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112-440: All systems, including molecular systems and particles, tend to vibrate at a natural frequency depending upon their structure; this frequency is known as a resonant frequency or resonance frequency . When an oscillating force, an external vibration, is applied at a resonant frequency of a dynamic system, object, or particle, the outside vibration will cause the system to oscillate at a higher amplitude (with more force) than when

224-557: A Bode plot . For the RLC circuit's capacitor voltage, the gain of the transfer function H ( iω ) is Note the similarity between the gain here and the amplitude in Equation ( 3 ). Once again, the gain is maximized at the resonant frequency ω r = ω 0 1 − 2 ζ 2 . {\displaystyle \omega _{r}=\omega _{0}{\sqrt {1-2\zeta ^{2}}}.} Here,

336-410: A 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 in ( t ). The voltage drop around the circuit is Rather than analyzing a candidate solution to this equation like in the mass on a spring example above, this section will analyze

448-400: A sequence of real numbers , oscillation of a real-valued function at a point, and oscillation of a function on an interval (or open set ). RLC circuit An RLC circuit is an electrical circuit consisting of a resistor (R), an inductor (L), and a capacitor (C), connected in series or in parallel. The name of the circuit is derived from the letters that are used to denote

560-420: A steady state solution that is independent of initial conditions and depends only on the driving amplitude F 0 , driving frequency ω , undamped angular frequency ω 0 , 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

672-447: A conversion factor. A more general measure of bandwidth is the fractional bandwidth, which expresses the bandwidth as a fraction of the resonance frequency and is given by The fractional bandwidth is also often stated as a percentage. The damping of filter circuits is adjusted to result in the required bandwidth. A narrow band filter, such as a notch filter , requires low damping. A wide band filter requires high damping. The Q factor

784-416: 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. Consider a damped mass on a spring driven by

896-477: A function proportional to the driving force with an induced phase change φ , where φ = arctan ⁡ ( 2 ω ω 0 ζ ω 2 − ω 0 2 ) + n π . {\displaystyle \varphi =\arctan \left({\frac {2\omega \omega _{0}\zeta }{\omega ^{2}-\omega _{0}^{2}}}\right)+n\pi .} The phase value

1008-533: A harmonic oscillator near equilibrium. An example of this is the Lennard-Jones potential , where the potential is given by: U ( r ) = U 0 [ ( r 0 r ) 12 − ( r 0 r ) 6 ] {\displaystyle U(r)=U_{0}\left[\left({\frac {r_{0}}{r}}\right)^{12}-\left({\frac {r_{0}}{r}}\right)^{6}\right]} The equilibrium points of

1120-428: A maximum impedance rather than a minimum impedance. For this reason they are often described as antiresonators ; it is still usual, however, to name the frequency at which this occurs as the resonant frequency. The resonance frequency is defined in terms of the impedance presented to a driving source. It is still possible for the circuit to carry on oscillating (for a time) after the driving source has been removed or it

1232-461: A mechanical oscillation. Oscillation, especially rapid oscillation, may be an undesirable phenomenon in process control and control theory (e.g. in sliding mode control ), where the aim is convergence to stable state . In these cases it is called chattering or flapping, as in valve chatter, and route flapping . The simplest mechanical oscillating system is a weight attached to a linear spring subject to only weight and tension . Such

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1344-848: A natural frequency and a damping ratio, ω 0 = 1 L C , {\displaystyle \omega _{0}={\frac {1}{\sqrt {LC}}},} ζ = R 2 C L . {\displaystyle \zeta ={\frac {R}{2}}{\sqrt {\frac {C}{L}}}.} The ratio of the output voltage to the input voltage becomes H ( s ) ≜ V out ( s ) V in ( s ) = ω 0 2 s 2 + 2 ζ ω 0 s + ω 0 2 {\displaystyle H(s)\triangleq {\frac {V_{\text{out}}(s)}{V_{\text{in}}(s)}}={\frac {\omega _{0}^{2}}{s^{2}+2\zeta \omega _{0}s+\omega _{0}^{2}}}} H ( s )

1456-418: A new restoring force in the opposite sense. If a constant force such as gravity is added to the system, the point of equilibrium is shifted. The time taken for an oscillation to occur is often referred to as the oscillatory period . The systems where the restoring force on a body is directly proportional to its displacement, such as the dynamics of the spring-mass system, are described mathematically by

1568-549: A quadratic equation. ( 3 k − m ω 2 ) ( k − m ω 2 ) = 0 ω 1 = k m , ω 2 = 3 k m {\displaystyle {\begin{aligned}&\left(3k-m\omega ^{2}\right)\left(k-m\omega ^{2}\right)=0\\&\omega _{1}={\sqrt {\frac {k}{m}}},\;\;\omega _{2}={\sqrt {\frac {3k}{m}}}\end{aligned}}} Depending on

1680-460: A resistor is not specifically included as a component. RLC circuits have many applications as oscillator circuits . Radio receivers and television sets use them for tuning to select a narrow frequency range from ambient radio waves. In this role, the circuit is often referred to as a tuned circuit. An RLC circuit can be used as a band-pass filter , band-stop filter , low-pass filter or high-pass filter . The tuning application, for instance,

