Debye Relaxation 1R-2C Model

The subscripts refer to frequency, a sine wave parameter. Doo is the surface charge density at t = 0+, which is after the step but so early that only apparently instantaneous polarization mechanisms have come to effect (high frequency e.g., electronic polarization). The capacitor charging current value at t = 0 is infinite, so the model has some physical flaws. Do is the charge density after so long time that the new equilibrium has been obtained and the charging current has become zero. With a single Debye dispersion, this low-frequency value is called the static value (see Section 6.2.1). t is the exponential time constant of the relaxation process. 

Here e and C are vectors in the time domain. We have already seen the ambiguity that e may mean e or e. In dispersion theory, this is less a problem, because the parameters used are at frequency extremes, where Cs = and Eoo = e . 

At low frequencies, we measure a frequency-independent capacitance Co = Coo + AC. The frequency must be low enough to guarantee that the polarization process can follow. In a sufficiently higher frequency range, we measure another frequency-independent capacitance Coo, lower than Cq. The frequency must be sufficiently high so that the polarization process in question cannot follow. 

The quadrature component dependence with increasing frequency is from a higher level to a lower, the in-phase component goes through a maximum. Coo and AC are ideal capacitors. At very high frequencies, Coo dominates the total admittance and capacitance. At very low frequencies, R is negligible with respect to the impedance of AC, and the two 

Maximum value of e is therefore not related to the resistor R, but R determines at what characteristic frequency the maximum value will occur. 

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