Lecture 161 EEE 302 Electrical Networks II Dr. Keith E. Holbert Summer 2001

Lecture 162 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 163 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 164 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 165 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 166 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 167 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 168 Capacitor (cont’d.) C + – vC(t)vC(t) Time Domain 1/sC + – VC(s)VC(s) +–+– v(0) s + – VC(s)VC(s) 1/sC Cv(0) Frequency Domain Equivalents IC(s)IC(s) IC(s)IC(s)

Lecture 169 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 1610 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 1611 Inductor (cont’d.) L + – vL(t)vL(t) Time Domain sLsL + – VL(s)VL(s) –+–+ i(0) s + – VL(s)VL(s) sLsL Li(0) Frequency Domain Equivalents i L (0) IL(s)IL(s) IL(s)IL(s)

Lecture 1612 Class Examples Extension Exercise E14.1