Analysis of RLC circuits: a)Use differential equations to calculate the voltage

Question

Analysis of RLC circuits:
a)Use differential equations to calculate the voltage response vc(t) for the network shown in Figure Q2-1.
Assume zero initial conditions.

b)State if the current is under-damped, critically damped or over-damped.

I would appreciate if working out and steps can be included

Explanation / Answer

Let us assume a current i1 in the first loop.. V(t)=i1R1+(1/c)int(i1)dt 4=i1+int(i1) diffentiating the above equation.,then we have 0=d(i1)/dt + i1 so the solution for the above equation is i1(D+1)=0 =>i1=c1exp(-1t) here c1 is the constant, provided initial conditions are zero when t=0,c1=0 therefore i1=0; Lets assume i2 current in second loop, 2i2+Ldi2/dt+(1/c)int(i2)=0 differentiate wrt i2 2di2/dt+d^2(i2)/dt+i2=0 i2(D+1)^2=0 i2=(c2+c3t)exp(-t) c2 and c3 are constants i2=0 for provided initial conditions.. So the voltage across capacitor is v=cxint(i2-i1) v=1x0 Vc=0 the roots for i2 are real and equal which means it gives critically damped response

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