L-C circuits

Question 1

what are LC oscillations?

Answer 1

LC oscillations refer to the repeated and natural exchange of energy between a capacitor (C) and an inductor (L) in an electrical circuit. When the charged capacitor is connected with an inductor and the circuit is closed, the charge on the capacitor decreases as current flows, storing energy in the inductor’s magnetic field. Once the capacitor is completely discharged, the inductor releases the stored magnetic energy by maintaining current flow and recharging the capacitor in the opposite sense. This cyclical transfer creates undamped, sinusoidal oscillations of current and voltage in an ideal LC circuit (with no resistance).

Such oscillations are analogous to the motion of a mass on a spring, where energy oscillates between kinetic and potential forms. They are crucial insights for understanding radio frequency circuits, tuned circuits, and the basis for oscillators widely used in electronics.

Question 2

The moment the circuit is completed, the following sequence takes place:

Answer 2

1. The capacitor starts discharging, generating a current through the inductor.
2. Magnetic energy builds up in the inductor as electric energy in the capacitor falls.
3. Once the capacitor’s charge drops to zero, the current reaches its peak.
4. The inductor keeps driving the current, recharging the capacitor with opposite polarity.
5. This continues repeatedly, causing current and voltage to oscillate.

Question 3

the natural frequency of LC oscillation

Answer 3

ω = 1/√(LC)
f = 1/(2π√(LC))

Question 4

derivation for LC oscillations

Answer 4

Consider the circuit at t = 0 with the capacitor charged to Q0. Once connected, the loop equation (using Kirchhoff’s voltage law) is:

VC + VL = 0
Where VC = Q/C (Q = charge on capacitor), and VL = L (dI/dt)
Also, I = -dQ/dt

Substituting, and rearranging:

Q/C + L(dI/dt) = 0
Q/C + L(d2Q/dt2) = 0
Thus, d2Q/dt2 + Q/LC = 0

This is the differential equation of simple harmonic motion.

Question 5

voltage phasor for LC circuits

Answer 5

V = VofL
~ VofC [ie, (VofL
- VofC) or (VofC - VofL
)]

Question 6

phase difference of voltage and current in LC circuit

Answer 6

if Xofl> Xofc - voltage leads by pi/2
if XofL < Xofc - current leads by pi/2
if XofL = Xofc, z = 0, i = infinity