M.S. SUKHIJA Formerly, Founder Principal Guru Nanak Dev Engineering College Bidar, Karnataka

Transcription

1 M.S. SUKHIJA Formerly, Founder Principal Guru Nanak Dev Engineering College Bidar, Karnataka T.K. NAGSARKAR Formerly, Professor and Head Department of Electrical Engineering Punjab Engineering College Chandigarh

2 Contents Preface List of Symbols. Definitions and Basic Circuit Concepts. Preamble.2 Conductors, Semiconductors, and Insulating Materials 2.3 Electric Charge and Current 2.4 Force and Work 4.5 Electric Potential, Potential Difference, and Electromotive Force 4.6 Electric Power and Energy 6.7 Circuit Elements 7.8 Energy Sources 6.9 Kirchhoff s Laws 22.0 Connection of Circuit Elements 26. Star (Y)Delta (D) Connections and their Transformations Nodal and Mesh Analyses Preamble Nodal Analysis Mesh Analysis Comparison of Node-voltage and Mesh-current Methods Signals and Waveforms Preamble Classification of Signals Periodic Functions Definitions Sinusoidal Functions Non-periodic Functions Random Functions Waveform Synthesis Fundamentals of Reactive Circuits 4 4. Preamble Circuit Responses Response of Source-free Circuits Forced Response Sinusoidal Steady-State Analysis Preamble Sinusoidal Source 53 v xiii

3 x Contents 5.3 Phasor Representation of Sinusoidal Functions Steady-state Response of Circuits to Sinusoidal Functions Resonance in AC Circuits Star (Y)Delta (D) Connections Nodal and Mesh Analyses Sudden Application of Sinusoidal Function Network Theorems Preamble Superposition Principle Thevenin and Norton Equivalent Circuits Maximum Power Transfer Theorem Reciprocity Theorem Compensation Theorem Tellegen s Theorem Millman s Theorem Alternating Current Power Circuit Analysis Preamble AC Power VoltAmpere and Complex Power Alternate Equations for Complex Power Computations Sign Convention for Complex Power Power Factor and Power Factor Angle Balanced Three-Phase Circuits Preamble Poly-phase Circuits Single-phase Systems and Three-phase Systems A Comparison Three-phase Systems Different Types of Three-phase Connections Three-phase Supply Analysis of Three-phase Circuits Three-phase Unbalanced Load Circuits Power in Three-phase Circuits Measurement of Three-phase Power Measurement of Reactive Power Mutually Coupled Circuits and Their Analyses Preamble Self-Inductance Mutual Inductance Analysis of Coupled Circuits Equivalent Circuit of Mutually Coupled Coils Energy in Two Linearly Coupled Coils Coupled Circuit as a Transformer Ideal Transformer Simulation Fundamentals of Graph Theory Preamble Graph Vocabulary 37

4 Contents xi 0.3 Matrix Representation of Graphs Formulation of Network Response Equations Using Incidence Matrices Duality in Networks 392. Analysis of Circuits by Laplace Transforms 402. Preamble Definition of Laplace Transform Definition of Inverse Laplace Transform Laplace Transforms of Common Forcing Functions Properties of Laplace Transforms Initial and Final Value Theorems 49.7 Partial Fractions 42.8 The Convolution Integral Application of Laplace Transform Techniques to Circuit Analysis Impedance and Admittance Functions Preamble Conceptualization of Complex Frequency Complex Frequency Plane Dimensions of Complex Frequency Impedance and Admittance Functions Network Functions Preamble Terminals and Ports Network Functions Computation of Network Functions Features of Network Functions Restrictions on Location of Poles and Zeros of Network Functions Response of a Circuit in the Time Domain from Pole and Zero Plots Amplitude and Phase Response from Pole and Zero Plots Performance of Active Networks RouthHurwitz Stability Criterion Two-Port Networks Preamble Restrictions on Simulation of Two-Port Networks Parameters of Two-Port Networks Correlation between different Two-Port Parameters Two-Port Reciprocal and Symmetric Networks Terminated Two-Port Networks Interconnection of Two-Port Networks Correlation between the Parameters and T- and p-representations Image Impedance 580

5 xii Contents 5. Fourier Series-Based Circuit Analysis Preamble Synopsis of Fourier Series Analysis Computation of the Coefficients of the Fourier Series Waveform Symmetry and Fourier Coefficients Line Spectra Synthesis of Waveforms Effective Values and Power Computations of Periodic Functions The Fourier Transform Filter and Attenuator Circuits Preamble Definitions Classification of Filters Filter Networks Concept of Working of LP and HP Filters Analysis of Filter Networks Categorisation of Pass Band and Stop Band Characteristic Impedance in the Pass Band and Stop Band Constant KLow Pass (LP) Filter Constant KHigh Pass (HP) Filter Constant KBand Pass (BP) Filter Constant KBand Elimination (BE) Filter m-derived Filters Attenuator Classification of Attenuators Insertion Loss Network Synthesis and Realizability Preamble Elements of Realizability Theory Hurwitz Polynomial Methodology for Obtaining Continued Fraction Expansion of D(s) Positive Real Functions Characteristics of a PRF Methodology for Simple Network Synthesis Synthesis of Two-Element Type One-Port Networks 752 Appendix A : MATLAB Applications in Linear Circuits 790 Appendix B : Linear Circuit Analysis with PSpice 809 Appendix C : Answers 828 Appendix D : Self Appraisal Test 848 Bibliography 868 Index 869

