Question

A series RLC circuit is in resonance when driven by a sinusoidal voltage at its resonant frequency, \(\omega_{0}=(L C)^{-1 / 2} .\) But if the same circuit is driven by a square-wave voltage (which is alternately on and off for equal time intervals), it will exhibit resonance at its resonant frequency and at \(\frac{1}{3}\), \(\frac{1}{5}, \frac{1}{7}, \ldots,\) of this frequency. Explain why.

Short answer

Answer: A series RLC circuit exhibits resonance at its natural resonant frequency and at fractions of this frequency when driven by a square-wave voltage because the square wave can be represented as an infinite series of sine waves (odd harmonics) using Fourier series representation. The circuit responds to each of the odd harmonics present in the square wave, resulting in resonance not only at its natural resonant frequency but also at fractions of this frequency, as the inductive and capacitive reactance can still cancel each other out at these frequencies, leading to a purely resistive impedance.

Step-by-step solution

Definition of Resonance

In a series RLC circuit, resonance occurs when the inductive and capacitive reactance cancel each other out, and the impedance of the circuit is purely resistive. The resonant frequency, \(\omega_0\) is given by the formula: $$\omega_0 = \frac{1}{\sqrt{LC}}$$

Fourier Series Representation of a Square Wave

A square wave can be represented as an infinite series of sine waves (odd harmonics) using Fourier series representation. The equation for the square wave is given by: $$V(t) = \frac{4V_0}{\pi}\sum_{n=1}^{\infty}\frac{(\sin{n\omega t})}{n}$$ where n is odd (i.e., \(n = 1, 3, 5, \ldots\))

Resonance in a Square Wave Driven RLC Circuit

When the RLC circuit is driven by a square wave, the circuit will respond to each of the odd harmonics (sine wave components) present in the square wave. As these harmonics have frequencies of \(n\omega\), the circuit will resonate at its natural resonant frequency (\(\omega_0\)) and at fractions \(\frac{1}{3}\), \(\frac{1}{5}\), \(\frac{1}{7}\ldots\) of this frequency.

Explanation for Resonance at Fractions of the Natural Resonant Frequency

When the RLC circuit is driven by a sinusoidal voltage at its resonant frequency, the circuit exhibits maximum energy transfer at \(\omega_0\), but when driven by a square wave, the RLC circuit must respond to all odd harmonics present in the square wave. Since these odd harmonics have frequencies of \(\omega, \frac{1}{3}\omega, \frac{1}{5}\omega, \ldots\), the circuit exhibits resonance not only at its natural resonant frequency \(\omega_0\), but also at fractions \(\frac{1}{3}\), \(\frac{1}{5}\), \(\frac{1}{7},\ldots\) of this frequency. This is because the inductive and capacitive reactance can still cancel each other out at these frequencies, resulting in a purely resistive impedance.