Editing Resonance (section)

=== RLC series circuits ===
{{summarize section|date=January 2021}}
{{See also|RLC Circuit#Series circuit}}
[[File:Rajz RLC soros.svg|thumb|An RLC series circuit]]
Consider a [[electrical network|circuit]] consisting of a [[resistor]] with resistance ''R'', an [[inductor]] with inductance ''L'', and a [[capacitor]] with capacitance ''C'' connected in series with current ''i''(''t'') and driven by a [[voltage]] source with voltage ''v''<sub>''in''</sub>(''t''). The voltage drop around the circuit is
{{NumBlk|:|<math>L \frac{di(t)}{dt} + Ri(t) + V(0)+\frac{1}{C} \int_{0}^t i(\tau)d\tau = v_\text{in}(t).</math>|{{EquationRef|4}}}}

Rather than analyzing a candidate solution to this equation like in the mass on a spring example above, this section will analyze the frequency response of this circuit. Taking the [[Laplace transform]] of Equation ({{EquationNote|4}}),
<math display="block">sLI(s) + RI(s) + \frac{1}{sC}I(s) = V_\text{in}(s),</math>
where ''I''(''s'') and ''V''<sub>''in''</sub>(''s'') are the Laplace transform of the current and input voltage, respectively, and ''s'' is a [[complex number|complex]] frequency parameter in the Laplace domain. Rearranging terms,
<math display="block">I(s) = \frac{s}{s^2L + Rs + \frac{1}{C}} V_\text{in}(s).</math>

====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
<math display="block">V_\text{out}(s) = \frac{1}{sC}I(s)</math>
or
<math display="block">V_\text{out}= \frac{1}{LC(s^2 + \frac{R}{L}s + \frac{1}{LC})} V_\text{in}(s).</math>

Define for this circuit a natural frequency and a damping ratio,
<math display="block"> \omega_0 = \frac{1}{\sqrt{LC}},</math>
<math display="block"> \zeta = \frac{R}{2}\sqrt{\frac{C}{L}}.</math>

The ratio of the output voltage to the input voltage becomes
<math display="block">H(s) \triangleq \frac{V_\text{out}(s)}{V_\text{in}(s)} = \frac{\omega_0^2}{s^2 + 2\zeta\omega_0s + \omega_0^2}</math>

''H''(''s'') is the [[transfer function]] between the input voltage and the output voltage. This transfer function has two [[zeros and poles|poles]]–roots of the polynomial in the transfer function's denominator–at
{{NumBlk||<math display="block">s = -\zeta\omega_0 \pm i\omega_0\sqrt{1-\zeta^2}</math>|{{EquationRef|5}}}}

and no zeros–roots of the polynomial in the transfer function's numerator. Moreover, for {{math|''ζ'' ≤ 1}}, the magnitude of these poles is the natural frequency ''ω''<sub>0</sub> and that for {{math|''ζ'' < 1/<math>\sqrt{2}</math>}}, 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 {{math|''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 ({{EquationNote|4}}) instead of the Laplace transform.  The transfer function, which is also complex, can be written as a gain and phase,
<math display="block"> H(i\omega) = G(\omega)e^{i\Phi(\omega)}.</math>

[[File:RLC Series Circuit Bode Magnitude Plot.svg|thumb|upright=1.35|Bode magnitude plot for the voltage across the elements of an RLC series circuit. Natural frequency {{math|''ω''<sub>0</sub> {{=}} 1 rad/s}}, damping ratio {{math|''ζ'' {{=}} 0.4}}. The capacitor voltage peaks below the circuit's natural frequency, the inductor voltage peaks above the natural frequency, and the resistor voltage peaks at the natural frequency with a peak gain of one. The gain for the voltage across the capacitor and inductor combined in series shows antiresonance, with gain going to zero at the natural frequency.]]

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
{{NumBlk||<math display="block"> G(\omega) = \frac{\omega_0^2}{\sqrt{\left(2\omega\omega_0\zeta\right)^2 + (\omega_0^2 - \omega^2)^2}}.</math>|{{EquationRef|6}}}}

Note the similarity between the gain here and the amplitude in Equation ({{EquationNote|3}}). Once again, the gain is maximized at the '''resonant frequency'''
<math display="block">\omega_r = \omega_0 \sqrt{1 - 2\zeta^2}.</math>

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.

==== 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
<math display="block">V_\text{out}(s) = sLI(s),</math>
<math display="block">V_\text{out}(s) = \frac{s^2}{s^2 + \frac{R}{L}s + \frac{1}{LC}} V_\text{in}(s),</math>
<math display="block">V_\text{out}(s) = \frac{s^2}{s^2 + 2\zeta\omega_0s + \omega_0^2} V_\text{in}(s),</math>

using the same definitions for ''ω''<sub>0</sub> and ''ζ'' as in the previous example. The transfer function between ''V''<sub>in</sub>(''s'') and this new ''V''<sub>out</sub>(''s'') across the inductor is
<math display="block">H(s) = \frac{s^2}{s^2 + 2\zeta\omega_0s + \omega_0^2}.</math>

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 {{Nowrap|''s'' {{=}} 0}}. Evaluating ''H''(''s'') along the imaginary axis, its gain becomes
<math display="block"> G(\omega) = \frac{\omega^2}{\sqrt{\left(2\omega\omega_0\zeta\right)^2 + (\omega_0^2 - \omega^2)^2}}.</math>

Compared to the gain in Equation ({{EquationNote|6}}) using the capacitor voltage as the output, this gain has a factor of ''ω''<sup>2</sup> in the numerator and will therefore have a different resonant frequency that maximizes the gain. That frequency is
<math display="block">\omega_r = \frac{\omega_0}{\sqrt{1 - 2\zeta^2}},</math>

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 ({{EquationNote|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.

==== 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.

==== 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.

==== Relationships between resonance and frequency response in the RLC series circuit example ====
These RLC circuit examples illustrate how resonance is related to the frequency response of the system. Specifically, these examples illustrate:
* How resonant frequencies can be found by looking for peaks in the gain of the transfer function between the input and output of the system, for example in a Bode magnitude plot
* How the resonant frequency for a single system can be different for different choices of system output
* The connection between the system's natural frequency, the system's damping ratio, and the system's resonant frequency
* The connection between the system's natural frequency and the magnitude of the transfer function's poles, pointed out in Equation ({{EquationNote|5}}), and therefore a connection between the poles and the resonant frequency
* A connection between the transfer function's zeroes and the shape of the gain as a function of frequency, and therefore a connection between the zeroes and the resonant frequency that maximizes gain
* A connection between the transfer function's zeroes and antiresonance

The next section extends these concepts to resonance in a general linear system.