6 CHAPTER Definitions and Basic Circuit Concepts Do not wait; the time will never be just right. Start where you stand, and work with whatever tools you may have at your command, and better tools will be found as you go along. George Herbert Key Concepts Introduction of electrical materials conductors, semiconductors, and insulators Defining the basic electrical terms charge, current, voltage, power, and energy Defining circuit components, linear, bilateral and unilateral elements, lumped and distributed parameter elements, passive and active branches, node, loop, and mesh Understanding the characteristics of the basic circuit elements, such as resistors, inductors, and capacitors Defining independent and dependent voltage and current sources Ability to transform a voltage source into a current source and vice versa without modifying the response in the network Defining Ohm s law, Kirchhoff s current law (KCL), and Kirchhoff s voltage law (KVL) and their applications in the determination of voltages and currents in circuits. Developing the ability to calculate equivalent resistance of seriesparallel combinations of resistances Reducing seriesparallel combination of inductances Understanding of voltage and current division Application of stardelta conversion for simplifying resistive circuits. PREAMBLE Generally speaking, network analysis is any structured technique used to mathematically analyze a circuit. A physical electrical network, or electrical circuit, is a system of interconnected components. The word components include sources of energy such as voltage sources or current sources; electrical elements such as resistors, inductors, and capacitors; electronic devices such as diodes,

7 2 Circuits and Networks: Analysis, Design, and Synthesis transistors to switches, loads, and connecting wires for interconnection of the components. These components can be as small as an integrated circuit on a silicon chip or as large as an electricity distribution network. In an electric circuit, transfer of charge takes place between different parts of the circuit. Energy transfer from a source to a load is accomplished by charge transfer. When running electric circuits, charge can neither be created nor destroyed and the total amount of charge remains constant. Charge in motion represents current. Based on well-defined electrical laws, an electrical circuit can be analyzed to compute voltages and current flows for all the elements of the network, and if desired, other quantities such as charge, fields, energy, power, etc. can be computed. Conversely, by employing the same electrical laws, a circuit may be synthesized to produce a given output from a known input. An electrical circuit may be analyzed to determine direct current (DC), alternating current (AC), and transient responses..2 CONDUCTORS, SEMICONDUCTORS, AND.2 INSULATING MATERIALS Based on the energy band theory, electrical materials are classified into conductors, semiconductors, and insulators. In metals such as copper and aluminium, the gap between the valence band and conduction band is very small or it may overlap. Thus, it is feasible to make the electrons move to the higher levels in the conduction band by applying a small amount of energy through an external (thermal or electric) source. This gives rise to freely moving electrons within a metal, which can be made to flow in a particular direction. In semiconductors, such as carbon, germanium, and silicon, due to a small but significant width of the forbidden zone between the valence and conduction bands, no free electrons are available at low temperatures. However, at room temperatures, it is feasible for some electrons to acquire sufficient energy and jump to the conduction band. Thus, in semiconductors, the density of electrons in the conduction band is not as high as in conductors and therefore, the former cannot conduct electric current as efficiently as the latter. In insulating materials, also referred to as dielectric materials, the forbidden gap between the valence and conduction bands is very large. This makes the electrons difficult to move to the conduction band by applying any amount of energy from a practically available source of energy. Examples of insulating materials are air, rubber, plastic, glass, and mica..3 ELECTRIC CHARGE AND CURRENT Atom is the smallest particle of matter. As per BohrRutherford s model of an atom, the mass of an atom and all its positive charge is concentrated in a nucleus, while negatively charged electrons revolve around the nucleus. A nucleus is made of protons, which have a positive charge and neutrons, which have no charge. There are exactly as many protons in the nucleus as electrons. Thus, an atom

8 Definitions and Basic Circuit Concepts 3 can be viewed as a core nucleus carrying positive charge, and the negative charge of encircling electrons is equal to the positive charge of protons. The basic unit of charge is the charge of the electron. In the SI system, the fundamental unit of charge is the coulomb (C). The accumulated charge on electrons equals coulomb. An electron carries a negative charge e = C. A single proton has a charge of C. In a metallic conductor, such as a copper wire, no current can flow through it, since the movement of the free electrons in the conduction band is necessarily random in nature. If, however, an external energy (say voltage) source is applied across the ends of the wire, the electrons within the conductor flow towards the positive terminal of the voltage source. The flow of charge constitutes the flow of current and is defined as the time rate of charge flow through a cross-sectional area of a conductor. Mathematically, the average current i av over a period of time, instantaneous current i, and the charge q transferred from time t 0 to t are respectively written as i av = D q Dt C/s (or ampere) (.a) i = dq dt C/s (or ampere) (.b) q = t ò t 0 idt C (.c) In Eq. (.a), Dq is the quantity of charge in coulombs that flows in Dt seconds. The unit of current is the ampere and is denoted by the symbol A. Thus, A current flow implies that the rate of flow of charge is C/s. Since current flow is due Current Energy source to the flow of electrons, there Fig.. Relative flow of electrons and current is a direction associated with the flow of current. Figure. shows the relative flow of electrons and current. It may be seen that the positive flow of current is assumed opposite to that of the direction of the flow of electrons. Example. A 5.0 A current is flowing through a conductor. Calculate the charge transferred in 5 seconds. Solution From Eq. (.c), 5 q = [ ] 5 ò (5.0) dt = 5t = 25 C 0 0

