Electric circuits lumped

As already mentioned, the lumped-circuit technique is the most widely used by the research community [16]. The dielectric response is measured in parallel-plate or coaxial geometry as an association of resistances (R) and capacitances (C) in parallel or series. In a.c. measurements only, the parallel and series electrical circuits can be forced to be equivalent [20] giving for the admittance, T((d) and impedance Z(co), the following equations ... 

Electrical circuits with lumped components are often used as models mimicking the electrical properties of tissue. The simplest models are with three components. If all components are ideal (not frequency-dependent values), the model is a Debye model. 

Time-resolved measurements on linear systems may be represented in many ways. For example, in the Nyquist representation, the transfer function H m) is plotted as a point in a two-dimensional plane having coordinates [Re(//) and Im(//)j, for each frequency measured. In specific cases, the transfer function may be represented also by an equivalent electrical circuit. This is a combination of lumped circuit elements (resistor, capacitor, etc.) having the same perturbation-response behavior as the system studied. 

The paper also presented an equivalent electric circuit for the Cole—Cole equation. The permittivity was modeled as two ideal, lumped capacitors and one frequency-dependent impedance (not physically realizable) modeled as a CPE. They stressed that this impedance was merely one way of expressing the experimental facts, and that it and its real and imaginary parts have no conventional meaning. The constant phase impedance was purely a descriptive model. 

ABSTRACT State determination of Li-ion cells is often accomplished with Electrochemical Impedance Spectroscopy (EIS). The measurement results are in frequency domain and used to describe the state of a Li-ion cell by parameterizing impedance-based models. Since EIS is a costly measurement method, an alternative method for the parameterization of impedance-based models with time-domain data easier to record is presented in this work. For this purpose the model equations from the impedance-based models are transformed from frequency domain into time domain. As an excitation signal a current step is applied. The resulting voltage step responses are the model equations in time domain. They are presented for lumped and derived for distributed electrical circuit elements, i.e. Warburg impedance, Constant Phase Element and RCPE. A resulting technique is the determination of the inner resistance from an impedance spectrum which is performed on measurement data. 

Figure 4.10. Electrical equivalent of microelectrode (adapted from Amatniek, 1958, Figure 12, p. 9). (a) Basic system, (b) model of microelectrode and single cell penetrated by electrode, (c) electrical circuit model of (b), (d) lumped parameter equivalent of (c).

Averaging methods. Equivalent resistance calculations in simple electric circuits is based on appropriate use of lumped or averaged properties. Similar results are desired in petroleum engineering, but in three widely used simulators we evaluated, averaging techniques are systematically abused. Formulas that are derived for linear (vs. cylindrical or spherical) flow under constant density, single-phase, identical-block-size assumptions are indiscriminately employed to process intermediate results in compressible, multiphase, variable grid block runs, leading to questionable results. 

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]. 

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]... 

Experimental results for Debye lumped circuit with Tq = 10 s (a) using l-coefficient transform, (b) using 5-coefficient transform (Reproduce by permission from Proc. Inst. Electrical a Electronic Eng., 1970, 117,1891)... 

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... 

While in lumped-circuit methods the dielectric response is measured in the frequency domain, following the ajpplication of a sinusoidal alternating electrical field, for frequencies below 10 Hz it is advantageous to cany out the measurements in the time domain because it is less time consuming. The polarization or depolarization current following the application of a step-like electrical field is measured as a function of time. 

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.

He discussed the three-component electric equivalent circuit with two resistors (one ideal, lumped, physically realizable electronic component one frequency-dependent not realizable) and a capacitor (frequency-dependent) in two different configurations. He discussed his model first as a descriptive model, but later discussed Philippson s explanatory interpretation (extra-/intracellular liquids and cell membranes). 

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). 

---