EIS Mathematical Theory

Last Updated: 1/4/23 by Alex Peroff

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Theory

1Theory

Impedance is defined as the measure of difficulty a circuit experiences related to the passage of an applied alternating electrical current. It is analogous to Ohm’s law, but unlike Ohm’s law impedance can change as a function of the frequency of the applied potential or current.

During an EIS experiment, if the applied signal is potential and the measured result is current, it is referred to as “potentiostatic EIS”. When the applied signal is current and the measured result is potential, it is referred to as “galvanostatic EIS”. For the case of potentiostatic EIS, a potential is applied with the form shown in Equation 1:

$\displaystyle{E(t)=E_o\cos{(\omega t)}}$

(1)

where $\displaystyle{E_o}$ is the potential sine wave amplitude, $\displaystyle{\omega}$ is the angular frequency, $\displaystyle{t}$ is the time, and the term $\displaystyle{\omega t}$ represents the phase of the waveform. If $\displaystyle{E_o}$ is small enough such that the system is linear, the resultant current waveform is also sinusoidal and will have the same frequency as the input signal; however, it may be shifted in phase, as shown in Equation 2:

$\displaystyle{i(t)=i_o\cos{(\omega t-\phi)}}$

(2)

where $\displaystyle{i_o}$ is the current sine wave amplitude and $\displaystyle{\phi}$ is the phase angle shift.

A more convenient mathematical rearrangement of the applied and resultant waveforms (Equations 1 and 2) is needed to clarify the measured impedance. This is accomplished by using complex coordinates via Euler’s formula, which is defined as:

$e^{jx}=\cos{x}+j\sin{x}$

(3)

where $\displaystyle{j}$ represents the imaginary unit (rather than the conventional symbol, $\displaystyle{i}$ , to prevent confusion with the symbol for current) and $\displaystyle{x}$ is any real number. If $\displaystyle{x}$ is substituted by an angle with the form $\displaystyle{\omega t+\phi_n}$ similarly to Equations 1 and 2, where $\displaystyle{\phi_n}$ is any given phase angle shift, and all terms are multiplied by a given constant amplitude, $\displaystyle{A_n}$ , Equation 3 is rearranged as follows:

$\displaystyle{A_n\cos{(\omega t+\phi_n)}=A_ne^{j(\omega t+\phi_n)}-jA_n\sin{(\omega t+\phi_n)}}$

(4)

The right-hand side of Equation 4 is a complex number while the left-hand side contains only a real quantity. Therefore, by equating the real portions of both sides of the equation, the following expression is obtained:

$\displaystyle{\text{Re}\{A_n\cos{(\omega t+\phi_n)}\}=\text{Re}\{A_ne^{j(\omega t+\phi_n)}-jA_n\sin{(\omega t+\phi_n)}\}}$

(5)

where $\text{Re}$ is the real portion of a number. The second term on the right-hand side of Equation 5 is eliminated because it is entirely imaginary, and the $\displaystyle{\text{Re}}$ operator is removed from the left-hand side because the cosine of a real number is also a real number:

$\displaystyle{A_n\cos{(\omega t+\phi_n)}=\text{Re}\{A_ne^{j(\omega t+\phi_n)}\}}$

(6)

Equation 6 is further simplified as follows:

$\displaystyle{A_n\cos{(\omega t+\phi_n)}=\text{Re}\{\tilde{X}e^{j\omega t}\}}$

(7)

where $\displaystyle{\tilde{X}}$ is a collection of all time-invariant terms and is equivalent to the following:

$\displaystyle{\tilde{X}=A_ne^{j\phi_n}}$

(8)

Using Equation 7, the applied potential and resultant current waveforms shown in Equations 1 and 2, respectively, are rewritten as:

$\displaystyle{E(t)=\text{Re}\{\tilde{E}e^{j\omega t}\}}$

(9)

$\displaystyle{i(t)=\text{Re}\{\tilde{i}e^{j\omega t}\}}$

(10)

where the time-invariant terms $\displaystyle{\tilde{E}}$ and $\displaystyle{\tilde{i}}$ are equivalent to:

$\displaystyle{\tilde{E}=E_o}$

(11)

$\displaystyle{\tilde{i}=i_oe^{-j\phi}}$

(12)

One condition of validity for impedance is stability (i.e., steady-state or time-invariant). Therefore, following Equations 9 through 12, an expression for impedance, $\displaystyle{Z}$ , analogous to Ohm’s law is shown below:

$\displaystyle{Z=\frac{\tilde{E}}{\tilde{i}}}$

(13)

Substituting Equations 11 and 12 into Equation 13 yields the expression:

$\displaystyle{Z=|Z|e^{j\phi}}$

(14)

where $\displaystyle{|Z|}$ is the impedance magnitude and is equivalent to $\displaystyle{\frac{E_o}{i_o}}$ . Using Euler’s formula (Equation 3), Equation 14 is finally expanded to the form:

$\displaystyle{Z=|Z|(\cos{\phi}+j\sin{\phi})=Z_r+j Z_i}$

(15)

This mathematical rearrangement allows the real ( $\displaystyle{Z_r}$ , equivalent to $\displaystyle{|Z|\cos{\phi}}$ ) and imaginary ( $\displaystyle{Z_i}$ , equivalent to $\displaystyle{|Z|\sin{\phi}}$ ) components of the impedance to be calculated separately.

During an EIS experiment, most commercial software packages automatically perform Fourier transform on the input and output signals at each frequency to extract corresponding values for $\displaystyle{|Z|}$ and $\displaystyle{\phi}$ . These values are then used with Equation 15 to calculate $\displaystyle{Z_r}$ and $\displaystyle{Z_i}$ . Hence, the result from a typical EIS experiment is a data table with five columns: $\displaystyle{f}$ , $\displaystyle{Z_r}$ , $\displaystyle{Z_i}$ , $\displaystyle{|Z|}$ , and $\displaystyle{\phi}$ .

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