---
title: Constant Phase Element (CPE)
url: https://www.emergentmind.com/topics/constant-phase-element-cpe
type: topic
---

# Constant Phase Element (CPE)

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 \(\mathrm{F\,s^{\alpha-1}}\) [2502.02608]. 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 [2007.05821]. 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 [2502.02608].

## 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 \(Y_{\mathrm{CPE}}=C_\alpha (j\omega)^\alpha\) [2007.05821]. 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” [2502.02608].

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 \(m\in\mathbb{N}\), \(m-1<\alpha<m\), and in the cited formulation \(m=1\) [2502.02608]. Under zero initial conditions, Laplace transformation recovers the impedance \(1/(C_\alpha s^\alpha)\), so the frequency-domain and time-domain descriptions are equivalent within the stated assumptions [2502.02608].

The memory structure is explicit in the voltage response to arbitrary current,
$$
v(t)=\frac{1}{C_\alpha\Gamma(\alpha)}\int_0^t (t-\tau)^{\alpha-1}i_c(\tau)\,d\tau,
$$
whose kernel \((t-\tau)^{\alpha-1}\) is algebraic rather than exponential [2502.02608]. This places the CPE in the class of fractional-order hereditary elements. For \(\alpha=1\) the model reduces to an ideal capacitor, whereas \(\alpha\to 0\) approaches resistive behavior [2502.02608].

The same formal structure also admits an inductive counterpart,
$$
\tilde Z(\omega)=(j\omega)^\alpha L_\alpha,
$$
with positive constant phase and \(L_\alpha\) carrying units \(\mathrm{H\,s^{\alpha-1}}\) [2007.05821]. 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 [2502.02608; 2411.17368]. In supercapacitor modeling, the same phenomenology is linked to surface effects, porous or fractal electrode structure, and slow ionic diffusion within pores [2301.13078].

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 [2007.05821], and another remarks that a plausible physical interpretation remains “obscure and perplexing” in supercapacitor applications [2301.13078]. 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 \(RC\) 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 [2301.13078]. 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 [1306.1949].

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 \(q=CV\). The status of the corresponding law for time-varying capacitance has been disputed. One candidate model uses time-domain multiplication,
$$
q(t)=c(t)v(t),
$$
which gives
$$
i(t)=c(t)\frac{dv(t)}{dt}+\frac{dc(t)}{dt}v(t).
$$
A competing model uses time-domain convolution,
$$
q(t)=c(t)\ast v(t).
$$
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 [2309.01701].

The same work emphasizes that a CPE is fundamentally different. Its constitutive law is fractional,
$$
i(t)=C_{\alpha'}\frac{d^{\alpha'}v(t)}{dt^{\alpha'}},
$$
and its charge–voltage relation is explicitly convolutional,
$$
q(t)=\tilde c(t)\ast v(t),
$$
with a power-law kernel
$$
\tilde c(t)\propto t^{-\alpha'}.
$$
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 [2309.01701].

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
$$
q(t)=c(t)\ast V(t)
$$
is dimensionally inconsistent if \(c(t)\) is interpreted as a capacitance in farads, because convolution introduces an extra factor of time and the right-hand side carries units \(\mathrm{C\cdot s}\), not \(\mathrm{C}\) [2203.04804]. The same comment observes that when the specialization
$$
c(t)=C\delta(t)
$$
is written, the Dirac delta contributes \(\mathrm{s}^{-1}\), so \(c(t)\) must actually have units \(\mathrm{F/s}\). On that reading, the convolution becomes dimensionally valid [2203.04804].

The proposed general correction is
$$
q(t)=C(t)\ast\frac{dV(t)}{dt}=\frac{dC(t)}{dt}\ast V(t),
$$
which preserves dimensional homogeneity while making the inverse-time factor explicit [2203.04804]. 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 [2203.04804]. 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 \(\alpha\). A broader class replaces that with a distribution of orders:
$$
i(t)=\int_{\beta_1}^{\beta_2}\phi(\alpha)\,C_\alpha\,{}_0D_t^\alpha v_c(t)\,d\alpha,
$$
where \(\phi(\alpha)\) is a non-negative, time-invariant weight function over an interval of orders [2502.02608]. Under zero initial conditions, the equivalent impedance becomes
$$
\tilde z(s)=\left(\int_{\beta_1}^{\beta_2}\phi(\alpha)\,C_\alpha s^\alpha\,d\alpha\right)^{-1}.
$$
For a uniform distribution of orders, the resulting impedance is not of the form \(1/(Cs^\alpha)\); the overall network is therefore not equivalent to a single CPE [2502.02608]. For discrete order distributions, parallel-connected elemental CPEs reduce to a single effective CPE only in the degenerate case where all orders are equal [2502.02608].