9 4 Circuits and Networks: Analysis, Design, and Synthesis.4 FORCE AND WORK According to the Coulomb s law of electrostatics, there exists an electrostatic force of magnitude F Newton (N) between two static bodies carrying charges of q and q 2 C, and separated by a distance of r metres (m). The magnitude of electrostatic force may be written as qq 2 F = N (.2) 2 4pe r where e = e 0 e r is the absolute permittivity of the medium, e 0 = F/m is the permittivity of free space, and e r represents the relative permittivity of the medium. It is given by the ratio of the permittivity constant e of the dielectric to the absolute permittivity constant e 0 of vacuum. For vacuum, e r has a value equal to unity, while for air e r is.0006, but for practical purposes, the value of e r for air is also assumed equal to unity. The force F is repulsive, if both the bodies carry the same charge and it is attractive, if the two bodies carry opposite charges. The electric field intensity E is defined as the force exerted on a unit positive charge placed at that point. Thus, from Eq. (.2) putting q 2 = C gives q E = N/C (.3) 2 4pe r and the electrostatic force F given by Eq. (.2) can be written as F = E q 2 N (.4) If an object is moved within the field F, work is done. If the movement of the object is in the direction of the force, the output is work. On the other hand, if the object is moved against the force, work is the input. Hence, if field F Newton acts for t seconds through a distance of d metres along a straight line, then the work done W is given by W = F d, N-m or joules (J) (.5).5 ELECTRIC POTENTIAL, POTENTIAL DIFFERENCE,.5 AND ELECTROMOTIVE FORCE In an isolated metallic conductor, numerous free electrons exist in the conduction band. These electrons can be made to flow in a given direction by applying an external energy source. For example, a battery, across the ends of the conductor. Due to chemical reactions inside a battery, negative charges accumulate at one terminal, i.e., the cathode, and positive charges accumulate at the other terminal, i.e., the anode. As the charges of unlike polarity attract each other, work has to be done by an external agency against these attractive forces to separate them. In a battery the work is done through electrochemical reactions. The greater the number of charges that are separated, the greater the work done to achieve this separation and greater is the potential energy of the separated charges. This

10 Definitions and Basic Circuit Concepts 5 situation finds an analogy in mechanics where work has to be done against the gravitational force in raising a mass to some height above sea level. The greater the mass, the greater is the potential energy possessed by that mass. The work done per unit charge is a measure of the amount of accumulated charge or a measure of potential energy that has been established. The work done per unit charge is known as potential. The unit of potential is volt, named after the Italian physicist Alessandro Volta, and volt is equal to joule/coulomb. The alternate name of this quantity is voltage. In mechanics, the work done in raising a mass from a height h to a height h 2 is equal to the difference in potential energies at these heights. Similarly, the work done per unit charge in a battery is the potential difference, (pd), between the terminals of the battery. Thus, charge moving in a circuit gives rise to current and it takes some work, or energy, for the charge to move between two points say, from point a to point b, in the circuit. The total work per unit charge associated with the motion of charge between two points is called voltage. If v is the voltage in volts, w is the energy in joules, and q is the charge in coulombs, then v = dw dq J/C or V (.6) The potential difference (pd) is defined as the energy transfer between two points of a conductor carrying a current of A, when the power dissipated between the points is W. As the pd is measured in volts, it is also termed as voltage drop. The pd between the battery terminals is known as the electromotive force (emf). The emf represents the driving influence that causes a current to flow. It is the energy that is used when a unit charge passes through the source. The term emf is always associated with energy conversion. The emf is usually represented by the symbol E and has the unit of volt. When the battery is connected externally through an ideal lossless conductor to a load, energy transfer to the load commences through the conductor. This is the potential difference across the load. When all the energy is transferred to the load unit, the pd across the load unit becomes equal to that of the battery emf. Thus, both emf and pd are similar entities and have the same units. However, the former is associated with energy while the latter causes the charge or current to flow. In Fig..2, emf and pd are represented by employing the convention that in both cases, the arrow head points to a higher potential. It may be noted that the current leaves the source of emf at the positive terminal and therefore, the direction of current flow is the same as that of the emf arrow. The current enters the load at the positive terminal, and thus the direction of current is opposite to pd arrow of the load.

11 6 Circuits and Networks: Analysis, Design, and Synthesis Current flow Source Source emf E Load pd V Load unit Fig..2 Diagrammatic representation of emf and pd.6 ELECTRIC POWER AND ENERGY Power is defined as the work done per unit time. If a field F Newton acts for t seconds through a distance of d metres along a straight line, then from Eq. (.5), the work done W = F d, N-m or J. The power p, either generated or dissipated by the circuit element, is given by W F d p = = = F u, (J/s) (.7) t t where, u is the velocity in m/s. The unit of power is joule/second or watt (after the Scottish engineer, James Watt). Also, by definition Power = Work = Work Charge = Voltage Current (.8) Time Charge Time Thus, if the current flowing between two points in the conductor is i, and the voltage across the two points is v, then the power, p, dissipated between two points in the conductor is given by p = v i (.9) Using Eqs (.b) and (.6) for current and voltage, Eq. (.9) can be written as dw dq joules coulombs p =, dq dt coulomb seconds dw joules =, or, watts (.0) dt seconds As in the case of voltage, power flow is directional. Conventionally, if current flows into the positive terminal of an element, power dissipated is positive and the element consumes power. On the other hand, if current flows out of the positive terminal of an element, the power dissipated is negative, that is, the element delivers power. Figure.2 shows the sign convention for power. Electric energy W is defined as the power consumed in a given time. Hence, if current i A flows in an element over a time period of t s when a voltage v volts is applied across it, the energy consumed is given by W = p t = v i t = v i t J or W s (.) The unit of energy W is joule or W s. However, in practice, the unit of energy is kilowatt-hour (kwh) which represents the work done at the rate of kw in a

