Constant Phase Element: Equivalent Capacitance

Impedance of a Constant Phase Element

In electrochemical impedance spectroscopy (EIS), constant phase elements (CPE) are used to model imperfect capacitive behavior of electrical systems. The impedance of a constant phase element (\(Z_{CPE}\)) is defined as

\[Z_{CPE} = \frac{1}{(i\omega)^{\alpha}Q} = \frac{1}{\omega^{\alpha}Q}\,e^{-\frac{\pi}{2}\alpha i} \tag{eq. 1}\]

Here, \(\omega\) is the angular frequency and \(\omega = 2\pi f\). \(i\) is the imaginary unit. \(Q\) and \(\alpha\) are frequency-independent constants.

The impedance of an ideal capacitor (\(Z_{C}\)) is defined as

\[Z_{C} = \frac{1}{i\omega C} \tag{eq. 2}\]

Here, \(C\) is the capacitance. In eq. 1, \(\alpha\) can have a value from 0 to 1. If \(\alpha = 1\), then eq. 1 becomes similar to eq. 2 and \(Q = C\); hence with \(\alpha = 1\) the constant phase element behaves like an ideal capacitor. For \(\alpha = 0\), eq. 1 becomes frequency independent. A frequency-independent impedance is the characteristic of a resistor, so with \(\alpha = 0\) the constant phase element behaves like an ideal resistor.

Fig. 1 shows qualitative Bode plots of a constant phase element with different values of \(\alpha\). For the CPE with \(\alpha = 0\), the phase will be 0° (red lines in the graphs) and for the CPE with \(\alpha = 1\), the phase will be -90° (black lines in the graphs). For a CPE with \(0 < \alpha < 1\), the phase can be calculated via -90° × \(\alpha\) (derived at the end of this application note). The phase of a CPE stays constant and does not change with frequency, hence the electrical element is known as a “constant phase element” (see Fig. 1 right).

Fig. 1: Qualitative Bode plots of a constant phase element with different values of \(\alpha\): impedance plot (left) and phase plot (right).

Equivalent Capacitance of a Single CPE

For an ideal capacitor (or a CPE with \(\alpha = 1\)), the impedance changes with frequency but the capacitance stays constant (see Fig. 2). On the other hand, the equivalent capacitance and the impedance of a CPE (with \(0 < \alpha < 1\)) change with frequency and \(\alpha\). The \(Q\) constant of the constant phase element has a unit of \(\text{Farad} \cdot s^{\alpha}\) and does not directly represent the equivalent capacitance of the constant phase element. Hence, when reporting the equivalent capacitance of a CPE it is necessary to also mention the frequency for which the equivalent capacitance is calculated. To compare different constant phase elements, the impedances of the constant phase elements are set equal to the impedance of an ideal capacitor (equating eq. 1 and eq. 2).

\[\frac{1}{i\omega C} = \frac{1}{(i\omega)^{\alpha}Q} \tag{eq. 3}\]

Considering only the impedance modulus in eq. 3, one gets eq. 3a for the relation between \(Q\) and \(C\). The complete derivation is provided at the end of this application note.

\[\frac{1}{\omega C} = \frac{1}{\omega^{\alpha}Q} \tag{eq. 3a}\]

Taking the logarithm on both sides,

\[-\log(\omega) - \log(C) = -\alpha\log(\omega) - \log(Q) \tag{eq. 3b}\]

For a frequency \(= \frac{1}{2\pi}\), the angular frequency \(\omega = 1\), hence \(\log(\omega) = 0\), then

\[C = Q\]

This shows that at a frequency of \(1/(2\pi)\), the equivalent capacitance of a single constant phase element becomes equal to \(Q\) and does not depend on \(\alpha\). This frequency of \(1/(2\pi)\) is very low for practical use (further explanation later), therefore, to calculate the equivalent capacitance of the constant phase element at a suitable frequency range, eq. 3a can be modified to the following:

\[C = \frac{(\omega_{norm})^{\alpha}}{\omega_{norm}}Q \tag{eq. 4}\]

Here, \(\omega_{norm}\) is the frequency at which the equivalent capacitance of the constant phase element is calculated. As a general rule, \(\omega_{norm}\) is usually taken from the high-frequency range (e.g. 1 kHz).

