Cell Capacitance

Calculating WFC Capacitance

Accurate calculation of WFC capacitance is essential for VIC circuit design. This page provides formulas and methods for determining the effective capacitance of various electrode configurations.

Total WFC Capacitance Model

The WFC has multiple capacitance contributions:

Series Model (simplified):

1/C_total = 1/C_edl,anode + 1/C_geo + 1/C_edl,cathode

For Practical VIC Frequencies:

At kHz frequencies, C_edl >> C_geo, so:

C_total ≈ C_geo

The geometric capacitance dominates for typical electrode gaps (>0.5 mm).

Geometric Capacitance Formulas

Parallel Plates

C = ε₀ε_rA / d

Quick Formula for Water:

C (nF) = 0.0708 × A(cm²) / d(mm)

Example:

Concentric Cylinders

C = 2πε₀ε_rL / ln(r_o/r_i)

Quick Formula for Water:

C (nF) = 4.45 × L(cm) / ln(r_o/r_i)

Thin Gap Approximation (when gap << radius):

C (nF) ≈ 0.0708 × 2πr_avg(cm) × L(cm) / d(mm)

Multiple Tubes (Array)

C_total = n × C_{single tube pair}

Where n is the number of tube pairs in parallel.

Meyer's 9-Tube Array Example:

Capacitance Calculator Table

Area (cm²)	Gap 0.5mm	Gap 1.0mm	Gap 1.5mm	Gap 2.0mm
25	3.54 nF	1.77 nF	1.18 nF	0.89 nF
50	7.08 nF	3.54 nF	2.36 nF	1.77 nF
100	14.2 nF	7.08 nF	4.72 nF	3.54 nF
200	28.3 nF	14.2 nF	9.44 nF	7.08 nF
500	70.8 nF	35.4 nF	23.6 nF	17.7 nF

Including EDL Effects

For more accurate modeling at lower frequencies or smaller gaps:

EDL Capacitance per Electrode:

C_edl = c_dl × A

Where c_dl ≈ 20-40 µF/cm² for stainless steel in water.

Total with EDL:

1/C_total = 1/C_geo + 2/C_edl

(Factor of 2 because both electrodes have EDL)

Example:

Measuring WFC Capacitance

Method 1: LCR Meter

• Most accurate method
• Measure at 1 kHz and 10 kHz (should be similar)
• Provides both C and R (ESR)
• Temperature affects reading

Method 2: RC Time Constant

1. Connect WFC in series with known resistor R
2. Apply step voltage
3. Measure time to reach 63% of final voltage
4. C = τ / R

Method 3: Resonant Frequency

1. Connect WFC with known inductor L
2. Drive with variable frequency
3. Find resonant peak
4. C = 1 / (4π²f₀²L)

Capacitance Variations

WFC capacitance can change during operation:

Factor	Effect on C	Typical Change
Temperature increase	C decreases (εr drops)	-0.4%/°C
Gas bubble formation	C decreases (less water)	-5% to -30%
Water level drop	C decreases	Proportional
Electrode coating	C may decrease	Variable
Applied voltage	Minor change	±5%

Design Workflow

1. Determine Required C

C_wfc = 1 / (4π²f₀²L₂)

2. Choose Electrode Gap

1-2 mm is typical. Smaller = higher C, larger = lower C.

3. Calculate Required Area

A = C × d / (ε₀ε_r) = C(nF) × d(mm) / 0.0708 (cm²)

4. Design Electrodes

Choose plate dimensions or tube sizes to achieve area.

5. Verify by Measurement

Build prototype and measure actual capacitance.

VIC Matrix Calculator: The Water Profile section calculates WFC capacitance automatically. Enter electrode type, dimensions, and gap. The calculator also shows how the capacitance affects resonant frequency and provides warnings if values are outside recommended ranges.

Next: Matching WFC to Circuit →