For RLC circuit determine and solve differential equation. R, L, C, E0 values are constants, E = E(t) = E0*sin(ω*t) (E is marked as V in the image). Then make program which calculates values of I(t) when R, L, C, E0, ω are given.
In short, I need to get function I(t), so I could get values at given time steps to plot graph.
Kirchoff's Voltage Law
The Attempt at a Solution
Most of this was provided by our teacher, but I'm still having troubles solving it (I generally suck at math).
We've got this equation which is built off R, L, C, E components ( I = I(t) ):
LI' + RI + 1/C * ∫Idt = E0*sin(ω*t)
Then got differencial equation, by finding derivative (to get rid of integral):
LI'' + RI' + I/C = E0*ω*cos(ω*t)
Tried to find solution for I by combining one solution and homogeneous equation's solution (I = Ip + Ih):
Ip = a*cos(ω*t) + b*sin(ω*t), where
a = -(E0S) / (R2 + S2)
b = (E0R) / (R2 + S2)
S = ω*L - 1 / (ω*C)
This leads to:
Ip = I0*sin(w*t - omega), where
I0 = sqrt(a2 + b2) = E0 / sqrt(R2 + S2)
tg(omega) = - a/b = S/R
And for homogeneous part we used characteristic equation:
Ih = C1*eX1*t + C2*eX2*t
X2 + R/L*X + 1/(L*C) = 0
Solving equation above using simple discriminant:
X1 = -R/(2*L) + sqrt(R2 / L2 - 4/(L*C))
X2 = -R/(2*L) - sqrt(R2 / L2 - 4/(L*C))
Assuming when t = 0 we have I(t) = 0, inserting this into Ih we get:
C1 + C2 = 0
But that's not enough to find C1 and C2 values and this is where I got stuck. I should be able to make program, but I need to at least solve it on paper first. Hopefully someone can help me out here.
Thank you in advance.