**Lumped Element Model**

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LumpedElement. Modeling. An ideal lumped element is not realizable even at lower microwave frequencies because of the associated parasitic reactances due to fringing fields. At RF and microwave frequencies, each component has associated electric and magnetic fields and finite dissipative loss. Thus, such components store or release electric and magnetic energies across them and their resistance accounts for the dissipated power. The relative values of the C, L, and R The frequency response function of the circuit is derived to obtain an expression for Q(sub out)/V(sub AC), the volume flow rate per applied voltage.Both a transmissionline model and its simpler variant, a lumpedelement model, can be used to predict the responses of a thicknessshearmode.quartz resonator sensor.The frequency response function of the circuit is derived to obtain an expression for the volume flow rate through the orifice per applied voltage across the piezoceramic.At this point it is worth noting that the passivity of the Efield finite element model follows from this result since (11.96) is derived from (11.117) through the elimination of an, . Passivity of the discrete model is important since, as shown earlier in this chapter, application of the PRIMA process to a passive statespace model leads to a guaranteed passive reducedorder model. The passive reduction of (11.1 17) using PRIMA is discussed next. 11.5.2 Incorporation of lumped elements The Specifically, distributed physical elements.must be represented by partial differential equations, which are in general very difficult to solve, while lumped elements can be represented by ordinary differential equations, which are relatively simple to solve. A singlelump model may be adequate. Consider, for example, the cantilever spring with a large mass on the end shown in Fig. 1.6o. If the end mass is much greater than that of the spring itself, it is common to consider the spring to be The simplest lumpedelement approximation of the quarter wavelength resonator is shown in Fig.7.1b. We assign compliance Creso to the left half of the resonator, be a) Q H T H Stack b) ∆x reso C reso L reso C reso L reso R visc R therm L rad R rad c) H T A d) Fig.7.1a–d Quarterwavelength.resonator. (a) A quarter wavelength resonator: a tube closed at one end and open at the other. (b) A simple lossless lumpedelement model of the resonator as a compliance Creso in series however, have come up with interesting variations on these lumped element models, and this section will outline those methods. 3.3.1 Lumped element Model The lumped elements outlined in the previous section can be used in creating a simple model useful in modeling PIN diode behavior in circuits. Figure 3.8 shows the models separately for the two device states: dc forward bias (or on) and dc reverse bias (or off). For the onstate, RS is usually much smaller than the open circuit