Question

LC and LRC circuits; simple harmonic and damped oscillations.

What role does the resistance play in the circuit? What happens when you remove the resistance from the oscillator (place the resistance in the branch of the circuit with the switch, to avoid charging with no resistance)?

How does Lenz’s Law lead directly to oscillation?

Is it possible to critically damp the RLC circuit? How would this be done, and what would its effect be?

Answer 1

A RLC circuit forms a harmonic oscillator for current, and oscillates in fashion quite similar to that of a LC circuit, with an exception though. The resistance in the circuit increases the decay of these oscillations, in other words the resistance acts as the reason for damping and hence the LCR circuit behaves like a damped oscillator, the resistor determines wether or not the circuit will resonate naturally( i.e without a driving source) such circuit are also called underdamped, while ones which require a driving source are called overdamped. The resistor also reduces the peak resonance frequency of the circuit.

Damping factor of the circuit is given as

If the resistance is removed and placed in the branch with the switch, the circuit will go into oscillating phase, and mimic a close approximation of an ideal LC circuit.

In an LC circuit, when an inductor is connected to a charged capacitor, current starts flowing. The current flowing through the inductor induces an EMF which opposes the electrons flow through it. This current flow set up a magnetic field around the inductor, the capacitor starts loosing energy from its electric field while the inductor starts storing energy in its magnetic field. Now once the capacitor is fully discharged,( at this point the magnetic field of the inductor has stored the full energy of the systm), now the magnetic field starts collapsing, and hence the varrying magnetic flux according to lens law sets up a back emf, which now starts charging the capacitor in reverse polarity. This continues untill the entire energy is converted back into electrical energy of the capacitor. So now the capacitor starts discharging, but note now that its polarity has been reverese, the flow of current also reverses. And the above process repeats and the circuit is driven into an oscillatory state, a harmonic oscillator for current. In an ideal LC circuit, this continues indefinitly.

Critical damping describes the scenario where the above stated damping factor of the circuit

Such a case can be acheived by selecting a specific resistance which has a value

This in turn gives,