Constant Phase Element Calculator

Calculate a constant phase element by finding impedance, Q, angular frequency, or phase angle from any other three values in selectable units.

Constant Phase Element Formula

The calculator uses the magnitude form of the constant phase element impedance relationship:

Z = 1 / (Q*^α)

Rearranged formulas are used when one of the other inputs is missing:

Q = 1 / (Z*^α)

= (1 / (Z*Q))⁽1 / α)

α = log(1 / (Z*Q)) / log()

Z = impedance magnitude, in ohms Ω
Q = constant phase element coefficient, in S·s^α
ω = angular frequency, in rad/s
α = phase angle or CPE exponent value used in the power term

To calculate impedance, the calculator divides 1 by the product of Q and ω raised to α. To calculate Q, it rearranges the same relationship and divides 1 by Z times ω raised to α. To calculate angular frequency, it solves the power expression for ω. To calculate α, it uses logarithms to isolate the exponent.

If you enter frequency in Hz, it is converted to angular frequency using ω = 2πf before the formula is applied. If you enter α in degrees, it is converted to radians before calculation.

Constant Phase Element Units and Conversions

Quantity	Accepted units	Base unit used in formula	Conversion
Impedance, Z	Ω, kΩ, MΩ	Ω	1 kΩ = 1,000 Ω; 1 MΩ = 1,000,000 Ω
CPE coefficient, Q	S·s^α, mS·s^α, μS·s^α	S·s^α	1 mS = 0.001 S; 1 μS = 0.000001 S
Angular frequency, ω	rad/s, Hz	rad/s	ω = 2πf
Alpha, α	radians, degrees	radians	radians = degrees × π / 180

Typical CPE Alpha Interpretation

α behavior	Common interpretation
α close to 0	More resistor-like behavior
α around 0.5	Often associated with diffusion-like behavior
α close to 1	More ideal capacitor-like behavior

Example Calculations

Example 1: Calculate impedance

Given Q = 0.0002 S·s^α, ω = 100 rad/s, and α = 0.8 radians:

Z = 1 / (0.0002*100⁰.8)

Z = 125.594322 Ω

The impedance is about 125.594 Ω.

Example 2: Calculate Q

Given Z = 2 kΩ, frequency = 50 Hz, and α = 45 degrees:

First convert the values: Z = 2,000 Ω, ω = 314.159 rad/s, and α = 0.785398 radians.

Q = 1 / (2000*314.159⁰.785398)

Q \approx 0.000005466 S*s^α

The CPE coefficient is about 5.466 μS·s^α.

FAQ

What is a constant phase element?

A constant phase element is a circuit element used to model non-ideal capacitive behavior, often in electrochemical impedance spectroscopy. It is useful when a real surface, coating, or interface does not behave like a perfect capacitor.

Why does impedance decrease when frequency increases?

In the formula Z = 1 / (Qω^α), ω is in the denominator. If Q and α stay constant, a larger angular frequency makes Qω^α larger, so the calculated impedance becomes smaller.

Is α the same as phase angle in every CPE model?

Not always. In many CPE references, α or n is a dimensionless exponent between 0 and 1, and the phase angle is related to that exponent. This calculator uses the entered α value directly as the exponent after converting degrees to radians if degrees are selected.