1792-706: A similar solution, but now there is a different equation for every direction. x ( t ) = A x cos ⁡ ( ω t − δ x ) , y ( t ) = A y cos ⁡ ( ω t − δ y ) , ⋮ {\displaystyle {\begin{aligned}x(t)&=A_{x}\cos(\omega t-\delta _{x}),\\y(t)&=A_{y}\cos(\omega t-\delta _{y}),\\&\;\,\vdots \end{aligned}}} With anisotropic oscillators, different directions have different constants of restoring forces. The solution

1904-417: A single sinusoid with phase shift, The underdamped response is a decaying oscillation at frequency ω d . The oscillation decays at a rate determined by the attenuation α . The exponential in α describes the envelope of the oscillation. B 1 and B 2 (or B 3 and the phase shift φ in the second form) are arbitrary constants determined by boundary conditions. The frequency ω d

2016-483: A sinusoidal position function: x ( t ) = A cos ⁡ ( ω t − δ ) {\displaystyle x(t)=A\cos(\omega t-\delta )} where ω is the frequency of the oscillation, A is the amplitude, and δ is the phase shift of the function. These are determined by the initial conditions of the system. Because cosine oscillates between 1 and −1 infinitely, our spring-mass system would oscillate between

2128-475: A sinusoidal, externally applied force. Newton's second law takes the form where m is the mass, x is the displacement of the mass from the equilibrium point, F 0 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 where Many sources also refer to ω 0 as the resonant frequency . However, as shown below, when analyzing oscillations of

2240-455: A system approaches continuity ; examples include a string or the surface of a body of water . Such systems have (in the classical limit ) an infinite number of normal modes and their oscillations occur in the form of waves that can characteristically propagate. The mathematics of oscillation deals with the quantification of the amount that a sequence or function tends to move between extremes. There are several related notions: oscillation of

2352-411: A system may be approximated on an air table or ice surface. The system is in an equilibrium state when the spring is static. If the system is displaced from the equilibrium, there is a net restoring force on the mass, tending to bring it back to equilibrium. However, in moving the mass back to the equilibrium position, it has acquired momentum which keeps it moving beyond that position, establishing

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2464-586: A value of resistor that causes it to be just on the edge of ringing is called critically damped . Either side of critically damped are described as underdamped (ringing happens) and overdamped (ringing is suppressed). Circuits with topologies more complex than straightforward series or parallel (some examples described later in the article) have a driven resonance frequency that deviates from   ω 0 = 1 / L C     {\displaystyle \ \omega _{0}=1/{\sqrt {L\,C~}}\ } , and for those

2576-418: Is ω r = ω 0 , {\displaystyle \omega _{r}=\omega _{0},} 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. Some systems exhibit antiresonance that can be analyzed in

2688-458: Is The critically damped response represents the circuit response that decays in the fastest possible time without going into oscillation. This consideration is important in control systems where it is required to reach the desired state as quickly as possible without overshooting. D 1 and D 2 are arbitrary constants determined by boundary conditions. The series RLC can be analyzed for both transient and steady AC state behavior using

2800-522: Is mechanical resonance , orbital resonance , acoustic resonance , electromagnetic resonance, nuclear magnetic resonance (NMR), electron spin resonance (ESR) and resonance of quantum wave functions . Resonant systems can be used to generate vibrations of a specific frequency (e.g., musical instruments ), or pick out specific frequencies from a complex vibration containing many frequencies (e.g., filters). The term resonance (from Latin resonantia , 'echo', from resonare , 'resound') originated from

2912-465: Is a widespread measure used to characterise resonators. It is defined as the peak energy stored in the circuit divided by the average energy dissipated in it per radian at resonance. Low- Q circuits are therefore damped and lossy and high- Q circuits are underdamped and prone to amplitude extremes if driven at the resonant frequency. Q is related to bandwidth; low- Q circuits are wide-band and high- Q circuits are narrow-band. In fact, it happens that Q

3024-474: Is also complex, can be written as a gain and phase, H ( i ω ) = G ( ω ) e i Φ ( ω ) . {\displaystyle H(i\omega )=G(\omega )e^{i\Phi (\omega )}.} 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

3136-417: Is an example of band-pass filtering. The RLC filter is described as a second-order circuit, meaning that any voltage or current in the circuit can be described by a second-order differential equation in circuit analysis. The three circuit elements, R, L and C, can be combined in a number of different topologies . All three elements in series or all three elements in parallel are the simplest in concept and

3248-442: Is defined as the ratio of these two; although, sometimes ζ is not used, and α is referred to as damping factor instead; hence requiring careful specification of one's use of that term. In the case of the series RLC circuit, the damping factor is given by The value of the damping factor determines the type of transient that the circuit will exhibit. The differential equation has the characteristic equation , The roots of

