Constant Phase Element (CPE)

• Constant Phase Element (CPE) is a fractional-order model capturing non-ideal capacitive behavior with a constant phase angle, effectively interpolating between resistive and capacitive responses.
• It uses Caputo fractional derivatives and convolution-based memory kernels to represent distributed relaxation times and anomalous interfacial dynamics in various materials.
• The CPE framework aids in quantifying energy partitioning between storage and loss, offering insights into porous-electrode modeling and supercapacitor performance.

Constant Phase Element (CPE) denotes the standard fractional-order model for non-ideal capacitive behavior in impedance spectroscopy. In its canonical capacitive form,

$$z_c(s)=\frac{1}{C_{\alpha}s^{\alpha}}, \qquad s=j\omega,\qquad C_{\alpha}>0,\qquad 0<\alpha<1,$$

where $\alpha$ is the order, also called the dispersion coefficient, and $C_\alpha$ is a pseudocapacitance with units $\mathrm{F\,s^{\alpha-1}}$ (Allagui et al., 2 Feb 2025). The phase is frequency independent, $\angle \tilde Z(\omega)=-\alpha\pi/2$, so the element interpolates between an ideal resistor and an ideal capacitor while retaining a constant phase angle (Holm et al., 2020). In electrochemistry, dielectric spectroscopy, porous-electrode theory, and supercapacitor modeling, the CPE is used to represent dispersive, lossy, and memory-bearing storage that cannot be reduced to a single ordinary capacitance (Allagui et al., 2 Feb 2025).

1. Canonical definition and constitutive equations

The standard frequency-domain definition writes the CPE impedance as

$$Z_{\mathrm{CPE}}(\omega)=\frac{1}{(j\omega)^\alpha C_\alpha},$$

or equivalently the admittance as $Y_{\mathrm{CPE}}=C_\alpha (j\omega)^\alpha$ (Holm et al., 2020). The real and imaginary parts scale as $\omega^{-\alpha}$, and the ratio of imaginary to real part remains fixed as frequency changes; this is the formal source of the term “constant phase element” (Allagui et al., 2 Feb 2025).

In the time domain, the same object is represented by a Caputo fractional differential constitutive law. For the capacitive CPE,

$i_c(t)=C_\alpha\,{}_0D_t^\alpha v_c(t),$

with

$${}_0D_t^\alpha f(t):=\frac{1}{\Gamma(m-\alpha)}\int_0^t (t-\tau)^{m-\alpha-1}f^{(m)}(\tau)\,d\tau,$$

where $\alpha$0, $\alpha$1, and in the cited formulation $\alpha$2 (Allagui et al., 2 Feb 2025). Under zero initial conditions, Laplace transformation recovers the impedance $\alpha$3, so the frequency-domain and time-domain descriptions are equivalent within the stated assumptions (Allagui et al., 2 Feb 2025).

The memory structure is explicit in the voltage response to arbitrary current,

$\alpha$4

whose kernel $\alpha$5 is algebraic rather than exponential (Allagui et al., 2 Feb 2025). This places the CPE in the class of fractional-order hereditary elements. For $\alpha$6 the model reduces to an ideal capacitor, whereas $\alpha$7 approaches resistive behavior (Allagui et al., 2 Feb 2025).

The same formal structure also admits an inductive counterpart,

$\alpha$8

with positive constant phase and $\alpha$9 carrying units $C_\alpha$0 (Holm et al., 2020). In most electrochemical usage, however, “CPE” refers to the capacitive form.

2. Phenomenology and physical interpretation

The CPE is widely used to model dispersive materials and electrode/electrolyte interfaces that do not behave like ideal capacitors. The cited works associate this non-ideal response with distributed surface reactivity, inhomogeneity, current and potential distributions, roughness, porosity, tortuous pore structure, possible fractal geometry, and broadly distributed local relaxation times (Allagui et al., 2 Feb 2025, Allagui et al., 2024). In supercapacitor modeling, the same phenomenology is linked to surface effects, porous or fractal electrode structure, and slow ionic diffusion within pores (Allagui et al., 2023).

A recurrent theme in the literature is that the CPE is physically suggestive but not uniquely interpreted. One source states that its physical meaning is only partially understood (Holm et al., 2020), and another remarks that a plausible physical interpretation remains “obscure and perplexing” in supercapacitor applications (Allagui et al., 2023). For that reason, the CPE often serves as a compact phenomenological model grounded in impedance data rather than a complete microscopic law.

