EEE 302 Electrical Networks II Dr Keith E

EEE 302 Electrical Networks II Dr. Keith E. Holbert Summer 2001 Lecture 16 1

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 Lecture 16 2

LC 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 3

Laplace 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 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 (this is very similar to how we derived an impedance relation for R also) Lecture 16 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) Lecture 16 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 step-function voltage source whose height is v(0) • A circuit representation of the Laplace transformation of the capacitor appears on the next page Lecture 16 7

Capacitor (cont’d. ) + Time Domain v. C(t) C – IC(s) + VC(s) – IC(s) + 1/s. C + – VC(s) – v(0) s 1/s. C Cv(0) Frequency Domain Equivalents Lecture 16 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) Lecture 16 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) • A circuit representation of the Laplace transformation of the inductor appears on the next page Lecture 16 10

Inductor (cont’d. ) + Time Domain v. L(t) i. L(0) L – IL(s) + VL(s) – IL(s) + s. L – + VL(s) – Li(0) s. L i(0) s Frequency Domain Equivalents Lecture 16 11

Class Examples • Extension Exercise E 14. 1 Lecture 16 12