3360-451: Is given by This is called the damped resonance frequency or the damped natural frequency. It is the frequency the circuit will naturally oscillate at if not driven by an external source. The resonance frequency, ω 0 , which is the frequency at which the circuit will resonate when driven by an external oscillation, may often be referred to as the undamped resonance frequency to distinguish it. The critically damped response ( ζ = 1 )

3472-1749: Is given by resolving the motion into normal modes . The simplest form of coupled oscillators is a 3 spring, 2 mass system, where masses and spring constants are the same. This problem begins with deriving Newton's second law for both masses. { m 1 x ¨ 1 = − ( k 1 + k 2 ) x 1 + k 2 x 2 m 2 x ¨ 2 = k 2 x 1 − ( k 2 + k 3 ) x 2 {\displaystyle {\begin{cases}m_{1}{\ddot {x}}_{1}=-(k_{1}+k_{2})x_{1}+k_{2}x_{2}\\m_{2}{\ddot {x}}_{2}=k_{2}x_{1}-(k_{2}+k_{3})x_{2}\end{cases}}} The equations are then generalized into matrix form. F = M x ¨ = k x , {\displaystyle F=M{\ddot {x}}=kx,} where M = [ m 1 0 0 m 2 ] {\displaystyle M={\begin{bmatrix}m_{1}&0\\0&m_{2}\end{bmatrix}}} , x = [ x 1 x 2 ] {\displaystyle x={\begin{bmatrix}x_{1}\\x_{2}\end{bmatrix}}} , and k = [ k 1 + k 2 − k 2 − k 2 k 2 + k 3 ] {\displaystyle k={\begin{bmatrix}k_{1}+k_{2}&-k_{2}\\-k_{2}&k_{2}+k_{3}\end{bmatrix}}} The values of k and m can be substituted into

Resonance - Misplaced Pages Continue

3584-449: Is its ability to resonate at a specific frequency, the resonance frequency , f 0 . Frequencies are measured in units of hertz . In this article, angular frequency , ω 0 , is used because it is more mathematically convenient. This is measured in radians per second. They are related to each other by a simple proportion, Resonance occurs because energy for this situation is stored in two different ways: in an electric field as

3696-418: Is measured in nepers per second. However, the unitless damping factor (symbol ζ , zeta) is often a more useful measure, which is related to α by The special case of ζ = 1 is called critical damping and represents the case of a circuit that is just on the border of oscillation. It is the minimum damping that can be applied without causing oscillation. The resonance effect can be used for filtering,

3808-652: Is similar to isotropic oscillators, but there is a different frequency in each direction. Varying the frequencies relative to each other can produce interesting results. For example, if the frequency in one direction is twice that of another, a figure eight pattern is produced. If the ratio of frequencies is irrational, the motion is quasiperiodic . This motion is periodic on each axis, but is not periodic with respect to r, and will never repeat. All real-world oscillator systems are thermodynamically irreversible . This means there are dissipative processes such as friction or electrical resistance which continually convert some of

3920-459: Is small, the resonant frequency is approximately equal to the natural frequency of the system, which is a frequency of unforced vibrations. Some systems have multiple and distinct resonant frequencies. A familiar example is a playground swing , which acts as a pendulum . Pushing a person in a swing in time with the natural interval of the swing (its resonant frequency) makes the swing go higher and higher (maximum amplitude), while attempts to push

4032-460: Is subjected to a step in voltage (including a step down to zero). This is similar to the way that a tuning fork will carry on ringing after it has been struck, and the effect is often called ringing. This effect is the peak natural resonance frequency of the circuit and in general is not exactly the same as the driven resonance frequency, although the two will usually be quite close to each other. Various terms are used by different authors to distinguish

4144-409: Is the resonant frequency for this system. Again, the resonant frequency does not equal the undamped angular frequency ω 0 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 0 , but

4256-468: Is the transfer function between the input voltage and the output voltage. This transfer function has two poles –roots of the polynomial in the transfer function's denominator–at and no zeros–roots of the polynomial in the transfer function's numerator. Moreover, for ζ ≤ 1 , the magnitude of these poles is the natural frequency ω 0 and that for ζ < 1/ 2 {\displaystyle {\sqrt {2}}} , our condition for resonance in

4368-843: Is the inverse of fractional bandwidth Q factor is directly proportional to selectivity , as the Q factor depends inversely on bandwidth. For a series resonant circuit ( as shown below ), the Q factor can be calculated as follows: where X {\displaystyle \,X\,} is the reactance either of L {\displaystyle \,L\,} or of C {\displaystyle \,C\,} at resonance, and Z o ≡ L C . {\displaystyle \,Z_{\text{o}}\equiv {\sqrt {{\frac {L}{\,C\,}}\,}}\;.} The parameters ζ , B f , and Q are all scaled to ω 0 . This means that circuits which have similar parameters share similar characteristics regardless of whether or not they are operating in

4480-469: Is the transient solution to the differential equation. The transient solution can be found by using the initial conditions of the system. Some systems can be excited by energy transfer from the environment. This transfer typically occurs where systems are embedded in some fluid flow. For example, the phenomenon of flutter in aerodynamics occurs when an arbitrarily small displacement of an aircraft wing (from its equilibrium) results in an increase in

