PREFACE These notes were mainly arranged from the lecture notes and the books written by the valuable scientists and teachers Ozay HSEYN,YurdakulCEYHUN,SleymanPENBEC andKhaldunAbdullah.Thefirsttwoofthemweremyteachersin70s.Ialsoaddedmy experience during my teaching period from 1984 to 1992 at the Military Academy (Kara Harp Okulu). Thesenotesarepreparedtakenintoaccountnotonlytheelectricalandelectronics engineering students, but also the mechatronics and control systems engineering students. Inmodernsystemsengineeringapplications,computersareunforgivablepartsofthe systems.Sincetheyareworkingmainlyinrealtime,statespaceapproachisadoptedasthe main tool for the analysis of the systems. Here only electrical systems are taken into account, butthisapproachcaneasilybeextendedtoanalysemechanical,hydrolic,biological, economical and fiscal systems as well. In general, this approach is a main tool for the analysis of dynamical systems that can be modelled by differential equations. The organization of these notes is as follows: Chapter 1 is an introductory chapter forthe definition of modelling, formulation and solution for the analysis and design of dynamical systems.InChapter2,mathematicalmodelsofideal2-terminalelectricalcomponentsare given.Chapter3givesthedefinitionsofbasictimefunctionsoraswecallthemassignals usedinelectricalsystemsasbasicvariablessuchascurrentandvoltage.Alsothemore advancedwaveformsisstudiedinthischapter.Chapter4isdevotedmainlytoresistive networks.Thefundamentalnetworktheorems,Krirchofflaws,basicsolutiontecghniques such as node, mesh and loop equation approaches for solving and analysing resistive circuits. Chapter 5 gives the state space representation of electrical systems. In this chapter the Graph theorythatisusedfortheformulationofelectricalsystemsintime-domainisgiven.Writng the state equations and solutions of state equations are given in detail. Chapter 5 is one of the mainpartsofthesenotesthatgivesthetimedomainformulationofelectricalsystems. Chapter 6 explains the responses of the systems in time domain. Chapter 7 is an introductory chapter for the transformation to the frequency domain. It gives the famous Laplace transform technique. In Chapter 8, electrical system analysis in s-domain is given. This can be achieved by using Laplace transform techniques for the formulation of electrical systems in s-domain. Chapter9isthesecondmainpartofthesenodesthatgivesthesinusoidalsteadystate responseofelectricalsystems.Inthischapter,jw-domainanalysisisintroducedandboth constant and variable frequency responses of systems by the use of phasor representations and frequency domain representations are teached. Also three-fhase systems are analysed. Chapter 10isachapterthatgivesthethree-phaseelectricalsystemsandconnectiontypes.The complex numbers are given in the Appendix.

