Transmission Line with Two Resistances

For the classical transmission line to be an appropriate model, one of the carriers, electrons or ions, must be much more mobile than the other. A more general model is shown in Fig. 4.4. In this transmission line there 

FIGURE 4.4. The dual transmission line with describing the motion of electrons in the polymer and Rx the motion of counterions in the pores is the combined distributed capacitance dominated by the Nemst and Donnan terms. 

When resistances are very unequal, p is small and the general model reduces to the classical line in Fig. 4.1. On the other hand when Rx = Re p has its maximum value of 1/4. The impedance is given by 

At high frequencies when p 3, we find the following simple expression for Z  

In the high-frequency limit the circuit behaves like the two resistances in parallel. The impedance is dominated by the smaller of the resistances as the current takes the most conducting pathway. The second term in Eqn. 23 describes a 45° Warburg line, but now it is displaced from the origin by Eqn. 24. It has the same type of dimensionless frequency term as the classical line. 

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Values of y can be calculated from experimental results at high frequency for Rx and with Eqn. 32 for bic from results at low frequency for Figure 4.11 shows a typical plot of Eqn. 30. A reasonable straight line is obtained. This combination of data for the two different resistances shows the value of the transmission line with two resistances. 

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