4592-805: Is then found, and used to be the effective potential constant: γ eff = d 2 U d r 2 | r = r 0 = U 0 [ 12 ( 13 ) r 0 12 r − 14 − 6 ( 7 ) r 0 6 r − 8 ] = 114 U 0 r 2 {\displaystyle {\begin{aligned}\gamma _{\text{eff}}&=\left.{\frac {d^{2}U}{dr^{2}}}\right|_{r=r_{0}}=U_{0}\left[12(13)r_{0}^{12}r^{-14}-6(7)r_{0}^{6}r^{-8}\right]\\[1ex]&={\frac {114U_{0}}{r^{2}}}\end{aligned}}} The system will undergo oscillations near

Resonance - Misplaced Pages Continue

4704-412: 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. 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

4816-663: The Laplace transform . If the voltage source above produces a waveform with Laplace-transformed V ( s ) (where s is the complex frequency s = σ + jω ), the KVL can be applied in the Laplace domain: where I ( s ) is the Laplace-transformed current through all components. Solving for I ( s ) : And rearranging, we have Solving for the Laplace admittance Y ( s ) : Simplifying using parameters α and ω 0 defined in

4928-419: The angle of attack of the wing on the air flow and a consequential increase in lift coefficient , leading to a still greater displacement. At sufficiently large displacements, the stiffness of the wing dominates to provide the restoring force that enables an oscillation. Resonance occurs in a damped driven oscillator when ω = ω 0 , that is, when the driving frequency is equal to the natural frequency of

5040-405: The simple harmonic oscillator and the regular periodic motion is known as simple harmonic motion . In the spring-mass system, oscillations occur because, at the static equilibrium displacement, the mass has kinetic energy which is converted into potential energy stored in the spring at the extremes of its path. The spring-mass system illustrates some common features of oscillation, namely

5152-415: The transient response of the circuit will die away after the stimulus has been removed. Neper occurs in the name because the units can also be considered to be nepers per second, neper being a logarithmic unit of attenuation. ω 0 is the angular resonance frequency. For the case of the series RLC circuit these two parameters are given by: A useful parameter is the damping factor , ζ , which

5264-868: 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

5376-458: 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 oscillators near their equilibria,

5488-404: The beating of the human heart (for circulation), business cycles in economics , predator–prey population cycles in ecology , geothermal geysers in geology , vibration of strings in guitar and other string instruments , periodic firing of nerve cells in the brain, and the periodic swelling of Cepheid variable stars in astronomy . The term vibration is precisely used to describe

5600-399: The capacitor combined in series. Equation ( 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 in minus the voltage drop across the resistor. The previous example showed that at the natural frequency of the system, the amplitude of

5712-434: The capacitor is charged and in a magnetic field as current flows through the inductor. Energy can be transferred from one to the other within the circuit and this can be oscillatory. A mechanical analogy is a weight suspended on a spring which will oscillate up and down when released. This is no passing metaphor; a weight on a spring is described by exactly the same second order differential equation as an RLC circuit and for all

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5824-432: The circuit has fallen to half the value passed at resonance. There are two of these half-power frequencies, one above, and one below the resonance frequency where Δ ω is the bandwidth, ω 1 is the lower half-power frequency and ω 2 is the upper half-power frequency. The bandwidth is related to attenuation by where the units are radians per second and nepers per second respectively. Other units may require

5936-432: 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,

6048-450: The circuit solves in three different ways depending on the value of ζ . These are overdamped ( ζ > 1 ), underdamped ( ζ < 1 ), and critically damped ( ζ = 1 ). The overdamped response ( ζ > 1 ) is The overdamped response is a decay of the transient current without oscillation. The underdamped response ( ζ < 1 ) is By applying standard trigonometric identities the two trigonometric functions may be expressed as

6160-434: The circuit. The resonant frequency is defined as the frequency at which the impedance of the circuit is at a minimum. Equivalently, it can be defined as the frequency at which the impedance is purely real (that is, purely resistive). This occurs because the impedances of the inductor and capacitor at resonant are equal but of opposite sign and cancel out. Circuits where L and C are in parallel rather than series actually have

6272-408: The constituent components of this circuit, where the sequence of the components may vary from RLC. The circuit forms a harmonic oscillator for current, and resonates in a manner similar to an LC circuit . Introducing the resistor increases the decay of these oscillations, which is also known as damping . The resistor also reduces the peak resonant frequency. Some resistance is unavoidable even if

6384-485: The coupled oscillators where energy alternates between two forms of oscillation. Well-known is the Wilberforce pendulum , where the oscillation alternates between the elongation of a vertical spring and the rotation of an object at the end of that spring. Coupled oscillators are a common description of two related, but different phenomena. One case is where both oscillations affect each other mutually, which usually leads to

