Resonating lumped circuits

The change in rf voltage at the detector due to a change in the loaded Q may be calculated easily from the equivalent lumped circuit for the resonator, where the resonance absorption may be modeled as a small series resistance 8R. The detector voltage is... 

It is our thesis that the loop-gap lumped circuit resonator introduced recently by us will eventually supplant microwave cavity resonators in ESR spectroscopy except for a few specialized applications [53,291-293], Figure 24 (from Ref. 291) shows this resonator. In a sense, this is a hybrid structure midway between low-frequency lumped circuits where a capacitor and an inductor are connected by a transmission line, and high-frequency distributed circuit cavity resonators where the electric and magnetic... 

Since the transverse shear wave may penetrate the damping surface layer and the viscous liquid, additivity of the equivalent electrical elements in the BVD circuit is only valid under certain particular conditions. Martin and Frye [53] studied the impedance near resonance of polymer film coated resonators in air with a lumped-element BVD model, modified to account for the viscoelastic properties of the film. In addition to the elements shown in Fig. 12.3 to describe the quartz crystal and the liquid, L/ and Rf were added to describe the viscoelastic film overlayer. For a small... 

One can show [42] that, when the surface mechanical impedance is not large, the distributed model in the vicinity of resonance (where we make measurements) can be reduced to the simpler lumped-element model of Fig. 13.8(b). This modified Butterworth-van Dyke (BVD) electrical equivalent circuit comprises parallel static and motional arms. The static... 

Figure 3.5 Equivalent-circuit models to describe the near-resonant electrical characteristics of the resonator (a) distributed model (b) lumped-element model. (Reprinted with permission. See Refs. [7 14J. (a) 1994 American Institute of Physics and (b) 1993 American Chemical Society.)...

The equivalent circuits (Figure 3.5) can be used to describe the electrical response of the perturbed device. The lumped-element model. Figure 3.Sb, is most convenient to use. When the resonator has a surface perturbation, the motional impedance increases, as represented by the equivalent-circuit model of Figure 3.7. This model contains the elements C , Li, C, and Ri corresponding to the unperturbed resonator. In addition, the surface perturbation causes an increase in the motional impedance Z(n as described by the complex electrical element Ze in Figure 3.7a. This element is given by [12]... 

Figure 3.7 Lumped-element equivalent-circuit models for die peituibed resonator [14] ...

We see that reducing the beam waist increases We note that is more weakly dependent on P. We will derive an expression for he field at the sample in a Fabry-Perot resonator in Section VIII after we have developed the appropriate lumped equivalent circuit for a transmission mode spectrometer. 

Fig. 6. Lumped equivalent circuit for a transmission mode spectrometer near a resonance. Kj is the output voltage of the millimeter wave source, Rj is the source resistance, and Cf are the equivalent inductance and capacitance of the resonator, is the resonator resistance, and the load resistance of the detector. The mutual inductances M, and M2 model the coupling into and out of the resonator.

We may write a lumped equivalent circuit for the resonator and coupled transmission lines following the prescription in RLS-8 (Montgomery et al., 1948) as shown in Fig. 6. At resonance, the power Pi into the load... 

Let us now calculate the that we can achieve at the sample. We may use the equivalent lumped parameter circuit in Fig. 6 for this purpose. The power dissipated in the resonator may be written in terms of measurable quantities as... 

Fig. 2 Electrical equivalent circuit models for a TSM resonator (a) transmission line model (TLM) and (b) Butterworth-vanDyke lumped element model (LEM). Circuit elements are defined in the main text.

Passive oscillator mode Impedance analysis of the forced oscillation of the quartz plate provides valuable information about the coating even if the active mode is not applicable anymore. For impedance analysis, a frequency generator is used to excite the crystal to a constraint vibration near resonance while monitoring the complex electrical impedance and admittance, respectively, dependent on the applied frequency (Figure 2B). For low load situations near resonance, an equivalent circuit with lumped elements - the so-called Butterworth—van-Dyke (BVD) circuit — can be applied to model the impedance data. The BVD circuit combines a parallel and series (motional branch) resonance circuit. The motional branch consists of an inductance Lq, a capacitance Cq, and a resistance Rq. An additional parallel capacitance Co arises primarily from the presence of the dielectric quartz material between the two surface electrodes (parallel plate capacitor) also containing parasitic contributions of the wiring and the crystal holder (Figure 2B). 

Equivalent circuit modeling is a useful and commonly applied analysis approach for designing piezoelectric microdispenser systems. The method has limited accuracy, and the lumped analysis approach requires the wavelength of the input signal to be larger than any characteristic length within the device. In a microdispenser with an internal cavity, for example, the Hemholtz resonance... 

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