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 [2502.02608]. 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,
$$
z_c(s)=\frac{1}{c_\alpha s^\alpha},
$$
turns the classical RC line into a time-fractional diffusion equation,
$$
{}_0D_t^\alpha v(x,t)=\frac{1}{r c_\alpha}\frac{\partial^2 v(x,t)}{\partial x^2},
$$
for a bounded pore with reflecting end condition [2411.17368]. The reduced finite-length impedance is then
$$
z_\alpha(s_n)=s_n^{-\alpha/2}\coth\!\left(s_n^{\alpha/2}\right),
$$
which reproduces the classical reflective finite-length Warburg when \(\alpha=1\) [2411.17368].

The asymptotics are especially informative. At high frequency, the finite line behaves as \(s_n^{-\alpha/2}\), so the distributed resistor-CPE line itself acts like a CPE of order \(\alpha/2\). At low frequency, the leading form is \(1/3+s_n^{-\alpha}\), corresponding to a resistance-like offset plus a CPE-like storage term of order \(\alpha\) [2411.17368]. The same model yields a broad distribution of relaxation times, which narrows toward Debye-like behavior as \(\alpha\to 1\) and broadens as \(\alpha\) decreases [2411.17368]. 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
$$
L(t)=L_0+\theta t
$$
yields, for the current response to a voltage impulse, a power-law form identical to the capacitive CPE impulse response when \(L_0=0\), and asymptotically identical when \(L_0>0\) and \(t\gg L_0/\theta\) [2007.05821]. The dual result holds for the inductive CPE using a parallel \(R\)-\(C(t)\) circuit with
$$
C(t)=C_0+\theta t.
$$
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 [2007.05821].

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 \(L_0=0\) or \(C_0=0\), with nonzero initial values giving only asymptotic CPE behavior [2007.05821]. These caveats matter because they separate constitutive equivalence from partial response matching.

In supercapacitor analysis, the CPE commonly appears in series with a resistor \(R_s\), producing
$$
Z_{\mathrm{tot}}(s)=R_s+\frac{1}{C_\alpha s^\alpha}.
$$
From this model,
$$
H_v(s)=\frac{1}{1+R_sC_\alpha s^\alpha}, \qquad
H_i(s)=\frac{C_\alpha s^\alpha}{1+R_sC_\alpha s^\alpha},
$$
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 [2301.13078]. Closed-form impulse responses involve Mittag-Leffler functions, and the method extends to instantaneous power \(p_c(t)=v_c(t)i_c(t)\) and accumulated energy [2301.13078].

A central practical consequence is that standard ideal-capacitor formulas such as \(E=\tfrac12 CV^2\) can be misleading for devices whose low-frequency branch is fitted by a CPE [2301.13078]. In the cited supercapacitor study, a fifth-order distributed \(RC\) circuit was used to emulate \(R_s\)-CPE behavior over a finite bandwidth, precisely because the CPE is fractional and not directly realizable as a single ideal element [2301.13078].

## 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 [2502.02608; 1306.1949]. 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 [2309.01701].

An important recent reinterpretation concerns the meaning of the dispersion coefficient \(\alpha\). Using an \(RC\)-network equivalency for the CPE,
$$
\tilde z(s)=\frac{1}{c_\alpha s^\alpha},
$$
one study decomposes the cumulative input energy
$$
E_{\rm in}(\tau)=\int_0^\tau v(t)i(t)\,dt
$$
into energy stored in capacitive modes, \(E_s(\tau)\), and energy dissipated in resistive modes, \(E_d(\tau)\), with
$$
E_{\rm in}(\tau)=E_s(\tau)+E_d(\tau)
$$
[2510.17812]. For a step voltage input, the ratios become
$$
\frac{E_s(\tau)}{E_{\rm in}(\tau)}=1-2^{-\alpha}, \qquad
\frac{E_d(\tau)}{E_{\rm in}(\tau)}=2^{-\alpha},
$$
so \(\alpha\) can be read as an energetic partition index rather than only a spectral-fit exponent [2510.17812]. The same paper reports that for ramp and quadratic inputs the energy ratios again reduce to pure functions of \(\alpha\), independent of excitation amplitude and material parameters [2510.17812].

This energetic reading is consistent with the limiting cases already built into the impedance definition. As \(\alpha\to 0\), the CPE becomes resistor-like and the dissipated fraction tends to unity; as \(\alpha\to 1\), it becomes capacitor-like, with the ideal-capacitor limits recovered for the respective excitation class [2510.17812]. 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 \(Z(s)\propto 1/s^\alpha\) [2110.12801]. This suggests a broader family of constant-phase concepts across disciplines, but the electrochemical CPE remains the fractional-order impedance element defined by \(1/(C_\alpha s^\alpha)\).

Source: https://www.emergentmind.com/topics/constant-phase-element-cpe