Enter Cole–Cole Plot Data
Use measured intercepts and the peak imaginary magnitude from one dominant arc. Optional geometry values estimate conductivity and effective relative permittivity.
Formula Used
| Quantity | Formula | Meaning |
|---|---|---|
| Resistance spread | ΔR = Z'₀ − Z'∞ | Arc width on the real axis |
| Center x-coordinate | xc = (Z'₀ + Z'∞) / 2 | Horizontal midpoint of the chord |
| Circle center y-coordinate | yc = (h² − (ΔR/2)²) / (2h) | Vertical shift of the depressed circle, where h = |Z''|max |
| Arc radius | r = h − yc | Radius of the fitted circular arc |
| Depression angle | θ = tan−1(|yc| / (ΔR/2)) | Measures arc depression below an ideal semicircle |
| Cole–Cole parameter | α ≈ θ / 90° | Geometry-based estimate of the depression factor |
| Relaxation time | τ = 1 / (2πfmax) | Time constant from the peak-frequency condition |
| Effective capacitance | Ceff = τ / ΔR | Equivalent capacitance for the dominant arc |
| DC conductivity | σ = t / (R0A) | Uses sample thickness t and electrode area A |
| Effective relative permittivity | εr = Cefft / (ε₀A) | Geometry-assisted dielectric estimate |
These relations work best for a dominant, isolated relaxation arc. Strong overlap, diffusion tails, or multiple processes can require full impedance fitting.
How to Use This Calculator
- Read the high-frequency and low-frequency intercepts directly from the real-axis crossings of one arc.
- Measure the arc peak as the maximum magnitude of the imaginary component, |Z''|.
- Enter the frequency at that peak to estimate the relaxation time.
- Choose the same impedance unit used in your dataset.
- Add sample thickness and electrode area only when you want conductivity and effective permittivity estimates.
- Press the calculate button to place the results above the form and draw the Cole–Cole plot.
- Use the export buttons to save a summary in CSV or PDF format.
Example Data Table
| Example item | Value |
|---|---|
| Low-frequency intercept Z'₀ | 1200 Ω |
| High-frequency intercept Z'∞ | 200 Ω |
| Peak |Z''| | 430 Ω |
| Peak frequency | 1590 Hz |
| Resistance spread ΔR | 1000 Ω |
| Relaxation time τ | 1.0005 × 10−4 s |
| Effective capacitance | 1.0005 × 10−7 F |
| Estimated α | 0.0956 |
Frequently Asked Questions
1. What does a Cole–Cole plot show?
It shows the relationship between the real and imaginary parts of impedance. A perfect Debye relaxation gives a semicircle, while depressed arcs indicate distributed relaxation times or non-ideal interfacial behavior.
2. Why must the low-frequency intercept be larger?
For a standard impedance arc, the curve starts near the high-frequency intercept and moves rightward toward the low-frequency intercept. That makes the low-frequency intercept larger on the real axis.
3. What happens if the peak is too high?
If the measured peak height exceeds half the intercept spacing, the geometry cannot form a valid depressed circular arc. That usually means the selected points are inconsistent or belong to different processes.
4. Is the calculated α an exact fitted Cole–Cole parameter?
No. It is a fast geometry-based estimate from the observed arc depression. Rigorous work should still use nonlinear fitting on the full impedance spectrum.
5. Why is τ obtained from the peak frequency?
For a dominant relaxation arc, the imaginary response reaches its maximum near the characteristic frequency. That gives the common approximation τ = 1/(2πfmax).
6. When should I enter thickness and area?
Enter them when you want material properties tied to geometry, such as conductivity or effective relative permittivity. Leave them blank if you only need arc parameters from the plot itself.
7. Can this calculator handle multiple arcs?
It is designed for one dominant arc at a time. For multiple arcs, isolate each process first or use a full equivalent-circuit fit across the complete frequency range.
8. What does the ideal reference arc mean on the graph?
The ideal arc is the semicircle expected when there is no depression. Comparing it with the measured arc highlights non-ideal broadening and gives intuition about the estimated α value.