Editing Resonance (section)

==Linear systems==
Resonance manifests itself in many [[Linear system|linear]] and [[nonlinear system]]s as oscillations around an equilibrium point. When the system is driven by a sinusoidal external input, a measured output of the system may oscillate in response. The ratio of the amplitude of the output's steady-state oscillations to the input's oscillations is called the gain, and the gain can be a function of the frequency of the sinusoidal external input. Peaks in the gain at certain frequencies correspond to resonances, where the amplitude of the measured output's oscillations are disproportionately large.

Since many linear and nonlinear systems that oscillate are modeled as [[harmonic oscillator]]s near their equilibria, a derivation of the resonant frequency for a driven, damped harmonic oscillator is shown. An [[RLC circuit]] is used to illustrate connections between resonance and a system's transfer function, frequency response, poles, and zeroes. Building off the RLC circuit example, these connections for higher-order linear systems with multiple inputs and outputs are generalized.

===The driven, damped harmonic oscillator===
{{main|Harmonic oscillator#Driven harmonic oscillators}}

Consider a damped mass on a spring driven by a sinusoidal, externally applied force. [[Newton's second law]] takes the form
{{NumBlk||<math display="block">m\frac{\mathrm{d}^2x}{\mathrm{d}t^2} = F_0 \sin(\omega t)-kx-c\frac{\mathrm{d}x}{\mathrm{d}t}, </math>|{{EquationRef|1}}}}
where ''m'' is the mass, ''x'' is the displacement of the mass from the equilibrium point, ''F''<sub>0</sub> is the driving amplitude, ''ω'' is the driving angular frequency, ''k'' is the spring constant, and ''c'' is the viscous damping coefficient. This can be rewritten in the form
{{NumBlk||<math display="block"> \frac{\mathrm{d}^2x}{\mathrm{d}t^2} + 2\zeta\omega_0\frac{\mathrm{d}x}{\mathrm{d}t} + \omega_0^2 x = \frac{F_0}{m} \sin(\omega t),</math>|{{EquationRef|2}}}}
where
* <math display="inline">\omega_0 = \sqrt{k /m}</math> is called the ''undamped [[angular frequency]] of the oscillator'' or the ''natural frequency'',
* <math>\zeta = \frac{c}{2\sqrt{mk}}</math> is called the ''damping ratio''.

Many sources also refer to ''ω''<sub>0</sub> as the ''resonant frequency''. However, as shown below, when analyzing oscillations of the displacement ''x''(''t''), the resonant frequency is close to but not the same as ''ω''<sub>0</sub>. In general the resonant frequency is close to but not necessarily the same as the natural frequency.{{sfn|Hardt|2004}} The RLC circuit example in the next section gives examples of different resonant frequencies for the same system.

The general solution of Equation ({{EquationNote|2}})  is the sum of a [[Transient (oscillation)|transient]] solution that depends on initial conditions and a [[steady state]] solution that is independent of initial conditions and depends only on the driving amplitude ''F''<sub>0</sub>, driving frequency ''ω'', undamped angular frequency ''ω''<sub>0</sub>, and the damping ratio ''ζ''. The transient solution decays in a relatively short amount of time, so to study resonance it is sufficient to consider the steady state solution.

It is possible to write the steady-state solution for ''x''(''t'') as a function proportional to the driving force with an induced [[phase (waves)|phase]] change, ''φ''.
{{NumBlk||<math display="block">x(t) = \frac{F_0}{m \sqrt{\left(2\omega\omega_0\zeta\right)^2 + (\omega_0^2 - \omega^2)^2}} \sin(\omega t + \varphi),</math>|{{EquationRef|3}}}}
where <math> \varphi = \arctan\left(\frac{2\omega \omega_0\zeta}{\omega^2 - \omega_0^2} \right) + n\pi.</math>

The phase value is usually taken to be between −180° and 0 so it represents a phase lag for both positive and negative values of the arctan argument.

[[File:Mplwp resonance zeta envelope.svg|thumb|300px|Steady-state variation of amplitude with relative frequency <math>\omega/\omega_0</math> and damping <math>\zeta</math> of a driven [[simple harmonic oscillator]]]]
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{{cite book
 | title = Optics, 3E
 | author = Ajoy Ghatak
 | author-link = Ajoy Ghatak
 | edition = 3rd
 | publisher = Tata McGraw-Hill
 | year = 2005
 | isbn = 978-0-07-058583-6
 | page = 6.10
 | url = https://books.google.com/books?id=jStDc2LmU5IC&pg=PT97 
 }}</ref>
-->

Resonance occurs when, at certain driving frequencies, the steady-state amplitude of ''x''(''t'') is large compared to its amplitude at other driving frequencies. For the mass on a spring, resonance corresponds physically to the mass's oscillations having large displacements from the spring's equilibrium position at certain driving frequencies.  Looking at the amplitude of ''x''(''t'') as a function of the driving frequency ''ω'', the amplitude is maximal at the driving frequency
<math display="block">\omega_r = \omega_0 \sqrt{1 - 2\zeta^2}.</math>

''ω''<sub>''r''</sub> is the '''resonant frequency''' for this system. Again, the resonant frequency does not equal the undamped angular frequency ''ω''<sub>0</sub> of the oscillator. They are proportional, and if the damping ratio goes to zero they are the same, but for non-zero damping they are not the same frequency. As shown in the figure, resonance may also occur at other frequencies near the resonant frequency, including ''ω''<sub>0</sub>, but the maximum response is at the resonant frequency.

Also, ''ω''<sub>''r''</sub> is only real and non-zero if <math display="inline">\zeta < 1 / \sqrt{2}</math>, so this system can only resonate when the harmonic oscillator is significantly underdamped. For systems with a very small damping ratio and a driving frequency near the resonant frequency, the steady state oscillations can become very large.

