Misplaced Pages

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79-403: The Q factor is a parameter that describes the resonance behavior of an underdamped harmonic oscillator (resonator). Sinusoidally driven resonators having higher Q factors resonate with greater amplitudes (at the resonant frequency) but have a smaller range of frequencies around that frequency for which they resonate; the range of frequencies for which the oscillator resonates is called

158-565: 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,

237-410: 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

316-420: 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

395-533: A 3 dB bandwidth of 10 kHz. In audio, bandwidth is often expressed in terms of octaves . Then the relationship between Q and bandwidth is Q = 2 B W 2 2 B W − 1 = 1 2 sinh ⁡ ( 1 2 ln ⁡ ( 2 ) B W ) , {\displaystyle Q={\frac {2^{\frac {BW}{2}}}{2^{BW}-1}}={\frac {1}{2\sinh \left({\frac {1}{2}}\ln(2)BW\right)}},} where BW

474-472: A body in circular motion travels a distance v T {\displaystyle vT} . This distance is also equal to the circumference of the path traced out by the body, 2 π r {\displaystyle 2\pi r} . Setting these two quantities equal, and recalling the link between period and angular frequency we obtain: ω = v / r . {\displaystyle \omega =v/r.} Circular motion on

553-451: A circuit where R , L , and C are all in parallel. The lower the parallel resistance is, the more effect it will have in damping the circuit and thus result in lower Q . This is useful in filter design to determine the bandwidth. In a parallel LC circuit where the main loss is the resistance of the inductor, R , in series with the inductance, L , Q is as in the series circuit. This is a common circumstance for resonators, where limiting

632-416: 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

711-440: A freely oscillating system's energy to fall off to e , or about 1 ⁄ 535 or 0.2%, of its original energy. This means the amplitude falls off to approximately e or 4% of its original amplitude. The width (bandwidth) of the resonance is given by (approximately): Δ f = f N Q , {\displaystyle \Delta f={\frac {f_{\mathrm {N} }}{Q}},\,} where f N

790-477: 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

869-437: A long time after being struck by a hammer. For an electrically resonant system, the Q factor represents the effect of electrical resistance and, for electromechanical resonators such as quartz crystals , mechanical friction . The 2-sided bandwidth relative to a resonant frequency of F 0 (Hz) is F 0 / Q . For example, an antenna tuned to have a Q value of 10 and a centre frequency of 100 kHz would have

SECTION 10

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948-848: 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 )

1027-458: 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 the same force is applied at other, non-resonant frequencies. The resonant frequencies of

1106-459: A perfect capacitor. Q L = X L R L = ω 0 L R L {\displaystyle Q_{L}={\frac {X_{L}}{R_{L}}}={\frac {\omega _{0}L}{R_{L}}}} where: Resonance Resonance is a phenomenon that occurs when an object or system is subjected to an external force or vibration that matches its natural frequency . When this happens,

1185-421: A series LC circuit equals the square root of the reciprocal of the product of the capacitance ( C , with SI unit farad ) and the inductance of the circuit ( L , with SI unit henry ): ω = 1 L C . {\displaystyle \omega ={\sqrt {\frac {1}{LC}}}.} Adding series resistance (for example, due to the resistance of the wire in a coil) does not change

1264-475: 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

1343-402: 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

1422-418: 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

1501-522: 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

1580-474: 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

1659-445: Is approximately the ratio of the stored energy to the energy dissipated over one radian of the oscillation; or nearly equivalently, at high enough Q values, 2 π times the ratio of the total energy stored and the energy lost in a single cycle. It is a dimensionless parameter that compares the exponential time constant τ for decay of an oscillating physical system's amplitude to its oscillation period . Equivalently, it compares

SECTION 20

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1738-572: Is important (such as dampers keeping a door from slamming shut) have Q near 1 ⁄ 2 . Clocks, lasers, and other resonating systems that need either strong resonance or high frequency stability have high quality factors. Tuning forks have quality factors around 1000. The quality factor of atomic clocks , superconducting RF cavities used in accelerators, and some high- Q lasers can reach as high as 10 and higher. There are many alternative quantities used by physicists and engineers to describe how damped an oscillator is. Important examples include:

1817-528: Is normally presented in the unit radian per second . The unit hertz (Hz) is dimensionally equivalent, but by convention it is only used for frequency  f , never for angular frequency  ω . This convention is used to help avoid the confusion that arises when dealing with quantities such as frequency and angular quantities because the units of measure (such as cycle or radian) are considered to be one and hence may be omitted when expressing quantities in terms of SI units. In digital signal processing ,

1896-548: Is referred to as the natural angular frequency (sometimes be denoted as ω 0 ). As the object oscillates, its acceleration can be calculated by a = − ω 2 x , {\displaystyle a=-\omega ^{2}x,} where x is displacement from an equilibrium position. Using standard frequency f , this equation would be a = − ( 2 π f ) 2 x . {\displaystyle a=-(2\pi f)^{2}x.} The resonant angular frequency in

1975-459: 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

2054-585: Is the angular frequency at which the stored energy and power loss are measured. This definition is consistent with its usage in describing circuits with a single reactive element (capacitor or inductor), where it can be shown to be equal to the ratio of reactive power to real power . ( See Individual reactive components .) The Q factor determines the qualitative behavior of simple damped oscillators. (For mathematical details about these systems and their behavior see harmonic oscillator and linear time invariant (LTI) system .) In negative feedback systems,

2133-936: Is the natural frequency , and Δ f , the bandwidth , is the width of the range of frequencies for which the energy is at least half its peak value. The resonant frequency is often expressed in natural units (radians per second), rather than using the f N in hertz , as ω N = 2 π f N . {\displaystyle \omega _{\mathrm {N} }=2\pi f_{\mathrm {N} }.} The factors Q , damping ratio ζ , natural frequency ω N , attenuation rate α , and exponential time constant τ are related such that: Q = 1 2 ζ = ω N 2 α = τ ω N 2 , {\displaystyle Q={\frac {1}{2\zeta }}={\frac {\omega _{\mathrm {N} }}{2\alpha }}={\frac {\tau \omega _{\mathrm {N} }}{2}},} and

2212-409: 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

2291-468: 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

2370-455: Is the bandwidth in octaves. In an ideal series RLC circuit , and in a tuned radio frequency receiver (TRF) the Q factor is: Q = 1 R L C = ω 0 L R = 1 ω 0 R C {\displaystyle Q={\frac {1}{R}}{\sqrt {\frac {L}{C}}}={\frac {\omega _{0}L}{R}}={\frac {1}{\omega _{0}RC}}} where R , L , and C are

2449-403: Is the frequency-to-bandwidth ratio of the resonator: Q = def f r Δ f = ω r Δ ω , {\displaystyle Q\mathrel {\stackrel {\text{def}}{=}} {\frac {f_{\mathrm {r} }}{\Delta f}}={\frac {\omega _{\mathrm {r} }}{\Delta \omega }},} where f r

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2528-430: Is the magnitude of the pseudovector quantity angular velocity . Angular frequency can be obtained multiplying rotational frequency , ν (or ordinary frequency , f ) by a full turn (2 π radians ): ω = 2 π rad⋅ ν . It can also be formulated as ω = d θ /d t , the instantaneous rate of change of the angular displacement , θ , with respect to time,  t . In SI units , angular frequency

2607-657: Is the ratio of the energy stored in the oscillating resonator to the energy dissipated per cycle by damping processes: Q = def 2 π × energy stored energy dissipated per cycle = 2 π f r × energy stored power loss . {\displaystyle Q\mathrel {\stackrel {\text{def}}{=}} 2\pi \times {\frac {\text{energy stored}}{\text{energy dissipated per cycle}}}=2\pi f_{\mathrm {r} }\times {\frac {\text{energy stored}}{\text{power loss}}}.} The factor 2 π makes Q expressible in simpler terms, involving only