Dr. Sedat NAZLIBLEK Ankara, 2009-2010 i INDEX CHAPER 1 Introduction..................................................................................................................1.. Contents........................................................................................................................1 1.1System Concept................................................................................................2 1.2System Classes..................................................................................................3 1.3System Properties and Modelling.....................................................................4 1.4Types of Formulation in Mathematical Modelling...........................................6 1.5Dynamical System............................................................................................7 1.6Types of Signals and Systems...........................................................................9 1.7Difference Equations.......................................................................................14 CHAPTER 2...............................................................................................................17 Electrical System Components..................................................................................17 Contents........................................................................................................................17 2.1Introduction......................................................................................................18 2.2Mathematical Model of Ideal Linear Time-Invariant Two Terminal Components.....................................................................................................18 2.2.1Linearity...........................................................................................................19 2.2.2Active Component...........................................................................................19 2.3Basic System Components and Their Models.................................................19 2.3.1Reference Directions........................................................................................20 2.3.2Energy Dissipating Components......................................................................21 2.3.3Power and Energy for Energy Dissipating Components..................................22 2.3.4Energy Storing Components............................................................................23 2.3.5The Types of Energy Storing Components......................................................24 2.3.6Ideal Sources (Generators)...............................................................................26 2.4Electrical Components.....................................................................................29 2.4.1A General Electrical Two-Terminal Component Associated Reference Directions for Terminal Voltage and Current..............................................................29 2.4.2Resistor, Linear Time-Invariant Electrical Energy Dissipating Component...31 2.4.3Inductor (Coil), Linear Time-Invariant Component Storing Magnetic Energy .........................................................................................................................35 2.4.4Capacitor, Linear Time-Invariant Component Storing Electrical Energy......37 2.4.5Ideal Sources...................................................................................................39 CHAPTER 3...............................................................................................................46 Basic Sysem Variables and Signals...........................................................................46 Contents.......................................................................................................................46 3.1Basic System Variables and Signals................................................................47 3.2Signal Types....................................................................................................48 3.3Graphical Differentiation and Integration.......................................................50 3.4Basic Signals...................................................................................................52 3.5The Relation Betwee Basic Signals................................................................58 3.6The Expression of Mixed Functions in terms of Basic Functions..................62 3.7Exponential Function......................................................................................69 ii 3.8Convolution Integral.......................................................................................70 3.8.1Convolution with Unit Impulse Signal............................................................71 3.8.2Impulse Response of Electrical System...........................................................72 3.9Periodic Functions (Signals)............................................................................76 3.9.1Types of Periodic Signals................................................................................77 3.9.2Average Value of a Signal (Direct Current DC Components).....................77 3.9.2.1 Average Value of Simple Sinusoidal Signals..................................................78 3.9.2.2 Sinusoidal Signals with Direct Current (dc) Components...............................79 3.9.3Effective (RMS) Value of a Signal..................................................................80 3.10Odd Function and Even Function....................................................................83 3.11Fourier Series Expansion.................................................................................85 QUESTIONS...............................................................................................................90 CHAPTER 4..............................................................................................................96 Fundamental Network Theorems.............................................................................96 Contents.......................................................................................................................96 4.1Introduction.....................................................................................................97 4.2Kirchoff Laws.................................................................................................97 4.3Terminal Equivalence and Reduction Techniques.........................................98 4.4Resistive Network Configurations..................................................................99 4.5Resistors in Series-Parallel............................................................................104 4.6 - Y Circuits and Their Transformations.....................................................106 4.6.1 - Y Conversion...........................................................................................107 4.6.2Y - Conversion...........................................................................................109 4.7Network Theorems........................................................................................110 4.7.1Superposition Theorem.................................................................................110 4.7.2Thevenin and Norton Theorems....................................................................112 4.8Maximum Power Transfer and Matching......................................................118 4.9Loop and Node Equations.............................................................................119 4.9.1Loop Equatins................................................................................................119 4.9.2Node Equations..............................................................................................121 CHAPTER 5.............................................................................................................127 State Space Representation of Electrical Systems.................................................127 5.1Graph Theory.................................................................................................128 5.2Basic Concepts of the Graph Theory.............................................................128 5.3System Graph and State Equations................................................................135 5.4Solution of State Equations............................................................................147 QUESTIONS.............................................................................................................161 CHAPTER 6............................................................................................................166 System Resposes......................................................................................................166 Contents.....................................................................................................................166 6.1Introduction....................................................................................................167 6.2Order of a System...........................................................................................167 6.3System Responses..........................................................................................168 6.3.1Zero-Input Response......................................................................................168 6.3.2Zero-State Response.......................................................................................168 6.3.3Steady-State Response...................................................................................168 6.3.4Transient Response........................................................................................168 iii 6.4Responses for Various Inputs (Zero-State Response)...................................169 6.4.1Step Response..............................................................................................169 6.4.2Impulse Response........................................................................................170 6.4.3Sinusoidal Response....................................................................................170 6.5Responses for Various Order Systems........................................................172 6.5.1Responses for First Order Systems..............................................................172 6.5.2Responses for Second Order Systems.........................................................179 CHAPTER 7..........................................................................................................188 Laplace Transform.................................................................................................188 7.1Introduction..................................................................................................189 7.2Definition and Existence of Laplace Transform..........................................190 7.3Laplace Transform of Basic Functions........................................................192 7.4Properties of Laplace Transform..................................................................195 7.5Inverse Laplave Transform...........................................................................208 CHAPTER 8...........................................................................................................221 Comlex Domain Analysis.......................................................................................221 s-domain Analysis...................................................................................................221 8.1Introduction...................................................................................................222 8.2Mathematical Model of Ideal Linear Time-Invariant Two-Terminal Components in s-domain...............................................................................223 8.2.1Basic System Components and Their Models...............................................223 8.2.2Reference Directions......................................................................................223 8.2.3Energy Dissipating Components....................................................................224 8.2.4Energy Storing Components..........................................................................225 8.2.5The Types of Energy Storing Components....................................................225 8.2.6Ideal Sources (Generators).............................................................................227 8.3Electrical Components...................................................................................228 8.3.1A General Electrical Two-Terminal Component and Associated ReferenceDirections for Terminal Voltage and Current ..............................................228 8.3.2Resistor, Linear Time-Invariant Electrical Energy Dissipating Component.229 8.3.3Inductor (Coil), Linear Time-Invariant Component Storing Magnetic Energy ........................................................................................................................231 8.3.4Capacitor, Linear Time-Invariant Component Storing ElectricalEnergy......233 8.3.5Ideal Sources..................................................................................................234 8.4Fundamental Network Theorems...................................................................235 8.5Network Formulations....................................................................................243 QUESTIONS..............................................................................................................259 CHAPTER 9.............................................................................................................264 Complex Domain Analysis.......................................................................................264 j-domain Analysis..................................................................................................264 Contents.....................................................................................................................264 9.1Introduction...................................................................................................265 9.2Mathematical Model of Ideal Linear Time-Invariant Two-Terminal Components in j-domain and Phasor representation..................................266 9.2.1Phasor Represntation of Sinusoidal Signals..................................................266 9.2.2Electrical Components...................................................................................267 iv 9.2.2.1 A General Electrical Two-Terminal Component and Associated ReferenceDirections for Terminal Voltage and Current ..............................................267 9.2.2.2 Resistor, Linear Time-Invariant Electrical Energy Dissipating Component.268 9.2.2.3 Inductor (Coil), Linear Time-Invariant Component Storing Magnetic Energy ........................................................................................................................269 9.2.2.4 Capacitor, Linear Time-Invariant Component Storing ElectricalEnergy......271 9.2.2.5 Ideal Sources..................................................................................................273 9.3Fundamental Network Theorems...................................................................274 9.3.1Terminal Equivalence.....................................................................................274 9.3.1.1 Series Connected 2-Terminal Components....................................................274 9.3.1.2 Parallel Connected 2-Terminal Components.................................................276 9.3.1.3 - Y Connected 2-Terminal Components....................................................277 9.4Complex Power.............................................................................................278 9.5Maximum Power Transfer and Impedance Matching...................................280 9.6Network Functions.........................................................................................282 9.7Resonance......................................................................................................283 9.7.1Series Resonant Circuit..................................................................................284 9.7.2Parallel Resonant Circuit................................................................................285 9.8Frequency Response.......................................................................................287 9.9Logarithmic Plots of Frequency Response and Decibel Concept (Bode Plots)...............................................................................................................292 9.9.1Sinusoidal Response Summary......................................................................292 9.9.2Frequency Response Curves..........................................................................293 9.10Filters.............................................................................................................304 QUESTIONS..............................................................................................................308 CHAPTER 10...........................................................................................................327 Three-Phase Systems................................................................................................327 10.1Introduction....................................................................................................328 10.2General Three-Phase System..........................................................................328 10.3Special Load Connections..............................................................................337 QUESTIONS..............................................................................................................338 APPENDIX...............................................................................................................340 Complex Numbers....................................................................................................340 A.Complex Numbers.........................................................................................341 A.1Rectangular Representation...........................................................................343 A.2Polar Representation......................................................................................343 A.3Exponential Representation...........................................................................344 A.4Exponential Form of Sinusoidal Functions....................................................345 CHAPTER 1 Introduction 1 CHAPTER 1 Introduction Contents CHAPTER 1 ............................................................................................................................... 1 Introduction ................................................................................................................................ 1 Contents1 1.1SYSTEM CONCEPT ..................................................................................................... 2 1.2SYSTEM CLASSES ...................................................................................................... 3 1.3SYSTEM PROPERTIES AND MODELLING ............................................................. 4 1.4TYPES OF FORMULATION IN MATHEMATICAL MODELLING ........................ 6 1.5DYNAMICAL SYSTEM ............................................................................................... 7 1.6TYPES OF SIGNALS AND SYSTEMS ....................................................................... 9 1.7DIFFERENCE EQUATIONS ...................................................................................... 14 CHAPTER 1 Introduction 2 1.1SYSTEM CONCEPT System An environment mapping input(s) to output(s) is called a system. Where Input:Force Output:Response A system according to the definition above is shown in Figure 1.1. Figure 1.1. A system. Thesystem,here,iscomposedofoneormoreelements(orcomponents)interacting with each other. That is, a sytem is a set of elements: } ..., , , {2 1 nE E E S Theinputsandoutputsofasystemarebehavioursrepresentingtheforcesand responses respectively, and they are expressed as mathematical time functions. Notation: Heavyarrowsrepresentmultipleinputsandoutputs,whereassinglelinearrows represent single input and sinle output. In Fig.1.2, a system having n inputs and m outputs is shown. System InputsOutputs CHAPTER 1 Introduction 3 Figure 1.2. n-input, m-output system. 1.2SYSTEM CLASSES The development of technology and changing the socio-economic life depending on it give rise to the appereance of complex systems in recent years. We may divide the systems into two general classes: (1)Physical systems (2)Socio-Economic systems Physical System Asystemrealizedbyinterconnectingdevicesandelementstoeachotheriscalleda PhysicalSystem.Examplesareelectrical,mechanical,chemical,optical,magnetic,hydrolic, pneumatic, thermic etc. Socio-Economic System AsystemcomposedofactivitiesoflivingsocietiesiscalledSocio-EconomicSystem. Examples are financial, economical, commercial, etc. Common Property between two Classes Two classes of systems have a common property such that they are composed of finite number of elements connected together with some interfaces. System Element AsubsetofasystemhavingatleastoneinputandoneoutputiscalledaSystem Element.} ..., , , {2 1 nE E E S Ei i=1, 2, ..., nInterface A point connecting at least two system elements is called an interface. Special case: Inputs and outputs are also considered to be interfaces. In Figure 1.3, a general system and its elements and interfaces are shown. System Inputs Outputs x1 x2 xn y1 y2 ym CHAPTER 1 Introduction 4 Where gi:input cj:output ek:element a:interface Figure 1.3. A general system. 1.3SYSTEM PROPERTIES AND MODELLING The systems in this context are systems which can be analysed and designed. Themeaningofanalysedanddesignedisthattheperformancesofthemcanbe measured numerically. System Properties A system should have the following properties: (i)It should be composed of finite number of elements. (ii)The connection points of the elements should be distinct. (iii)Eachelementcanbecharacterisedtotallyindependentlyfromtheother elements connected to it. Analyticalapproachforsystemanalysisanddesignhasthreesequentialsteps(see Figure 1.4): Modelling, formulation, and solution g1 c1 e1 e2 e3 e4 e5 a1a2 a3 a6 CHAPTER 1 Introduction 5 Figure 1.4. Analytical approach. Modelling: Mathematical representation of system elements is called modelling. Mathematicalmodelcanbedevelopedbyisolatingeachelementfromtheothers. (Fundamental axiom of the system theory) There are two methods for obtaining a mathematical model: (1)Experimental, (2)Physical laws. Thatis,mathematicalmodelcanbeobtainedeitherbyapplyingasequenceoftestsonthe isolatedelementorbyutilizingknownphysicallawsdependinguponthephysicalnatureof the element, for example, Newtons law or Ohms law, etc. System parameter Aquantitycharacterizingasystemelementiscalledsystemparameter(e.g., resistance R, capacitance C, inductance L, mass m, spring coefficient k, etc.) Formulation Formulationisaproceduretoobtainsystemequationsbyuseoftherelationsof connections of elements and physical laws governing them. System variables (State variables) Unknown time functions within the system equation determining the behaviour of the system are called system variables or state variables. Itsisimportanttoselectminimaldimensionformathematicalequations. Minimal dimensional equation is capable of determining all behaviour of the system. Modelling Formulation Solution CHAPTER 1 Introduction 6 Solution The calculation of unknown system variables is called solution. 1.4TYPES OF FORMULATION IN MATHEMATICAL MODELLING There are three types of formulation in mathematical modelling: (1)Algebraic equations. (2)Differential equations. (3)Difference equations. Algebraic equation Anequationincludingonlythesystemvariablesthemselvesiscalledanalgebraic equation in which system variables can be calculated only byalgebraic operations (addition, subtraction, multiplication and division). Differential equation Anequationincludingsystemvariablesthemselvesandtheirderivativesiscalleda differentialequationinwhichsystemvariablescanbecalculatedbysolvingdifferential equations. Difference equation An equation including system variables themselves and their shifted states indiscrete instancesiscalledadifferenceequationinwhichsystemvariablescanbecalculatedby recursively solving difference equations. In this course, we mainly deal with these kind of equations. Example 1.1 Someexamplesofalgebraic,differentialanddifferenceequatiosaregiveninthe following: (1)Algebraic equation: A x(t) = B Where x(t) is the system variable, (A, B) . (2)Differential equation CHAPTER 1 Introduction 7 ) ( ) ( ) (0t Ax t x t xdtd Where x(t) is the system variable, A . (3)Difference equation x(k+1)=Ax(k),k=0, 1, 2, .... Wherex(k) is the system variable at discrete instant k, x(k+1) is the system variable at discrete instant (k+1). A . In all these equations A is a scalar parameter. Wegaveexamplesoflinearequationsabove.Inreallife,systemsandtheequations describingthemareingeneralnon-linearequations.However,welinearizethemduring analysis and design phases around the operating points. 1.5DYNAMICAL SYSTEM We can give a coarse definition for a dynamical system. It is not a complete definition, but in general sufficient for most applications. A system having elements storing some sort of energy is called a dynamical system. Dynamicalsystemsareformulatedbyeitherdifferentialordifferenceequations depending on the nature of system variables. We can group the system equations into two types (1)Linear equations, (2)Non-linear equations. Linear Equation and Linear System Ifeachtermofannthorderequationincludesonlythefirstpowerofisolated(not multiplied or divided by any other system variable) dependent variable and its derivatives (or shifted ones), it is called a linear equation. Asystemdefinedbythelinearequationsiscalledalinearsystem(amoreformal definition for a linear system will be given later in this chapter). Linear systems can be divided into two types: (1)Linear time-invariant systems, CHAPTER 1 Introduction 8 (2)Linear time-varying systems. Iftheparametersofsystemelementsarenotchangingwithtime,thesesystemsarecalled linear time-invariant systems. We call the systems having parameters changing with time as linear time-varying systems. Non-linear Equation and Non-linear System If an equation is not linear, it is called non-linear equation. A system defined by a non-linear equation is called a non-linear system. Example 1.2 Examples of mathematical models related to linear time-invariant, linear time-varying and non-linear systems are given in the following: (1)Linear time-invariant system: ) ( ) ( ) ( ) (0t bu t ax t x t xdtd where a and b are scalar constants composed of system element parameters,x(t) is system (or state) variable, u(t) is the input (external force) variable of the system. Thisequationisafirst-order(sincethederivativeofthesystemvariablehasbeen taken only once) linear time-invariant differential equation. It is time invariant, since a and b are constant coefficients. (2)Linear time-varying system: ) ( ) ( ) ( ) (0t x t a t x t xdtd This is a first-order linear time-varying equation. Its time varying property comes from the fact that the parameter a is also a function of t. (3)Non-linear system: ) ( ) ( ) (20t ax t x t xdtd This is not a linear system, since the variable at the right hand side is quadratic. CHAPTER 1 Introduction 9 1.6TYPES OF SIGNALS AND SYSTEMS Here, we use the term signal for mathematical time functions. Analog signal Ananalogsignalisasignalwhosetargetset(amplitude)mayassumecontinuous range of values, Target . Quantized signal Aquantizedsignalisasignalwhosetargetset(amplitude)mayassumeonlyfinite number of distinct values. The process of representing a varible by a set of distinct values is called quantization, andtheresultingdistinctvaluesarecalledquantizedvalues.Thequantizedvariablechanges onlybyasetofdistinctsteps.Asignalassumingquantizedvaluesinamplitudeiscalleda quantized signal. A continuous-time signal Acontinuous-timesignalisasignaldefinedoveracontinuousrangeoftime.More formally, Domain . Theamplitudemayassumeacontinuousrangeofvaluesormayasumeonlyafinite number of distinct values. Depending on it, we can classify the continuous time signals into two groups: (1)Continuous time analog signal,(2)Continuous - time quantized signal. Continuous time analog signal A continuous time analog signal is a signal defined over a continuous range of time whose amplitude can assume a continuous range of values. More formally, Domain . and Target . Continuous - time quantized signal Acontinuoustimequantizedsignalisdefinedoveracontinuousrangeoftimebut whose amplitude is assuming quantized values. CHAPTER 1 Introduction 10 Fig.1.5(a)showsacontinuous-timeanalogsignal,andFig.1.5(b)showsacontinuous-time quantized signal. Figure 1.5.Types of signals. 001 010 111 110 101 100 011 000 (a) Continuous-time analog signal (b) Continuous-time quantized signal (c) Discrete (Sampled data) signal (d) Digital signal t t t t CHAPTER 1 Introduction 11 A Discrete-time signal A discrete-time signal is a signal defined only at discrete instants of time (that is, one in which the independent variable t is quantized). Discrete (Sampled data) signal In a discrete time signal, if the amplitude can assume a continouos range of values, then the signal is called a sampled data signal or shortly discrete signal. Asampleddatasignalcanbegeneratedbysamplingananalogsignalatdiscrete instants of time. It is an amplitude modulated pulse signal. Fig.1.5(c ) shows a sampled data signal. Digital signal A digital signal is a discrete time signal with quantized amplitude. Such a signal can be represented by a sequence of numbers, for example, in the form of binary numbers. (In practice, many digital signals are obtained by sampling analog signals and then quantizing them; it is the quantization that allows these analog signals to be read as finite binary words.) Fig.1.5(d)depictsadigitalsignal.Clearly,itisasignalquantizedbothinamplitude and in time. The use of the digital computer requires quantization of signals both in amplitude and in time. Here, we use the term quantization for only in amplitude and discretization for time. Linear System Lets take the system shown in Fig.1.6. Figure 1.6A system. If x1 givesy1 and x2 gives y2 then the system is linear if and only if, for every scalar and , the response to the input x1 + x2 gives y1 + y2 This is called principle of superposition. System xy CHAPTER 1 Introduction 12 As I said before, a linear system may be described by a linear differential equation or a linear difference equation. Alineartimeinvariantsystemisoneinwhichthecoefficicentsinthedifferential equation or difference equation do not vary with time, that is, one in which the properties of the system do not change with time. More formally, Asysteminwhichtherelationship(dynamicalbehaviour)betweentheinputandthe outputsignalsisunchangingwithrespecttotimeiscalledatimeinvariantsystem.For example,ifc(t)istheoutputvectorinresponsetotheinputvectorr(t)foracontinuous system, that is, c(t) = S[r(t)] then a time invariant system is characterized by the relationship c(t-) = S[r(t-)] for all andr(t). Inwords,ifc(t)istheresponsetor(t),thenthesystemsresponsetothesameinput signal applied seconds later is the same output delayed by seconds. Notation: Inthiscourse,scalarvaluedfunctionsarewrittenbysmallnormalletterslikex(t), whereas vector valued functions by small boldface letters like x(t), vectors by boldface small letters like v, and matrices by capital boldface letters like A. Continuous and Discrete Systems Asysteminwhichthecharacterizingsignals(e.g.inputandoutputsignals)areall continuousiscalledacontinuoussystem.Mostcontinuoussystemsaregovernedby differential equations. When the characterizing signals are all discrete type signals, the system is said to be adiscretesystem.Ingeneral,adiscretesystemischaracterizedbydifferenceequationsthat relatethevariousinputandoutputsignals.Asanexample,anydynamicalprocess implemented by a digital computer is discrete. Some systems posses both continuous and discrete signals. Such systems are referred to as hybrid systems. CHAPTER 1 Introduction 13 Sampling The process of sampling is a very basic concept in digital control and data processing theory. In a very idealistic sense, a sampler is a device that transforms a continuous signal into a discrete signal. Let r(t) be the continuous input signal to a sampler as shown in Fig.1.7(a). Theoutputofthissamplerwillbeasequenceofnumbers,spacedintime,which appear at sampling instances tk and have the values equal to the continuous input signal at the sampling instances. A simple model of the sampler is shown in Fig.1.7(b), where the switch is thoughtofasbeingopenforalltimeexceptatthesamplingindtancestk,whenitcloses instantaneouslytopasstheinputsignalr(t).Normally,thesamplingintervalsarespaced equidistant in time so that tk = kT where T is the sampling time (period). (a) (b) Figure 1.7Process of sampling. Example 1.3 Let the input to the sampler depicted in Fig.1.7(a) be given by r(t) = 2 + 3t + sin5t Then its output will be a sequence of numbers that appear at times tk and have values r(tk) = 2 + 3 tk + sin5 tk Sampler r(t)r(tk) tk r(t)r(tk) CHAPTER 1 Introduction 14 1.7DIFFERENCE EQUATIONS Weshallconcentrateoureffortstowardstudyingsystemsthatarecharacterizedby differenceequations.Atruediscretesystemisinherentlyrepresentablebydifference equations. Inordertoobtainafeelforwhatexactlyadifferenceequationis,letusconsiderthe fololowing simple example. Suppose that it is desired to perform the operation of integration numerically, that is, to evaluate td r t c0) ( ) ((1) by some iterative technique. This may be desirable if the function to be integrated, r(t), has no closed form integral, for example. A standard technique is first to approximate the function ra(t) as shown in Fig.1.8. The piecewise constant function ra(t) is given by ra(t) = r(nT) for nT t < nT+T Figure 1.8Numerical integration process. It is easy that Eqn.(1) is approximated by kTkmmT r T d r kT c010) ( ) ( ) ( (2) =Tr(0) + Tr(1) + ... + Tr(k-1) Infact,thisistheordinarydefinitionoftheintegrationprocessasTapproacheszero.This relationship requires theretention of all past values of r(mT) in order to determine the value of the integral t=kT. t ra(t) r(t) T 02T3T CHAPTER 1 Introduction 15 A much simpler and systematic method is obtained by first determining T kTkmmT r T d r T kT c00) ( ) ( ) ( = Tr(0) + Tr(1) + ... + Tr(k-1) + Tr(kT) and subtarcting Eqn.(2) from this expression, which gives c(kT+T) c(kT) = T r(kT)(3) Eqn.(3) is a first-order difference equation. Now,itisnecessaryonlytoretainthepreviousvalueoftheintegralc(kT)andthe present sampled value of r(t)[i.e., r(kT)] in order to determine the value of the integral at t = (k + 1)T. Incidentally, this represents substantial savings in storage space of a digital computer is used in evaluating Eqn.(1) and k is large. The dynamics of this first-order difference equation work as follows: 1.One firstnotes the initial condition 00 ) ( ) 0 (od r c 2.Next, one evalutaes Eqn.(3) first for k=0, then k=1, and so forth; that is, c(T) = Tr(0) + c(0) = Tr(0) = c(1) c(2T) = Tr(1) + c(1) c(2T) = Tr(1) + c(2) ............................... c(mT) = Tr(mT-T) + c(mT-T) At each state of the iteration, the new value of c(mT) is evaluated by adding its previous value to the new input r(mT-T) multiplied by T. WecaneasilydropthecomputationtimeTandwriteafirstorderdifference equation as c(k+1) + a c(k) = b r(k)(4) CHAPTER 1 Introduction 16 A more general nth order, linear difference equation has the form: c(n+k) + a1 c(nk-1) + .... + an c(k) = = b0 r(n+k) + b1 r(n+k-1) + ... + bn r(k)(5) In this case, one computes c(n+k) by retaining its previous values c(n+k-1), c(n+k-2),..., c(k) and the input sequence r(n+k-1), r(n+k-2),..., r(k) and then performs the multiplications and additions prescribed by Eqn.(5). It is seen that the input signal [Sequence of numbers ..., r(-1), r(0), r(1), ...] influences in a definite way the output sequence of numbers c(k). This has ita obvious parallel in continuous systems, where the input signal [e.g. r(t)=sint] influences the systems output in a similar way. The coefficients a1 , ..., an , b0 , ..., bn

