Editing Resonance (section)

==== Antiresonance ====
{{Main|Antiresonance}}

Some systems exhibit antiresonance that can be analyzed in the same way as resonance. For antiresonance, the amplitude of the response of the system at certain frequencies is disproportionately ''small'' rather than being disproportionately large. In the RLC circuit example, this phenomenon can be observed by analyzing both the inductor and the capacitor combined.

Suppose that the output voltage of interest in the RLC circuit is the voltage across the inductor ''and'' the capacitor combined in series. Equation ({{EquationNote|4}}) showed that the sum of the voltages across the three circuit elements sums to the input voltage, so measuring the output voltage as the sum of the inductor and capacitor voltages combined is the same as ''v''<sub>''in''</sub> minus the voltage drop across the resistor. The previous example showed that at the natural frequency of the system, the amplitude of the voltage drop across the resistor ''equals'' the amplitude of ''v''<sub>''in''</sub>, and therefore the voltage across the inductor and capacitor combined has zero amplitude. We can show this with the transfer function.

The sum of the inductor and capacitor voltages is
<math display="block">V_\text{out}(s) = (sL+\frac{1}{sC})I(s),</math>
<math display="block">V_\text{out}(s) = \frac{s^2+\frac{1}{LC}}{s^2 + \frac{R}{L}s + \frac{1}{LC}} V_\text{in}(s).</math>

Using the same natural frequency and damping ratios as the previous examples, the transfer function is
<math display="block">H(s) = \frac{s^2+\omega_0^2}{s^2 + 2\zeta\omega_0s + \omega_0^2}.</math>

This transfer has the same poles as the previous examples but has zeroes at
{{NumBlk||<math display="block">s = \pm i\omega_0.</math>|{{EquationRef|7}}}}

Evaluating the transfer function along the imaginary axis, its gain is
<math display="block">G(\omega) = \frac{\omega_0^2-\omega^2}{\sqrt{\left(2\omega\omega_0\zeta\right)^2 + (\omega_0^2 - \omega^2)^2}}.</math>

Rather than look for resonance, i.e., peaks of the gain, notice that the gain goes to zero at ''ω'' = ''ω''<sub>0</sub>, which complements our analysis of the resistor's voltage. This is called '''antiresonance''', which has the opposite effect of resonance. Rather than result in outputs that are disproportionately large at this frequency, this circuit with this choice of output has no response at all at this frequency. The frequency that is filtered out corresponds exactly to the zeroes of the transfer function, which were shown in Equation ({{EquationNote|7}}) and were on the imaginary axis.