The capacitor shown in the circuit below initially holds a
charge q_{0}. The switch is closed at t = 0. Find the charge on the
capacitor as a function of time, if R^{2}/4 < L/C. What is the
oscillation frequency of the circuit when R --> 0?

Solution:

- Concepts:

RLC circuits - Reasoning:

We are asked to analyze the transient behavior of an RLC circuit. - Details of the calculation:

Q/C – IR – LdI/dt = 0. I = - dQ/dt. d^{2}Q/dt^{2}+ (R/L)dQ/dt + Q/(LC) = 0.

Let Q(t) = Aexp(-bt). Then b^{2}- b (R/L) + 1/(LC) = 0.

b = R/(2L) ± (R^{2}/(4L^{2}) - 1/(LC))^{½}.

If R^{2}/4 < L/C,^{2}/(4L^{2}))^{½}= α ± iω.

α = R/(2L), ω = (1/(LC)) - R^{2}/(4L^{2}))^{½}. Q(t) = q_{0}exp(-αt)cos(ωt).

When R --> 0 then α --> 0 and ω**-->**1/(LC)^{½}.

Consider the circuit below, where C_{1} is initially charged to 75 V.
Assume that C_{1} = 0.01 F, C_{2} = 0.003 F, and L = 15 H. Explain how to open
and close the switches so as to discharge C_{1} and charge C_{2}. Starting at t
= 0 give explicit times for opening and closing each switch. What is the
final voltage across C_{2}?

Solution:

- Concepts:

RL transient circuits - Reasoning:

We are asked to analyze the transient behavior of an RL circuit. - Details of the calculation:

Assume that at t = 0 the switch S_{1}is closed and the switch S_{2}stays open.

The equation of the circuit is Q/C_{1}= –LdI/dt, Ld^{2}Q/dt^{2}= –Q/C_{1}.

Q = Q_{1}exp(iω_{1}t), where ω_{1}= 1/(LC_{1})^{½}.

After time t_{1}= π/(2ω_{1}) the capacitor C_{1}will be discharged, all the energy will be stored in the inductor.

½Q_{1}^{2}/C_{1}= ½LI_{max}^{2}.

Q_{1}= (0.01*75) = 0.75 C, t_{1}= 0.6 s, ½Q_{1}^{2}/C_{1}= 28.13 J, I_{max}= 1.94 A.

At t_{1}we close S_{2}and open S_{1}. The equation of the circuit now is Q'/C_{2}= –LdI/dt, Ld^{2}Q'/dt^{2}= –Q'/C_{2}.

I' = I_{max}exp(iω_{2}(t-t_{1})), where ω_{2}= 1/(LC_{2})^{½}.

At time t_{2}= π/(2ω_{2}) + t_{1}the capacitor C_{2}will hold the maximum charge Q_{2}and no energy will be stored in the inductor.

At time t_{2}we open switch S_{2}.

½Q_{2}^{2}/C_{2}= ½LI_{max}^{2}. V_{2}(final) = Q_{2}/C_{2}.

t_{2}= 0.94 s, Q_{2}= 0.41 C. V_{2}(final) = 137 V.

A mass m fixed to a spring of spring constant k and
immersed in a fluid provides a model for a damped harmonic oscillator. A
circuit with inductance L, capacitance C and resistance R provides an electric
analog to such an oscillator.

(a) Write out the circuit equation for the LCR circuit and Newton's second law
of motion for the damped oscillator. What assumption must be made about the
dependence of the damping of the mass on velocity for the two equations to have
the same mathematical form?

(b) How are the constants m, k, and b (damping constant) related to the circuit
constants L, C and R? To what parameters of the electric circuit are the
mechanical quantities x (displacement) and v (velocity) related?

(c) Derive and expression for the displacement and velocity in the limit of
weak damping.

(d) What voltages measured in the circuit would give values proportional to the
displacement and velocity of the mechanical oscillator?

Solution:

- Concepts:

RLC circuit, damped harmonic oscillator - Reasoning:

We are asked to compare the differential equation describing the behavior of a series LRC circuit with the equation of motion for a damped harmonic oscillator. - Details of the calculation:

(a) LRC circuit: Q/C = -IR – LdI/dt. Ld^{2}Q/dt^{2}= -RdQ/dt – Q/C

Damped oscillator: md^{2}x/dt^{2}= -bdx/dt – kx. We must assume that the magnitude of the damping force is proportional to the speed.(b) d

^{2}Q/dt^{2}= -(R/L)dQ/dt – Q/(LC), d^{2}x/dt^{2}= -(b/m)dx/dt – (k/m)x.

R/L --> b/m, 1/(LC) --> k/m.

Possible substitutions: L --> m, R --> b, k --> 1/C.

Q --> x, I = dQ/dt --> v = dx/dt.(c) Try x = x

_{0}exp(at+ d).

a^{2}+ ba/m + k/m = 0. a = -b/2m ± ( b^{2}/4m^{2}– k/m)^{½}.

x = x_{0}exp(-βt) cos(ωt + d), β = b/2m, ω^{2}= k/m – β^{2}= ω_{0}^{2}– β^{2}.

If the damping is weak then x ≈ x_{0}exp(-βt) cos(ω_{0}t + d).(d) The voltage across the capacitor is proportional to Q --> x, The voltage across the resistor is proportional to I --> v.