Impedance lumped elements

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... 

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... 

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]... 

To discuss the results, the sensor is represented as a lossy capacitor, with both the capacitance C and the resistance R depending on frequency (Fig. 4 the frequency dependence of the equivalent-circuit elements is a consequence of the distributed nature of the processes in the sensor, which cannot be modeled appropriately by only two lumped elements with frequency-independent element values.). That simplifies the recognition of even small changes in the impedance, as changes at low frequencies become easily visible in the representation of the resistance R(f) and changes at higher frequencies become even more visible in the representation of the capacitance C(f). 

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... 

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). 

As a first example of CNLS fitting, the circuit shown in Figure 3.3.5 was constructed with lumped elements whose values were measured on an impedance bridge (top figures) (Macdonald, Schoomnan, and Lehnen [1982]). This circuit leads to very little structure in either the Z or T 3-D plots shown in Figures 3.3.6 and 3.3.7. The... 

Figure 1.62. (a) Diagrammatic view of the correlation circuit for photoconductivity cross-correlation measurement in (CH) on broadband microstrip lines, (b) Corresponding circuit diagram. The switches are modeled using the approximation of lumped elements with a static capacitance Cg and a resistor having a time-dependent conductance G(/) in parallel. Z is the characteristic impedance of the transmission lines. (Reprinted with permission from ref. 149)... 

For lumped elements, e.g. resistors, capacitors or combinations of these elements, the differential equations, impedances and VSR are well-known [4]. Distributed elements, i.e. Warburg impedance. Constant Phase Element, or parallel connections like RCPE, also known as ZARC or Cole-Cole element, have non-integer exponents a of the complex frequency s in frequency domain. This corresponds to fractional differential equations in time domain and thus the calculation of the VSR requires fractional calculus, as can be seen in the following derivations. 

Figure 5. The figure shows the R-RC-RC-WB model. Besides the Warburg impedance, this model contains only lumped elements. Its impedance is shown in Equation (19), the VSR in frequency domain is shown in Equation (20) and the VSR in time domain is shown in Equation (21).

A method was proposed for the parameterization of impedance based models in the time domain, by deriving the corresponding time domain model equation with inverse Laplace transform of the frequency domain model equation assuming a current step excitation. This excitation signal has been chosen, since it can be easily applied to a Li-ion cell in an experiment, allows the analytical calculation of the time domain model equation and is included in the definition of the inner resistance. The voltage step responses of model elements were presented for lumped elements and derived for distributed model elements that have underlying fractional differential equations using fractional calculus. The determination of the inner resistance from an impedance spectrum was proposed as a possible application for this method. Tests on measurement data showed that this method works well for temperatures around room temperature and current excitation amplitudes up to 10 C. This technique can be used for comparisons of measured impedance spectra with conventionally determined inner resistances. 

Modeling of High-Speed Interconnections. Modeling the electrical behavior of an interconnection involves two steps. First, the transmission line characteristics, such as the characteristic impedance, propagation constant, capacitance, resistance, dielectric conductance, and coupling parameters, must be calculated from the physical dimensions and material properties of the interconnection. In addition, structures, such as wire bonds, vias, and pins, must be represented by lumped resistance (R), inductance (L), and capacitance (C) elements. 

The disadvantages of IS are primarily associated with possible ambiguities in interpretation. An important comphcation of analyses based on an equivalent circuit (e.g. Bauerle [1969]) is that ordinary ideal circuit elements represent ideal lumped-constant properties. Inevitably, aU electrolytic cells are distributed in space, and their microscopic properties may be also independently distributed. Under these conditions, ideal circuit elements may be inadequate to describe the electrical response. Thus, it is often found that Z/to) cannot be well approximated by the impedance of an equivalent circuit involving only a finite number of ordinary lumped-constant elements. It has been observed by many in the field that the use of distributed impedance elements [e.g. constant-phase elements (CPEs) (see Section 2.2.2.2)] in the equivalent circuit greatly aids the process of fitting observed impedance data for a cell with distributed properties. 

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. 

For commonly used lumped and distributed elements, the differential equations, impedances and VSR are summarized in Table 2. 

This model contains lumped circuit elements and one Warburg impedance, and its parameter set has six elements P= [ R2,R i,R, Ci,C, Q. In the Nyquist plot this model results in two symmetric semicircles and a -45° diffusion branch shifted on real axis with the value of the series resistance R. Figure 5 shows the electrical equivalent circuit and the impedance is given... 

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