12 Definitions and Basic Circuit Concepts 7 period of hour (h). For billing purposes, electric utilities consider one unit equal to kwh..7 CIRCUIT ELEMENTS An electrical network is composed of a number of basic elements, such as resistors, inductors, capacitors, etc. interconnected in a suitable manner and energized by one or more voltage/current sources. The elements of the network may be of different kinds such as linear or non-linear, bilateral or unilateral, active or passive, lumped or distributed, time dependent or time invariant and so on. In this section, the basic elements which constitute an electric circuit are discussed. Prior to proceeding with the discussion of the characteristics of basic circuit elements, it would be appropriate to define terms frequently employed in circuits. Circuit element An individual component such as a resistor, inductor, capacitor, diode, transistor, energy source, etc. which constitutes a circuit, is known as a circuit element. Network and circuit A network is a connection of two or more simple circuit elements. A circuit is a network that has at least one closed path. Every circuit is a network, but all networks may not be circuits. Branch A branch is an element of the network having only two terminals. Passive and active branch A branch is said to be active when it contains one or more energy sources. A passive branch does not contain an energy source. Linear element When the current and voltage relationship in an element can be simulated by a linear equation either algebraic, differential, or integral type, the element is said to be a linear element. Bilateral and unilateral element A bilateral element conducts equally well in either direction. Resistors and inductors are examples of bilateral elements. When the currentvoltage relations are different for the two directions of current flow, the element is said to be unilateral. Diode is a unilateral element. Lumped and distributed parameter elements Lumped parameter elements are those, which for the purpose of analysis may be treated as physically separate elements such as resistance, inductance, capacitance, etc. The distributed parameter element cannot be modelled as a combination of a physically identifiable separate resistor, inductor, or capacitor. Node The junction point of two or more branches is known as a node. Loop and mesh Any closed path, formed by the branches in a network, is known as a loop. A mesh is a kind of loop, which does not contain any network branch enclosed within it..7. Resistors According to Ohm s law enunciated by the German mathematician Georg Simon Ohm, potential difference v across the ends of a conductor is proportional to the

13 8 Circuits and Networks: Analysis, Design, and Synthesis current i flowing through the conductor at a constant temperature. Mathematically, Ohm s law is expressed as v µ i or v = R i (.2) or i = v R (.2a) where R is the proportionality constant and is designated as the conductor resistance and has the unit of ohm (W). Resistance may be interpreted as a characteristic of a conductor which obstructs the flow of current and in the process, electric energy is lost which appears in the form of heat. The free electrons in a conductor can be made to flow in a particular direction by applying an external voltage source. The application of voltage source produces an electric field within the conductor, which produces a directed motion of the free electrons. The motion of these free electrons is opposite to the electric field and during its motion through the conductor, collides with other particles. These collisions result in the production of irreversible heat loss. A resistor is said to be linear if it satisfies Ohm s law, that is, the current through the resistor is proportional to the pd across it. If the magnitude of resistance varies with the variation of voltage or current, the resistor is said to be non-linear. Resistors made of semiconductor materials are non-linear resistors. The resistance R of a conductor is a function of its geometry, that is, it is directly proportional to its length l m and inversely proportional to its crosssection a m 2. Thus, R = r l a W (.3) where r is the proportionality constant and is called the specific resistance or resistivity of the conductor. From Eq. (.3), resistivity is expressed as, 2 R a metre r =, ohm or, ohm-metre (.4) l metre The inverse of resistance is called conductance G and may be written from Eq. (.3) as follows: G = = a a = s mho or siemens (S) (.5) R r l l where s is called the specific conductance or conductivity of the material. From Eq. (.5), s is obtained as follows: s = l = metre or, mho per metre or, S/m (.6) 2 r R a ohm metre The power consumed by the conductor, in terms of the applied voltage v is obtained by substituting for the current i (Eq..2a) in Eq. (.9) as follows:

14 Definitions and Basic Circuit Concepts 9 2 v v p = vi = v = = Gv 2 W (.7a) R R The power consumed in the conductor, in terms of the current flow i is given by 2 2 p = vi = ir i = i R = i (.7b) G Resistance depends not only on the material composition of a conductor, but also on the temperature. For most metallic conductors, with the exception of carbon and insulating materials, the resistance increases with increase in temperature. Resistance of all current carrying conductors increases due to heat dissipation and associated temperature rise in the resistance. However, the resistance of some metallic alloys such as constantan (60% copper and 40% nickel) and manganin (84% copper, 2% manganese, and 4% nickel) shows no variation over a considerable range of temperature variation and is employed in the manufacturing of resistance boxes which requires it to be precise. When the rate of heat produced, due to the current flow, becomes equal to the rate of heat dissipated, steady-state condition is attained and the conductor acquires a constant temperature. Thus, the capacity of a resistor to dissipate maximum heat, without getting damaged or burnt, is defined as its power rating. For example, if a 50 W resistor has a current rating of A, then its specification for maximum heat rating would be 50 W. Similarly, if the same 50 W resistor had a current rating of 0.5 A, its maximum heat dissipation rating would be 2.5 W. Example.2 A 0 m long aluminium wire and a 5 m long copper wire are connected across a battery source of 2 V. If the current flowing through the composite wire is 00 ma, determine (i) the resistance of, and (ii) voltage drop across each wire. Also calculate the diameter of the wires. What is the power consumed by each wire? Assume that the wires have the same cross-section. Take the resistivity of aluminium, r Al = mw-m and that of copper, r Cu = 0.07 mw-m. Solution Assume the cross-section of the wire to be a m 2. Then the resistance of aluminium R Al and the resistance of copper R Cu are obtained as R Al = ral = = W a a a R Cu = rcu = 0.07 = W a a a Dividing R Al by R Cu gives RAl = R = Cu By Ohm s law, the total resistance = = W Then, R Al R Cu = ( ) R Cu = 20 W

