuit, the algebraic sum of the products of the current and the resistance of each part of the circuit is equal to the resultant emf in the circuit”. We can also say that the sum of the emfs in any closed circuit is equal to the sum of the potential differences in that circuit. Module 1 • Network Theorems 2 If there are two sources of supply in a closed circuit, the net emf is the sum (or difference) of the individual emfs, depending on their polarities. E R I r E = Ir + IR Figure 1.2 If two batteries are connected in parallel to a load it is usual to assume currents and their directions and label them with a minimum number of unknown values. An equation may be set up for each closed circuit, but it will only be necessary to set up as many equations as there are unknown currents. The equations are then solved simultaneously. A negative quantity for any current implies that an incorrect direction was assumed and that the current flows in a direction opposite to that originally assumed.
eBook - PDFNetwork and Switching Theory
A NATO Advanced Study Institute
- Giuseppe Biorci(Author)
- 2014(Publication Date)
- Academic Press(Publisher)
Chapter I GENERAL PROPERTIES AND REPRESENTATION OF NETWORKS The Relationship between Various Energy Distribution Theorems S. DUINKER Philips Zentrallaboratorium GmbH, Hamburg, West Germany I. Introduction Network theory, in its development toward an important separate branch of mathematical physics during the past 50 years, has yielded a large number of very interesting and important theorems, unveiling the behavior of networks under a great variety of more or less specialized conditions. Unfortunately, some of these theorems are found in particular contexts or at rather obscure places in the literature and, therefore, have escaped general attention for a long time. For this reason, it more than once happened that the same theorem was discovered independently by different network theorists. Moreover, in cases where a network theorem was found in connection with a particular problem, its validity was sometimes either unnecessarily restricted or illegitimately generalized, whereas the efforts to supply an independent proof of the theorem often obscured its relationship to other theorems. It is the purpose of this paper to show the connection existing between a category of fairly general Network Theorems pertaining to the energy distribution in networks. Some of these theorems are well known; others have found little notice so far or can be considered as being the network counterparts of principles already known for a long time in other branches of physics. One result, concerning the transient behavior of nonlinear networks, is thought to be novel. 1 2 S. DUINKER II. Basic Network Principles The distribution of voltages v and currents i in any network, however complicated, always obeys Kirchhofes laws, viz., ΣΗ = Ο (i) for all branch currents i h joining at a node n, and » > = 0 (2) for all branch voltages v b contributing to a closed mesh m.
eBook - ePub- Wai-Kai Chen(Author)
- 2018(Publication Date)
- CRC Press(Publisher)
2 Network Laws and Theorems 2.1 Kirchhoff’s Voltage and Current LawsNodal Analysis • Mesh Analysis • Fundamental Cutset-Loop Circuit Analysis • References2.2 Network TheoremsThe Superposition Theorem • The Thévenin Theorem • The Norton Theorem • The Maximum Power Transfer Theorem • The Reciprocity TheoremRay R. Chen San Jose State University Artice M. Davis San Jose State University Marwan A. Simaan University of Pittsburgh 2.1 Kirchhoff’s Voltage and Current Laws Ray R. Chen and Artice M. DavisCircuit analysis, like Euclidean geometry, can be treated as a mathematical system; that is, the entire theory can be constructed upon a foundation consisting of a few fundamental concepts and several axioms relating these concepts. As it happens, important advantages accrue from this approach — it is not simply a desire for mathematical rigor, but a pragmatic need for simplification that prompts us to adopt such a mathematical attitude.The basic concepts are conductor, element, time, voltage, and current. Conductor and element are axiomatic; thus, they cannot be defined, only explained. A conductor is the idealization of a piece of copper wire; an element is a region of space penetrated by two conductors of finite length termed leads and pronounced “leeds”. The ends of these leads are called terminals and are often drawn with small circles as in Figure 2.1 .Conductors and elements are the basic objects of circuit theory; we will take time, voltage, and current as the basic variables. The time variable is measured with a clock (or, in more picturesque language, a chronometer). Its unit is the second, s. Thus, we will say that time, like voltage and current, is defined operationally, that is, by means of a measuring instrument and a procedure for measurement. Our view of reality in this context is consonant with that branch of philosophy termed operationalism [1 ].Voltage is measured with an instrument called a voltmeter, as illustrated in Figure 2.2 . In Figure 2.2 , a voltmeter consists of a readout device and two long, flexible conductors terminated in