6496-469: The current and input voltage, respectively, and s is a complex frequency parameter in the Laplace domain. Rearranging terms, I ( s ) = s s 2 L + R s + 1 C V in ( s ) . {\displaystyle I(s)={\frac {s}{s^{2}L+Rs+{\frac {1}{C}}}}V_{\text{in}}(s).} An RLC circuit in series presents several options for where to measure an output voltage. Suppose

6608-737: The different dynamics of each circuit element make each element resonate at a slightly different frequency. Suppose that the output voltage of interest is the voltage across the resistor. In the Laplace domain the voltage across the resistor is V out ( s ) = R I ( s ) , {\displaystyle V_{\text{out}}(s)=RI(s),} V out ( s ) = R s L ( s 2 + R L s + 1 L C ) V in ( s ) , {\displaystyle V_{\text{out}}(s)={\frac {Rs}{L\left(s^{2}+{\frac {R}{L}}s+{\frac {1}{LC}}\right)}}V_{\text{in}}(s),} and using

6720-426: The displacement x ( t ), the resonant frequency is close to but not the same as ω 0 . In general the resonant frequency is close to but not necessarily the same as the natural frequency. The RLC circuit example in the next section gives examples of different resonant frequencies for the same system. The general solution of Equation ( 2 ) is the sum of a transient solution that depends on initial conditions and

6832-401: The energy stored in the oscillator into heat in the environment. This is called damping. Thus, oscillations tend to decay with time unless there is some net source of energy into the system. The simplest description of this decay process can be illustrated by oscillation decay of the harmonic oscillator. Damped oscillators are created when a resistive force is introduced, which is dependent on

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6944-407: The equation above yields: For the case where the source is an unchanging voltage, taking the time derivative and dividing by L leads to the following second order differential equation: This can usefully be expressed in a more generally applicable form: α and ω 0 are both in units of angular frequency . α is called the neper frequency , or attenuation , and is a measure of how fast

7056-486: The equation in s -domain are, The general solution of the differential equation is an exponential in either root or a linear superposition of both, The coefficients A 1 and A 2 are determined by the boundary conditions of the specific problem being analysed. That is, they are set by the values of the currents and voltages in the circuit at the onset of the transient and the presumed value they will settle to after infinite time. The differential equation for

7168-419: The equilibrium point. The force that creates these oscillations is derived from the effective potential constant above: F = − γ eff ( r − r 0 ) = m eff r ¨ {\displaystyle F=-\gamma _{\text{eff}}(r-r_{0})=m_{\text{eff}}{\ddot {r}}} This differential equation can be re-written in

7280-766: The existence of an equilibrium and the presence of a restoring force which grows stronger the further the system deviates from equilibrium. In the case of the spring-mass system, Hooke's law states that the restoring force of a spring is: F = − k x {\displaystyle F=-kx} By using Newton's second law , the differential equation can be derived: x ¨ = − k m x = − ω 2 x , {\displaystyle {\ddot {x}}=-{\frac {k}{m}}x=-\omega ^{2}x,} where ω = k / m {\textstyle \omega ={\sqrt {k/m}}} The solution to this differential equation produces

7392-474: The field of acoustics, particularly the sympathetic resonance observed in musical instruments, e.g., when one string starts to vibrate and produce sound after a different one is struck. Resonance occurs when a system is able to store and easily transfer energy between two or more different storage modes (such as kinetic energy and potential energy in the case of a simple pendulum). However, there are some losses from cycle to cycle, called damping . When damping

7504-802: The first derivative of the position, or in this case velocity. The differential equation created by Newton's second law adds in this resistive force with an arbitrary constant b . This example assumes a linear dependence on velocity. m x ¨ + b x ˙ + k x = 0 {\displaystyle m{\ddot {x}}+b{\dot {x}}+kx=0} This equation can be rewritten as before: x ¨ + 2 β x ˙ + ω 0 2 x = 0 , {\displaystyle {\ddot {x}}+2\beta {\dot {x}}+\omega _{0}^{2}x=0,} where 2 β = b m {\textstyle 2\beta ={\frac {b}{m}}} . This produces

7616-1022: The form of a simple harmonic oscillator: r ¨ + γ eff m eff ( r − r 0 ) = 0 {\displaystyle {\ddot {r}}+{\frac {\gamma _{\text{eff}}}{m_{\text{eff}}}}(r-r_{0})=0} Thus, the frequency of small oscillations is: ω 0 = γ eff m eff = 114 U 0 r 2 m eff {\displaystyle \omega _{0}={\sqrt {\frac {\gamma _{\text{eff}}}{m_{\text{eff}}}}}={\sqrt {\frac {114U_{0}}{r^{2}m_{\text{eff}}}}}} Or, in general form ω 0 = d 2 U d r 2 | r = r 0 {\displaystyle \omega _{0}={\sqrt {\left.{\frac {d^{2}U}{dr^{2}}}\right\vert _{r=r_{0}}}}} This approximation can be better understood by looking at

7728-401: The frequency response of this circuit. Taking the Laplace transform of Equation ( 4 ), s L I ( s ) + R I ( s ) + 1 s C I ( s ) = V in ( s ) , {\displaystyle sLI(s)+RI(s)+{\frac {1}{sC}}I(s)=V_{\text{in}}(s),} where I ( s ) and V in ( s ) are the Laplace transform of

