Impedance mathematical expression

The mathematical expressions which describe the impedance of some passive circuits are shown below, where a passive circuit is one that does not generate current or potential [129], In this regard, the impedance response of simple passive circuit elements, such as a pure resistor with resistance R, a pure capacitor with capacitance C, and a pure inductor with inductance L, are given, respectively ... 

Let us derive the mathematical expression for the impedance applicable to this circuit, as a function of (0 ... 

Basically, Impedance spectra and admittance spectra contain the same Information It is a matter of experimental and Interpretational convenience to decide which one to use. Moreover, the Imaginmy and real parts of Z and y are pairwise coupled by Kramers-Kron equations (variants of (1.4.4.31 and 321) so that in principle one can be obtained when the other Is known Such a transformation requires very accurate data over a large rringe of frequencies, which for Inorganic solids are not always available, so that for practical purposes these equations have only limited impact. However, mathematical expressions for Z and y should always agree with the Kramers-Kronlg relations, a fact that can be used to validate electrospectroscopic data ). 

The physical imderstanding of the current paths and potential drops in the system serves to guide the structure of the corresponding electrical circuit. The mathematical expression for the interfaciai impedance can be obtained following the development presented in the subsequent chapters. Several examples are given in the following sections to illustrate the procedure. 

For the first time, there was a mathematical expression for the impedance dispersion corresponding to the circular arc found experimentally. The equation introduced a new parameter the somewhat enigmatic constant a. He interpreted a as a measure of molecular interactions, with no interactions a = 1 (ideal capacitor). Comparison was made with the impedance of a semiconductor diode junction (selenium barrier layer photocell). 

Though in the general case, mathematical expressions of the Nernst model are more complicated than of those semi-infinite diffusion, stationary mass transport is described by a rather simple Eq. (3.12). In this connection, there occurs an interesting possibility to use superposition of both models, which is convenient to apply when i is the periodic time function. Perturbation signals of this type are considered in the theory of electrochemical impedance spectroscopy. In this case, i(t)... 

Under this electrochemical configuration, it is commonly accepted that the system can be expressed by the Randles-type equivalent circuit (Fig. 6, inset) [23]. For reactions on the bare Au electrode, mathematical simsulations based on the equivalent circuit satisfactorily reproduced the experimental data. The parameters used for the simulation are as follows solution resistance, = 40 kS2 cm double-layer capacitance, C = 28 /xF cm equivalent resistance of Warburg element, W — R = 1.1 x 10 cm equivalent capacitance of Warburg element, IF—7 =l.lxl0 F cm (

charge-transfer resistance, R = 80 kf2 cm. Note that these equivalent parameters are normalized to the electrode geometrical area. On the other hand, results of the mathematical simulation were unsatisfactory due to the nonideal impedance behavior of the DNA adlayer. This should... 

An important example of the system with an ideally permeable external interface is the diffusion of an electroactive species across the boundary layer in solution near the solid electrode surface, described within the framework of the Nernst diffusion layer model. Mathematically, an equivalent problem appears for the diffusion of a solute electroactive species to the electrode surface across a passive membrane layer. The non-stationary distribution of this species inside the layer corresponds to a finite - diffusion problem. Its solution for the film with an ideally permeable external boundary and with the concentration modulation at the electrode film contact in the course of the passage of an alternating current results in one of two expressions for finite-Warburg impedance for the contribution of the layer Ziayer = H(0) tanh(icard)1/2/(iwrd)1/2 containing the characteristic - diffusion time, Td = L2/D (L, layer thickness, D, - diffusion coefficient), and the low-frequency resistance of the layer, R(0) = dE/dl, this derivative corresponding to -> direct current conditions. 

However, the length of the sample, if it is comparable to the length of the acoustic waves can cause wave interference effects to arise at the sample-water interface and these must be taken into account in a theoretical description. The mathematical treatment of these effects can be found in the literature of transmission line theory which provides the following expression for the impedance of a length 1 of material... 

As described in the subsequent chapters in Part m, models for the impedance response can be developed from proposed hypotheses involving reaction sequences (e.g., Chapters 10 and 12), mass transfer (e.g., Chapters 11 and 15), and physical phenomena (e.g.. Chapters 13 and 14). These models can often be expressed in the mathematical formalism of electrical circuits. Electrical circuits can also be used to construct a framework for accounting for the phenomena that influence the impedance response of electrochemical systems. A method for using electrical circuits is presented in this chapter. 

The mathematical models for the convective-diffusion impedance associated with convective diffusion to a disk electrode are developed here in the context of a generalized framework in which a normalized expression accoxmts for the influence of mass transfer. 

Based upon Faraday s work, James Clerk Maxwell published his famous equations in 1873. He more specifically calculated the resistance of a homogeneous suspension of uniform spheres (also coated, two-phase spheres) as a function of the volume concentration of the spheres. This is the basic mathematical model for cell suspensions and tissues still used today. However, it was not Maxwell himself who in 1873 formulated the four equations we know today as Maxwell s equations. Maxwell used the concept of quaternions, and the equations did not have the modern form of compactness he used 20 equations and 20 variables. It was Oliver Heaviside (1850—1925) who first expressed them in the form we know today. It was also Heaviside who coined the terms impedance (1886), conductance (1885), permeability (1885), admittance (1887), and permittance, which later became susceptance. 

The complete expression of the impedance contains 11 parameters. Based on the mathematical structure of (3.32), the parameters are expected to be strongly correlated. It was therefore indeed found that the number of parameters was decreased basis on reasonable assumptions. However, this was achieved in such a way that the contributions of the individual branches to the total capacity of the film could be determined. Figure 3.13 illustrates the goodness-of-fit. It was concluded that the low-lfequency distortion effect (CPE behavior) is most likely connected with the film s nonuniformity however, the surface roughness of the underlying metal substrate influences the ratio of the long to the short polymer chains. At low frequencies the characteristics of the impedance spectra are mainly determined by the long polymer chains. With the help of these models, reasonable values for the different parameters that characterize the polymer film electrodes can be derived. 

The basic approach consists of defining on a purely mathematical basis a CPE by an impedance or an admittance expressed as a complex number proportional to a power p of the prodnct of the imaginary nnmber times the angular frequency (O, which is eqnivalent to say that the argnment of an exponential fnnction is a multiple of the same product... 

As in Example 2.13, the current is shifted in phase by angle (p and the modulus of the impedance is the same. Such an exercise is mathematically simple but time consuming. It should be added that the program Maple, which allows for symbolic calculations, allows for automatic separation of expressions into simple fractions using the function convert (f, parfrac, s), in which function f of the parameter s is transformed into simple fractions, for example ... 

The evaluation of the non-linear ip(> )-characteristics given by equation (33) is mathematically difficult. Therefore, the static polarisation curves are usually linearised and subjected to Fourier transformation that yields an expression of the Faradaic impedance Zp of the interface for the given operating point. 

The impedance is then represented as a complex number that can also be expressed in complex mathematics as a combination of "real," or in-phase ( reaJ "imaginary," or out-of-phase (Z, ), parts (Figure 1-3B) ... 

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