points called probes that can be held against other conductors, thereby making electrical contact with them. These conductors are usually covered with an insulating material. One is often colored red and the other black. The one colored red defines the positive polarity of voltage, and the other the negative polarity. Thus, voltage is always measured between two conductors. If these two conductors are element leads, the voltage is that across the corresponding element. Figure 2.3 is the symbolic description of such a measurement; the variable v, along with the corresponding plus and minus signs, means exactly the experimental procedure depicted in Figure 2.2 , neither more nor less. The outcome of the measurement, incidentally, can be either positive or negative. Thus, a reading of v = −12 V, for example, has meaning only when viewed within the context of the measurement. If the meter leads are simply reversed after the measurement just described, a reading of vʹ = +12 V will result. The latter, however, is a different variable; hence, we have changed the symbol to v
eBook - PDF- J. David Irwin, R. Mark Nelms(Authors)
- 2022(Publication Date)
- Wiley(Publisher)
With the current level of computational power available to us, we can solve the node or loop equations that define the network in a flash. With regard to the theorems, we have found that in some cases the theorems do not neces- sarily simplify the problem and a straightforward attack using node or loop analysis is as good an approach as any. This is a valid point, provided that we are simply looking for some particu- lar voltage or current. However, the real value of the theorems is the insight and understanding that they provide about the physical nature of the network. For example, superposition tells us what each source contributes to the quantity under investigation. However, a computer solu- tion of the node or loop equations does not tell us the effect of changing certain parameter values in the circuit. It does not help us understand the concept of loading a network or the ramifications of interconnecting networks or the idea of matching a network for maximum power transfer. The theorems help us to understand the effect of using a transducer at the input of an amplifier with a given input resistance. They help us explain the effect of a load, such as a speaker, at the output of an amplifier. We derive none of this information from a node or loop analysis. In fact, as a simple example, suppose that a network at a specific pair of terminals has a Thévenin equivalent circuit consisting of a voltage source in series with a 2-kΩ resistor. If we connect a 2-Ω resistor to the network at these terminals, the voltage across the 2-Ω resistor will be essentially nothing. This result is fairly obvious using the Thévenin theorem approach; however, a node or loop analysis gives us no clue as to why we have obtained this result. We have studied networks containing only dependent sources. This is a very important topic because all electronic devices, such as transistors, are modeled in this fashion. Motors in power systems are also modeled in this way.
eBook - PDFNetwork and Switching Theory
A NATO Advanced Study Institute
- Giuseppe Biorci(Author)
- 2014(Publication Date)
- Academic Press(Publisher)
Chapter I GENERAL PROPERTIES AND REPRESENTATION OF NETWORKS The Relationship between Various Energy Distribution Theorems S. DUINKER Philips Zentrallaboratorium GmbH, Hamburg, West Germany I. Introduction Network theory, in its development toward an important separate branch of mathematical physics during the past 50 years, has yielded a large number of very interesting and important theorems, unveiling the behavior of networks under a great variety of more or less specialized conditions. Unfortunately, some of these theorems are found in particular contexts or at rather obscure places in the literature and, therefore, have escaped general attention for a long time. For this reason, it more than once happened that the same theorem was discovered independently by different network theorists. Moreover, in cases where a network theorem was found in connection with a particular problem, its validity was sometimes either unnecessarily restricted or illegitimately generalized, whereas the efforts to supply an independent proof of the theorem often obscured its relationship to other theorems. It is the purpose of this paper to show the connection existing between a category of fairly general Network Theorems pertaining to the energy distribution in networks. Some of these theorems are well known; others have found little notice so far or can be considered as being the network counterparts of principles already known for a long time in other branches of physics. One result, concerning the transient behavior of nonlinear networks, is thought to be novel. 1 2 S. DUINKER II. Basic Network Principles The distribution of voltages v and currents i in any network, however complicated, always obeys KirchhofFs laws, viz., Z 4 = 0 (1) for all branch currents i b joining at a node n> and 5 > & = 0 (2) for all branch voltages v h contributing to a closed mesh m.
