Transmission line finite length

For porous electrodes, an additional frequency dispersion appears. First, it can be induced by a non-local effect when a dimension of a system (for example, pore length) is shorter than a characteristic length (for example, diffusion length), i.e. for diffusion in finite space. Second, the distribution characteristic may refer to various heterogeneities such as roughness, distribution of pores, surface disorder and anisotropic surface structures. De Levie used a transmission-line-equivalent circuit to simulate the frequency response in a pore where cylindrical pore shape, equal radius and length for all pores were assumed [14]. 

The equivalent circuit analog of this situation is a finite-length transmission line terminated with an open circuit. A constant activity or concentration is also a common condition for the interface removed from x = 0. In this case the finite-length transmission line would be terminated in a resistance, and the impedance is given by the expression... 

It is therefore apparent that in passing from high to low freqnency in a system of this kind, there is an additional impedance dne to concentration polarization. Macdonald and HnU [1984] considered this effect on the electrical response of this type of system. Under many drcnmstances, the presence of concentration polarization might be confnsed with an interface impedance. At different ratios of mobiU-ties of anions and cations, either diffnsion-Iike response (finite-length transmission line behavior) or parallel capacitative-resistive behavior may appear. Ac impedance methods have been nsed to determine ionic transference nnmbers in polymeric electrolytes using this principle (Sorensen and Jacobsen [1982]). 

But an electrolytic cell or dielectric test sample is always finite in extent, and its electrical response often exhibits two generic types of distributed response, requiring the appearance of distributed elements in the equivalent circuit used to fit IS data. The first type, that discussed above, appears just because of the finite extent of the system, even when all system properties are homogeneous and space-invariant. Diffusion can lead to a distributed circuit element (the analog of a finite-length transmission line) of this type. When a circuit element is distributed, it is found that its impedance cannot be exactly expressed as the combination of a finite number of ideal circuit elements, except possibly in certain limiting cases. 

Although we shall not discuss the general Z fu)) further here, there is one additional specific case which follows from it and deserves mention. Suppose that the finite-length transmission line analog is open-circuited (see Franceschetti and Macdonald [1979c]). Then no direct current can flow in the actual system, as it could with Zw (but not Zw, and the concentration of the diffusing particle increases at the... 

For the sake of analysis, we consider the behavior of an oxidized organic pol5mier material to be represented as a single pore that is either of infinite or finite length. This type of system can be treated from a mathematical viewpoint as a transmission line in series with the uncompensated solution resistance Ru (see Fig. 1.54). The electrode is characterized by the electronic resistance Re oi its solid phase, the ionic... 

This is the expression for the impedance of a dual transmission line of finite length, and it is very similar to Eqn. 377. In terms of the RC model, we note that the low-frequency expression in Eqn. 424 reduces to... 

Figure 6. Impedance plot corresponding to a uniform, finite-length rc transmission line with an open termination.

Figures 14.7a,b show the measured transmission (S j) and reflection (S ) coefficients of one particular CPW with length of 40 mm. Ripples can be observed in Figure 14.7a, which are a byproduct of minor impedance mismatch arising from the difference in initially estimated PDMS dielectric constant to the true dielectric constant. This minor mismatch is to be expected, as one of the aims of this process is to determine the true dielectric constant starting with a value defined in the data sheet. Since the reflection magnitude of Figure 14.7b remains below 10 dB at aU frequencies, it can be concluded that the impedance of the transmission line remains close to 50 O over the entire bandwidth. The transmission magnitude attenuates across the bandwidth at approximately 5.5 dB with additional loss at 20 GHz. This loss is high, but not excessively so. The components of loss would be associated with the conductor loss due to finite conductivity, this is small as gold behaves close to PEC at microwave frequencies, and dielectric losses originate from the PDMS substrate. The loss will have the form ...

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