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EEE 302 Electrical Networks IILaplace Circuit SolutionsLC BehaviorLaplace Circuit Element ModelsResistorCapacitorCapacitor (cont’d.)Slide 8InductorInductor (cont’d.)Slide 11Slide 12Lecture 16 1EEE 302Electrical Networks IIDr. Keith E. HolbertSummer 2001Lecture 16 2Laplace Circuit Solutions•In this chapter we will use previously established techniques (e.g., KCL, KVL, nodal and loop analyses, superposition, source transformation, Thevenin) in the Laplace domain to analyze circuits•The primary use of Laplace transforms here is the transient analysis of circuitsLecture 16 3LC Behavior•Recall some facts on the behavior of LC elements•Inductors (L):–The current in an inductor cannot change abruptly in zero time; an inductor makes itself felt in a circuit only when there is a changing current–An inductor looks like a short circuit to d.c.•Capacitors (C):–The voltage across a capacitor cannot change discontinuously; a capacitor makes itself felt only when there exists a changing potential (voltage) difference–A capacitor looks like an open circuit to d.c.Lecture 16 4Laplace Circuit Element Models•Here we develop s-domain models of circuit elements•Voltage and current sources basically remain unchanged except that we need to remember that a dc source is really a constant, which is transformed to a 1/s function in the Laplace domain•Note on subsequent slides how without initial conditions, we could have used the substitution s=jLecture 16 5Resistor•We start with a simple (and trivial) case, that of the resistor, R•Begin with the time domain relation for the elementv(t) = R i(t)•Now Laplace transform the above expressionV(s) = R I(s)•Hence a resistor, R, in the time domain is simply that same resistor, R, in the s-domain (this is very similar to how we derived an impedance relation for R also)Lecture 16 6Capacitor•Begin with the time domain relation for the element•Now Laplace transform the above expressionI(s) = s C V(s) - C v(0)•Interpretation: a charged capacitor (a capacitor with non-zero initial conditions at t=0) is equivalent to an uncharged capacitor at t=0 in parallel with an impulsive current source with strength C·v(0)dttvdCti)()( Lecture 16 7Capacitor (cont’d.)•Rearranging the above expression for the capacitor•Interpretation: a charged capacitor can be replaced by an uncharged capacitor in series with a step-function voltage source whose height is v(0)•A circuit representation of the Laplace transformation of the capacitor appears on the next pagesvCsss)0()()( IVLecture 16 8Capacitor (cont’d.)C+–vC(t)Time Domain1/sC+–VC(s)+–v(0)s+–VC(s)1/sCCv(0)Frequency Domain EquivalentsIC(s)IC(s)Lecture 16 9Inductor•Begin with the time domain relation for the element•Now Laplace transform the above expressionV(s) = s L I(s) - L i(0)•Interpretation: an energized inductor (an inductor with non-zero initial conditions) is equivalent to an unenergized inductor at t=0 in series with an impulsive voltage source with strength L·i(0)dttidLtv)()( Lecture 16 10Inductor (cont’d.)•Rearranging the above expression for the inductor•Interpretation: an energized inductor at t=0 is equivalent to an unenergized inductor at t=0 in parallel with a step-function current source with height i(0)•A circuit representation of the Laplace transformation of the inductor appears on the next pagesiLsss)0()()( VILecture 16 11Inductor (cont’d.)L+–vL(t)Time DomainsL+–VL(s)–+i(0)s+–VL(s)sLLi(0)Frequency Domain EquivalentsiL(0)IL(s)IL(s)Lecture 16 12Class Examples•Extension Exercise


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