15 0 Circuits and Networks: Analysis, Design, and Synthesis 20 (i) \ R Cu = = W and R Al = = W (ii) Voltage drop across copper wire = ( ) = V Voltage drop across aluminium wire = ( ) = V Now, the cross-section of the wire, a = d p 2 = = m. 4 \ d = = m or d = 6.8 cm p Power consumed by copper wire = ( ) = W Power consumed by aluminium wire = ( ) = W.7.2 Inductors In 820, Danish physicist, Hans Christian Oersted discovered that a magnetic field is produced around a current carrying conductor. In 83, an Englishman Michael Faraday and the American inventor Joseph Henry discovered almost simultaneously, that a voltage is induced in a stationary conductor when placed in a varying magnetic field. They showed that the magnitude of the induced voltage e is proportional to the time rate of change of current, di/dt, producing the magnetic field. Then e µ di dt di or e = L (.8) dt In Eq. (.8), e and i are both functions of time; if needed, this fact may be stressed by writing e(t) and i(t) instead. The proportionality constant L is called inductance. The unit of inductance is Henry (H) named after Joseph Henry. An inductor is represented symbolically as shown in Fig..3. Fig..3 Symbolic representation of an inductor An inductor is said to be linear if it satisfies Eq. (.8), that is, the voltage is proportional to the derivative of current passing through it. Physically, an inductor is constituted of several turns of thin wire wound on a magnetic or an air core. Thus, the effect of an inductor, in a circuit, is realized when the current is varying. From Eq. (.8) inductance is given by e volt-sec L = or Henry (H) (.9) di/ dt ampere An inductor is said to have an inductance of H, when a current, uniformly varying at the rate of A/s, induces a voltage of V. Integration of Eq. (.8) yields the current in the circuit. Hence,

16 Definitions and Basic Circuit Concepts i = L t ò 0 edt i(0) A (.20) In Eq. (.20), i(0) is the current in the circuit at t = 0. It may also be observed from the equation that the current in an inductor cannot change suddenly in zero time. The instantaneous power p at any instant in the inductor is given by di p = ei = Li W (.2) dt It is clear from Eq. (.2) that when i is constant, p = 0, since di/ dt = 0. Also, when the current is increasing, (di/dt) is positive and hence the inductor stores energy. Conversely, when the current is decreasing, (di/dt) is negative, and the inductor releases the stored energy. Therefore, an inductor may be seen as a static storage device which stores energy, from the source, when the current is increasing and releases the same, to the energy source, when the current is decreasing. The energy stored W L, in an inductance, in a given time t may be computed as under W L = t t ò pdt = ò ei dt (.22) 0 0 Substituting for e from Eq. (.8), and assuming that at t = 0, i = 0, and at t = t, i = i, in Eq. (.22) gives W L = i i æ di ö 2 òç L i dt = L i di = Li J è dt ø ò (.23) Example.3 A voltage source v = V m sin wt is applied across an inductor of 0.5 H. Derive an expression for the current flow in the inductor. If the effective value of the applied voltage source is 2 V, determine the instantaneous value of the current and energy in the inductor at t = 0, 0.004, 0.0, 0.05, and 0.02 s. What is the effective value of the current? Assume (i) frequency of the voltage source equal to 50 Hz; and (ii) current flow in the inductor at t = 0 is zero. Solution The instantaneous current in the inductor i = L t ò 0 vdt t 0.5 ò é ë ù û dt = 2 2 sin ( 2p ft) 0 = cos(34) t =- 0.08cos(34) t A Energy stored = Li = 0.5 i = 0.25 i J 2 2

17 2 Circuits and Networks: Analysis, Design, and Synthesis t in s i in A W L in J Effective value of current = 0.08 = A 2 i Example.4 Calculate the Amperes 5 voltage across a 5 H inductor at t = s and t = 3 s, when a current having the variation O shown in Fig..4 is flowing 2 3 t,s through the inductor. Plot the voltage across the conductor 5 against time. Fig..4 Solution For the period between t = 0 to t = 2 s, the rate of change of current is di dt = Di -0 = Dt 2 = 5 A The voltage across the inductor v L at t = s is obtained as follows: di v L = L = 5 (- 5) =- 25 V dt For the period between t = 2 to t = 3 s, the rate of change of current is di D i 5 = = = 5 A dt Dt The voltage across the inductor at t = 3 s is obtained as follows: di v L = L = 5 (5) = 25 V dt Figure.5 shows the variation of voltage versus time across the inductor. v, Volts 25 O 2 3 t, s Capacitors Fig..5 A capacitor is a physical device, which when polarized by an electric field by applying a suitable voltage across it, stores energy in the form of a charge

