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...unction at ansarbitrary frequency with arbitrary peak magnitude.sSolving (6) for Z2 and replacing Q2/P2 with a constant, xo,swhich specifies the height of the resonant peak, andsletting s = jωCsgivess=-=(7)-=-swhere ωC is a scalar value that represents the frequencysof the desired resonant peak.sThis impedance can besseparated into real and imaginary parts by multiplyingsthe complex conjugate of the denomi...

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...ntroller.sNote the controller outputs a Q2 for an inputsP2, therefore the controller transfer function is thesinverse of Z2.sThe control law is given bys(8)swhere the gains KP and KI are given bys(9)s=-=(10)-=-sThis produces an analytical controller design based onscomputing the controller gains KP and KI from thesrelation for (7) for desired values of ωC and x0.sThe closed loop transfer function in terms o...

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...esponse tosthe wave as a solid body with effective inertias(2)swhere le and S are the effective length and crossssectional area of the neck. When dissipation is small,sresonance occurs at a frequencys=-=(3)-=-sSumming the forces on the inertia produces a secondsorder differential equation relating the pressure, P1s(N/m2), at the entrance of the neck to the volumetric flowsrate, or "volume velocity,"s(m3/s)...

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... co is the speed of sound in the medium.sFigure 1: Ideal Helmholtz Resonator (Birdsong, 1999)sThe mass of air in the neck will oscillate in response tosthe wave as a solid body with effective inertias=-=(2)-=-swhere le and S are the effective length and crossssectional area of the neck. When dissipation is small,sresonance occurs at a frequencys(3)sSumming the forces on the inertia produces a secondsorder ...

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...(N/m2), at the entrance of the neck to the volumetric flowsrate, or "volume velocity,"s(m3/s). Temkin's model issextended here by converting the differential equationsinto the transfer function models=-=(4)-=-swhere Ra is the resistance due to radiation losses andsviscous damping of the medium.sThe ideal Helmholtz resonator model can be modified bysadding a boundary condition to the cavity interiorssurface...

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...on that occurs with frequency in an integralscontroller.sNote the controller outputs a Q2 for an inputsP2, therefore the controller transfer function is thesinverse of Z2.sThe control law is given bys=-=(8)-=-swhere the gains KP and KI are given bys(9)s(10)sThis produces an analytical controller design based onscomputing the controller gains KP and KI from thesrelation for (7) for desired values of ωC and ...

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...lscontroller.sNote the controller outputs a Q2 for an inputsP2, therefore the controller transfer function is thesinverse of Z2.sThe control law is given bys(8)swhere the gains KP and KI are given bys=-=(9)-=-s(10)sThis produces an analytical controller design based onscomputing the controller gains KP and KI from thesrelation for (7) for desired values of ωC and x0.sThe closed loop transfer function in te...

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... dimensions of the cavity weressmall compared with the wavelength of the incidentswave.sThe cavity creates an acoustic compliance thatscan be computed from the physical dimensions of thesresonator ass=-=(1)-=-swhere V is the cavity volume, ρo is the density of thesmedium, and co is the speed of sound in the medium.sFigure 1: Ideal Helmholtz Resonator (Birdsong, 1999)sThe mass of air in the neck will oscill...

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...odel can be modified bysadding a boundary condition to the cavity interiorssurface, relating the surface volume velocity, Q2, to thespressure acting on the surface, P2 (Radcliffe andsGogate, 1994) ass=-=(5)-=-swhere Z2 is an arbitrary acoustic impedance.sFigure 2sshows a Helmholtz resonator with this boundaryscondition added.sFigure 2.sSemi-Active Helmholtz Resonator (SHR) withsimpedance boundary condition...

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...Figure 2.sSemi-Active Helmholtz Resonator (SHR) withsimpedance boundary condition, Z2 (Birdsong, 1999)sThe SHR transfer function resulting from thissmodification can then be found (Birdsong, 1999) iss=-=(6)-=-sThe SHR uses a feedback control system to modify thesresonant dynamic response of the system by specifyingsZ2.sIn particular, a specific value of Z2 can be found thatsproduces a resonance in the Q1/P...