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© The scientific sentence. 2010

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:

(Vb - Vc) + (Va - Vb) + (Vc - Va) = 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


Equation of LC oscillation circuit:

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) = Qm cos (ωot + φ)

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

The current is

I = dQ/dt = - Im sin (ωot + φ)
With Im = Qmωo the maximum current.


LC oscillation circuit:
ωo = [(1/LC)]1/2, Im = Qmωo


Q(t) = Qm cos (ωot + φ)
I(t) = - Im sin (ωot + φ)


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

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

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

Q(0) = Qo = Qm cos φ
0 = - Im sin φ

Hence φ = 0 and Qo = Qm. With Im = Qoωo , The equations becomes


Q(t) = Qo cos (ωot)
I(t) = - Im sin (ωot)



2. Energy of LC circuit

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

At time t,

Q(t) = Qm cos (ωot + φ) , and
I (t) = - Im sin (ωot + φ)


Therefore

U = UE + UB = (1/2)Q2/C + (1/2) L I2

We have

(1/C) Q2 = (1/C) Qm2 cos2ot + φ) , and
L I2 = L Im2 sin2 ωot + φ)

Since Im = Qmωo, and ωo2 = 1/LC, we can write

L Im2 = L Qm2ωo2 = (1/C) Qm2

Therefore

U = (1/2)Q2/C + (1/2) L I2 = (1/2C) Qm2 cos2ot + φ) + (1/2C) Qm2 sin2 ωot + φ) = (1/2C) Qm2[cos2ot + φ) + sin2 ωot + φ] = (1/2C) Qm2 = (L/2) Im2


Electromagnetic energy in LC circuit:

U = UE + UB = (1/2C) Qm2 = (L/2) Im2



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.







 


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