7840-556: The function are then found: d U d r = 0 = U 0 [ − 12 r 0 12 r − 13 + 6 r 0 6 r − 7 ] ⇒ r ≈ r 0 {\displaystyle {\begin{aligned}{\frac {dU}{dr}}&=0=U_{0}\left[-12r_{0}^{12}r^{-13}+6r_{0}^{6}r^{-7}\right]\\\Rightarrow r&\approx r_{0}\end{aligned}}} The second derivative

7952-444: 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

8064-419: The gain, notice that the gain goes to zero at ω = ω 0 , 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

8176-573: The general solution. ( k − M ω 2 ) a = 0 [ 2 k − m ω 2 − k − k 2 k − m ω 2 ] = 0 {\displaystyle {\begin{aligned}\left(k-M\omega ^{2}\right)a&=0\\{\begin{bmatrix}2k-m\omega ^{2}&-k\\-k&2k-m\omega ^{2}\end{bmatrix}}&=0\end{aligned}}} The determinant of this matrix yields

8288-608: The general solution: x ( t ) = e − β t ( C 1 e ω 1 t + C 2 e − ω 1 t ) , {\displaystyle x(t)=e^{-\beta t}\left(C_{1}e^{\omega _{1}t}+C_{2}e^{-\omega _{1}t}\right),} where ω 1 = β 2 − ω 0 2 {\textstyle \omega _{1}={\sqrt {\beta ^{2}-\omega _{0}^{2}}}} . The exponential term outside of

8400-451: The harmonic oscillator example, the poles are closer to the imaginary axis than to the real axis. Evaluating H ( s ) along the imaginary axis 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 ( 4 ) instead of the Laplace transform. The transfer function, which

8512-927: The mass on a spring example, the resonant frequency remains ω r = ω 0 1 − 2 ζ 2 , {\displaystyle \omega _{r}=\omega _{0}{\sqrt {1-2\zeta ^{2}}},} but the definitions of ω 0 and ζ change based on the physics of the system. For a pendulum of length ℓ and small displacement angle θ , Equation ( 1 ) becomes m ℓ d 2 θ d t 2 = F 0 sin ⁡ ( ω t ) − m g θ − c ℓ d θ d t {\displaystyle 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}}} and therefore Consider

8624-628: The matrices. m 1 = m 2 = m , k 1 = k 2 = k 3 = k , M = [ m 0 0 m ] , k = [ 2 k − k − k 2 k ] {\displaystyle {\begin{aligned}m_{1}=m_{2}=m,\;\;k_{1}=k_{2}=k_{3}=k,\\M={\begin{bmatrix}m&0\\0&m\end{bmatrix}},\;\;k={\begin{bmatrix}2k&-k\\-k&2k\end{bmatrix}}\end{aligned}}} These matrices can now be plugged into

8736-548: The maximum response is at the resonant frequency. Also, ω r is only real and non-zero if ζ < 1 / 2 {\textstyle \zeta <1/{\sqrt {2}}} , 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. For other driven, damped harmonic oscillators whose equations of motion do not look exactly like

8848-407: The most straightforward to analyse. There are, however, other arrangements, some with practical importance in real circuits. One issue often encountered is the need to take into account inductor resistance. Inductors are typically constructed from coils of wire, the resistance of which is not usually desirable, but it often has a significant effect on the circuit. An important property of this circuit

8960-432: The object. Light and other short wavelength electromagnetic radiation is produced by resonance on an atomic scale , such as electrons in atoms. Other examples of resonance include: Resonance manifests itself in many linear and nonlinear systems 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

9072-465: The occurrence of a single, entrained oscillation state, where both oscillate with a compromise frequency . Another case is where one external oscillation affects an internal oscillation, but is not affected by this. In this case the regions of synchronization, known as Arnold Tongues , can lead to highly complex phenomena as for instance chaotic dynamics. In physics, a system with a set of conservative forces and an equilibrium point can be approximated as

9184-586: The oscillation is said to be driven . The simplest example of this is a spring-mass system with a sinusoidal driving force. x ¨ + 2 β x ˙ + ω 0 2 x = f ( t ) , {\displaystyle {\ddot {x}}+2\beta {\dot {x}}+\omega _{0}^{2}x=f(t),} where f ( t ) = f 0 cos ⁡ ( ω t + δ ) . {\displaystyle f(t)=f_{0}\cos(\omega t+\delta ).} This gives

9296-445: The others. This leads to a coupling of the oscillations of the individual degrees of freedom. For example, two pendulum clocks (of identical frequency) mounted on a common wall will tend to synchronise. This phenomenon was first observed by Christiaan Huygens in 1665. The apparent motions of the compound oscillations typically appears very complicated but a more economic, computationally simpler and conceptually deeper description