Fig. 2: Calculated equivalent capacitance of a CPE element as a function of the frequency for different exponents (\(\alpha\)).

Parallel R-CPE Circuit

In practice, the constant phase element is used together with other electrical elements in an electrical circuit. In such cases, determining the equivalent capacitance is not very straightforward. For calculating the equivalent capacitance of a constant phase element in an electrical circuit, it is essential to choose a frequency range where the constant phase element is dominant (the measured impedance is primarily because of the constant phase element, i.e. contributions of other elements to the overall impedance can be neglected).

Fig. 3 shows a Bode plot of a parallel R-CPE. From Fig. 3 it is clear that at the frequency of \(1/(2\pi)\), the phase angle is close to zero, indicating the dominance of the resistor at this frequency. Only at frequencies above 1 kHz is the phase angle close to -81° (the maximum phase angle for a CPE with \(\alpha = 0.9\)), illustrating the dominance of the constant phase element in this frequency range. Therefore, the equivalent capacitance of the constant phase element must be calculated in this frequency range (using eq. 4), in particular with \(\omega_{norm} = 2\pi f = 6.2\) kHz as the normalizing angular frequency.

Fig. 3: Bode plot of a parallel R-CPE circuit. The CPE is dominant at frequencies above 1 kHz.

Even at high-frequency range, the phase is not exactly -81° (but -80.1° at 1 kHz), indicating a small contribution from the parallel resistance, so the calculated equivalent capacitance contains some marginal error.

Hsu and Mansfeld’s Formula

Hsu and Mansfeld derived a formula which provides the equivalent capacitance of a constant phase element in a parallel R-CPE circuit without having to consider the appropriate frequency (\(\omega_{norm}\)). For this, Hsu et al. equated the impedance of the complete R-CPE circuit with the Cole-Cole impedance expression for a circuit containing a parallel resistor and a capacitive-type element. The total impedance of a parallel R-CPE (\(Z_{R-CPE}\)) can be defined as

\[\frac{1}{Z_{R-CPE}} = \frac{1}{R} + (i\omega)^{\alpha}Q\]

\[Z_{R-CPE} = \frac{R}{1 + (i\omega)^{\alpha}QR} \tag{eq. 5}\]

The Cole-Cole expression for the impedance (\(Z_{CC}\)) of a parallel resistor-capacitive-type element circuit is

\[Z_{CC} = \frac{R}{1 + (i\omega\lambda)^{\alpha}} \tag{eq. 6}\]

Here, the time constant \(\lambda = CR\). Equating eq. 5 and eq. 6,

\[(i\omega)^{\alpha}QR = (i\omega CR)^{\alpha}\]

\[C = R^{\frac{1-\alpha}{\alpha}}\,Q^{\frac{1}{\alpha}} \tag{eq. 7}\]

From eq. 7, the equivalent capacitance of a constant phase element in a parallel R-CPE circuit can be determined. Here, as the effect of the parallel resistor is also considered, choosing an appropriate frequency range is not required. The equivalent capacitance of the constant phase element (from eq. 7) corresponds to the frequency at which the imaginary impedance of the R-CPE circuit exhibits a maximum.

Brug’s Formula

Brug et al. also provided a formula to calculate the equivalent capacitance of the constant phase element. This formula includes the effect of the series as well as the parallel resistance on the calculation of the equivalent capacitance.

\[C = \left(\frac{R_{s}R_{p}}{R_{s} + R_{p}}\right)^{\frac{1-\alpha}{\alpha}}\,Q^{\frac{1}{\alpha}} \tag{eq. 8}\]

Here, \(R_{s}\) is the series resistance and \(R_{p}\) is the parallel resistance.

Which formula to use depends on the test object. For a normal distribution, the Hsu formula is appropriate, whereas for a surface distribution, Brug’s formula is the better choice.

All formulas discussed above rely on \(Q\) and \(\alpha\) obtained from a fit to a measured impedance spectrum. Since \(\alpha\) is determined by the phase angle and the CPE-dominated range often lies at high frequencies, an accurate equivalent capacitance requires a spectrum that is reliable in both modulus and phase over a wide frequency range — as provided by the Zahner IM7 potentiostat with its integrated frequency response analyzer. The Zahner Analysis software already includes different tools (according to the normalization method and the Hsu/Brug et. al. formulas) to calculate the equivalent capacitance of constant phase elements.

