linguistic:voice_change, quality:corruption
Voltage across the inductor

The resonant frequency need not always take the form given in the examples above. For the RLC circuit, suppose instead that the output voltage of interest is the voltage across the inductor. As shown above, in the Laplace domain the voltage across the inductor is V out (s) = s L I (s), {\displaystyle V_{\text{out}}(s)=sLI(s),} V out (s) = s 2 s 2 + R L s + 1 L C V in (s), {\displaystyle V_{\text{out}}(s)={\frac {s^{2}}{s^{2}+{\frac {R}{L}}s+{\frac {1}{LC}}}}V_{\text{in}}(s),} V out (s) = s 2 s 2 + 2 ζ ω 0 s + ω 0 2 V in (s), {\displaystyle V_{\text{out}}(s)={\frac {s^{2}}{s^{2}+2\zeta \omega _{0}s+\omega _{0}^{2}}}V_{\text{in}}(s),}

using the same definitions for ω and ζ as in the previous example. The transfer function between V (s) and this new V (s) across the inductor is H (s) = s 2 s 2 + 2 ζ ω 0 s + ω 0 2. {\displaystyle H(s)={\frac {s^{2}}{s^{2}+2\zeta \omega _{0}s+\omega _{0}^{2}}}.}

This transfer function has the same poles as the transfer function in the previous example, but it also has two zeroes in the numerator at s = 0. Evaluating H (s) along the imaginary axis, its gain becomes G (ω) = ω 2 (2 ω ω 0 ζ) 2 + (ω 0 2 − ω 2) 2. {\displaystyle G(\omega)={\frac {\omega ^{2}}{\sqrt {\left(2\omega \omega _{0}\zeta \right)^{2}+(\omega _{0}^{2}-\omega ^{2})^{2}}}}.}

Compared to the gain in Equation (6) using the capacitor voltage as the output, this gain has a factor of ω in the numerator and will therefore have a different resonant frequency that maximizes the gain. That frequency is ω r = ω 0 1 − 2 ζ 2, {\displaystyle \omega _{r}={\frac {\omega _{0}}{\sqrt {1-2\zeta ^{2}}}},}

So for the same RLC circuit but with the voltage across the inductor as the output, the resonant frequency is now larger than the natural frequency, though it still tends towards the natural frequency as the damping ratio goes to zero. That the same circuit can have different resonant frequencies for different choices of output is not contradictory. As shown in Equation (4), the voltage drop across the circuit is divided among the three circuit elements, and each element has different dynamics. The capacitor's voltage grows slowly by integrating the current over time and is therefore more sensitive to lower frequencies, whereas the inductor's voltage grows when the current changes rapidly and is therefore more sensitive to higher frequencies. While the circuit as a whole has a natural frequency where it tends to oscillate, the different dynamics of each circuit element make each element resonate at a slightly different frequency.