Resonance
Voltage across the capacitor

An RLC circuit in series presents several options for where to measure an output voltage. Suppose the output voltage of interest is the voltage drop across the capacitor. As shown above, in the Laplace domain this voltage is V out (s) = 1 s C I (s) {\displaystyle V_{\text{out}}(s)={\frac {1}{sC}}I(s)} or V out = 1 L C (s 2 + R L s + 1 L C) V in (s). {\displaystyle V_{\text{out}}={\frac {1}{LC(s^{2}+{\frac {R}{L}}s+{\frac {1}{LC}})}}V_{\text{in}}(s).}

Define for this circuit a natural frequency and a damping ratio, ω 0 = 1 L C, {\displaystyle \omega _{0}={\frac {1}{\sqrt {LC}}},} ζ = R 2 C L. {\displaystyle \zeta ={\frac {R}{2}}{\sqrt {\frac {C}{L}}}.}

The ratio of the output voltage to the input voltage becomes H (s) ≜ V out (s) V in (s) = ω 0 2 s 2 + 2 ζ ω 0 s + ω 0 2 {\displaystyle H(s)\triangleq {\frac {V_{\text{out}}(s)}{V_{\text{in}}(s)}}={\frac {\omega _{0}^{2}}{s^{2}+2\zeta \omega _{0}s+\omega _{0}^{2}}}}

H (s) is the transfer function between the input voltage and the output voltage. This transfer function has two poles –roots of the polynomial in the transfer function's denominator–at

and no zeros–roots of the polynomial in the transfer function's numerator. Moreover, for ζ ≤ 1, the magnitude of these poles is the natural frequency ω and that for ζ < 1/ 2 {\displaystyle {\sqrt {2}}}, our condition for resonance in the harmonic oscillator example, the poles are closer to the imaginary axis than to the real axis.

Evaluating H (s) along the imaginary axis s = iω, the transfer function describes the frequency response of this circuit. Equivalently, the frequency response can be analyzed by taking the Fourier transform of Equation (4) instead of the Laplace transform. The transfer function, which is also complex, can be written as a gain and phase, H (i ω) = G (ω) e i Φ (ω). {\displaystyle H(i\omega)=G(\omega)e^{i\Phi (\omega)}.}

A sinusoidal input voltage at frequency ω results in an output voltage at the same frequency that has been scaled by G (ω) and has a phase shift Φ (ω). The gain and phase can be plotted versus frequency on a Bode plot. For the RLC circuit's capacitor voltage, the gain of the transfer function H (iω) is

Note the similarity between the gain here and the amplitude in Equation (3). Once again, the gain is maximized at the resonant frequency ω r = ω 0 1 − 2 ζ 2. {\displaystyle \omega _{r}=\omega _{0}{\sqrt {1-2\zeta ^{2}}}.}

Here, the resonance corresponds physically to having a relatively large amplitude for the steady state oscillations of the voltage across the capacitor compared to its amplitude at other driving frequencies.
keywords:existence, detectable_format:title