Laplace Transform Solutions of Transient Circuits Dr Holbert

Laplace Transform Solutions of Transient Circuits Dr. Holbert March 5, 2008 Lect 13 EEE 202 1

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 Lect 13 EEE 202 2

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 Lect 13 EEE 202 3

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 Lect 13 EEE 202 4

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 Lect 13 EEE 202 5

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) Lect 13 EEE 202 6

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 stepfunction voltage source whose height is v(0) • Circuit representations of the Laplace transformation of the capacitor appear on the next page Lect 13 EEE 202 7

Capacitor (cont’d. ) + Time Domain i. C(t) v. C(t) C – IC(s) + VC(s) – IC(s) + 1/s. C + – VC(s) – v(0) s 1/s. C Cv(0) Laplace (Frequency) Domain Equivalents Lect 13 EEE 202 8

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) Lect 13 EEE 202 9

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 Lect 13 EEE 202 10

Inductor (cont’d. ) + Time Domain v. L(t) i. L(0) L – IL(s) + s. L VL(s) – IL(s) + – + VL(s) – Li(0) s. L i(0) s Laplace (Frequency) Domain Equivalents Lect 13 EEE 202 11

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 Lect 13 EEE 202 12

Transient Analysis • Sometimes we not only must Laplace transform the circuit, but we must also find the initial conditions Lect 13 Element Capacitor DC Steady-State I = 0; open circuit Inductor V = 0; short circuit EEE 202 13

Class Examples • Drill Problems P 6 -4, P 6 -5 Lect 13 EEE 202 14