At the same time, several works give the element more concrete interpretive content. In one direction, the CPE is treated as a compact representation of a distribution of relaxation times: a finite distributed $C_\alpha$1 network can emulate a CPE over a finite bandwidth, and the observed current is then the combined contribution of multiple capacitive branches with different time constants (Allagui et al., 2023). In another direction, anomalous Poisson–Nernst–Planck analysis shows that low-frequency equivalent circuits containing one or more CPEs can arise from integro-differential boundary conditions with temporal memory at the electrodes, so the CPE becomes the circuit signature of anomalous interfacial dynamics rather than a purely abstract fitting element (Lenzi et al., 2013).

This dual status—empirical fitting element on one hand, compact surrogate for distributed or anomalous physics on the other—is central to the modern understanding of the CPE. A measured CPE-like response need not imply one unique mechanism; it may indicate the presence of temporally nonlocal interfacial dynamics, distributed relaxation spectra, heterogeneous subdomains, or several of these simultaneously.

3. Memory kernels, charge–voltage relations, and dimensional consistency

For an ordinary fixed capacitor, the defining relation is $C_\alpha$2. The status of the corresponding law for time-varying capacitance has been disputed. One candidate model uses time-domain multiplication,

$C_\alpha$3

which gives

$C_\alpha$4

A competing model uses time-domain convolution,

$C_\alpha$5

Measurements on an ordinary time-varying capacitor implemented with a motor-driven potentiometer and op-amps matched a power-law response over about two decades of time rather than an exponential, which was reported as confirmation of the multiplication model for the ordinary, non-fractional case (Marthins et al., 2023).

The same work emphasizes that a CPE is fundamentally different. Its constitutive law is fractional,

$C_\alpha$6

and its charge–voltage relation is explicitly convolutional,

$C_\alpha$7

with a power-law kernel

$C_\alpha$8

Accordingly, a CPE is not an ordinary capacitor whose capacitance merely varies in time or frequency; it is a memory element whose present state depends on the history of the voltage through a nonlocal kernel (Marthins et al., 2023).

A second debate concerns dimensional homogeneity in time-domain CPE formulations. A pointed comment on a charge–voltage equation used by Fouda et al. argued that

$C_\alpha$9

is dimensionally inconsistent if $\mathrm{F\,s^{\alpha-1}}$0 is interpreted as a capacitance in farads, because convolution introduces an extra factor of time and the right-hand side carries units $\mathrm{F\,s^{\alpha-1}}$1, not $\mathrm{F\,s^{\alpha-1}}$2 (Pandey, 2022). The same comment observes that when the specialization

$\mathrm{F\,s^{\alpha-1}}$3

is written, the Dirac delta contributes $\mathrm{F\,s^{\alpha-1}}$4, so $\mathrm{F\,s^{\alpha-1}}$5 must actually have units $\mathrm{F\,s^{\alpha-1}}$6. On that reading, the convolution becomes dimensionally valid (Pandey, 2022).

The proposed general correction is

$\mathrm{F\,s^{\alpha-1}}$7

which preserves dimensional homogeneity while making the inverse-time factor explicit (Pandey, 2022). The same source argues that this formulation is more general for memory laws described by polynomials and connects, in the light of fractional calculus, to the Curie–von Schweidler law (Pandey, 2022). Taken together with the experimental distinction between ordinary time-varying capacitors and CPEs, this suggests that the essential constitutive feature of the CPE is not mere parameter variation but hereditary convolution with a correctly dimensioned kernel.

4. Networks, distributed order, and porous-electrode transmission lines

A single-order CPE assumes one fixed exponent $\mathrm{F\,s^{\alpha-1}}$8. A broader class replaces that with a distribution of orders:

$\mathrm{F\,s^{\alpha-1}}$9

where $\angle \tilde Z(\omega)=-\alpha\pi/2$0 is a non-negative, time-invariant weight function over an interval of orders (Allagui et al., 2 Feb 2025). Under zero initial conditions, the equivalent impedance becomes

$\angle \tilde Z(\omega)=-\alpha\pi/2$1

For a uniform distribution of orders, the resulting impedance is not of the form $\angle \tilde Z(\omega)=-\alpha\pi/2$2; the overall network is therefore not equivalent to a single CPE (Allagui et al., 2 Feb 2025). For discrete order distributions, parallel-connected elemental CPEs reduce to a single effective CPE only in the degenerate case where all orders are equal (Allagui et al., 2 Feb 2025).