18 Definitions and Basic Circuit Concepts 3 separation. In its simplest form, a capacitor is made up of two parallel conducting plates separated by an insulating material or air. The symbolic representation of a capacitor is shown in Fig..6. i C S i C Fig..6 (a) v (b) (a) Symbolic representation of a capacitor; (b) capacitor connected to a voltage source The ability of the capacitor to store charge is measured in terms of capacitance. Capacitance of a capacitor is defined as the charge stored per volt applied and its unit is farad (F). The practical unit used is microfarad (mf) since farad is too large a unit. In Fig..6(b), assume that the charge on the capacitor at any time t after the switch S is closed is q coulombs and the voltage across it is v volts, then by definition, q coulomb C = Farad v = volt = (.24) Current i, flowing through the capacitor can be obtained by applying Eq. (.b) to Eq. (.24) as follows: dq dv i = = C amperes (.25) dt dt Equation (.25) shows that the current flows in the circuit only when the applied voltage is varying. In other words, if a DC voltage is applied across a capacitor in the steady state, no current flows through the circuit, since the charge on the capacitor is equal to source voltage, that is, the capacitor behaves like an open circuit. The voltage across a capacitor may be obtained by integrating Eq. (.25) with respect to time as v = C t ò 0 idt v(0) volts (.26) The constant v(0) is an integration constant which defines the initial voltage across the capacitor when the switch S is closed. It may also be observed from the equation that voltage in a capacitor cannot change instantaneously, that is, in zero time.

19 4 Circuits and Networks: Analysis, Design, and Synthesis Power p in the capacitor is written as dv p = vi = Cv watts (.27) dt The energy W C, in the capacitor at any time t is given by W C = t ò 0 pdt J (.28) Substituting for p from Eq. (.27) in Eq. (.28), and assuming that at t = 0, v = 0, and at t = t, v = v, gives W C = v ò æ dvö ç Cv dt = C = è ø òvdv Cv dt v 2 J (.29) From the foregoing, it is evident that the voltage stored in a capacitor depends upon the instantaneous value of the voltage. Further, when the voltage is increasing, the capacitor stores the energy from the source and when the voltage is decreasing, it returns the energy to the source. Therefore, like an inductor, a capacitor is also a storage device which manifests itself in a circuit when the voltage is varying. Example.5 A varying voltage represented by Fig..7 is applied across a capacitor having a capacitance of 50 mf. (a) Calculate the current flow 5 during the period (i) 0 t 0.5 s and (ii) 0.5 t.0 s. (b) What is the charge accumulated, power, and O energy stored in the capacitor at t,s t = 0.25 s, and t = 0.8 s? Fig..7 Solution V (a) (i) For the period 0 t 0.5, v = 5 30 V 0.5 t = t The current flow during this period, i = (ii) For the period 0.5 t.0, v = The current flow during this period, i = dv C dt = ( ) 30 =.5 ma V 0.5 t = - t =.5 ma (b) At t = 0.25 s, v = = 7.5 V \ q = vc = = C p = v i = = 0.03 W dv C dt = ( ) [ 30]

20 Definitions and Basic Circuit Concepts 5 W C 0.25 = Cv = 50 0 (7.5) = J 2 2 The charge at t = 0.8 s is given by q = vc = ( ) = C p = v i = ( ) ( ) = 6 ( ) = W W C 0.8 = Cv = 50 0 ( ) = J 2 2 Example.6 A varying current represented by the curve shown in Fig..8 is flowing through an ideal capacitor having a capacitance of 500 mf. Derive expressions for the voltage and energy developed across the capacitor. Plot the variation of voltage versus time. Assume that the capacitor is initially uncharged. i,ma Fig..8 2 t, s Solution For the period 0 ³ t ³.0, the current flowing through the capacitor is written as i = 0. A. The voltage developed across the capacitor is given by v = òidt = (0.0) dt v(0) = 200 t v(0)v ò - 6 C where v(0) is an integration constant. Since the capacitor is uncharged at t = 0, v(0) = 0. Hence, v = 200t V (.6.) Cv = t = 0t J Energy stored W C 0 t.0 = ( ) ( ) 2 For the period.0 t 2.0, the current flowing through the capacitor is written as i = 0. A. Thus voltage developed is written as v = ( ) dt = t v () - 6 ò V where v() is an integration constant. From Eq. (.6.), at t =.0 s, v = 200 V. Hence, 200 = 200 (.0) v() or v() = 400 V Thus the expression for the voltage, for the period.0 ³ t ³ 2.0, is as under v = 200 t 400 = 200 (2 t) V

21 6 Circuits and Networks: Analysis, Design, and Synthesis Energy stored W C.0 t = 2 Cv -6 = ( ) [ 200(2 - t) ] 2 2 = 0 (2 t) 2 J The variation of voltage, across the capacitor, with time is shown in Fig..9. v,v 200 O 2 t,s Fig..9.8 ENERGY SOURCES The active element in an electric circuit is the energy source. An electric circuit must be energised by at least one energy source. However, it may have more than one energy source. Energy sources may be categorized as (a) voltage and (b) current sources. Broadly, the voltage and current sources may be classified as, (i) Ideal independent sources (ii) Dependent or Controlled sources (iii) Practical sources.8. Ideal Independent Sources Ideal independent sources are of two types:. Voltage source 2. Current source.8.. Independent Voltage Sources Ideal independent voltage sources have the feature of supplying constant designed voltage independent of the magnitude or direction of the flow of current. The magnitude of the current, however, is determined by the conditions at the terminals of the voltage source. In other words, an ideal independent voltage source is capable of delivering or absorbing unlimited energy endlessly at a constant voltage. From the foregoing, it may be concluded that the internal resistance/impedance of such a source is zero. The voltage of an ideal independent source may be constant (DC) or it may be a function of time. Figure.0 shows the symbolic representation and the vi characteristic of an ideal independent voltage source. I i() t vs () t V S vs () t V Fig..0 (a) (b) (c) i() t Symbolic representation of an ideal independent voltage source (a) DC voltage source; (b) time-dependent voltage source; (c) vi characteristics

