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Characteristics Equations, Overdamped-, Underdamped-, …

Kevin D. Donohue, University of Kentucky1 Characteristics Equations, overdamped -, underdamped -, and Critically Damped CircuitsKevin D. Donohue, University of Kentucky2In previous work, circuits were limited to one energy storage element, which resulted in first-order differential equations. Now, a second independent energy storage element will be added to the circuits to result in second order differential equations:xadtdxadtxdty2122)(++=Kevin D. Donohue, University of Kentucky3 Find the differential equation for the circuit below in terms of vc and also terms of iLShow:vs(t)RLC+vc(t)_iL(t)cccscccsvLCdt dvLRdtvdLCtvvdtdvRCdtvdLCtv1)( )(2222++= ++= ++= ++=tLLLstLLLsdiLCiLRdtdiLtvdiCRidtdiLtv )(1)( )(1)(Kevin D.

RLC + vc(t) _ iL(t) Kevin D. Donohue, University of Kentucky 5 The method for determining the forced solution is the same for both first and second order circuits. The new aspects in solving a second order circuit are the possible forms of natural solutions and the

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