Derivation of Modulus and Phase

Starting with eq. 1,

\[Z_{CPE} = \frac{1}{(i\omega)^{\alpha}Q} = \frac{1}{\omega^{\alpha}Q}\,e^{-\frac{\pi}{2}\alpha i}\]

\[Z_{CPE} = \frac{1}{\omega^{\alpha}Q}\left[\cos\left(\frac{\pi\alpha}{2}\right) - i\,\sin\left(\frac{\pi\alpha}{2}\right)\right]\]

Here, the real and imaginary parts are

\[Z_{real} = \frac{1}{\omega^{\alpha}Q}\cos\left(\frac{\pi\alpha}{2}\right)\]

\[Z_{imag} = -\frac{1}{\omega^{\alpha}Q}\sin\left(\frac{\pi\alpha}{2}\right)\]

Modulus

\[Z = \sqrt{Z_{real}^{2} + Z_{imag}^{2}}\]

Inserting the values for the real and imaginary parts,

\[Z = \sqrt{\left[\frac{1}{\omega^{\alpha}Q}\cos\left(\frac{\pi\alpha}{2}\right)\right]^{2} + \left[-\frac{1}{\omega^{\alpha}Q}\sin\left(\frac{\pi\alpha}{2}\right)\right]^{2}}\]

Taking \(\frac{1}{\omega^{\alpha}Q}\) common,

\[Z = \sqrt{\left(\frac{1}{\omega^{\alpha}Q}\right)^{2}\left[\cos^{2}\left(\frac{\pi\alpha}{2}\right) + \sin^{2}\left(\frac{\pi\alpha}{2}\right)\right]}\]

Since \(\cos^{2}(x) + \sin^{2}(x) = 1\),

\[Z = \sqrt{\left(\frac{1}{\omega^{\alpha}Q}\right)^{2}}\]

\[Z = \frac{1}{\omega^{\alpha}Q}\]

Phase

\[\varnothing = 2\tan^{-1}\left[\frac{Z_{imag}}{\sqrt{Z_{real}^{2} + Z_{imag}^{2}} + Z_{real}}\right]\]

Inserting the values for the real and imaginary parts,

\[\varnothing = 2\tan^{-1}\left[\frac{-\dfrac{1}{\omega^{\alpha}Q}\sin\left(\frac{\pi\alpha}{2}\right)}{\dfrac{1}{\omega^{\alpha}Q} + \dfrac{1}{\omega^{\alpha}Q}\cos\left(\frac{\pi\alpha}{2}\right)}\right]\]

\(\frac{1}{\omega^{\alpha}Q}\) cancels out from the numerator and denominator,

\[\varnothing = 2\tan^{-1}\left[\frac{-\sin\left(\frac{\pi\alpha}{2}\right)}{1 + \cos\left(\frac{\pi\alpha}{2}\right)}\right]\]

Here, \(\sin(x) = 2\sin(x/2)\cos(x/2)\) and \(1 + \cos(x) = 2\cos^{2}(x/2)\), so

\[\varnothing = 2\tan^{-1}\left[-\tan\left(\frac{\pi\alpha}{4}\right)\right] = -\frac{\pi\alpha}{2}\]

\[\varnothing = -90^{\circ} \cdot \alpha\]

References

• K.S. Cole, R.H. Cole; J. Chem. Phys. 9 (1941) 341-352

• K.S. Cole, R.H. Cole; J. Chem. Phys. 10 (1942) 98-105

• G.J. Brug, A. L. G. van den Eeden, M. Sluyters-Rehbach, J. H. Sluyters; Journal of Electroanalytical Chemistry 176 (1984) 275-295

• C.H. Hsu, F. Mansfeld; Corrosion 57 / No. 9 (2001) 747-748

• M.R. Shoar Abouzari, F. Berkemeier, G. Schmitz, D. Wilmer; Solid State Ionics 180 (2009) 922-927

• B. Hirschorn, M. Orazem, B. Tribollet, V. Vivier, I. Frateur, M. Musiani; El. Acta 55 (2010) 6218-6227