EE201 Lecture 22 P 1 Introduction to Second Order Circuits LC Circuits Combining an inductor and capacitor iC iL KVL vC t vL t vC vL KCL C L ic t iL t iL t ic t C dvc t Recall diL t dt 1 L vL t Take derivative of Eqn 22 1 2 d vc t diL t 1 v t C dt dt2 L c dt 22 1 EE201 d2vc t dt2 Lecture 22 1 v t c LC 22 2 Similarly L diL t dt vL t vC t Differentiate w r t time d2iL t dt2 1 L dvc t 1 ic t dt LC d2iL t 1 iL t 2 dt LC 22 3 P 2 EE201 Lecture 22 P 3 Eqns 22 2 and 22 3 are often written as d2vc t 1 vc t 0 22 2a 2 LC dt d2iL t 1 i t 0 22 3a L 2 LC dt General differential equation for LC circuits 1 d2x t x t 0 22 4 2 LC dt Eqn 22 4 describes the behavior of the capacitor voltage or inductor current in an LC circuit Possible solutions for Eqn 22 4 x t A cos t x t B sin t x t A cos t B sin t All valid for specific values of A B and EE201 Lecture 22 P 4 Strategy for finding A B Solve for second derivative of trial solution d2 A cos t B sin t dt2 2 A cos t B sin t 0 Comparing this to Eqn 22 4 1 1 2 LC LC From trigonometry if initial condition x t0 is known constants A B can be found Suppose t0 0 x 0 A cos 0 B sin 0 x 0 A EE201 If Lecture 22 P 5 dx 0 is known dt dx 0 A sin 0 B cos 0 dt dx 0 x 0 B dt Note if to 0 then solve simultaneous equations for x to and x to Example Find vc t and iL t The switch opens at t 0 s t 0 10 0 2A 10 iL t 0 9H 0 1F vc t EE201 Lecture 22 P 6 Step1 Establish initial conditions Circuit configuration at t 0 sec 10 10 iL t 0 9H 0 2A 0 1F vc t From inspection vc 0 vc 0 0 V 1 10 0 2A 0 1A iL 0 1 10 1 10 Step2 Solve for constants A B for capacitor voltage when t 0 sec iL 0 vc t A cos t B sin t A 0 t0 0 EE201 dvc t Lecture 22 A sin t B cos t dt dvc t dt iL t C iL t C A sin t C B cos t iL 0 C B iL 0 B C 1 LC B 0 3 V 0 1 0 1 1 0 09 3 33 s 1 P 7 EE201 Lecture 22 P 8 Write vc t expression vc t A cos t B sin t vc t 0 3sin 3 33t V Step 3 Repeat steps for inductor current when t 0 sec iL t A cos t B sin t iL 0 A 0 1 A diL 0 1 1 vL 0 vc 0 0 L dt L diL 0 0 A sin t B cos t dt B 0 iL t 0 1 cos 3 33t A Note A and B are different for vc t and iL t solutions
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