Lumped-element model

Figure 2-75. Lumped element models of transmission line electrical characteristics.

The suggested procedure to arrive at this goal is presented in Fig. 3.1. It starts with the transfer of a certain microhotplate layout into a geometry model for a complex FEM simulation. This step is shown in Fig. 3.2 and will be explained in more detail in one of the next sections. A complex 3-d FEM simulation is then performed. The results of this simulation are used to produce a lumped-element model. This model is translated into a hardware description language (HDL). Using the resistances of the device elements such as the heater resistance, Rheat> and the resistance of the temperature sensor, Rx. co-simulations with the circuitry can be performed. 

Using the thermal resistance and the total heat capacitance, the dynamic equation for a lumped-element model in the linear regime can be written as ... 

In the case of viscoelastic loaded QCM two approaches have been followed one methodology is to treat the device as an acoustic transmission line with one driven piezo-electric quartz layer and one or more surface mechanical load (TLM) [50, 51]. A simpler approach is to use a lumped-element model (LEM) that represents mechanical inter-actions by their equivalent electrical BVD circuit components [52, 53]. 

In most electrochemical applications the lumped element model (LEM) is a good approximation within 1% of the transmission line model (TLM) provided the quartz and film impedance condition ZfIZQ < 1 is fulfilled [54]. 

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... 

For the films and conditions we have used, the transmission line and lumped element models give indistinguishable results. Fitting of the data of Fig. 13.7 yields G as a function of time. These values increase at short times (due to nucleation phenomena) to long time limiting values of G = 1.9 x 106 dyne cm-2 and G" = 3.0 x 108 dyne cm-2. These values of the shear modulus components show that, in dichloromethane, the PVF film is a very rubbery polymer in which there is considerable viscoelastic loss when the film thickness exceeds 1 p.m. 

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]... 

There are two electrical equivalent circuits in common usage, the transmission line model (TLM) and a lumped element model (LEM) commonly referred to as the Butterworth-van Dyke (BvD) model these are illustrated in Figs. 2(a and b), respectively. In the TLM, there are two acoustic ports that represent the two crystal faces one is exposed to air (i.e. is stress-free, indicated by the electrical short) and the other carries the mechanical loading (here, a film and the electrolyte solution, represented below by the mechanical loading Zs). These acoustic ports are coimected by a transmission line, which is in turn connected to the electrical circuitry by a transformer representing the piezoelectric coupling. For the TLM, one can show [18, 19] that the motional impedance (Zj ) associated with the surface loading can be related to the mechanical impedances of... 

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.

The realism of the lumped element models is not easy to assert in view of modem theories and measurement techniques, and these models can be regarded as pragmatic explanatory models. They are, however, the only models that differentiate between substmctures of the skin and have clinical value in that they aid in choosing measuring technique. 

Rathnasingham and Breuer [4] developed the first low-order model of a synthetic jet using a control volume model for the flow and an empirical model for the stmctural dynamics of the diaphragm. A lumped element model of a piezoelectric-driven synthetic jet actuator was derived in [5]. 

Gallas Q, Hohnan R, Nishida T, Carroll B, Sheplak M, Cattafesta L (2003) Lumped element modeling of piezoelectric-driven synthetic jet actuators. AIAA J41(2) 240-247... 

Beginning with fundamentals of fluid dynamics, correlations for the pressure loss in channel elements are presented, which are concatenated to fluidic networks to distribute fluid homogeneously over a certain area. Computational fluid dynamic (CFD) simulations of single elements are exploited for analytical pressure loss correlations. These are employed in lumped element modeling of networks and manifolds, which are too complex for direct simulations. Design strategies and methods are presented for charmel networks, manifolds for parallel channels on a plate and headers for stacked-plate devices. 

Murray s law applied to cooling systems results in structures similar to Figure 2.5 and leads to branched systems and devices as displayed in Figure 2.6. The pressure loss in a network can be calculated with lumped element modeling and with the help of electronic circuit layout routines see the next section and Sack et al. [21[. 

Fig. 3(a) Lumped-element model of dry nanogap electrodes (b) Lump-element model of spacer and gap... 

A test method for examining the effect of flux residues at frequencies as high as 10 GHz was developed (Ref 40). A bidirectional coupler was used (Fig. 18) and a lumped element model to... 

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