2686-445: Is the resonant frequency Δ f is the resonance width or full width at half maximum (FWHM) i.e. the bandwidth over which the power of vibration is greater than half the power at the resonant frequency, ω r = 2 πf r is the angular resonant frequency, and Δ ω is the angular half-power bandwidth. Under this definition, Q is the reciprocal of fractional bandwidth . The other common nearly equivalent definition for Q

2765-412: 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

2844-708: The damping ratio , relative bandwidth , linewidth and bandwidth measured in octaves . The concept of Q originated with K. S. Johnson of Western Electric Company 's Engineering Department while evaluating the quality of coils (inductors). His choice of the symbol Q was only because, at the time, all other letters of the alphabet were taken. The term was not intended as an abbreviation for "quality" or "quality factor", although these terms have grown to be associated with it. The definition of Q since its first use in 1914 has been generalized to apply to coils and condensers, resonant circuits, resonant devices, resonant transmission lines, cavity resonators, and has expanded beyond

2923-479: The resistance , inductance and capacitance of the tuned circuit, respectively. Larger series resistances correspond to lower circuit Q values. For a parallel RLC circuit, the Q factor is the inverse of the series case: Q = R C L = R ω 0 L = ω 0 R C {\displaystyle Q=R{\sqrt {\frac {C}{L}}}={\frac {R}{\omega _{0}L}}=\omega _{0}RC} Consider

3002-923: 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

3081-458: 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,

3160-467: The bandwidth. Thus, a high- Q tuned circuit in a radio receiver would be more difficult to tune, but would have more selectivity ; it would do a better job of filtering out signals from other stations that lie nearby on the spectrum. High- Q oscillators oscillate with a smaller range of frequencies and are more stable. The quality factor of oscillators varies substantially from system to system, depending on their construction. Systems for which damping

3239-399: 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

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3318-432: 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,

3397-428: The coefficients of the second-order differential equation describing most resonant systems, electrical or mechanical. In electrical systems, the stored energy is the sum of energies stored in lossless inductors and capacitors ; the lost energy is the sum of the energies dissipated in resistors per cycle. In mechanical systems, the stored energy is the sum of the potential and kinetic energies at some point in time;

3476-469: 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

3555-1082: The damping ratio can be expressed as: ζ = 1 2 Q = α ω N = 1 τ ω N . {\displaystyle \zeta ={\frac {1}{2Q}}={\alpha \over \omega _{\mathrm {N} }}={1 \over \tau \omega _{\mathrm {N} }}.} The envelope of oscillation decays proportional to e or e , where α and τ can be expressed as: α = ω N 2 Q = ζ ω N = 1 τ {\displaystyle \alpha ={\omega _{\mathrm {N} } \over 2Q}=\zeta \omega _{\mathrm {N} }={1 \over \tau }} and τ = 2 Q ω N = 1 ζ ω N = 1 α . {\displaystyle \tau ={2Q \over \omega _{\mathrm {N} }}={1 \over \zeta \omega _{\mathrm {N} }}={\frac {1}{\alpha }}.} The energy of oscillation, or

3634-737: 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

3713-427: 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

3792-633: The dominant closed-loop response is often well-modeled by a second-order system. The phase margin of the open-loop system sets the quality factor Q of the closed-loop system; as the phase margin decreases, the approximate second-order closed-loop system is made more oscillatory (i.e., has a higher quality factor). Q Ω = R w δ 1 − m 2 v m , p 2 , {\displaystyle Q_{\Omega }={\frac {R_{\mathrm {w} }}{\delta }}{\frac {1-m^{2}}{v_{m,p}^{2}}},} Physically speaking, Q

3871-417: The electronics field to apply to dynamical systems in general: mechanical and acoustic resonators, material Q and quantum systems such as spectral lines and particle resonances. In the context of resonators, there are two common definitions for Q , which are not exactly equivalent. They become approximately equivalent as Q becomes larger, meaning the resonator becomes less damped. One of these definitions

