Editing Resonance (section)

==== Voltage across the resistor ====
Suppose that the output voltage of interest is the voltage across the resistor. In the Laplace domain the voltage across the resistor is
<math display="block">V_\text{out}(s) = RI(s),</math>
<math display="block">V_\text{out}(s) = \frac{Rs}{L\left(s^2 + \frac{R}{L}s + \frac{1}{LC}\right)} V_\text{in}(s),</math>

and using the same natural frequency and damping ratio as in the capacitor example the transfer function is
<math display="block">H(s) = \frac{2\zeta\omega_0s}{s^2 + 2\zeta\omega_0s+\omega_0^2}.</math>

This transfer function also has the same poles as the previous RLC circuit examples, but it only has one zero in the numerator at ''s'' = 0. For this transfer function, its gain is
<math display="block"> G(\omega) = \frac{2\zeta\omega_0\omega}{\sqrt{\left(2\omega\omega_0\zeta\right)^2 + (\omega_0^2 - \omega^2)^2}}.</math>

The resonant frequency that maximizes this gain is
<math display="block">\omega_r = \omega_0,</math>
and the gain is one at this frequency, so the voltage across the resistor resonates ''at'' the circuit's natural frequency and at this frequency the amplitude of the voltage across the resistor equals the input voltage's amplitude.