eBook - PDF- G R Jones(Author)
- 2013(Publication Date)
- Newnes(Publisher)
4.2.4 Network Theorems Network Theorems can simplify complicated networks, facil-itate the solution of specific network branches and deal with particular network configurations (such as two-ports). They are applicable to linear networks for which superposition is valid, and to any form (scalar, complexor, or operational) of voltage, current, impedance and admittance. In the following, the Ohm and Kirchhoff laws, and the reciprocity and compen-sation theorems, are basic; star-delta transformation and the Millman theorem are applied to network simplification; and the Helmholtz-Thevenin and Helmholtz-Norton theorems deal with specified branches of a network. Two-ports are dealt with in Section 4.2.5. Basic network analysis 4/5 4.2.4.1 Ohm (Figure 4.2(a)) For a branch of resistance R or conductance G, I=V/R = VG V = I/R = I/G; R = V/l = 1/G Summation of resistances R { , R 2 , ..., in series or parallel gives Series: R = R x + R 2 +... or G = l/(l/G { + l/G 2 + ...) Parallel: R = l/(l/R x + l/R 2 +...) or G = G { + G 2 +... The Ohm law is generalised for a.c. and transient cases by / = V/Z or /(p) = V(p)/Z(p), where p is the operator a/at. 4.2.4.2 Kirchhoff (Figure 4.4) The node and mesh laws are Node: i { + i 2 +... = 2/ = 0 Mesh: e x + e 2 +... = i{Z x + i 2 Z 2 + ... or Xe = XiZ 4.2.4.3 Reciprocity If an e.m.f. in branch P of a network produces a current in branch Q, then the same e.m.f. in Q produces the same current in P. The ratio of the e.m.f. to the current is then the transfer impedance or admittance. 4.2.4.4 Compensation For given circuit conditions, any impedance Z in a network that carries a current / can be replaced by a generator of zero internal impedance and of e.m.f. E = -IZ. Further, if Z is changed by , then the effect on all other branches is that which would be produced by an e.m.f.
- W K Chen(Author)
- 1997(Publication Date)
- WSPC(Publisher)
CHAPTER 2 FOUNDATIONS OF ELECTRICAL NETWORK THEORY One of the most important applications of graph theory in physical science is its use in electrical network theory. The main purpose of this chapter is to provide a rigorous mathematical foundation, based on the theory of graphs, for the discipline of electrical network theory. Apart from the electrical network problems, much of the discussion is sufficiently general to be applicable to general linear systems (see, for example, TRENT [1955]). From a physical viewpoint, the network problem deals with predicting the behavior of a system of interconnected physical elements in terms of the char-acteristics of the elements and the manner in which these elements are inter-connected. The geometrical properties of a network are independent of the constituents of its branches, and so in topological discussions it is usual to replace each branch by a line segment. In other words, from an abstract view-point, any lumped electrical network can be represented by a graph with edges denoting, to some extent at least, electrical components and weights repre-senting the constituents of the components. The graph is referred to as a model of the physical network. Since the networks that we deal with and analyze are models consisting of an interconnection of idealized physical elements such as inductors, capacitors, resistors, and generators, we cannot always assume the existence and uniqueness of their solutions. Thus, it is important to discuss conditions under which a unique solution can be obtained for an electrical network. KIRCHHOFF [1847] made the first comprehensive investigation of the electrical network problem, and proved the existence of a solution to a resistive network. MAXWELL [1892] pointed out, however, that Kirchhoff's formulation omitted the concept of potential.
eBook - PDF- F.H. Jr. Branin(Author)
- 2012(Publication Date)
- Academic Press(Publisher)
THE NETWORK CONCEPT AS A UNIFYING PRINCIPLE IN ENGINEERING AND THE PHYSICAL SCIENCES FRANKLIN H. BRANIN, JR. System Communications Division Laboratory IBM Corporation Kingston, New York SUMMARY The concept of a network of interconnected elements per-vades many areas of both engineering and the physical sci-ences. Basic to this concept are the existence of: (1) a pair of functionally related complementary or conjugate vari-ables that characterize each individual network element and (2) topological constraints imposed on these conjugate var-iables by the very act of interconnecting the network ele-ments. The underlying mathematical description of the net-work problem belongs to algebraic topology which explains clearly why network analogies are so ubiquitous. An under-standing of the fundamentals of the network problem and its extension to the 3-network, in which surface and volume ele-ments are involved, provides a unified basis for analyzing networks of all kinds -whether electrical, mechanical, struc-tural, molecular, acoustic, hydraulic, or thermal, to mention a few. 41 42 F. H. BRANIN, JR. 1. INTRODUCTION Many physical phenomena can be described by means of a net-work of interconnected elements, each of which is characterized by a pair of functionally related complementary or conjugate variables. The properties of each network element are described by constitutive relations involving these conjugate variables. In electrical networks, current and voltage are the appropriate variables whereas force and velocity are the conjugate variables for mechanical, structural and molecular networks. In hydraulic and acoustic systems, the complementary variables are fluid flow and pressure while in thermal systems they are heat flow (or en-tropy flow) and temperature.
