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project2 - differential equation LC d 2 V c RC dV c V c = V...

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Periodically Forced RLC Circuit In this particular project, we are given an RLC circuit with the components in series. In addition the Vt(t) is time dependent forcing term. Thus our goal is to find the voltage across the capacitor. Part 1, we are interested in the long term behavior of the general solutions. Part 2 we would limit the possible values of R,L,C and see what effect it has on the general solutions. Part 3, we will assign values to R,L,C and vary the frequency and the amplitude in the forcing term. Part 1: As it was stated in the textbook (page 432) our analysis of this circuit leads to a
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Unformatted text preview: differential equation: LC * d 2 V c + RC * dV c + V c = V T Dt 2 dt If , V T = a sin (wt) , where a = V MAX (the amplitude of the forcing function) Then, our equation now becomes: LC * d 2 V c + RC * dV c + V c = V MAX * sin (wt) Dt 2 dt In determining the short term behavior (can be consider the homogenous portion) of our equation, we find the following: LC * d 2 V c + RC*dV c + V c = Dt 2 dt d 2 V c + R *dV c + 1 *V c = Dt 2 L dt L C Let, P = R/L , Q = 1/LC and V c (t) = e st Then, d 2 e st + P*d(e st ) + Q (e st ) = dt 2 dt Which leads to eigenvalues, S 2 e st + P S e st...
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