determine the characteristic of this input output dynamic behaviour. How one selects these coefficients is one of the main topics of this course. Nonanticipative System A system in which the present output c does not depend on the values of the input r in future time is nonanticipative. Nonanticipative systems are characterized by the outputs being dependent completely on present and previous values of the input signal. Anticipative systems are physically nonrealizable, since the present output depends on the future values of the input. CHAPTER 2 Electrical System Components 17 CHAPTER 2 Electrical System Components Contents CHAPTER 2 ............................................................................................................................. 17 Electrical System Components ................................................................................................. 17 Contents ................................................................................................................................ 17 2.1Introduction .............................................................................................................. 18 2.2Mathematical Model of Ideal Linear Time-Invariant Two-Terminal components .. 18 2.2.1Linearity ............................................................................................................... 19 2.2.2Active component ................................................................................................ 19 2.3Basic System Components and Their Models .......................................................... 19 2.3.1Reference Directions ............................................................................................ 20 2.3.2Energy Dissipating Components .......................................................................... 21 2.3.3Power and Energy for Energy Dissipating Components ...................................... 22 2.3.5The Types of Energy Storing Components .......................................................... 24 2.3.6Ideal Sources (Generators) ................................................................................... 26 Input power of a source: ....................................................................................................... 26 2.4Electrical Components ............................................................................................. 29 2.4.1A General Electrical Two-Terminal Component and Associated reference Directions for Terminal Voltage and Current ...................................................................... 29 2.4.2Resistor, Linear Time-Invariant Electrical Energy Dissipating Component ....... 31 2.4.3Inductor (Coil), Linear Time-Invariant Component Storing Magnetic Energy ... 35 2.4.4Capacitor, Linear Time-Invariant Component Storing Electrical Energy ........... 37 2.4.5Ideal Sources ........................................................................................................ 39 CHAPTER 2 Electrical System Components 18 2.1Introduction In this chapter we give the mathematical models of the ideal linear time-invariant two-terminal(or2-terminal)components.First,wegivegeneraltwo-terminalcomponents,then wefocusontheelectricalideallineartime-invarianttwo-terminalcomponentssuchas resistor,capacitorandinductor.Wecandefineamulti-terminalcomponentsuchasathree-terminalcomponent,afour-terminalcomponentand,ingeneralan-terminalcomponent. However,withinthiscontent,wewillonlygivethetwo-terminalcomponentsindetail.The components that we deal with in this book are ideal linear time-invariant components. The properties of this type of components are as follows: (1)They are lumped components, i.e., the electrical behaviour produced at their terminalsislumpedatapoint.Inotherwords,theelectricalenergyentersorleavesthe component at the terminals. Unlike a lumped component, Distributed components arealso available.However,theydonthavedefiniteterminals.Electricalbehaviourisdistributed alongthecomponent.Thistypeofcomponentscanbefoundmainlyinmicrowavesystems suchaswaveguides,antennasetc.Distributedcomponentsarenotasubjectofthiscourse. We only concentrate on the lumped components in this book. (2)They cannot be decomposed further into other 2-terminal components. (3)They are idealized components that cannot be found in real life. (4)The models, that is, mathematical equations (or we will call these equations as terminalequationsinsubsequentchapters)thatcharacterizethebehaviourofthese components are independent of how the component is interconnected with other components toformanelectricalsystem.Thisisactuallytheaxiomofthesystemtheory.Thisproperty impliesthatthevariouscomponentscanberemovedeitherliterallyorconceptuallyfrom the remaining part of the network and can be studied in isolation to establish the mathematical models of their characteristics. (5)Theyare linear, time-invariant components, therefore, theycan be modeled as linear algebraic and/or linear differential equations. (6)Thesecomponentswillbepassivethatmeansthattheydonotamplifyany energy. 2.2Mathematical Model of Ideal Linear Time-Invariant Two-Terminal components Themathematicalmodelof2-terminalcomponentsconstitutingasystemisthe building block of that system in design and analysis phases. Asitismentionedinpreviouschapter,themathematicalmodelof2-terminal component can be created by using two methods called as the experimental method andthe mathematical method that uses physics laws. CHAPTER 2 Electrical System Components 19 Herewedealwith,asismentionedbefore,onlylinearandpassivecomponents.The phrase that acomponent is linear means that theinput-output relation of this component can be represented by a linear equation. The definition of the linearity" is given in the following paragraph. 2.2.1Linearity Lets assume that xii = 1,2,..., ninputs yj j = 1,2,..., moutputs If the following expressions are true then the system is linear: xi +x2 + ... + xnyj +y2 + ... +ym (2.1) a x a y(2.2) Passivitymeansthatthesystemdoesnotamplifyanyenergy.Themathematical definition of the passivity is given below: For a passive component, lets assume that x(t) is the flow variable y(t) is the across variable and p(t) = x(t) y(t) is the instantaneous power. If t p() d+ W(t0) 0(2.3) t0 then this component is called apassive component. WhereW(t0) is the limited energy stored initially inside the component. 2.2.2Active component An active component is an element that is not passive. That is, if it satisfies t p() d+ W(t0) 0)whenevertheelectricalpotentialof terminalAattimetisgreaterthantheelectricalpotentialofterminalBattimet,withboth potentials being measured with respect to the same reference.If we callthese two potentials vA and vB, respectively, then v(t) =vA - vB (2.29) GiventhereferencedirectionforthecurrentshowninFig.2.4,byconvention, theterminalcurrentiispositiveattimet(thatisi(t)>0)wheneverattimetanetflowof positive charges enters the branch at terminal A and leaves it at terminal B. Thereferencedirectionschosenasdiscussedabovearecalledassociated reference directions. The reference direction for the terminal voltage and the reference direction for theterminalcurrentaresaidbeassociatedifapositivecurrententersthecomponentatthe A B + - i(t) v(t)Figure 2.4 A general two-terminal electrical component and reference directions of terminal variables.CHAPTER 2 Electrical System Components 30 terminal marked with a plus (+) sign and leaves the component at the terminal marked with a minus(-)sign.Whentheassociatedreferencedirectionsareused,thecomponentissaidto consumepowerattimetwhenevertheactualterminalvoltageandactualterminalcurrent productyieldsapositivequantity,thatisp(t)=v(t)i(t)>0.Ifontheotherhand,p(t)0 there exist a >0, such that for t - t0 < , there is x(t) x( t0) < , the signal x(t) is said to be continuous at t = t0 . Thatis,incontinuoussignals,ifasmallvariationoccursint,thenasmallvariation occurs in the signal x(t). t x(t) t0t x(t0) x(t) ekil 3.2. Continuous signal CHAPTER 3 Basic System Variables and Signals 49 Piecewise Continuous Signal Ifinasignaltherearefinitenumberofdiscontinuitiesandthesignalpiecesare continuous in between the discontinuities, then this signal is said to bepiecewise continuous signal. ApiecewisecontinuoussignalisshowninFig.3.3.Whereattimest1,t2,andt3 discontinuities can be seen. Discrete Signal If a signal has a certain value in some time instances and zero in the intervals between these instanes, then it is called a discrete time signal. More formally, for a discrete time signal H = { }and T = { } where H : Target domain T : Definition domain (time domain). A discrete time signal is shown in Fig. 3.4. t x(t) t1 Figure 3.3. Piecewise continuous signal t2t3 k x(k) -1 Figure 3.4. Discrete signal 012 CHAPTER 3 Basic System Variables and Signals 50 3.3Graphical Differentiation and Integration Inelectricalsystems,differentiation(derivative)andintegrationoperationscanbe encountered frequently. In this section, graphical differentiation and integration operations in time domain will be given. Differentiation or Derivative Derivative gives a measure for the variation (increase or decrease) in a variable. If we look at the Fig. 3.5, we see that if we proceed in t from time t , the value of the signalincreasesfromx(t)tox(t+t),i.e.,achangeinthevalueofthesignalhappens.Ifwe look at the rate of this change, x(t) = tan (3.5) t If t approaches zero, the hypotenuseAB becomes tangent to the signal at the timet. In this case, x(t) lim= m (Tangents slope)(3.6) t0t This limit is defined as the derivative and can be written as follows: x(t+ t) x(t) = lim(3.7) t0t d o x(t)=x(t) dt t x(t) t+t Figure 3.5. Derivative operation t x(t) x(t+t) t x(t) A B Tangent CHAPTER 3 Basic System Variables and Signals 51 Integral The integral of a signal in a certain time means that it is thearea collected under the signal up to that time t. Lets take the signal in Fig.3.6. Lets look at the area under the signal from t = 0to t. Theeasiestwaytodeterminethisareaistodividetheregionunderthesignalintosmall squares or into stripes having a width of and then count the small areas and summing them. Since a stripe is a rectangle, its area simply is the multiplication of its width by length, i.e., Ai = x(ti ) t(3.8) If we take the summation of the areas of all stripes, n n Atotal = Ai= x(ti ) t(3.9) tn=t i=0 i=0 nt Atotal=lim x(ti ) t= x(t) dt(3.10) tn=tt 0 i=00 In general the integral at time t: t y(t)= x() d (3.11) t0 In order to take into account the old history of the signal, it is taken as t0 = - : t y(t)= x() d (3.12) - t x(t) tn = t x(t1) Figure 3.6. Integral operation t1 t2x(t2) x(tn) t ti AiCHAPTER 3 Basic System Variables and Signals 52 3.4Basic Signals Intheanalysis,synthesisandthedesignphaseofthesystems,threetypesofsignals are used frequently. These are: (i)Unit Step Function (Signal) (ii)Unit Ramp Function (Signal) (iii)Unit Impulse (Dirac) Function (Signal) 3.4.1Unit Step Function (Signal), u(t) The Unit Step Function (Signal), u(t), is shown in Fig. 3.7 and it can be defined as: (3.13) If you notice that for t 0, the magnitude of the unit step function is constant and takes the value of 1. For this reason this is called a unit step function. Shifted Step Function, u(t-t0) Thefunctionobtainedbyshiftingtheunitstepfunctiontotheleftortotherightin timeaxisiscalledshiftedstepfunctionanditisdefinedanalyticallyasinthefollowing (Fig.3.8): (3.14) t u(t) 1 0 Figure 3.7.Unit step function (Signal). u(t) = 0fort < 0 1 fort 0 u(t-t0) = 0 fort < t0

