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.

Fourier series is a mathematical technique to represent any periodic function in terms of an infinite sum of sine and cosine functions. A square-wave voltage can be expressed as a sum of sinusoidal waveforms with multiple frequencies – each being an odd multiple of the fundamental frequency. Here is the Fourier series expansion of a square-wave function with amplitude A and fundamental period T: \(V(t) = \frac{4A}{\pi}\left(\sum_{n=1,3,5,...}^{\infty} \frac{1}{n} \sin\left(\frac{2\pi nt}{T}\right)\right)\) You can see from this equation that the square wave consists of a series of frequency components: the fundamental frequency (\(\frac{1}{T}\)) and its odd harmonics (\(\frac{1}{3}\cdot\frac{1}{T}\), \(\frac{1}{5}\cdot\frac{1}{T}\), \(\frac{1}{7}\cdot\frac{1}{T}\), ...).

A resonant circuit is one in which the reactive power of the inductor and capacitor cancel each other out, leaving only the resistive part of the circuit. This occurs when the impedance of the inductor and capacitor are equal in magnitude and opposite in phase. The resonant frequency is given by: \(\omega_0 = \frac{1}{\sqrt{LC}}\)

Now, let's analyze the RLC circuit driven by the square-wave voltage, which has a frequency spectrum based on its Fourier series. At the fundamental frequency and the odd multiples of this frequency, the series RLC circuit is subject to sinusoidal components of the supplied voltage. When the resonance condition is met, it means the impedance of the inductor and capacitor is balanced, and the circuit will resonate at that particular frequency. As the square-wave voltage consists of the fundamental frequency and its odd harmonics, the RLC circuit will exhibit resonance at all these frequencies, including the resonant frequency, \(\omega_0\), and its odd fractions (\(\omega_0/3\), \(\omega_0/5\), \(\omega_0/7\),...).

In conclusion, a series RLC circuit driven by a square-wave voltage exhibits resonance at the resonant frequency and its odd harmonics. This behavior occurs because the square wave can be represented by a Fourier series as a sum of sinusoidal components consisting of the fundamental frequency and odd harmonics, and the circuit is subject to these sinusoidal components. When the resonance condition is met for each frequency component, the circuit resonates at that frequency, leading to the observed phenomenon.