This result is significant because it constrains the interpretation of fitted CPE exponents. A single-CPE fit may be a useful approximation when the order distribution is narrow, but a broad distribution produces phase variation with frequency and cannot be collapsed to one unique order without loss of structure (Allagui et al., 2 Feb 2025). The same point also reframes heterogeneous electrochemical interfaces: a CPE-like spectrum does not necessarily imply one fractional process.

A related generalization appears in porous-electrode transmission-line theory. Replacing the usual distributed capacitance per unit length by a CPE per unit length,

$\angle \tilde Z(\omega)=-\alpha\pi/2$3

turns the classical RC line into a time-fractional diffusion equation,

$\angle \tilde Z(\omega)=-\alpha\pi/2$4

for a bounded pore with reflecting end condition (Allagui et al., 2024). The reduced finite-length impedance is then

$\angle \tilde Z(\omega)=-\alpha\pi/2$5

which reproduces the classical reflective finite-length Warburg when $\angle \tilde Z(\omega)=-\alpha\pi/2$6 (Allagui et al., 2024).

The asymptotics are especially informative. At high frequency, the finite line behaves as $\angle \tilde Z(\omega)=-\alpha\pi/2$7, so the distributed resistor-CPE line itself acts like a CPE of order $\angle \tilde Z(\omega)=-\alpha\pi/2$8. At low frequency, the leading form is $\angle \tilde Z(\omega)=-\alpha\pi/2$9, corresponding to a resistance-like offset plus a CPE-like storage term of order $$Z_{\mathrm{CPE}}(\omega)=\frac{1}{(j\omega)^\alpha C_\alpha},$$0 (Allagui et al., 2024). The same model yields a broad distribution of relaxation times, which narrows toward Debye-like behavior as $$Z_{\mathrm{CPE}}(\omega)=\frac{1}{(j\omega)^\alpha C_\alpha},$$1 and broadens as $$Z_{\mathrm{CPE}}(\omega)=\frac{1}{(j\omega)^\alpha C_\alpha},$$2 decreases (Allagui et al., 2024). This provides a mechanistic basis for the empirical dispersed finite-length Warburg form in porous electrodes.

5. Equivalent circuits, realizations, and time-domain computation

One strand of research gives exact or asymptotic circuit realizations of CPE behavior using ordinary components whose values vary linearly in time. For the capacitive CPE, a series circuit composed of a resistor and an inductor

$$Z_{\mathrm{CPE}}(\omega)=\frac{1}{(j\omega)^\alpha C_\alpha},$$3

yields, for the current response to a voltage impulse, a power-law form identical to the capacitive CPE impulse response when $$Z_{\mathrm{CPE}}(\omega)=\frac{1}{(j\omega)^\alpha C_\alpha},$$4, and asymptotically identical when $$Z_{\mathrm{CPE}}(\omega)=\frac{1}{(j\omega)^\alpha C_\alpha},$$5 and $$Z_{\mathrm{CPE}}(\omega)=\frac{1}{(j\omega)^\alpha C_\alpha},$$6 (Holm et al., 2020). The dual result holds for the inductive CPE using a parallel $$Z_{\mathrm{CPE}}(\omega)=\frac{1}{(j\omega)^\alpha C_\alpha},$$7-$$Z_{\mathrm{CPE}}(\omega)=\frac{1}{(j\omega)^\alpha C_\alpha},$$8 circuit with

$$Z_{\mathrm{CPE}}(\omega)=\frac{1}{(j\omega)^\alpha C_\alpha},$$9

These constructions do not claim that all observed CPEs are physically caused by linearly increasing inductance or capacitance; rather, they provide exact mathematical realizations of the relevant input–output laws (Holm et al., 2020).

The same source is explicit about the limitations. Time-varying realizations are not standard LTI systems; the equivalence is input–output specific; and exact matching requires $Y_{\mathrm{CPE}}=C_\alpha (j\omega)^\alpha$0 or $Y_{\mathrm{CPE}}=C_\alpha (j\omega)^\alpha$1, with nonzero initial values giving only asymptotic CPE behavior (Holm et al., 2020). These caveats matter because they separate constitutive equivalence from partial response matching.