1 fort t0 CHAPTER 3 Basic System Variables and Signals 53 Amplified Step Function, X u(t) Thestepfunctionwhosemagnitudeisamplifiedtoacertainleveliscalledthe amplified step function and shown as in the following (Fig.3.9). (3.15) Both Shifted and Amplified Step Function, X u(t-t0) Thestepfunctionwhosemagnitudeisamplifiedtoacertainlevelandalsoshiftedin time axis is called the both amplified and shifted step function and shown as in the following (Fig.3.10). (3.16) X u(t) = 0 fort < 0 X fort 0Xu(t-t0) = 0fort < t0 X fort t0

t u(t- t0 ) 1 0 Figure 3.8.Shifted Unit step function (Signal). t0 t Xu(t) X 0 Figure 3.9. Amplified step function (Signal). CHAPTER 3 Basic System Variables and Signals 54 3.4.2Unit Ramp Function (Signal), r(t) =t u(t) The Unit Ramp Function written as r(t) = t u(t) is shown in Fig. 3.11 and is defined as in the following. (3.17) r(t) =t u(t) = 0 fort < 0 tfor t 0 t r(t)=tu(t) 1 0 Figure 3.11.The Unit Ramp Function (Signal). 1 t Xu(t- t0 ) X 0 Figure 3.10.Both amplified and shifted step function (Signal). t0 CHAPTER 3 Basic System Variables and Signals 55 Shifted Ramp Function, r(t-t0) =(t-t0) u(t-t0) Thefunctionobtainedbyshiftingtheunitrampfunctiontotheleftortotherightin timeaxisiscalledshiftedrampfunctionanditisdefinedanalyticallyasinthefollowing (Fig.3.12): (3.18) Amplified and Shifted Ramp Function, Xr(t-t0) =X (t-t0) u(t-t0) The ramp function whose magnitude is amplified to a certain level and also shifted in time axis is called the both amplified and shifted step function and shown as in the following (Fig.3.13). Notice that the amplified ramp function has a slope greater than unity. (3.19) r(t-t0) =(t-t0) u(t-t0) = 0 fort < t0

t-t0 for t t0 t r(t-t0) =(t-t0) u(t-t0) 1 0 Figure 3.12.Shifted Unit Ramp Function (Signal). t0 Xr(t-t0) =X(t-t0) u(t-t0) = 0fort < t0

