Question: Use differential equations to calculate the voltage response vc(t) for

Question

Question:

Use differential equations to calculate the voltage response vc(t) for the network shown in Figure Q2-1.
Assume zero initial conditions. State if the current is under-damped, critically damped or over-damped.

Hi there,

hopefully the pic is clear;

I have been stuck on this problem for a long time and gone through tutorials. From what I know this circuit is going to require a KVL and KCL equation, therefore it should be a second order system. The problem is that I'am not sure about the steps involved in translating these KVL and KCL equations to the end differential equation.

It would help me out so much if the solution and procedure to this problem can be explained at each stage. Then I can relate this to other RLC questions for the future.

Thank you for any help,

Lilly

Explanation / Answer

you can solve the problem using laplace transform method which is easy compared to DF method but essentially both ate the same i am just doing it in frequency domain IN laplace domain V(t) becomes 4/s (since LT of u(t) = 1/s) and R1(s)=1 R2(s) = 2 L(s) = sL=s C(s) = 1/sC = 1/s now R2 and L are in series so equivalent impedance is (2+s) which is in parallel with 1/s=> equivalent impedance is (s+2)/(s^2+2s+1) Vc(s) =I(s)/s I(s) = (4/s)/((s+2)/(s^2+2s+1) + 1)) = 4(s^2+3s+3)/s(s+2) Vc(s)=4(s^2+3s+3)/s^2(s+2) applying partial fractions we get Vc(s) = 3/s+6/s^2+1/(s+2) taking inverse laplace transform we get Vc(t) ={3+6t+e^(-2t)}u(t) in this way you can solve for any circuit by taking equivalent laplace transform to each component and write your required thing in frequency domain then taking Inverse LT you get the required ans in time domain(dont forget to apply initial conditions if they are given)

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