A simple series circuit has an inductor of 1henry , a capacitor of 10^-6 farads and a resistor of 1000ohms . The initial charge on the capacitor is zero. If a 12volt battery is connected to the circuit and the circuit is closed at t=0, find the charge on the capacitor 1sevond later and the steady state charge.
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Expert's answer
2020-03-16T10:56:31-0400
The voltage and the current relationships in circuit analysis
i=dtdq
voltage drop across the inductor
VL=Ldtdi=Ldt2d2q
voltage drop across the resistor
VR=Ri=Rdtdq
voltage drop across the capacitor
VC=Cq
Kirchhoff's Second Law
Ldt2d2q+Rdtdq+Cq=E(t)
Given that L=1H,R=1000ohm,C=10−6F,E(t)=12V
1dt2d2q+1000dtdq+10−61q=12
dt2d2q+1000dtdq+1000000q=12
The characteristic equation is
r2+1000r+1000000=0
D=(1000)2−4(1000000)=−3(1000000)
r=2−1000±i10003
qc=e−500t(c1cos(5003t)+c2sin(5003t))
Find the particular solution of the non-homogeneous differential equation using the method of the undetermined coefficients.
Assume that Q(t)=A is a solution of the non-homogeneous differential equation where E(t)=12.
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