If Q(t) = charge on a capacitor at time t in an RLC circuit (with R, L
Question:
If Q(t) = charge on a capacitor at time t in an RLC circuit (with R, L and C being the resistance, inductance and capacitance, respectively) and E(t) = applied voltage,then Kirchhoff’s Laws give the following 2nd order differential equation for Q(t):
LQ''(t) + RQ'(t) + Q(t)*(1/C) = E(t) (∗)
Assume L = 1, C = 1/5, R = 4 and E(t) = 10 cos ωt.
1. Use ode45 (and plot routines) to plot the solution of (∗) with Q(0) = 0 and Q'(0)=0 over the interval 0 ≤ t ≤ 80 for ω = 0, 0.5, 1, 2, 4, 8, 16.
2. Let A(ω) = maximum of |Q(t)| over the interval 30 ≤ t ≤ 80 (this approximates
the amplitude of the steady-stat solution). Experiment with various values of ω and
discuss what appears to happen to A(ω) as ω → ∞ and as ω → 0. Also, interpret your
findings in terms of an equivalent spring-mass system.
tip:
Remark: There is an analogy between spring-mass system and RLC circuits given by:
Spring-mass system RLC circuit
mu'' + cu' + ku = F(t) LQ'' + RQ' + Q*(1/C) = E(t)
u = Displacement Q = Charge
u' = Velocity Q'= I = Current
m = Mass L = Inductance
c = Damping constant R = Resistance
k = Spring constant (1/C) = (Capacitance)^−1
F(t) = External force E(t) = Voltage
Fundamentals of Physics
ISBN: 978-0471758013
8th Extended edition
Authors: Jearl Walker, Halliday Resnick