Lect13EEE 2021 Laplace Transform Solutions of Transient Circuits Dr. Holbert March 5, 2008

Lect13EEE 2022 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

Lect13EEE 2023 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

Lect13EEE 2024 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

Lect13EEE 2025 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

Lect13EEE 2026 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)

Lect13EEE 2027 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

Lect13EEE 2028 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) Laplace (Frequency) Domain Equivalents IC(s)IC(s) IC(s)IC(s) iC(t)iC(t)

Lect13EEE 2029 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)

Lect13EEE 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

Lect13EEE 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) Laplace (Frequency) Domain Equivalents i L (0) IL(s)IL(s) IL(s)IL(s)

Lect13EEE 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

Lect13EEE Transient Analysis Sometimes we not only must Laplace transform the circuit, but we must also find the initial conditions ElementDC Steady-State CapacitorI = 0; open circuit InductorV = 0; short circuit

Lect13EEE Class Examples Drill Problems P6-4, P6-5