X(t-t0 )for t t0 iin t Xr(t-t0) =X(t-t0) u(t-t0) X 0 Figure 3.13.Amplified and shifted ramp function (Signal). t0 t0+1 t0+1 CHAPTER 3 Basic System Variables and Signals 56 3.4.3Unit Impulse (Dirac) Function (Signal), (t) The UnitImpulse (Dirac) Function (Signal),(t), is shown in Fig.3.14 and is defined as in the following: (3.20) and (t) dt = 1(3.21) - Theunitimpulsefunctioncannotbephysicallyrealizable,butitsatisfiesvery important benefits for the evaluation of the system behaviour and analysis. The unit impulse function can be obtained mathematically by the use of the unit pulse function defined below. Unit Pulse Function, p(t) The Unit Pulse Function, p(t), is shown in Fig.3.15 and defined as follows: (3.22) t (t) 1 0 Figure 3.14.The Unit Impulse (Dirac) Function (Signal). (t) = 0for t 0 fort= 0 p(t) = 0for t - iin, lim e - t A e - t 0 t ohalde, f(t) stel mertebedendir ve- yaknsama absisidir. (b)g(t) = ( e t )3 3 2 e - t f(t) = e - t ( e t )3 = e - t( e t )3 = e( t - t ) = e t ( t - t )