- John Bird(Author)
- 2013(Publication Date)
- Butterworth-Heinemann(Publisher)
Part 2 Electrical Principles This page intentionally left blank 11 D.c. circuit theorems At the end of this chapter you should be able to: • state and use Kirchhoff s laws to determine unknown currents and voltages in d.c. circuits • understand the superposition theorem and apply it to find currents in d.c. circuits • understand general d.c. circuit theory • understand Thevenin's theorem and apply a procedure to determine unknown currents in d.c. circuits • recognize the circuit diagram symbols for ideal voltage and current sources • understand Norton's theorem and apply a procedure to determine unknown currents in d.c. circuits • appreciate and use the equivalence of the Thevenin and Norton equivalent networks • state the maximum power transfer theorem and use it to determine maximum power in a d.c. circuit 11.1 Introduction The laws which determine the currents and voltage drops in d.c. networks are: (a) Ohm's law (see Chapter 2); (b) the laws for resistors in series and in parallel (see Chapter 5); and (c) Kirchhoff s laws (see section 11.2). In addition, there are a number of circuit theorems which have been developed for solving problems in electrical networks. These include: (i) the superposition theorem (see section 11.3); (ii) Thevenin's theorem (see section 11.5); (iii) Norton's theorem (see section 11.7); and (iv) the maximum power transfer theorem (see section 11.8). 11. 2 Kirchhofes laws Kirchhoff s laws state: (a) Current law. At any junction in an electric circuit the total current flowing towards that junction is equal to the total current flowing away from the junction, i.e. / = 0. Thus, referring to Fig. 11.1: h + h = h + h + h or h + h -h -h ~ h = ° (b) Voltage law. In any closed loop in a network, the algebraic sum of the voltage drops (i.e. products of current and resistance) taken around the loop is equal to the resultant e.m.f acting in that loop. Thus, referring to Fig. 11.2:
eBook - PDFElectrical and Electronic Principles 3 Checkbook
The Checkbook Series
- J O Bird, A J C May(Authors)
- 2016(Publication Date)
- Newnes(Publisher)
1 Circuit theorems A. MAIN POINTS CONCERNED WITH D.C. CIRCUIT ANALYSIS 1 The laws which determine the currents and voltage drops in d.c. networks are: (a) Ohm's law, (b) the laws for resistors in series and in parallel, and (c) Kirchhoffs laws. In addition, there are a number of circuit theorems which have been devel-oped for solving problems in electrical networks. These include: (i) the superposition theorem, (ii) Thévenin's theorem, (iii) Norton's theorem, and (iv) the maximum power transfer theorem. 2 The superposition theorem states: 7/1 any network made up of linear resistances and containing more than one source ofemf, the resultant current flowing in any branch is the algebraic sum of the currents that would flow in that branch if each source was considered separately y all other sources being replaced at that time by their respective internal resistances. * (See Problems 1 and 2) 3 The following points involving d.c. circuit analysis need to be appreciated before proceeding with problems using Thevenin's and Norton's theorems: (i) The open-circuit voltage, E, across terminals AB in Fig 1 is equal to 10 V, since no current flows through the 2 Ω resistor and hence no voltage drop occurs. (ii) The open-circuit voltage, E, across terminals AB in Fig 2(a) is the same as the voltage across the 6 Ω resistor. The ciruit may be redrawn as shown in Fig 2(b) 'tèi)™ by voltage division in a series circuit, i.e. E = 30 V 1 / (iii) (iv) (v) (vi) For the circuit shown in Fig 3(a) represent-ing a practical source supplying energy, V = E—Ir, where E is the battery emf, V is the battery terminal voltage and r is the internal resistance of the battery. For the circuit shown in Fig 3(b), V V = E-{-Dr, i.e. V = E+Ir The resistance 'looking-in' at terminals AB in Fig 4(a) is obtained by reducing the circuit in stages as shown in Figs 4(b) to (d). Hence the equivalent resistance across AB is 7 Ω. For the circuit shown in Fig 5(a), the 3 Ω resistor carries no current and the p.d.
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