In supercapacitor analysis, the CPE commonly appears in series with a resistor $Y_{\mathrm{CPE}}=C_\alpha (j\omega)^\alpha$2, producing

$Y_{\mathrm{CPE}}=C_\alpha (j\omega)^\alpha$3

From this model,

$Y_{\mathrm{CPE}}=C_\alpha (j\omega)^\alpha$4

so the CPE-side voltage and current can be reconstructed in the time domain by convolution or, for periodic and sampled signals, by Fourier decomposition and harmonic-by-harmonic filtering (Allagui et al., 2023). Closed-form impulse responses involve Mittag-Leffler functions, and the method extends to instantaneous power $Y_{\mathrm{CPE}}=C_\alpha (j\omega)^\alpha$5 and accumulated energy (Allagui et al., 2023).

A central practical consequence is that standard ideal-capacitor formulas such as $Y_{\mathrm{CPE}}=C_\alpha (j\omega)^\alpha$6 can be misleading for devices whose low-frequency branch is fitted by a CPE (Allagui et al., 2023). In the cited supercapacitor study, a fifth-order distributed $Y_{\mathrm{CPE}}=C_\alpha (j\omega)^\alpha$7 circuit was used to emulate $Y_{\mathrm{CPE}}=C_\alpha (j\omega)^\alpha$8-CPE behavior over a finite bandwidth, precisely because the CPE is fractional and not directly realizable as a single ideal element (Allagui et al., 2023).

6. Interpretive debates, energetic meaning, and related extensions

Several current debates concern what a fitted CPE parameter set actually means. One caution is that a measured CPE-like response does not prove a single underlying fractional mechanism: distributed-order superposition, heterogeneous pore charging, or anomalous interfacial boundary conditions can all produce spectra that resemble a CPE over part of the frequency range [(Allagui et al., 2 Feb 2025); (Lenzi et al., 2013)]. Another caution is constitutive: an ordinary time-varying capacitor and a CPE may both exhibit power-law behavior in selected transients, but they are not the same object because the former is modeled by time-domain multiplication and the latter by convolutional fractional memory (Marthins et al., 2023).

An important recent reinterpretation concerns the meaning of the dispersion coefficient $Y_{\mathrm{CPE}}=C_\alpha (j\omega)^\alpha$9. Using an $\omega^{-\alpha}$0-network equivalency for the CPE,

$\omega^{-\alpha}$1

one study decomposes the cumulative input energy

$\omega^{-\alpha}$2

into energy stored in capacitive modes, $\omega^{-\alpha}$3, and energy dissipated in resistive modes, $\omega^{-\alpha}$4, with

$\omega^{-\alpha}$5

(Allagui et al., 26 Sep 2025). For a step voltage input, the ratios become

$\omega^{-\alpha}$6

so $\omega^{-\alpha}$7 can be read as an energetic partition index rather than only a spectral-fit exponent (Allagui et al., 26 Sep 2025). The same paper reports that for ramp and quadratic inputs the energy ratios again reduce to pure functions of $\omega^{-\alpha}$8, independent of excitation amplitude and material parameters (Allagui et al., 26 Sep 2025).

This energetic reading is consistent with the limiting cases already built into the impedance definition. As $\omega^{-\alpha}$9, the CPE becomes resistor-like and the dissipated fraction tends to unity; as $i_c(t)=C_\alpha\,{}_0D_t^\alpha v_c(t),$0, it becomes capacitor-like, with the ideal-capacitor limits recovered for the respective excitation class (Allagui et al., 26 Sep 2025). The proposal does not replace structural explanations based on roughness, porosity, or heterogeneity, but it supplies a physically interpretable quantity directly tied to storage versus loss.

Outside classical electrochemistry, the phrase constant-phase behavior also appears in control. Reset-based “Constant in Gain, Lead in Phase” elements and their continuous-reset variants aim at broadband phase shaping with nearly unchanged gain and are described as capable of complex-order behavior, although that work does not explicitly define the classical CPE impedance $i_c(t)=C_\alpha\,{}_0D_t^\alpha v_c(t),$1 (Karbasizadeh et al., 2021). This suggests a broader family of constant-phase concepts across disciplines, but the electrochemical CPE remains the fractional-order impedance element defined by $i_c(t)=C_\alpha\,{}_0D_t^\alpha v_c(t),$2.