2 lim e t ( t - t ) (snrsz bir ekilde) t Dolaysyla,g(t) stel mertebeden deildir. CHAPTER 7 Laplace Transform 192 7.3BAST FONKSYONLARIN LAPLACE DNM Bu alt blmde sistem teoride sklkla karmza kan stel fonksiyon, sins, cosins, basamakfonksiyonu,yokufonksiyonuvedrtfonksiyonugibitemelfonksiyonlarn Laplace dnmlerini greceiz. stel Fonksiyonea t E.7.1 de verilen Laplace dnm tanmn kullanarak, { e a t} =e a t e- s tdt 0 =e ( a - s) tdt 0 1 = e ( a - s) t a s 0 1 = ( e ( a - s) - e 0 ) a s s=+jwolduunubiliyoruz.Ohalde,a0)whenevertheelectricalpotentialof terminalAattimetisgreaterthantheelectricalpotentialofterminalBattimet,withboth potentials being measured with respect to the same reference.If we callthese two potentials vA and vB, respectively, then v(t) =vA - vB (8.20) In s-domain, we can write it as V(s) = VA - VB (8.21) GiventhereferencedirectionforthecurrentshowninFig.8.3,byconvention, theterminalcurrentiispositiveattimet(thatisi(t)>0)wheneverattimetanetflowof positive charges enters the branch at terminal A and leaves it at terminal B.Thereferencedirectionschosenasdiscussedabovearecalledassociated reference directions.A B + - I(s) V(s)Figure 8.3 A general two-terminal electrical component and reference directions of terminal variables in s-domain.Z(s) Or Y(s) CHAPTER 8 Complex Domain Analysiss-domain Analysis 229 The reference direction for the terminal voltage and the reference direction for theterminalcurrentaresaidbeassociatedifapositivecurrententersthecomponentatthe terminal marked with a plus (+) sign and leaves the component at the terminal marked with a minus (-) sign. The relation between the variables V(s) and I(s) are algebraic and can be written as V(s) = Z(s) I(s)(8.22) or I(s) = Y(s) V(s)(8.23) where Z(s): impedance, Ohm () Y(s): Admittance, Mho () I(s): Current variable, Ampere (A) V(s): Voltage variable, Volt (V) Notice that the OhmsLaw is still valid. However, the terms in front of current and voltage, namely,Z(s)andY(s)arenotconstantbutdependontheindependentvariables.Theseare Ohm () and Mho () respectively. We call Z(s) as impedance and Y(s) as admittance. The currentandvoltagevariablesarealsofunctionofsbuttheyareinampereandvolt respectively again. 8.3.2Resistor, Linear Time-Invariant Electrical Energy Dissipating Component The component dissipating electrical energy is called resistor. The flow variable of the resistor is called current i(t) and across variable is called voltage v(t). The terminal equation of the resistor can be written as: v(t) = R i(t)(8.24)or i(t) = G v(t)(8.25) If we take the Laplace transform of Eqn.8.22 and 8.23, V(s) = R I(s)(8.26)or I(s) = G V(s)(8.27) Where CHAPTER 8 Complex Domain Analysiss-domain Analysis 230 R: impedance, Ohm () G: admittance, Mho () I(s): Current variable, Amper (A) V(s): Voltage variable, Volt (V) Theschematicrepresentationofaresistor,itsreferencedirectionsandinput-output characteristic are shown in Fig. 8.4. Two Special Types of Resistors: There are two special types of resistors: (1)Open circuit, (2)Short circuit. Open Circuit: Atwo-terminalcomponentiscalledanopencircuitifitsterminal currentisidenticallyequaltozero,whateveritsterminalvoltagemaybe.Thesymbolofan open circuit, its mathematical model and v i characteristic are shown in Fig. 8.5. A B + - V(s) = R I(s) I(s) R Figure 8.4. Electrical resistor, its reference directions and theinput-output characteristic. I(s) V(s) Slope = R =1/G 0 A B + - V(s)I(s)=0 R= Figure 8.5. Open circuit and theinput-output characteristic. I(s) V(s) Slope = 0 CHAPTER 8 Complex Domain Analysiss-domain Analysis 231 Theterminalcharacteristicofanopencircuitisastraightlinepassingthroughthe origin of the V Iplane with an infinite slope; that is R= or, equivalently, G = 0. Short Circuit: Atwo-terminalcomponentiscalledashortcircuitifitsterminal voltageisidenticallyequaltozero,whateveritsterminalcurrentmaybe.Thesymbolofa short circuit, its mathematical model and v i characteristic are shown in Fig. 8.6. Theterminalcharacteristicofanopencircuitisastraightlinepassingthroughthe origin of the V I plane with a zero slope; that is R= 0 or, equivalently, G = . Notice that the terminal equation of a resistive element is again an algebraic equation. It is also similar to the expression in time domain. The resistance in s-domain is constant impedance. 8.3.3Inductor (Coil), Linear Time-Invariant Component Storing Magnetic Energy Alineartime-invarianttwo-terminalcomponentstoringmagneticenergyiscalledan inductor (or coil). The flow variable of the inductor is current, iL(t), and the across variable is voltagevL(t). The terminal equation of the inductor: d vL(t)= LiL(t)(2.28) dt 0 =L iL(t)(2.29) A B + - v(t) =0 I(s) R=0 Figure 8.6. Short circuit and theinput-output characteristic. I(s) V(s) Slope = 0 0 CHAPTER 8 Complex Domain Analysiss-domain Analysis 232 If we take the Laplace transform of Eqn.2.28 assuming the initial condition is zero: )} ( { )} ( { t idtdL L t v LL L By using the real derivative theorem and linearity of the Laplace transform, we can obtain ) ( ) ( s I Ls s VL L(8.30) sL s ZL) ( (8.31) where L: Inductance, Henry ZL(s)= sL: impedance, Ohm () IL (s): Current variable, Amper (A) VL(s): Voltage variable, Volt (V) Aschematicrepresentationofalineartime-invarianttwo-terminalinductorandits reference directions are shown in Fig.8.7. Figure 8.7Inductor in s-domain. A B ZL(s)=sLIL(s) + - VL(s) CHAPTER 8 Complex Domain Analysiss-domain Analysis 233 8.3.4Capacitor, Linear Time-Invariant Component Storing Electrical Energy Alineartime-invarianttwo-terminalcomponentstoringelectricalenergyis called capacitor. The flow variable is the current iC(t), across variable is the voltage vC(t). The terminal equation of the capacitor: d iC(t)= CvC(t)(8.32) dt 0 =C vC(t)(8.33) If we take the Laplace transform of Eqn.8.32 assuming the initial condition is zero: )} ( { )} ( { t vdtdC L t i LC C By using the real derivative theorem and linearity of the Laplace transform, we can obtain ) ( ) ( s V Cs s IC C(8.34) ) (1) ( s IsCs VC C(8.35) sCs ZC1) ( (8.36) where C: Capacitance, Farad ZC(s)= 1/sC: impedance, Ohm () IC (s): Current variable, Amper (A) VC(s): Voltage variable, Volt (V) Aschematicrepresentationofalineartime-invarianttwo-terminalcapacitorandits reference directions are shown in Fig.8.8.CHAPTER 8 Complex Domain Analysiss-domain Analysis 234 Figure 8.8.Capacitor in s-domain. 8.3.5Ideal Sources The Ideal Independent Voltage Source A two-terminal component is called an independent voltage source or simply a voltage source (driver), if it maintains a specified voltageVs(s)across its terminals whatever its terminal current I(s) may be. Thus, the complete description of the voltage source requires the specification of the function Vs(s). The symbol for the ideal independent voltage source is shown in Fig. 8.9. Figure 8.9.The symbol of an ideal independent voltage source The Ideal Independent Current Source A two-terminal component is called an independent current source or simply a current source (driver), if it maintains a specified currentIs(s) through its terminals whatever its terminal voltage V(s) may be. Thus, the complete description of the current source requires the specification of the function Is(s). The symbol for the ideal independent current source is shown in Fig. 8.10.A B Vs(s) + - I(s) A B + - IC(s) VC(s) C ZC(s)=1/sC CHAPTER 8 Complex Domain Analysiss-domain Analysis 235 Figure 8.10.The symbol of an ideal independent current source 8.4Fundamental Network Theorems Thetime-domainistheonlyphysicaldomain.Thes-domainispurelyconceptual.A transformation from a physical domain to a conceptual domain may sometimes bring a better understanding of formulation or easier analysis procedure to a physical problem. Some of the concepts and definitions can be created in this unphysical domain such as frequency response, transferfunction,impedance,etc.Inthissection,wegiveterminalequivalence,series connection of two-terminal components and parallel connection of two-terminal components, -Yconnection,superposition,TheveninsandNortonstheoremsandmaximumpower transfer and impedance matching in s-domain. 8.4.1Terminal Equivalence We say that n-terminal networks are terminal equivalent if their mathematical models are identical, that is, their terminal equations corresponding to the same terminals. It should be understood that this equivalence is defined only at the terminals. According to this definition, two 2-port networks are said to be terminal equivalent if their characterization in terms of port currents and port voltages are identical. Supposewearegivenann-terminalnetworkN0withanarbitrarynumberof componentsconnectedinanarbitrarymanner.Ingeneral,thisnetworkcanbereplacedby infinitelymanydifferentnetworksNisuchthateachNiisterminalequivalenttotheoriginal network N0 at its terminals. We will refer to the process of obtaining an equivalent network having less number of components than the original network as a reduction process. A B Is(s) V + - CHAPTER 8 Complex Domain Analysiss-domain Analysis 236 8.4.1.1 Series Connected 2-Terminal Components Assumethatthecomponentshavetheterminalequationsins-domainasfollows (Fig.8.11): Vi(s) = Zi(s) Ii(s)(8.37) where Zi(s): impedance of the ith component, Ohm () Ii (s): Current variable through the ith component, Amper (A) Vi(s): Voltage variable across the ith component, Volt (V) We assume that the initial conditions of the components are zero. That is, the energy storing elements are initially empty. Figure 8.11A two-terminal component. The impedance Zi(s) is the open circuit impedance of the element. TheseriesconnectionofnimpedancesbetweentwopointsAandBareshownin Fig.8.12. Figure 8.12Series connection of impedances + - A B Zi(s)Vi(s) Ii(s) + - A B Z1(s) Vi(s) Ii(s) Z2(s)Zn(s) ++ + - - - V1(s) V2(s) Vn(s) + - A B Zi(s)Vi(s) Ii(s) CHAPTER 8 Complex Domain Analysiss-domain Analysis 237 From the KVL, we have ) ( ) ( ) ( ) (2 1s V s V s V s Vn i(8.38) ) ( ) ( ) ( ) (2 1s I s I s I s In i(8.39) We can write Eqn.8.38 as follows: ) ( ) ( ) ( ) ( ) ( ) ( ) (2 2 1 1s I s Z s I s Z s I s Z s Vn n i By using Eqn.3.39 that all currents are equal to each other: ) ( )] ( ) ( ) ( [ ) (2 1s I s Z s Z s Z s Vi n i Since, ) ( ) ( ) ( s I s Z s Vi i i we can see that the equivalent impedance for series impedances is ) ( ) ( ) ( ) (2 1s Z s Z s Z s Zn i(8.40) 8.4.1.2 Parallel Connected 2-Terminal Components Assumethatthecomponentshavetheterminalequationsins-domainasfollows (Fig.8.13): Ii(s) = Yi(s) Vi(s)(8.41) where Yi(s):admittance of the ith component, Mho Ii (s): Current variable through the ith component, Amper (A) Vi(s): Voltage variable across the ith component, Volt (V) We assume that the initial conditions of the components are zero. That is, the energy storing elements are initially empty. CHAPTER 8 Complex Domain Analysiss-domain Analysis 238 Figure 8.13A two-terminal component. The admittance Yi(s) is the short circuit admittance of the element. TheparallelconnectionofnadmittancesbetweentwopointsAandBareshownin Fig.8.14. Figure 8.14Parallel connection of admittances From the KCL, we have ) ( ) ( ) ( ) (2 1s I s I s I s In i(8.42) ) ( ) ( ) ( ) (2 1s V s V s V s Vn i(8.43) We can write Eqn.8.42 as follows: ) ( ) ( ) ( ) ( ) ( ) ( ) (2 2 1 1s V s Y s V s Y s V s Y s In n i By using Eqn.3.43 that all voltages are equal to each other: ) ( )] ( ) ( ) ( [ ) (2 1s V s Y s Y s Y s Ii n i + - A B Yi(s)Vi(s) Ii(s) + - A B Yi(s)Vi(s) Ii(s) + - A B Vi(s) Ii(s) Y1(s)Y2(s) Yn(s) CHAPTER 8 Complex Domain Analysiss-domain Analysis 239 Since, ) ( ) ( ) ( s V s Y s Ii i i we can see that the equivalent admittance for parallel impedances is ) ( ) ( ) ( ) (2 1s Y s Y s Y s Yn i(8.44) Since, ) (1) (s Zs Yii ) (1) (1) (1) (12 1s Z s Z s Z s Zn i (8.45) 8.4.1.3 -Y Connected 2-Terminal Components Since the terminal equations and the other relations in s-domain are algebraic, the rules valid for resistive network is also valid for networks defined in s-domain. Therefore, the -Y Connections and all the other rules are the same in s-domain. Yand Y transformations are done by usingFig.8.15. Figure 8.15 Yand Y transformations in s-domain A B C ZA ZC ZB ZZZY ZX CHAPTER 8 Complex Domain Analysiss-domain Analysis 240 We can make the Ytransformation as follows: Z Y XZ YAZ Z ZZ ZZ (8.46a) Z Y XZ XBZ Z ZZ ZZ (8.46b) Z Y XY XCZ Z ZZ ZZ (8.46c) We can make the Y transformation as follows: AC A C B B AXZZ Z Z Z Z ZZ (8.47a) BC A C B B AYZZ Z Z Z Z ZZ (8.47b) CC A C B B AZZZ Z Z Z Z ZZ (8.47c) 8.4.1.4 Superposition Principle The output of a linear system with two or more inputs is the sum of responses for each individual input acting alone. Fortheelectricalsystems,wecanexpressthesuperpositionprincipleas follows: Theresponseofalinearnetworkwhentwoormoresourcesacting simultaneouslyisthesumoftheresponsesforeachindividualsourceactingalonewiththe remaining sources being dead. Itisimportanttonotethatthesuperpositionprinciplehasbeenconsideredonlyfor linearsystemshavingnoinitialenergy,thatiswhentheinitialstatevector,) 0 ( x ,iszero. However, the superposition principle can be extended to cover linear systems when0 ) 0 ( x , by just considering the initial state vector) 0 ( xas an excitation vector. CHAPTER 8 Complex Domain Analysiss-domain Analysis 241 8.4.1.5 Thevenins and Nortons theorems Ifweareinterestedintheresponseofatwo-terminal(one-port)networkatits terminals, then it will be shown that the replacement of this two-terminal network by a simple equivalentnetworkwillreducetheamountofcomputations.TheveninsandNortons theorems are useful in this respect. 8.4.1.5.1Definition: (Thevenins equivalent) Consider a one-port linear network N, that contains any number of independent drivers internally. N can be replaced by an independent voltage source VT connected in series withanimpedanceZTasshowninFig.8.16a.Thisequivalentrepresentationiscalledthe Thevenins equivalent of N and VT and ZT are called the Thevenins equivalent voltage and impedance respectively. 8.4.1.5.1Definition: (Nortons equivalent) Consider a one-port linear network N, that contains any number of independent drivers internally. N can be replaced by an independent current source INconnected in parallel withanimpedanceZNasshowninFig.8.16b.Thisequivalentrepresentationiscalledthe NortonsequivalentofNandINandZNarecalledtheNortonsequivalentcurrentand impedance respectively. Fromthetwodefinitionsabove,itfollowsthattheTheveninsequivalentnetworkis the dual of the Nortons equivalent network or vice versa. (a)Thevenins equivalent network (b)Nortons equivalent network Figure 8.16(a)Thevenins equivalent network; (b) Nortons equivalent network + - V0(s) I0(s) N + - V0(s) I0(s) ZT(s) +-VT(s) + - V0(s) I0(s) N + - V0(s) I0(s) ZT(s) IN(s) ZN CHAPTER 8 Complex Domain Analysiss-domain Analysis 242 AssumethatforagivennetworkN,bothTheveninsandNorton2sequivalentsexist. Then ZT, ZN , VT , IN are interrelated: N TZ Z (8.48) N N TI Z V (8.49) TTNZVI (8.50) ForanalternatemethodofobtainingTheveninsandNortonsequivalentnetworks, notice that if all of the internal sources of N are dead, then VT and IN will identically be zero. Hence,incomputingeitheroneoftheequivalentnetworks,twostepsincalculationare required. ForTheveninsequivalentnetwork,thefirststepistoobtaintheThevenin impedance: 000TVTIVZ (8.51) i.e., Thevenins equivalent impedance is the impedance seen at the terminals when all of the internal sources are dead. The second step is to obtain the Thevenins voltage: 000ITV V (8.52) i.e.,Theveninsequivalentvoltageisequaltothepotentialdifferenceacrosstheterminals when the external current source is dead which means that the terminals left open-circuited. Similarly,forNortonsequivalentnetwork,thefirststepistoobtaintheNortons impedance: 000NINIVZ (8.53) i.e.,Nortonsequivalentimpedanceistheimpedanceseenattheterminalswhenallofthe internal sources are dead. CHAPTER 8 Complex Domain Analysiss-domain Analysis 243 The second step is to obtain the Nortons current: 000VNI I (8.54) i.e., Nortons equivalent current is equal to the current passing through the terminals when the external current source is dead which means that the terminals left short-circuited. 8.5Network Formulations In this section, we give two network formulations, namely mesh formulation and node formulation.Theseformulationscangiveoverallsolutionsoftheelementsconstitutingthe network instead of only single element variables. 8.5.1Mesh Formulation Inmeshformulation,wecanusethegraphshowninFig.17.Thegraphconsistsofa set of meshes {m1, m2, , m(n+1)m}. If the mesh mi contains at least one new edge which does not exist in the subset of meshes {m1, m2, , mi-1}, then all n meshes are called independent meshes. The orientation of the currents is chosen as CW direction. The main idea used in mesh formulation is to assign a mesh current Ij j=1, , (n+1)m toeachmeshinthesetofindependentmeshesandtodeterminethesemeshcurrents.Since these currents are independent, their linear combinations will yield all the edge currents of the network. We can also determine all the voltages of the elements. Figure 17Independent mashes Let N be a network containing 2-terminal R, L, and C components. Lets assume that to each current source in the network N, a single mesh is assigned such that no other current sourcesexitinthismesh.TheorientationsofallmeshcurrentsarechosenasCWdirection. Then we obtain the mesh equation as follows by using KVL for each mesh: ) ( ) ( ) ( ) ( s F s B s I s Zm(8.55) I1I2 I3 Im+1 Im+2Im+3 Ii-1Ii+1 Inm+1 Ii Im I2m I(n+1)m Inm+2Inm+3 CHAPTER 8 Complex Domain Analysiss-domain Analysis 244 where) (s Z isasymmetricnxnimpedancecoefficientmatrix,) (s Imisnx1meshcurrents vector,) (s Bis the nxm source coefficient matrix, and) (s F is mx1 sources vector. Note that in the mesh formulation, it is not necessary to choose the set of independent meshesthesameasthesetoffundamentalcircuitsingraphformulation.Anysetofmeshes will be sufficientfor our purpose as long as they are independent.That is, the set of meshes selected is not unique. In mesh formulation, it is not also necessary to draw the network graph of the network under consideration and select a formulation tree. Eqn.8.60 can be obtained very easily by inspection as follows: Thecoefficientmatrix) (s Z appearingatthelefthandsideofEqn.8.55issimpleto write.Itisobtainedbyjustconsideringthepropertiesofthemeshimpedancematrix.To obtaintheentriesinthesourcecoefficientmatrixattheright,weconsidertheedges corresponding to the sources Vis and Ijs. 8.5.2Node Formulation Node formulation is analogous to the mesh formulation. Many of the steps involved in theestablishmentofnodeformulationfollowtheproceduressimilartothoseinconnection with the mesh formulation. In node formulation, the unknown variables are node voltages. The node voltages may be thought of as a set of voltages incident on vertices that are identified as nodes with respect to a reference node (ground). Followingtheaboveproperties,weseethatoncethenodevoltagesareobtained,the solution of the network is completed. The problem then reduces to the formulation of network equations in which only the node voltages are unknown variables. This is achieved by writing a set of incidence equations for each node and solving for the node voltages. This will be the basic idea that will be used while establishing the node formulation. LetGbethegraphofanelectricalnetworkNcontaining2-terminalR,L,C components and let N be a consistent network. It is known that if the network N has v nodes (vertices),thenithas(v-1)branchvoltages.Sinceeachbranchdefinesacut-setandeach branch voltage can be expressed as the sum of node voltages, then the number of independent node voltages is equal to (v-1). However, if in N there are nv voltage sources, then the number ofunknownbranchandnodevoltageswillbereducedbynv.Insuchacase,thenumberof independent node voltages can be associated with the voltage sources. As an illustration, consider the network given in Fig.8.18. First of all we determine the nodes, and we assign one of the nodes as the ground. Then we write KCL for each node. We obtainthefollowingnodeequationinmatrix-vectorform.Andthenwecansolvethis equation easily. ) ( ) ( ) ( ) ( s F s D s V s Yn(8.56) CHAPTER 8 Complex Domain Analysiss-domain Analysis 245 where) (s Y isasymmetricnxnadmittancecoefficientmatrix,) (s V nisnx1nodevoltages vector,) (s Dis the nxm source coefficient matrix, and) (s F is mx1 sources vector. Figure 8.18Independent nodes 8.5.3Transforming State Equations into s-domain Before doing the transformation of state equation into s-domain, lets make ageneral definitionthesocalledtransferfunctionincomplexdomain(s-domainorfrequency domain). Definition: Transfer Function The ration between the output function of the system to the input function in complex domain while the initial conditions (states) are zero is called the transfer function. Transfer function is defined for single-input and single-output (SISO) systems. (1)TransferFunctionofSingle-InputandSingle-Outputandsinglestatevariable Systems Thestateequationofasingle-inputsingle-output(SISO)withsinglestatevariable system is a scalar type equation as seen in the following: o x(t)= a x(t) + b v(t)(8. 57) y(t)= c x(t) + d v(t)(8. 58) While the initial state value is zero, ie.,x0 = 0, taking the laplace transform of Eqn.8.57 and Eqn.8.58, { x(t)}= { a x(t)} + { b v(t)} V1(s) +-I8

