RLC Oscillation Frequency And Damping

Ideal Oscillation

For negligible resistance, the modern LC oscillation frequency is:

f_0 = \frac{1}{2\pi\sqrt{LC}}

This corresponds to the condition where inductive reactance and capacitive reactance are equal in magnitude:

2\pi f L = \frac{1}{2\pi f C}

Damping

With resistance present, the oscillation decays. Modern form:

i(t) = A e^{-\alpha t}\cos(\omega_d t + \phi)

where:

\alpha = \frac{R}{2L}

and:

\omega_d = \sqrt{\frac{1}{LC} - \left(\frac{R}{2L}\right)^2}

Physical Meaning

The oscillation is an exchange between magnetic energy in inductance and electric energy in capacity. Resistance removes energy each cycle, reducing amplitude.

Worked Example

If:

L = 10\ \text{mH},\quad C = 1\ \mu\text{F}

then:

f_0 = \frac{1}{2\pi\sqrt{0.01 \cdot 10^{-6}}} \approx 1591.5\ \text{Hz}

Steinmetz Reading

The OCR candidate discusses frequency of oscillation, decreasing amplitude, critical resistance, and decrement. The formulas on this page use modern notation while preserving the source concept: oscillation depends on inductance and capacity, while resistance damps it.