The constant phase element calculator helps compute the impedance of a constant phase element (CPE), which is used to model non-ideal capacitors in electrical circuits. It is widely used in electrochemistry, impedance spectroscopy, and material science to analyze systems with distributed time constants. This tool calculates impedance values and separates the real and imaginary parts for further analysis.
Formula of Constant Phase Element Calculator
Step 1: Define the formula
The impedance of a constant phase element is calculated using:
Z = 1 / (Q * (j * omega)^n)
Where:
Z is the impedance in ohms
Q is the constant phase element coefficient in F·s^(n-1)
j is the imaginary unit, where j = square root of -1
omega is the angular frequency in radians per second
n is the phase exponent, ranging between 0 and 1
Step 2: Calculate angular frequency
Angular frequency is computed as:
omega = 2 * pi * f
Where:
f is the frequency in hertz
Step 3: Substitute values into the formula
Substitute the values for Q, omega, and n into the formula:
Z = 1 / (Q * (j * omega)^n)
Step 4: Separate real and imaginary components
To analyze the impedance further, split Z into its real and imaginary parts:
Real part: Re(Z) = |Z| * cos(phi)
Imaginary part: Im(Z) = |Z| * sin(phi)
Where:
|Z| is the magnitude of the impedance
phi is the phase angle, calculated as -n * pi / 2
Table of Common Calculations
| Parameter | Formula | Example Value |
|---|---|---|
| Angular frequency | omega = 2 * pi * f | 628.3 rad/s |
| Impedance | Z = 1 / (Q * (j * omega)^n) | 100 + j50 ohms |
| Phase angle | phi = -n * pi / 2 | -45 degrees |
| Magnitude of Z | Z |
Example of Constant Phase Element Calculator
Problem
A CPE has a coefficient Q = 0.01 F·s^(n-1), a phase exponent n = 0.8, and is operated at a frequency of 100 Hz. Calculate the impedance and separate its real and imaginary parts.
Solution
- Calculate angular frequency:
omega = 2 * pi * f
omega = 2 * 3.1416 * 100 = 628.32 rad/s - Calculate impedance:
Z = 1 / (Q * (j * omega)^n)
Z = 1 / (0.01 * (j * 628.32)^0.8)First, compute (j * 628.32)^0.8:
Magnitude = 628.32^0.8 = 89.8
Phase angle = 0.8 * pi / 2 = 1.2566 radiansZ = 1 / (0.01 * 89.8 * exp(j * 1.2566))
Z = 1 / (0.898 * exp(j * 1.2566))
Z = 1.113 – j1.113 ohms (approximated) - Separate real and imaginary parts:
Real part = Re(Z) = 1.113 ohms
Imaginary part = Im(Z) = -1.113 ohms
Result
The impedance of the CPE is 1.113 – j1.113 ohms.
Most Common FAQs
It calculates the impedance of a CPE, allowing for the analysis of non-ideal capacitive behavior in systems such as batteries and corrosion studies.
The phase exponent determines the deviation of the CPE from an ideal capacitor. A value of 1 corresponds to a perfect capacitor.
Yes, by varying the frequency input, the calculator can compute the impedance of a CPE over a range of frequencies, which is useful for impedance spectroscopy.