==== The pendulum ====
For other driven, damped harmonic oscillators whose equations of motion do not look exactly like the mass on a spring example, the resonant frequency remains
<math display="block">\omega_r = \omega_0 \sqrt{1 - 2\zeta^2},</math>
but the definitions of ''ω''<sub>0</sub> and ''ζ'' change based on the physics of the system. For a pendulum of length ''ℓ'' and small displacement angle ''θ'', Equation ({{EquationNote|1}}) becomes
<math display="block"> m\ell\frac{\mathrm{d}^2\theta}{\mathrm{d}t^2} = F_0 \sin(\omega t)-mg\theta-c\ell\frac{\mathrm{d}\theta}{\mathrm{d}t}</math>

and therefore
* <math>\omega_0 = \sqrt{\frac{g}{\ell}},</math>
* <math>\zeta = \frac{c}{2m}\sqrt{\frac{\ell}{g}}.</math>

=== 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.

=== Generalizing resonance and antiresonance for linear systems ===

Next consider an arbitrary linear system with multiple inputs and outputs. For example, in [[state-space representation]] a third order [[linear time-invariant system]] with three inputs and two outputs might be written as
<math display="block">\begin{bmatrix} \dot{x}_1 \\ \dot{x}_2 \\ \dot{x}_3 \end{bmatrix} =
 A \begin{bmatrix} x_1(t) \\ x_2(t) \\ x_3(t) \end{bmatrix} + B \begin{bmatrix} u_1(t) \\ u_2(t) \\ u_3(t) \end{bmatrix},</math>
<math display="block">\begin{bmatrix} y_1(t) \\ y_2(t) \end{bmatrix}
= C \begin{bmatrix} x_1(t) \\ x_2(t) \\ x_3(t) \end{bmatrix} + D \begin{bmatrix} u_1(t) \\ u_2(t) \\ u_3(t) \end{bmatrix},</math>
where ''u''<sub>''i''</sub>(''t'') are the inputs, ''x''<sub>''i''</sub>(t) are the state variables,  ''y''<sub>''i''</sub>(''t'') are the outputs, and ''A'', ''B'', ''C'', and ''D'' are matrices describing the dynamics between the variables.

This system has a [[transfer function matrix]] whose elements are the transfer functions between the various inputs and outputs. For example,
<math display="block">
\begin{bmatrix} Y_1(s) \\ Y_2(s) \end{bmatrix} = 
\begin{bmatrix} 
H_{11}(s) & H_{12}(s) & H_{13}(s) \\
H_{21}(s) & H_{22}(s) & H_{23}(s)
\end{bmatrix}
\begin{bmatrix} U_1(s) \\ U_2(s) \\ U_3(s) \end{bmatrix}.
</math>

Each ''H''<sub>''ij''</sub>(''s'') is a scalar transfer function linking one of the inputs to one of the outputs. The RLC circuit examples above had one input voltage and showed four possible output voltages–across the capacitor, across the inductor, across the resistor, and across the capacitor and inductor combined in series–each with its own transfer function. If the RLC circuit were set up to measure all four of these output voltages, that system would have a 4×1 transfer function matrix linking the single input to each of the four outputs.

Evaluated along the imaginary axis, each ''H''<sub>''ij''</sub>(''iω'') can be written as a gain and phase shift,
<math display="block">H_{ij}(i\omega) = G_{ij}(\omega)e^{i\Phi_{ij}(\omega)}.</math>

Peaks in the gain at certain frequencies correspond to resonances between that transfer function's input and output, assuming the system is [[exponential stability|stable]].

Each transfer function ''H''<sub>''ij''</sub>(''s'') can also be written as a fraction whose numerator and denominator are polynomials of ''s''.
<math display="block">H_{ij}(s) = \frac{N_{ij}(s)}{D_{ij}(s)}.</math>

The complex roots of the numerator are called zeroes, and the complex roots of the denominator are called poles. For a stable system, the positions of these poles and zeroes on the complex plane give some indication of whether the system can resonate or antiresonate and at which frequencies. In particular, any stable or [[marginal stability|marginally stable]], complex conjugate pair of poles with imaginary components can be written in terms of a natural frequency and a damping ratio as
<math display="block">s = -\zeta\omega_0 \pm i\omega_0\sqrt{1-\zeta^2},</math>
as in Equation ({{EquationNote|5}}). The natural frequency ''ω''<sub>0</sub> of that pole is the magnitude of the position of the pole on the complex plane and the damping ratio of that pole determines how quickly that oscillation decays. In general,{{sfn|Hardt|2004}}
* Complex conjugate pairs of ''poles'' near the imaginary axis correspond to a peak or resonance in the frequency response in the vicinity of the pole's natural frequency. If the pair of poles is ''on'' the imaginary axis, the gain is infinite at that frequency.
* Complex conjugate pairs of ''zeroes'' near the imaginary axis correspond to a notch or antiresonance in the frequency response in the vicinity of the zero's frequency, i.e., the frequency equal to the magnitude of the zero. If the pair of zeroes is ''on'' the imaginary axis, the gain is zero at that frequency.

In the RLC circuit example, the first generalization relating poles to resonance is observed in Equation ({{EquationNote|5}}). The second generalization relating zeroes to antiresonance is observed in Equation ({{EquationNote|7}}). In the examples of the harmonic oscillator, the RLC circuit capacitor voltage, and the RLC circuit inductor voltage, "poles near the imaginary axis" corresponds to the significantly underdamped condition ζ < 1/<math>\sqrt{2}</math>.