22 Definitions and Basic Circuit Concepts 7 From the figure, it may be noted that the convention adopted is that the current flowing out of the positive terminal is positive. Further, the magnitude of the DC voltage source is represented by the upper case symbol V S, while the magnitude of the time-dependent voltage source is represented by the lower case symbol v S (t). From the vi characteristic in Fig..0 (c), it is seen that the magnitude V of the voltage source is independent of the magnitude of the current supplied by it. Open-circuit and short-circuit conditions in a voltage source are identified respectively by i S = 0 and v S = Independent Current Sources Independent current sources are characterized by their ability to supply constant designed current irrespective of the direction and magnitude of the voltage. The resistance/impedance condition at the terminals of the current source establishes its voltage. Figure. shows the symbolic representation and vi characteristic of an independent current source. i S () t I S is () t I Fig.. (a) (b) (c) v() t Symbolic representation of an ideal independent current source (a) DC current source; (b) time-dependent current source; (c) vi characteristics In this case also, the convention for the current flow is that it flows out of the positive terminal and small case letter i S (t) is used to represent time-dependent current while upper case letter I S is used for DC current. Correspondingly, open and short circuit conditions in a current source are identified by v S = 0 and i S = 0, respectively. It is a common practice to employ voltage sources in electric circuit analysis. Current sources find wider applications in electronic circuits. However, with the help of additional circuitry, it is feasible to transform a voltage source into a current source and vice-versa. High voltage DC transmission engineering is based on constant current flow through transmission lines. Example.7 An 8 W resistor is connected across a DC 2 V independent voltage source. (i) Determine the current and voltage across the resistor. (ii) If the 8 W resistor and a 4 W resistor were to be connected end to end (in series) across the voltage source, what will be the current and voltage across each resistor. (iii) Repeat (i) and (ii) with the independent voltage source replaced by a DC 2 A independent current source. Solution (i) Current I through 8 W resistor is given by

23 8 Circuits and Networks: Analysis, Design, and Synthesis I = = A and voltage V across it is V =.5 8 = 2 V (ii) 2 V is now applied to the combination of the 8 W and 4 W resistors. The current is given by = A. I = ( ) Then, voltage across the 8 W resistor = 8 = 8 V And voltage across 4 W resistor = 4 = 4 V (iii) When the DC 2 A current source is connected across the 8 W resistor, full current flows through it. Therefore, voltage across the resistor 2 8 = 96 V. When the current source is connected across the 8 W and 4 W combination, then too, full 2 A current flows through the combination. Then, voltage across the 8 W resistor = 2 8 = 96 V. voltage across the 4 W resistor = 2 4 = 48 V. voltage across the combination of the resistors = = 44 V..8.2 Dependent or Controlled Sources Dependent or controlled sources are uni-directional sources whose output (voltage or current) are controlled by the input variable (voltage or current) and are independent of any other variables in the circuit. Dependent sources facilitate the simulation of electronic circuits. Four types of controlled sources are symbolically represented as shown in Fig..2 and are described below: Fig..2 (a) Symbolic representation of dependent sources; (a) voltage source, (b) current source (b).8.2. Voltage Controlled Voltage Sources (VCVS) Figure.3 shows a VCVS in which the voltage output of the source is dependent on the input voltage. Mathematically, the output voltage is written as v S = Av i (.30)

24 Definitions and Basic Circuit Concepts 9 i i =0 i v i vs=av i Fig..3 VCVS In Eq. (.30), A is a dimensionless proportionality constant or voltage gain, and v i is the open circuit input terminal voltage Voltage Controlled Current Sources (VCCS) In this case, the current source output is controlled by the open circuit input terminal voltage v i. Figure.4 shows the circuit diagram for a VCCS. Here, the output of the current source is mathematically given by i i =0 i v i i S =Av i Fig..4 VCCS i S = Av i (.3) In Eq. (.3), A is a proportionality constant having the dimension of conductance (mho) Current Controlled Current Sources (CCCS) The output in a CCCS is controlled by short circuiting the input terminals (see Fig..5) and is expressed as i S = Ai i (.32) In Eq. (.32), A is a proportionality constant or current gain. i i i v i =0 i S =Ai i Fig..5 CCCS

25 20 Circuits and Networks: Analysis, Design, and Synthesis Current Controlled Voltage Sources (CCVS) In a CCVS also, as shown in Fig..6, the voltage at the output terminals is controlled by short circuiting the input voltage terminals. Equation (.33) provides an expression for controlling the output of a voltage source. v S = Ai i (.33) In Eq. (.33), A is a proportionality constant having the dimension of resistance (ohm). i i i v i =0 v S =Ai i.8.3 Practical Sources Fig..6 CCVS Consider a load resistor of R L W connected across the terminals of an ideal voltage source v S (t) as shown in Fig..7(a). By Ohm s law, the current through the load resistor i L (t), which is equal to the current i S (t) delivered by the source, is given by vs () t i L (t) = i S (t) = A (.34) RL Equation (.34) shows that as R L 0, i L which implies that the idealized voltage source should have the capacity to supply unlimited current, a condition, impossible to meet in actual practice. Next consider Fig..7 (b) which shows a resistor r S connected in series with an ideal voltage source and the combination is connected across a load resistor R L W. Then, the current through the load resistance R L is given by i S () t il () t r S i t S () il () t v S (t ) R L vl () t vs () t vl () t R L (a) is max () t (b) v S () t vl () t R L Fig..7 (c) Simulation of a practical voltage source

