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Laplace Transform Solutions of Transient Circuits

Laplace Transform Solutions of Transient Circuits. Dr. Holbert March 5, 2008. Introduction. In a circuit with energy storage elements, voltages and currents are the solutions to linear, constant coefficient differential equations

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Laplace Transform Solutions of Transient Circuits

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  1. Laplace Transform Solutions of Transient Circuits Dr. Holbert March 5, 2008 EEE 202

  2. Introduction • In a circuit with energy storage elements, voltages and currents are the solutions to linear, constant coefficient differential equations • Real engineers almost never solve the differential equations directly • It is important to have a qualitative understanding of the solutions EEE 202

  3. Laplace 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 circuits EEE 202

  4. Laplace Circuit Element Models • Here we develop s-domain models of circuit elements • DC 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 EEE 202

  5. Resistor • We start with a simple (and trivial) case, that of the resistor, R • Begin with the time domain relation for the element v(t) = R i(t) • Now Laplace transform the above expression V(s) = R I(s) • Hence a resistor, R, in the time domain is simply that same resistor, R, in the s-domain EEE 202

  6. Capacitor • Begin with the time domain relation for the element • Now Laplace transform the above expression I(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) EEE 202

  7. Capacitor (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) • Circuit representations of the Laplace transformation of the capacitor appear on the next page EEE 202

  8. Capacitor (cont’d.) iC(t) + Time Domain vC(t) C – IC(s) IC(s) + + 1/sC Cv(0) 1/sC VC(s) VC(s) + – v(0) s – – Laplace (Frequency) Domain Equivalents EEE 202

  9. Inductor • Begin with the time domain relation for the element • Now Laplace transform the above expression V(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) EEE 202

  10. Inductor (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) • Circuit representations of the Laplace transformation of the inductor appear on the next page EEE 202

  11. Inductor (cont’d.) + Time Domain vL(t) L iL(0) – IL(s) IL(s) + + sL sL VL(s) i(0) s VL(s) – + Li(0) – – Laplace (Frequency) Domain Equivalents EEE 202

  12. Analysis Techniques • In this section we apply our tried and tested analysis tools and techniques to perform transient circuit analyses • KVL, KCL, Ohm’s Law • Voltage and Current division • Loop/mesh and Nodal analyses • Superposition • Source Transformation • Thevenin’s and Norton’s Theorems EEE 202

  13. Transient Analysis • Sometimes we not only must Laplace transform the circuit, but we must also find the initial conditions EEE 202

  14. Class Examples • Drill Problems P6-4, P6-5 EEE 202

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