Electromagnetic oscillations LC and RLC circuits

Capacitors and inductors are energy-storage devices. A capacitor stores electric energy and an inductor stores magnetic energy. Resistors just cause energy to be dissipated as a heat.

1. LC circuit: Oscillations

we examine the behavior of an idealized circuit (negligible resistance), which contains only an inductance L and a capacitance C, an LC circuit.



The capacitor in the circuit is charged by an external battery and then the battery is taken away.

When the switch is closed (position 1), the capacitor will begin to discharge through the inductor.

During the discharging, there is a current I(t) crossing the circuit and charge Q(t) on the plates of the capacitor.

The current I is the rate at which charge is transferred from a plate to the other. In the figure, the sense of the current (positive) is counterclockwise. The potential difference between the plates is positive. The potential through the inductor decreases then the difference between the terminals of the inductor is negative.

Kirchhoff's loop rule gives:

(V_b - V_c) + (V_a - V_b) + (V_c - V_a) = 0 . That is

+ Q/C - L (dI/dt) + 0 = 0     (Eq. 1)

We have I = ± dQ/dt

During the discharging of the capacitor, Q decreases, hence dQ is negative so is I = dQ/dt and d(dQ/dt)/dt = d²Q/dt².

Therefore

I = - dQ/dt. The equation (Eq. 1) becomes:

Q/C + L d²Q/dt² = 0 . Rearranging, we find

d²Q/dt² + (1/LC)Q = 0

This differential equation is the same as the differential equation of a simple harmonc oscillator, like the mass-spring without friction system. The mass-spring harmonic oscillator provides a mechanical analog to the LC circuit

The solution of this equation is of the form:

Q(t) = Q_m cos (ω_ot + φ)

Where ω_o=[(1/LC)]^1/2, is the angular frequency of the oscillation, φ is the phase constant, and (ω_ot + φ) is the phase.

The current is

I = dQ/dt = - I_m sin (ω_ot + φ)
With I_m = Q_mω_o the maximum current.

These two equations describe the oscillation of the charge and the current, with the frequency ν_o = ω_o/2π.

The constants Q_m and φ are determined from the initial conditions. Suppose the capacitor was given a charge Q_o while the switch was open, and the switch was closed at t = 0. Then the initial conditions are Q = Q_o, and I = 0 at t = 0.

Substituting I = 0 andwith t = 0 into the above equations, we find:

Q(0) = Q_o = Q_m cos φ
0 = - I_m sin φ

Hence φ = 0 and Q_o = Q_m. With I_m = Q_oω_o , The equations becomes

2. Energy of LC circuit

The electric energy stored in a charged capacitor is U_E = (1/C) Q², and the magnetic energy stored in a current-carrying inductor is U_B = (1/2) L I². The electromagnetic energy U in the LC circuit is U = U_E + U_B.

At time t,

Q(t) = Q_m cos (ω_ot + φ) , and
I (t) = - I_m sin (ω_ot + φ)

Therefore

U = U_E + U_B = (1/2)Q²/C + (1/2) L I²

We have

(1/C) Q² = (1/C) Q_m² cos² (ω_ot + φ) , and
L I² = L I_m² sin² ω_ot + φ)

Since I_m = Q_mω_o, and ω_o² = 1/LC, we can write

L I_m² = L Q_m²ω_o² = (1/C) Q_m²

Therefore

U = (1/2)Q²/C + (1/2) L I² = (1/2C) Q_m² cos²(ω_ot + φ) + (1/2C) Q_m² sin² ω_ot + φ) = (1/2C) Q_m²[cos²(ω_ot + φ) + sin² ω_ot + φ] = (1/2C) Q_m² = (L/2) I_m²

The electromagnetic energy in the LC circuit remains constant. Continually changing back and foorth between electric energy in the capacitor and the magnetic energy in the inductor.