3950-474: 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

4029-406: The frequency at which a system oscillates to the rate at which it dissipates its energy. More precisely, the frequency and period used should be based on the system's natural frequency, which at low Q values is somewhat higher than the oscillation frequency as measured by zero crossings. Equivalently (for large values of Q ), the Q factor is approximately the number of oscillations required for

SECTION 50

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4108-403: The frequency may be normalized by the sampling rate , yielding the normalized frequency . In a rotating or orbiting object, there is a relation between distance from the axis, r {\displaystyle r} , tangential speed , v {\displaystyle v} , and the angular frequency of the rotation. During one period, T {\displaystyle T} ,

4187-455: 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

4266-445: 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

4345-419: 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

4424-456: 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

4503-484: The lost energy is the work done by an external force , per cycle, to maintain amplitude. More generally and in the context of reactive component specification (especially inductors), the frequency-dependent definition of Q is used: Q ( ω ) = ω × maximum energy stored power loss , {\displaystyle Q(\omega )=\omega \times {\frac {\text{maximum energy stored}}{\text{power loss}}},} where ω

4582-927: 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

4661-548: 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

4740-496: The object or system absorbs energy from the external force and starts vibrating with a larger amplitude . Resonance can occur in various systems, such as mechanical, electrical, or acoustic systems, and it is often desirable in certain applications, such as musical instruments or radio receivers. However, resonance can also be detrimental, leading to excessive vibrations or even structural failure in some cases. All systems, including molecular systems and particles, tend to vibrate at

4819-434: 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

SECTION 60

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4898-622: 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

4977-754: The power dissipation, decays twice as fast, that is, as the square of the amplitude, as e or e . For a two-pole lowpass filter, the transfer function of the filter is H ( s ) = ω N 2 s 2 + ω N Q ⏟ 2 ζ ω N = 2 α s + ω N 2 {\displaystyle H(s)={\frac {\omega _{\mathrm {N} }^{2}}{s^{2}+\underbrace {\frac {\omega _{\mathrm {N} }}{Q}} _{2\zeta \omega _{\mathrm {N} }=2\alpha }s+\omega _{\mathrm {N} }^{2}}}\,} For this system, when Q > ⁠ 1 / 2 ⁠ (i.e., when

5056-405: The resistance of the inductor to improve Q and narrow the bandwidth is the desired result. The Q of an individual reactive component depends on the frequency at which it is evaluated, which is typically the resonant frequency of the circuit that it is used in. The Q of an inductor with a series loss resistance is the Q of a resonant circuit using that inductor (including its series loss) and

5135-413: 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

5214-400: 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

5293-470: 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

5372-455: 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

5451-451: 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

5530-669: 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

5609-671: 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

5688-623: 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

5767-410: 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

5846-413: 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

5925-453: 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

6004-455: The system is underdamped), it has two complex conjugate poles that each have a real part of −α . That is, the attenuation parameter α represents the rate of exponential decay of the oscillations (that is, of the output after an impulse ) into the system. A higher quality factor implies a lower attenuation rate, and so high- Q systems oscillate for many cycles. For example, high-quality bells have an approximately pure sinusoidal tone for

6083-519: The unit circle is given by ω = 2 π T = 2 π f , {\displaystyle \omega ={\frac {2\pi }{T}}={2\pi f},} where: An object attached to a spring can oscillate . If the spring is assumed to be ideal and massless with no damping, then the motion is simple and harmonic with an angular frequency given by ω = k m , {\displaystyle \omega ={\sqrt {\frac {k}{m}}},} where ω

6162-835: 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

6241-486: The zeroes of the transfer function, which were shown in Equation ( 7 ) and were on the imaginary axis. Angular frequency In physics , angular frequency (symbol ω ), also called angular speed and angular rate , is a scalar measure of the angle rate (the angle per unit time) or the temporal rate of change of the phase argument of a sinusoidal waveform or sine function (for example, in oscillations and waves). Angular frequency (or angular speed)

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