Transmission line impedance

Bisquert, J., Gratzel, M., Wang, Q., Fabregat-Santiago, F. Three-channel transmission line impedance model for mesoscopic oxide electrodes functitmalized with a conductive coating. J. Phys. Chem. B 110, 11284-11290 (2006)... 

The expression transmission line impedance apphed to a point on a transmission line signifies the vector ratio of line voltage to line current at that particular point. This is the impedance that would be obtained if the transmission line were cut at the point in question, and the impedance looking toward the receiver were measured. 

Adjustment of the loading capacitor couples the 50-t2 transmission-line impedance to the impedance of the cavity. Heavy loading lowers the plate impedance presented to the tube by the cavity. Light loading reflects a higher load impedance to the amplifier plate. 

The overall expression for impedance is generally given by Eqn. 423. The behavior exhibited by transmission line impedance is governed by the absolute size of the dimensionless product 2N /i,. If 2N /i, 1, a condition guaranteed by the joint existance of a thin film and a large impedance y, then impedance reduces to... 

Fig. 15 (a) Transmission line model for the generalized diffusion impedance, (b) Transmission line model for a porous electrode, (c) Transmission line impedance model for diffusion-recombination in a mesoporous Ti02 electrode, including also interfacial impedances and mass transport impedance in electrolyte... 

Series capacitor - connected in series at the far end of a long transmission or FTP distribution line to offset the reactive component of the line impedance, contain the voltage drop and enhance the receiving-end voltage. It can support a transmission or distribution system in the following ways ... 

Figure 15 shows a set of complex plane impedance plots for polypyr-rolein NaC104(aq).170 These data sets are all relatively simple because the electronic resistance of the film and the charge-transfer resistance are both negligible relative to the uncompensated solution resistance (Rs) and the film s ionic resistance (Rj). They can be approximated quite well by the transmission line circuit shown in Fig. 16, which can represent a variety of physical/chemical/morphological cases from redox polymers171 to porous electrodes.172... 

In studies of these and other items, the impedance method is often invoked because of the diagnostic value of complex impedance or admittance plots, determined in an extremely wide frequency range (typically from 104 Hz down to 10 2 or 10 3 Hz). The data contained in these plots are analyzed by fitting them to equivalent circuits constructed of simple elements like resistances, capacitors, Warburg impedances or transmission line networks [101, 102]. Frequently, the complete equivalent circuit is a network made of sub-circuits, each with its own characteristic relaxation time or its own frequency spectrum. 

A variety of transmission line structures can be fabricated in planar layers of conductor and dielectric (Figure 9). The stripline and offset stripline are best suited for multilayer structures. The offset stripline, with two orthogonal signal layers between a pair of reference voltage planes, eliminates one intermediate plane and achieves higher characteristic impedance for a given dielectric thickness than do two stripline layers but increases the possibility for crosstalk between layers. 

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. 

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. 

In general, the quantities being determined by microwave measurements are complex reflection and transmission coefficients or complex impedances normalized to the impedances of the transmission lines connecting a network analyser and the device-under-test (dut). In addition to linear frequency domain measurements by means of a network analyser the determination of possible non-linear device (and thus material) properties requires more advanced measure-... 

Non resonant techniques are only of limited use to determine microwave losses with high precision, in particular when the losses are very small. Flowever, for the investigation of nonlinear absorption phenomena (i.e. rf power dependent on surface impedance or loss tangent) by intermodulation distortion measurements broad-band test devices are more common. Typically, a planar transmission line with an impedance of 50 Ohms can be employed for intermodulation... 

The model to describe the electrochemical behavior of the porous electrode was first treated by De Levie.160-162 He represented a pore surface by a transmission line as shown in Figure 8, and derived the following expression for the impedance of the pore, Z0... 

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) . 

Two types of electrical analogy model for the interpretation of impedance data can be used based on combinations of resistances and capacitances, or based on transmission lines. These possibilities are now described. 

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