Transmission line models

Figure 16. General transmission-line model for a conducting polymer-coated electrode. CF is the faradaic pseudo-capacitance of the polymer film, while Rt and Rt are its electronic and ionic resistance, respectively. R, is the uncompensated solution resistance.

Surface roughness is also expected to result in depression of the capacitance semi-circle. This phenomenon, which is indeed apparent in both Figures 1 and 2, is, however, unrelated to surface area. Rather, it is attributable to surface heterogeneity, i.e. the surface is characterized by a distribution of properties. Macdonald (16) recently reviewed techniques for representing distributed processes. A transmission line model containing an array of parallel R/C units with a distribution of values is physically attractive, but not practical. An alternative solution is introduction of an element which by its very nature is distributed. The Constant Phase Element (CPE) meets such a requirement. It has the form P = Y0 wn... 

Hoefer, U. Steiner, K. Wagner, E. Contact and sheet resistances of Sn02 thin films from transmission line model measurements. Sensors and Actuators B (1995), p. 59-63. 

Many interesting phenomena can arise in nonlinear periodic structures that possess the Kerr nonlinearity. For analytic description of such effects, the slowly varying amplitude (or envelope) approximation is usually applied. Alternatively, in order to avoid any approximation, we can use various numerical methods that solve Maxwell s equations or the wave equation directly. Examples of these rigorous methods that were applied to the modelling of nonlinear periodical structures are the finite-difference time-domain method, transmission-line modelling and the finite-element frequency-domain method." ... 

Physical Modelling. The last method of synthesis, physical modeling, is the modeling of musical instruments by their simulating their acoustic models. One popular model is the acoustic transmission line (discussed by Smith in his chapter), where a non-linear source drives the transmission line model. Waves are propagated down the transmission line until discontinuities (represented by nodes of impedance mismatches) are found and reflected waves are introduced. The transmission lines can be implemented with lattice filters. 

Song et al.172 theoretically calculated the impedance spectra based upon Eq. (34) with such distribution functions of PSD as normal, lognormal, Lorcntzian, log Lorentzian distributions. They concluded that the impedance spectra simulated based upon the transmission line model (TLM) with different PSD functions share a common point that the wider PSD leads to the more frequency dispersion in the impedance spectra. 

Figure 10. Nyquist plot of the impedance spectrum experimentally measured on the ACFCE at an applied potential of 0.1 V (vs. SCE) in a 30 wt % H2SO4 solution. Dotted and solid lines represent the impedance spectra theoretically calculated based upon the transmission line model (TLM) in consideration of pore size distribution (PSD) and pore length distribution (PLD), respectively. Reprinted with permission from G. -J. Lee, S. -I. Pyun, and C. -H. Kim, J. Solid State Electrochem., 8 (2004) 110. Copyright 2003, with kind permission of Springer Science and Business Media.

The first resistance Rs is the resistance of the electrolyte outside the pores the R, elements are the electrolyte resistances inside the pores of the electrode and are the double layer capacitances along the pores. This model is called the Transmission Line Model (TLM) by De Levie. A careful selection of a set of Rv C values allows to calculate back the experimental plot such as the one presented in Figure 1.23 [28]. It can be noted that constant phase element (CPE) can be used to replace the capacitance C for better fitting, the CPE impedance ZCPE being ZCPE = l//(Cco) . 

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]. 

Quantitatively, we proceed via the use of equivalent circuit models. The most general model is the distributed transmission line model of Fig. 

Fig. 13.8. Equivalent circuit models for crystal impedance responses (a) transmission line model (b) lumped clement (modified Butterworth van Dyke) model.

Albery WJ, Elliot CM, Mount AR (1990) A transmission line model for modified electrodes and thin layer cells. J Electroanal Chem 288 15-34... 

Eloot etal. suggested a new general matrix method for calculations involving noncylindrical pores, in which the pore is divided into sections and for each section a transmission line model with constant impedances is used. Direct simulations of the impedances for porous electrodes were also carried out using a random walk method. ... 

C. Christopoulos, The Transmission-Line Modeling Method TLM. Piscataway, NJ IEEE Press, 1995. 

Naturally, electrical engineers have designed equivalent circuits for nonelectrical wave phenomena. The waves may or may not be confined to cables. For simple propagating waves, the equivalent circuits are often called transmission line models. The transmission line has two ports representing input and output. The input-output relation can be predicted by applying the Kirchhoff laws to the set of elements located in between. The circuit elements may be simple resistors or capacitors, but their electrical impedance may also be a more complicated function of frequency (see, for instance. Fig. 6)... 

The frequency dispersion of porous electrodes can be described based on the finding that a transmission line equivalent circuit can simulate the frequency response in a pore. The assumptions of de Levi s model (transmission line model) include cylindrical pore shape, equal radius and length for all pores, electrolyte conductivity, and interfacial impedance, which are not the function of the location in a pore, and no curvature of the equipotential surface in a pore is considered to exist. The latter assumption is not applicable to a rough surface with shallow pores. It has been shown that the impedance of a porous electrode in the absence of faradaic reactions follows the linear line with the phase angle of 45° at high frequency and then... 

MattiKarjalainen, Unto K. Laine, Timo I. Laakso, and Vesa Valimaki. Transmission-Line Modeling and Real-Time Synthesis of String and Wind Instruments. Proceedings of the International Computer Music Conference 293-296 (1991). 

Reeves, G. and Harrison, H., Obtaining the specific contact resistance from transmission line model measurements , IEEE Electron Device Lett., 1982, 3, 111-13. 

As mentioned above, reduction in model order is typically achieved by combining segmental and transmission line models. The idea here is to represent critical, nonlinear components by segments, while the less critical, more linear components are lumped together and represented by Westkessel terminations. 

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