Question #107558
A series RLC circuit with R = 6 ohm, C = 0.02 Farad and L = 0.1 has no applied voltage. Find the subsequent current in the circuit if the initial charge, on the capacitor is q0 and the initial current is zero.
1
Expert's answer
2020-04-02T13:43:13-0400

A series RLCRLC circuit with  R=6 ohm, C=0.02 Farad and L=0.1 has no applied voltage. Find the subsequent current in  the circuit if the initial charge, on the capacitor is q0q_0 and the initial current is zero. 

Kirchhoff's voltage law:

uR+uL+uC=0u_R+u_L+u_C=0

where uR,uL,uCu_R,u_L,u_C are the voltages across R,L and C respectively.


Substituting in the constitutive equations:

Ri(t)+Ldi(t)dt+1Cti(τ)dτ=0Ri(t)+L\frac{di(t)}{dt}+\frac{1}{C}\int \limits_{-\infty}^{t}i(\tau)d\tau=0

Differentiating and dividing by L :

d2i(t)dt2+RLdi(t)dt+1LCi(t)=0\frac{d^2i(t)}{dt^2}+\frac{R}{L}\frac{di(t)}{dt}+\frac{1}{LC}i(t)=0

This can usefully be expressed in a more generally applicable form:

d2i(t)dt2+2αdi(t)dt+ω02i(t)=0α=R2L,ω0=1LC\frac{d^2i(t)}{dt^2}+2\alpha\frac{di(t)}{dt}+\omega_0^2i(t)=0\\ \alpha=\frac{R}{2L}, \omega_0=\frac{1}{\sqrt{LC}}

The differential equation has the characteristic equation: 

s2+2αs+ω02=0s^2+2\alpha s+\omega_0^2=0

The roots of the equation in s are:

i(t)=A(es1tes2t)==Aeαt(eα2ω02teα2ω02t)α=60.2=30,ω0=10.002α2ω02=90010.002=400=20i(t)=A(e^{s_1t}-e^{s_2t})=\\ =Ae^{-\alpha t}(e^{\sqrt{\alpha^2-\omega_0^2}t}-e^{-\sqrt{\alpha^2-\omega_0^2}t})\\ \alpha=\frac{6}{0.2}=30, \omega_0=\frac{1}{\sqrt{0.002}}\\ \sqrt{\alpha^2-\omega_0^2}=\sqrt{900-\frac{1}{0.002}}=\sqrt{400}=20

therefore

i(t)=Ae30t(e20te20t)=A(e10te50t)i(t)=Ae^{-30 t}(e^{20t}-e^{-20t})=A(e^{-10t}-e^{-50t})

The initial charge on the capacitor is q0q_0 and initial current is zero: 

Ldi(t)dtt=0+q0C=0di(t)dtt=0=A(10e10t+50e50t)di(t)dtt=0=40Aq0LC=40AA=q040LC=12.5q0L\frac{di(t)}{dt}|_{t=0}+\frac{q_0}{C}=0\\ \frac{di(t)}{dt}|_{t=0}=A(-10e^{-10t}+50e^{-50t})\\ \frac{di(t)}{dt}|_{t=0}=40A\\ \frac{q_0}{LC}=-40A\\ A=-\frac{q_0}{40LC}=-12.5q_0

Therefore:

i(t)=12.5q0(e10te50t)i(t)=-12.5q_0(e^{-10t}-e^{-50t})




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