AB C D E Y5

Y2

Y3

Y7

Y6

Y4

VA

VB

VC

VD

CHAPTER 8 Complex Domain Analysiss-domain Analysis 246 sX(s)=a X(s) + b V(s)(8. 59) { y(t)}= { c x(t)} + { d v(t)} Y(s)= c X(s) + d V(s)(8. 60) Arranging Eqn.8.59: sX(s) a X(s) = b V(s) (s a) X(s) = b V(s) b X(s) =V(s)(8. 61) s a Putting Eqn.8.61 into Eqn.8.60, b Y(s) =( c+ d ) V(s)(8. 62) s a Eqn.8.62isgivinganalgebraicrelationbetweentheinputV(s)andtheoutputY(s). According to the definition on transfer function given above, the transfer function of a single-input single-output with single state sytem can be obtained by Eqn.8.62 as follows: Y(s)cb = G(s) = + d (8. 63) V(s)s a (2)TransferFunctionofaSingle-InputSingle-Output(SISO)withMultipleState Variable System The state equationand response equation for a single-input single-outputwith multiple state variable system is a matrix-vector type equation given below: o x(t)= A x(t) + B v(t)(8. 64) y(t)= C x(t) + d v(t)(8. 65) where x(t) nx1, A nxn, B nx1, v(t) 1x1, y(t) 1x1, C 1xn and d 1x1. CHAPTER 8 Complex Domain Analysiss-domain Analysis 247 If we take the Laplace transform of Eqn.8.64 and Eqn.8.65, o {x(t)}={ A x(t)} + {B v(t)} {y(t)}= {C x(t)} + {d v(t)} sX(s)=A X(s) + B V(s)(8. 66) Y(s)= C X(s) + d V(s)(8. 67) Lets arrange Eqn.8.66, sX(s) AX(s) = B V(s) (s I A ) X(s) = B V(s)(8. 68) Since Eqn.8.68 is a matrix-vektr type equation, if we multiply both sides by (s I A ) -1 , X(s) = (s I A ) -1 B V(s)(8. 69) Putting Eqn.8.69 into Eqn.8.67, Y(s) = [ C (s I A ) -1 B+ d ] V(s) UsingthedefinitionofTransferfunction,wecanwritethetransferfunctionforsingle-input single-output system: Y(s) = G(s)= C (s I A ) -1 B+ d (8. 70) V(s) The transfer function in Eqn.8.70 is a complex function with the following structure: am sm + am -1 sm -1 + ... + a1 s + a0 G(s)= (8. 71) bn sn + bn -1 sn -1 + ... + b1 s + b0 CHAPTER 8 Complex Domain Analysiss-domain Analysis 248 Impulse Response Function As you know, we defined the transfer function of a system in complex domain as the ratio between the input function and the output function. That is, Y(s) am sm + am -1 sm -1 + ... + a1 s + a0 G(s) = =V(s)bn sn + bn -1 sn -1 + ... + b1 s + b0 This is a rational function. The system response: Y(s) = G(s) V(s)(8. 72) If the input function is unity, that is, V(s) = 1 then, Y(s) = G(s)(8. 73) Inverse Laplace transform of the input function is -1{ 1 } = (t) Therefore, the output in Eqn.8.73 is the Laplace transform of the impulse response. If we take inverse Laplace transform of Eqn.8.73: -1{ Y(s) } = -1{ G(s) } = g(t)(8. 74) The function g(t) in Eqn.8.74 is called impulse response function. The impulse response function in time-domain corresponds to the transfer function in complex-domain (a very important concept). g(t)G(s) Foranyinputfunction,thesystemresponseistheinverseLaplacetarnsformof Eqn.8.70. That is,y(t) = -1{ Y(s) } = -1{ G(s) V(s)} CHAPTER 8 Complex Domain Analysiss-domain Analysis 249 From the convolution integral (Borel) theorem, y(t) = g() v(t ) d(8. 75) = g(t) * v(t)(8. 76) where * represents the convolution integral. Therefore,ingeneral,theoutputofasystemistheconvolutionintgraloftheinput with the impulse response of the system. (3)SystemTransferMatrixofaMultiple-InputMultiple-Output(MIMO)and Multiple State System Thestateequationandresponseequationofamultiple-inputmultiple-output and multiple state variable system are vectr-matrix type equations shown in the following.

