A series RLC circuit withR = 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.
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Expert's answer
2020-03-16T13:16:02-0400
Task:
R=6Ω;L=0.1H;C=0.02F;i(0)=0A;qC(0)=q0
Find:i(t)
Solution:
Current through the capacitor and voltage across the inductance can be described as:
iC=CdtdvC(1)vL=LdtdiL(2)
In this case all the elements are connected in series, so iC=iL=i . According to the second Kirchhoff’s circuit law:
vR+vL+vC=0(3)
After inserting (1) and (2) into (3) the last equation can be rewritten as:
vR+vL+vC=Ri+Ldtdi+C10∫t0idt=0(4)
After differentiation of the left and right parts of equation (4) it is possible to get:
It is a second order ordianary differential equation with constant coefficients. To solve this equation it is necessary to find the roots of the characteristic polynomial:
To get the partial solution it is necessary to solve the Cauchy's problem. First of all its is necessary to get initial condition for dtdi . According to (4):
Ldtdi=−Ri−uC⇒dtdi(0)=L1(−Ri(0)−uC(0))(6)
To get uC(0) it is convenient to use the defenition of capacity:
C=uCqC⇒uC(0)=CqC(0)=Cq0(7)
After inserting (7) into (6) the latter can be rewritten as:
dtdi(0)=L1(−Ri(0)−Cq0)(8)
According to the task, i(0)=0, so, according to (8):
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