9408-622: The output voltage of interest is the voltage drop across the capacitor. As shown above, in the Laplace domain this voltage is V out ( s ) = 1 s C I ( s ) {\displaystyle V_{\text{out}}(s)={\frac {1}{sC}}I(s)} or V out = 1 L C ( s 2 + R L s + 1 L C ) V in ( s ) . {\displaystyle V_{\text{out}}={\frac {1}{LC(s^{2}+{\frac {R}{L}}s+{\frac {1}{LC}})}}V_{\text{in}}(s).} Define for this circuit

9520-404: The parenthesis is the decay function and β is the damping coefficient. There are 3 categories of damped oscillators: under-damped, where β < ω 0 ; over-damped, where β > ω 0 ; and critically damped, where β = ω 0 . In addition, an oscillating system may be subject to some external force, as when an AC circuit is connected to an outside power source. In this case

9632-500: The positive and negative amplitude forever without friction. In two or three dimensions, harmonic oscillators behave similarly to one dimension. The simplest example of this is an isotropic oscillator, where the restoring force is proportional to the displacement from equilibrium with the same restorative constant in all directions. F → = − k r → {\displaystyle {\vec {F}}=-k{\vec {r}}} This produces

9744-423: The potential curve of the system. By thinking of the potential curve as a hill, in which, if one placed a ball anywhere on the curve, the ball would roll down with the slope of the potential curve. This is true due to the relationship between potential energy and force. d U d t = − F ( r ) {\displaystyle {\frac {dU}{dt}}=-F(r)} By thinking of

9856-432: The potential in this way, one will see that at any local minimum there is a "well" in which the ball would roll back and forth (oscillate) between r min {\displaystyle r_{\text{min}}} and r max {\displaystyle r_{\text{max}}} . This approximation is also useful for thinking of Kepler orbits . As the number of degrees of freedom becomes arbitrarily large,

9968-428: The properties of the one system there will be found an analogous property of the other. The mechanical property answering to the resistor in the circuit is friction in the spring–weight system. Friction will slowly bring any oscillation to a halt if there is no external force driving it. Likewise, the resistance in an RLC circuit will "damp" the oscillation, diminishing it with time if there is no driving AC power source in

10080-414: The rapid change in impedance near resonance can be used to pass or block signals close to the resonance frequency. Both band-pass and band-stop filters can be constructed and some filter circuits are shown later in the article. A key parameter in filter design is bandwidth . The bandwidth is measured between the cutoff frequencies , most frequently defined as the frequencies at which the power passed through

10192-413: 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. 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

10304-400: 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

10416-470: The same definitions for ω 0 and ζ as in the previous example. The transfer function between V in ( s ) and this new V out ( 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

10528-493: The same force is applied at other, non-resonant frequencies. The resonant frequencies of a system can be identified when the response to an external vibration creates an amplitude that is a relative maximum within the system. Small periodic forces that are near a resonant frequency of the system have the ability to produce large amplitude oscillations in the system due to the storage of vibrational energy . Resonance phenomena occur with all types of vibrations or waves : there

10640-442: The same frequency band. The article next gives the analysis for the series RLC circuit in detail. Other configurations are not described in such detail, but the key differences from the series case are given. The general form of the differential equations given in the series circuit section are applicable to all second order circuits and can be used to describe the voltage or current in any element of each circuit. In this circuit,

10752-455: The same natural frequency and damping ratio as in the capacitor example the transfer function is H ( s ) = 2 ζ ω 0 s s 2 + 2 ζ ω 0 s + ω 0 2 . {\displaystyle H(s)={\frac {2\zeta \omega _{0}s}{s^{2}+2\zeta \omega _{0}s+\omega _{0}^{2}}}.} This transfer function also has

10864-451: The same natural frequency and damping ratios as the previous examples, the transfer function is H ( s ) = s 2 + ω 0 2 s 2 + 2 ζ ω 0 s + ω 0 2 . {\displaystyle H(s)={\frac {s^{2}+\omega _{0}^{2}}{s^{2}+2\zeta \omega _{0}s+\omega _{0}^{2}}}.} This transfer has

10976-669: 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 G ( ω ) = 2 ζ ω 0 ω ( 2 ω ω 0 ζ ) 2 + ( ω 0 2 − ω 2 ) 2 . {\displaystyle G(\omega )={\frac {2\zeta \omega _{0}\omega }{\sqrt {\left(2\omega \omega _{0}\zeta \right)^{2}+(\omega _{0}^{2}-\omega ^{2})^{2}}}}.} The resonant frequency that maximizes this gain

11088-671: The same poles as the previous examples but has zeroes at Evaluating the transfer function along the imaginary axis, its gain is G ( ω ) = ω 0 2 − ω 2 ( 2 ω ω 0 ζ ) 2 + ( ω 0 2 − ω 2 ) 2 . {\displaystyle G(\omega )={\frac {\omega _{0}^{2}-\omega ^{2}}{\sqrt {\left(2\omega \omega _{0}\zeta \right)^{2}+(\omega _{0}^{2}-\omega ^{2})^{2}}}}.} Rather than look for resonance, i.e., peaks of