o x(t)= A x(t) + B v(t)(8. 77) y(t)= C x(t) + D v(t)(8. 78) where x(t) nx1, A nxn, B nxm, v(t) mx1, y(t) rx1, C rxn and D rxm. If we take the Laplace transform of Eqn.8.77 and Eqn..8.78, o {x(t)}={ A x(t)} + {B v(t)} {y(t)}= {C x(t)} + {D v(t)} sX(s)=A X(s) + B V(s) Y(s) = C X(s) +D V(s) Lets arrange them, sX(s) AX(s) = B V(s) (s I A ) X(s) = B V(s) CHAPTER 8 Complex Domain Analysiss-domain Analysis 250 Since the last equation is a vector-matrix equation, we can multiply both sides by (s I A ) -1 , X(s) = (s I A ) -1 B V(s) and Y(s) = [ C (s I A ) -1 B+ D ] V(s)(8. 79) SinceinEqn.8.79,Y(s)andV(s)arevectors,theirratioisnotdefined.But,wecan define the transfer matrix here: G(s) = [ C (s I A ) -1 B+ D ] (8. 80) The G(s) matrix in Eqn.8.80 is rxm. Poles and Zeros of the Transfer Function The roots of the numerator and the denominator of a general transfer function given in Eqn.8.15 are called the zeros and the poles of the system respectively. Now, first lets take a single-input single-output and single state variable system, and then take single-input single-output but multiple state variable system. Thedenominatorofasingle-inputsingle-outputandsinglestatevariablesystem transfer function is a first-degree polynomial. If we equate it to zero: s a = 0 hence s = a = - 1/(8. 81) isthesystempole.Therefore,thereisonlyonepoleofaasingle-inputsingle-outputand single state variable system. This is equal to the inverse of the time constant. The number of poles of a single-input single-output but multiple state variable system transferfunctiongiveninEqn.8.70isn,becauseifwewritetheEqn.8.70asfollows (assuming that d=0 for simplicity), G(s) = C (s I A ) -1 B C adj(s I A ) B = (8. 82) det(s I A ) Since the denominator polynomial of Eqn.8.82 is also the charactersitic polynomial of the matrix A and when we equate it to zero it is the characteristic equation of the matrix A, det(s I A ) = 0 CHAPTER 8 Complex Domain Analysiss-domain Analysis 251 sn + bn -1 sn -1 + ... + b1 s + b0 = 0 (s s1 ) (s s2 ) ... (s sn ) = 0 therootsobtainedsi(i=1,2,...,n),thatis,thepolesofthetransferfunctionarethe eigenvalues of the matrix A. In short, Poles = Eigenvalues(8. 83) Example 8.1 Determine the equivalent impedance seen from the Port AB for the circuit in Fig.8.19. Figure 8.19 Solution: First of all, we have to transform the network into s-domain. Since the network is linear, we can transform it component by component. So, 4 ) (s ZR s sL s ZL) ( Since these two impedances are in parallel seen from Port AB, sss Z s Z s Z s ZL R Eq AB44) ( // ) ( ) ( ) ( Example 2: ForthenetworkshowninFig.8.20a,findtheinputimpedanceseenbythecurrent source. A + - L=1 H i(t) R=4ohms B CHAPTER 8 Complex Domain Analysiss-domain Analysis 252 (a)(b) Figure 8.20 Solution: First, we have to transform the network into s-domain as seen in Fig.8.20b. Notice that there is neither a series nor parallel configuration seen at the input side. However, if we apply aYtransform to the part of the circuit composed of 2 1,L L RZ and Z Z , we obtain Z Y XZ and Z Z,from Eqn.8.47 as ) 3 2 (3 2 2 1 2 1212 2 1 1sss sss s s sZZ Z Z Z Z ZZLL R L L L RX ohms ) 3 / 2 (2 3 2222 2 1 1sss sZZ Z Z Z Z ZZLL R L L L RY ohms ) 3 2 (2 2 2 1 1s sZZ Z Z Z Z ZZRL R L L L RZ ohms Now, as seen, the input impedance can be written as Y Z C X inZ Z Z Z s Z // ) // ( ) (1 (HW: Obtain) (s Zin). Example 3: Determine the Thevenins equivalent of the circuit given in Fig.8.20 as seen by the impedance ZC2(s). C1=1 F R=1ohm C2=2 F L2=2 H L1=1 H I=1/s ZC1 =1/s ohm ZR=1ohm ZC2=1/2s ohm ZL2=2s ohm ZL1=sohm I=1/s ZX ZY ZZ CHAPTER 8 Complex Domain Analysiss-domain Analysis 253 Solution: We can use the resultant network afterYtransformation. In this case, ) // ( ) (1 Z C X THZ Z Z s Z For Thevenins voltage, lets take ZZ out for simplicity. Then, the open-circuit voltage seen from the terminals where the impedance ZZ is to be connected is the Thevenins voltage: ssss I Z s VY TH3 / 2 1)32( ) (volts (HW: Obtain) (s ZTH and draw the Thevenins equivalent circuit). Example 8.4. Transformthenetworkgivenintimedomain(Fig.8.21)intos-domainanddetermine the load voltage and current. Figure 8.21Network given in time domain Solution: Sincethenetworkisalinearsystem,itcanbedirectlytransformedintos-domain (Fig.8.22). Figure 8.22Network in s-domain vi(t)=cost +-C1=1F R1=10 +-C2=1F R2=5 R3=10 L=1H v L(t) i L(t) Vi(s) +-Z2=1/s Z1=10 +-Z3=1/s Z4=5 Z5=10 ZL=s V L(s) I L(s) CHAPTER 8 Complex Domain Analysiss-domain Analysis 254 The input voltage is transformed to s-domain by Laplace transform: 1} {cos )} ( {2s st L t v Li volts(8. 84) Thevenins equivalent of the network seen by the load (Fig.8.23) : Figure 8.23Thevenins equivalent of the network seen by the load TheTheveninsimpedancecanbedeterminedbyshort-circuitedtheinput voltage source. ) (s ZT[(Z1//Z2)+Z3]//Z4+Z5 10 5 // )1110110( ) (ssss ZT 101 25 505 10010 5 //101 2010 5 // )11 1010(2 2s sss sss s By simplifying the expression: 1 25 5015 350 500) (22s ss ss ZT(8. 85) Now, from Fig.8.24 and using Eqn.8.84 and 85, we determine the load voltage and current as follows: TT LLLVZ Z Zs V ) (Therefore, 11 25 5015 350 500) (222s ss ss ssss VL +-Load ZT(s) ZL(s)=sVT(s) CHAPTER 8 Complex Domain Analysiss-domain Analysis 255 Simplifying the above expression: 1 15 351 525 501 25 50) (2 2 32s ss s ss ss VL volts and the load current is ) ( ) () (s Zs Vs ILLL Hence, 1115 351 525 501 25 50) (2 2 32s s s ss ss ILampere Example 8.5 For the network in Fig.8.24, write the mesh equations. Figure 8.24 Solution: For the first mesh, we write KVL: 0 ) (2 5 1 R RV V s V (8. 86) Using terminal equations, we write the Eqn8.86 as follows: 0 ) ( ) (2 1 2 1 5 1 m m mI I R I R s V (8. 87) For the second mesh, we write KVL: 06 3 2 C L RV V V (8. 88) v1(t) +R5

-R2

L3 R4

C6

C7

I8

Im1

Im2

Im3

Im4

CHAPTER 8 Complex Domain Analysiss-domain Analysis 256 Using terminal equations, we write the Eqn8.88 as follows: 01) ( ) (22 64 3 2 3 1 2 2 m m m m m mIsCI I I sL I I R (8. 89) For the third mesh, we write KVL: 07 4 3 C R LV V V (8. 90) Using terminal equations, we write the Eqn8.90 as follows: 01) ( ) (374 3 4 4 2 3 3 m m m m m mIsCI I R I I I sL(8. 91) Note that the current of the fourth mesh is chosen in the direction of the current source I8, that is, 8 4I Im(8. 92) ArrangingEqn.8.87,8.89,8.91and8.92,weobtainthematrix-vectormeshequationas follows: 813 4332173 4 3363 2 22 5 2) (00011( 01(0 ) (IVsL RsLIIIsCsL R sLsLsCsL R RR R Rmmm(8. 93) NotethatEqn.8.93isamatrix-vectorequationwiththecoefficientmatrixissymmetric.We can easily solve the matrix, and find the mesh currents as follows: ) ( ) ( ) ( ) (1s F s B s Z s Im(8. 94) Homework: Find the vector) (s Im. Example 8.6 For the network in Fig.8.25, write the node equations. CHAPTER 8 Complex Domain Analysiss-domain Analysis 257 Figure 8.25 Solution: There are four nodes in this network. Node D is chosen as the ground. Every node is considered one by one. Lets take Node A. There are four currents coming into and going out to the Node A, namely,I1, IY2, IY3 and IY6. We consider the component currentsgoing out to the node as (+) and the source current as it is (in this case, I1 is coming into the node therefore we take it as (-)). Then we write the KCL for this node. For Node A, KCL is -I1 +IY2 +IY3 + IY6.=0(8. 95) Similarly, for Node B and C, we can write KCLs respectively: IY3 +IY4 +IY5 .=0(8. 96) IY4 + IY6. -I7 =0(8. 97) Using terminal equations in s-domain, we can write these equations in terms of node voltages as: -I1 +Y2VA +Y3(VA-VB) +Y6 (VA-VC).=0(8.98) Y3(VB-VA) +Y4(VB-VC) +Y5VB.=0(8. 99) Y4(VC-VB) + Y6 (VC-VA)-I7 =0(8. 100) ArrangingEqn8.98-8.100andputtingthemintomatrix-vectorform,weobtainnode equations: 716 4 4 64 5 4 3 36 3 6 3 20) () () (IIVVVY Y Y YY Y Y Y YY Y Y Y YCBA(8. 101) Notice that the admittance matrix is symmetric and it can easily bewritten directly from the network. I1(s) I7 A B C D Y2

Y3

Y6

Y5

Y4

VA

VB

VC

CHAPTER 8 Complex Domain Analysiss-domain Analysis 258 Homework: SolveEqn.8.72forI1=1ampere,I7=1/sampere,R2=10ohm,L3=0.1Henry,R4=20 ohm, C5=0.01 Farad, and C6=0.1 Farad. CHAPTER 8 Complex Domain Analysiss-domain Analysis 259 CHAPTER 8 QUESTIONS 1.For the circuit given in Fig.P.8.1, find the input impedance Zin(s). Fig.P.8.1 2.For the circuit given in Fig.P.8.2, calculate the equivalent impedance between the terminals A and B. Fig.P.8.2 3.For the circuit given in Fig.P.8.3, find the input impedance Zin(s). Fig.P.8.3 R1=1 ohmL=0.5 H C=0.1 F R2=2 ohms A B Zin(s) 1/s ohm s ohm AB s ohm 10 ohms + - V0(s) 1 ohm s ohm A B C D 1 ohm 2s ohm 1/s ohm 1/2s ohm + - Vi(s) 1 ohm Z0ut(s) CHAPTER 8 Complex Domain Analysiss-domain Analysis 260 4.For the circuit given in Fig.P.8.4, find the transfer function ) ( ) () (s Vs Vs Gio. Fig.P.8.4 5.For the circuit given in Fig.P.8.5, find the transfer function ) ( ) () (s Is Is Gio. Fig.P.8.5 6.For the circuit given in Fig.P.8.6, find the transfer function ) ( ) () (s Vs Vs Gio. Fig.P.8.6 7.For the circuit given in Fig.P.8.7, find the transfer function ) ( ) () (s Vs Vs Gio. Fig.P.8.7 R C vi(t) + - vo(t) + - ii(t) R io(t) C C vi(t) + - vo(t) + - R R L vi(t) + - vo(t) + - CHAPTER 8 Complex Domain Analysiss-domain Analysis 261 8.For the circuit given in Fig.P.8.8, find the transfer function ) ( ) () (s Vs Vs Gio. Fig.P.8.8 9.Compare the transfer function of the circui