Step Function Excitation and Time Constant

Immittance theory is based upon sinusoidal excitation and sinusoidal response. In relaxation theory (and cell excitation studies), a step waveform excitation is used, and the time constant is then an important concept. If the response of a step excitation is an exponential curve, the time constant is the time to reach 63% of the final, total response. Let us for instance consider a series resistor-capacitor (RC)-connection, excited with a controlled voltage step, and record file current response. The current as a function of time I(t) after the step is I(t) = (V/R)e , file time constant x = RC, and I( oo) = 0. 

However, if we excite the same series RC-circuit with a controlled current step and record the voltage across the RC circuit, the voltage will increase linearly with time ad infinitum. The time constant is infinite. Clearly, the time constant is dependent not only on the network itself, but on how it is excited. The time constant of a network is not a parameter uniquely defined by the network itself. Just as immittance must be divided between impedance and admittance dependent on voltage or current driven excitation, there are two time constants dependent on how the circuit is driven. The network may also be a three-or four-terminal network. The time constant is then defined with a step excitation signal at the first port, and the possibly exponential response is recorded at the second port. 

Many dielectrics do not show exponential discharge curves, but fractional power curves. This has led to new models of Universality (see Section 9.2.12). 

The step waveform contains an infinite number of frequencies, and the analysis with such nonsinusoids is done with Laplace transforms. 

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