The basic principles of Electrochemical Impedance Spectroscopy (EIS).
For any electric circuit, which consists of various passive elements (i.e. resistors, capacitors and inductors) the behaviour of the whole circuit to an applied ac voltage, is dependent upon both the behaviour of individual elements, and also on their arrangement in the circuit with respect to each other. If a dc direct voltage is applied to the elements that comprise the equivalent circuit, the resulting current can be measured using Ohms law.
For the case where a low amplitude sine wave Eac, of a particular frequency, is applied across a passive element, then:
E ac = E0 sin(ω t) (1)
Where:
Eac = potential at time t;
E0 = maximum voltage amplitude; ω = is the angular frequency, ω = 2πf; t = is the time.
Under these conditions, the resulting current response of a sine wave Iac will be given by:
I ac = Eac /X …. (2)
Where:
Iac = current at time t;
X = the reactance of the particular passive element in the electrical circuit.
When the applied signal is a sinusoidal voltage wave and the resulting signal is a sinusoidal current wave, then X is called the impedance Z; conversely, when the applied signal is a sinusoidal current wave, the resulting signal is a sinusoidal voltage wave, X which is called the admittance Y.
The value of the reactance of a capacitor or an inductor can be expressed as a complex quantity by the complex operator j, j = −1 [11], and using this notation the reactance of the elements are given by [12]:
For a resistor: XR = R
For a capacitor: XC = 1/-jωC
For an inductor: XL = jωL ………… (3-4)
For the impedance, Z(ω), as mentioned above...
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...ooking at the response of actual real-world systems. In some systems, the Nyquist plot was expected to be a semicircle with the center on the X -axis. However, the observed plot was indeed the arc of a circle, but with the center some distance below the x-axis.” “These depressed semicircles have been explained variously by a number of phenomena depending on the nature of the system being investigated. However the common thread among these explanations is that some property of the system is not homogeneous or that there is some distribution (dispersion) of the value of some physical property of the system. The CPE is usually represented by two parameters, Q° and n”. “It is tempting to simply associate the value of Q° for a CPE with the capacitance value, C, for an equivalent capacitor. The value range of n is between 0 and 1. When n = 0, Q° = R. When n = 1, Q = C.”
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where I is the current through the conductor in units of amperes, V is the potential difference