26 () t i L (t) = i S (t) = ( r R ) S v S and the voltage across the load resistance R L is given by L A Definitions and Basic Circuit Concepts 2 (.35a) v L (t) = il() t RL = éëvs () t - il() t rsùû (.35b) Equation (.35a) shows that as R L 0, the maximum current required to be supplied by the voltage source is limited by the series resistor r S [see Fig..7(c)] and is given by vs () t i S max (t) = A (.36) r From Eq. (.35b), it is seen that as R L, that is open circuit, i L (t) 0, the load voltage approaches the voltage of the source. Equation (.35b) can thus be used to plot the vi characteristic of a practical voltage source and is shown in Fig..8. A practical voltage source has a drooping vi characteristic. The foregoing discussion provides the basis for introducing the concept of the S il () t i S max () t vs () t = rs Practical source O Fig..8 vl () t = v S () t Ideal source vl () t vi characteristics of a practical voltage source internal resistance r S of a voltage source, and it may be seen as a resistance, which is internal to the energy source and limits the source current to a maximum value. A desirable feature for an ideal voltage source to possess would be that its internal resistance be as small as possible, so that the internal voltage drop i L (t) r S is a minimum. Often, the effective internal resistance of a voltage source is quoted in the technical specifications for the source, so that the user may take this parameter into account. Figure.9 shows a diagrammatic representation of a practical current source. Based on the same analogy as used for a practical voltage source, it can be shown that the vi characteristic is also drooping and is drawn in Fig..20: r S is () t v L () t R L r S is () t vl () t (a) Fig..9 (b) Simulation of a practical current source It is left as a tutorial exercise for the reader to verify the drooping nature of the vi characteristic and prove that the internal resistance of a practical current source be as high as possible in order that it approaches the behaviour of an ideal current source.

27 22 Circuits and Networks: Analysis, Design, and Synthesis il () t is () t Ideal source Practical source Fig..20 O v L() t = r i ( t) S S v t L () vi characteristics of a practical current source.8.4 Source Transformation Since characteristics of the voltage and current sources are dependent on the terminal conditions, it is feasible to transform a voltage source into a current source and vice-a-versa without modifying the response in the network to which it might be connected. The transformation is accomplished in the following manner: v S () t r S il () t vl () t Network N i S () t v = () t S rs vl () t r S r S il () t vl () t Network N Fig..2 (a) Source transformation: (a) Representation of a practical voltage source connected to a network N; (b) representation of a practical current source connected to a network N (b) Figure.2(a) simulates a practical voltage source by an ideal voltage source whose output voltage is v S (t) and a series finite resistance r S. The voltage v L (t) at the terminals of the network N is given by v L (t) = v S (t) i L (t) r S (.37) where, i L (t) is the current supplied to the load. Dividing Eq. (.37) by r S leads to vl () t vs () t = - il () t (.38) rs rs It is easily seen, by referring to Fig..2(b) that Eq. (.38) holds true. In other words, the magnitude of the current supplied by the energy (current) source is v S (t)/r S A and its resistance is r S W. Thus, Fig..2(b) shows the simulation of a voltage source transformed into an equivalent current source. It can be easily proved that the reverse transformation, that is, replacing a current source by a voltage source is feasible..9 KIRCHHOFF S LAWS The foundation of circuit analysis is based on the laws of current distribution and voltage division in a network. These two laws, named after the German physicist,

28 Definitions and Basic Circuit Concepts 23 Gustav Robert Kirchhoff (824887) who first enunciated them, are known as Kirchhoff s Current Law and Kirchhoff s Voltage Law..9. Kirchhoff s Current Law (KCL) The attribute of an electric charge, that it can neither be created nor destroyed but must be conserved forms the basis of Kirchhoff s current law (KCL). It states that the sum of the currents at a junction (node) in a circuit is zero, at all instants of time. Mathematically, KCL is written as m= n å i = 0 (.39) m m = where i m is the current in the mth element connected to node k and the total number i 4 of elements connected to the node is n. This i k law applies equally well to circuits driven by constant DC sources, and time variable sources. Figure.22 demonstrates the application of KCL. Fig..22 i 5 Application of KCL Since a direction is associated with the flow of current, it is necessary to define directions. Assume that the currents flowing into the junction are positive and currents flowing out of the junction are negative. Based on this assumption, KCL may be applied to the junction in Fig..22 as under i i 2 i 3 i 4 i 5 = 0 (.40a) or i i 2 = i 3 i 4 i 5 (.40b) Example.8 In Fig..23, calculate v at the node and i. The data is as follows: i 2 = 6 A, v 3 = 0e 2t, v 4 = e 2t. Solution Using Eq. (.2a) gives v 4-2t 0e - 2t i 3 = = 2e A 5 Fig..23 The current i 4 is computed by employing Eq. (.25) as d 2t 2t i 4 = ( e - ) 0.5 e - =- A 4 dt Application of KCL to the node yields i i 2 i 3 i 4 = 0 or i = i 2 i 3 i 4 = 6 2e 2t ( 0.5e 2t ) = (6 2.5e 2t ) A The negative sign shows that the direction of flow of current is opposite to the assumed direction. In order to calculate v, Eq. (.8) is used as under - 2t d -( 6 2.5e ) v = = e 5 dt -2t V i 2 i i 2 v 5 H 3 v i 4 i 3 5 W 4 F i 3