11200-623: 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

11312-410: 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

11424-1065: The solution: x ( t ) = A cos ⁡ ( ω t − δ ) + A t r cos ⁡ ( ω 1 t − δ t r ) , {\displaystyle x(t)=A\cos(\omega t-\delta )+A_{tr}\cos(\omega _{1}t-\delta _{tr}),} where A = f 0 2 ( ω 0 2 − ω 2 ) 2 + 4 β 2 ω 2 {\displaystyle A={\sqrt {\frac {f_{0}^{2}}{(\omega _{0}^{2}-\omega ^{2})^{2}+4\beta ^{2}\omega ^{2}}}}} and δ = tan − 1 ⁡ ( 2 β ω ω 0 2 − ω 2 ) {\displaystyle \delta =\tan ^{-1}\left({\frac {2\beta \omega }{\omega _{0}^{2}-\omega ^{2}}}\right)} The second term of x ( t )

11536-413: 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 ω r = ω 0 1 − 2 ζ 2 . {\displaystyle \omega _{r}=\omega _{0}{\sqrt {1-2\zeta ^{2}}}.} ω r

11648-410: The starting point of the masses, this system has 2 possible frequencies (or a combination of the two). If the masses are started with their displacements in the same direction, the frequency is that of a single mass system, because the middle spring is never extended. If the two masses are started in opposite directions, the second, faster frequency is the frequency of the system. More special cases are

11760-451: The swing at a faster or slower tempo produce smaller arcs. This is because the energy the swing absorbs is maximized when the pushes match the swing's natural oscillations. Resonance occurs widely in nature, and is exploited in many devices. It is the mechanism by which virtually all sinusoidal waves and vibrations are generated. For example, when hard objects like metal , glass , or wood are struck, there are brief resonant vibrations in

11872-439: The system. When this occurs, the denominator of the amplitude is minimized, which maximizes the amplitude of the oscillations. The harmonic oscillator and the systems it models have a single degree of freedom . More complicated systems have more degrees of freedom, for example, two masses and three springs (each mass being attached to fixed points and to each other). In such cases, the behavior of each variable influences that of

11984-1087: The three components are all in series with the voltage source . The governing differential equation can be found by substituting into Kirchhoff's voltage law (KVL) the constitutive equation for each of the three elements. From the KVL, where V R , V L and V C are the voltages across R , L , and C , respectively, and V ( t ) is the time-varying voltage from the source. Substituting V R = R   I ( t ) , {\displaystyle V_{R}=R\ I(t)\,,} V L = L d I ( t ) d t {\displaystyle \,V_{\mathrm {L} }=L{\frac {\mathrm {d} I(t)}{\mathrm {d} t}}\,} and V C = V ( 0 ) + 1 C ∫ 0 t I ( τ ) d τ {\displaystyle \,V_{\mathrm {C} }=V(0)+{\frac {1}{\,C\,}}\int _{0}^{t}I(\tau )\,\mathrm {d} \tau \,} into

12096-407: The two, but resonance frequency unqualified usually means the driven resonance frequency. The driven frequency may be called the undamped resonance frequency or undamped natural frequency and the peak frequency may be called the damped resonance frequency or the damped natural frequency. The reason for this terminology is that the driven resonance frequency in a series or parallel resonant circuit has

12208-413: The undamped resonance frequency, damped resonance frequency and driven resonance frequency can all be different. Damping is caused by the resistance in the circuit. It determines whether or not the circuit will resonate naturally (that is, without a driving source). Circuits that will resonate in this way are described as underdamped and those that will not are overdamped. Damping attenuation (symbol α )

12320-509: The value. This is exactly the same as the resonance frequency of a lossless LC circuit – that is, one with no resistor present. The resonant frequency for a driven RLC circuit is the same as a circuit in which there is no damping, hence undamped resonant frequency. The resonant frequency peak amplitude, on the other hand, does depend on the value of the resistor and is described as the damped resonant frequency. A highly damped circuit will fail to resonate at all, when not driven. A circuit with

12432-835: The voltage drop across the resistor equals the amplitude of v in , 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 V out ( s ) = ( s L + 1 s C ) I ( s ) , {\displaystyle V_{\text{out}}(s)=(sL+{\frac {1}{sC}})I(s),} V out ( s ) = s 2 + 1 L C s 2 + R L s + 1 L C V in ( s ) . {\displaystyle V_{\text{out}}(s)={\frac {s^{2}+{\frac {1}{LC}}}{s^{2}+{\frac {R}{L}}s+{\frac {1}{LC}}}}V_{\text{in}}(s).} Using

12544-641: The zeroes of the transfer function, which were shown in Equation ( 7 ) and were on the imaginary axis. Oscillation Oscillation is the repetitive or periodic variation, typically in time , of some measure about a central value (often a point of equilibrium ) or between two or more different states. Familiar examples of oscillation include a swinging pendulum and alternating current . Oscillations can be used in physics to approximate complex interactions, such as those between atoms. Oscillations occur not only in mechanical systems but also in dynamic systems in virtually every area of science: for example

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