Transient Response for Second Order Circuits Characteristics Equations

Second-Order Circuits: In previous work, circuits were limited to one energy storage element, which

Example Find the differential equation for the circuit below in terms of vc and

Solving Second-Order Systems: Ø Ø The method for determining the forced solution is the

Natural Solutions Ø Find characteristic equation from homogeneous equation: Ø Convert to polynomial by

Overdamped Ø If roots are real and distinct ( solution becomes ), natural a

Critically Damped Ø If roots are real and repeated ( natural solution becomes ),

Underdamped Ø If roots are complex ( becomes: ), natural solution or Kevin D.

Example Ø Find the unit step response for vc and i. L for the

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Transient Response for Second. Order Circuits Characteristics Equations, Overdamped-, Underdamped-, and Critically Damped Circuits Kevin D. Donohue, University of Kentucky 1

Second-Order Circuits: In 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: Kevin D. Donohue, University of Kentucky 2

Example Find the differential equation for the circuit below in terms of vc and also terms of i. L Show: Kevin D. Donohue, University of Kentucky 3

Example Find the differential equation for the circuit below in terms of vc and also terms of i. L(t) is(t) R C + vc(t) _ L Show: Kevin D. Donohue, University of Kentucky 4

Solving Second-Order Systems: Ø Ø 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 requirement for two independent initial conditions to resolve the unknown coefficients. In general the natural response of a second-order system will be of the form: Kevin D. Donohue, University of Kentucky 5

Natural Solutions Ø Find characteristic equation from homogeneous equation: Ø Convert to polynomial by the following substitution: to obtain Ø Based on the roots of the characteristic equation, the natural solution will take on one of three particular forms. Roots given by: Kevin D. Donohue, University of Kentucky 6

Overdamped Ø If roots are real and distinct ( solution becomes ), natural a 1=2 a 1=-3 Kevin D. Donohue, University of Kentucky 7

Critically Damped Ø If roots are real and repeated ( natural solution becomes ), a 1=2 a 1=-3 Kevin D. Donohue, University of Kentucky 8

Underdamped Ø If roots are complex ( becomes: ), natural solution or Kevin D. Donohue, University of Kentucky 9

Example Ø Find the unit step response for vc and i. L for the circuit below when: a) R=16 , L=2 H, C=1/24 F b) R=10 , L=1/4 H, C=1/100 F c) R=2 , L=1/3 H, C=1/6 F Show: a) b) c) Kevin D. Donohue, University of Kentucky 10