# 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<sub>total</sub> = 1/C<sub>edl,anode</sub> + 1/C<sub>geo</sub> + 1/C<sub>edl,cathode</sub>

#### For Practical VIC Frequencies:

At kHz frequencies, C<sub>edl</sub> &gt;&gt; C<sub>geo</sub>, so:

C<sub>total</sub> ≈ C<sub>geo</sub>

The geometric capacitance dominates for typical electrode gaps (&gt;0.5 mm).

## Geometric Capacitance Formulas

### Parallel Plates

C = ε₀ε<sub>r</sub>A / d

#### Quick Formula for Water:

C (nF) = 0.0708 × A(cm²) / d(mm)

#### Example:

<div class="formula-box" id="bkmrk-a-%3D-50-cm%C2%B2%2C-d-%3D-1-mm" style="background: #e7f3ff; padding: 20px; border-left: 4px solid #17a2b8; margin: 20px 0;">- A = 50 cm², d = 1 mm
- C = 0.0708 × 50 / 1 = 3.54 nF

</div>### Concentric Cylinders

C = 2πε₀ε<sub>r</sub>L / ln(r<sub>o</sub>/r<sub>i</sub>)

#### Quick Formula for Water:

C (nF) = 4.45 × L(cm) / ln(r<sub>o</sub>/r<sub>i</sub>)

#### Thin Gap Approximation (when gap &lt;&lt; radius):

C (nF) ≈ 0.0708 × 2πr<sub>avg</sub>(cm) × L(cm) / d(mm)

### Multiple Tubes (Array)

C<sub>total</sub> = n × C<sub>single tube pair</sub>

Where n is the number of tube pairs in parallel.

#### Meyer's 9-Tube Array Example:

<div class="formula-box" id="bkmrk-9-concentric-tube-pa" style="background: #e7f3ff; padding: 20px; border-left: 4px solid #17a2b8; margin: 20px 0;">- 9 concentric tube pairs
- Each pair: C ≈ 5 nF
- Total: C = 9 × 5 = 45 nF

</div>## Capacitance Calculator Table

<table id="bkmrk-area-%28cm%C2%B2%29-gap-0.5mm" style="width: 100%; border-collapse: collapse; margin: 20px 0;"><thead><tr style="background: #28a745; color: white;"><th style="padding: 10px; border: 1px solid #ddd;">Area (cm²)</th><th style="padding: 10px; border: 1px solid #ddd;">Gap 0.5mm</th><th style="padding: 10px; border: 1px solid #ddd;">Gap 1.0mm</th><th style="padding: 10px; border: 1px solid #ddd;">Gap 1.5mm</th><th style="padding: 10px; border: 1px solid #ddd;">Gap 2.0mm</th></tr></thead><tbody><tr><td style="padding: 10px; border: 1px solid #ddd;">25</td><td style="padding: 10px; border: 1px solid #ddd;">3.54 nF</td><td style="padding: 10px; border: 1px solid #ddd;">1.77 nF</td><td style="padding: 10px; border: 1px solid #ddd;">1.18 nF</td><td style="padding: 10px; border: 1px solid #ddd;">0.89 nF</td></tr><tr><td style="padding: 10px; border: 1px solid #ddd;">50</td><td style="padding: 10px; border: 1px solid #ddd;">7.08 nF</td><td style="padding: 10px; border: 1px solid #ddd;">3.54 nF</td><td style="padding: 10px; border: 1px solid #ddd;">2.36 nF</td><td style="padding: 10px; border: 1px solid #ddd;">1.77 nF</td></tr><tr><td style="padding: 10px; border: 1px solid #ddd;">100</td><td style="padding: 10px; border: 1px solid #ddd;">14.2 nF</td><td style="padding: 10px; border: 1px solid #ddd;">7.08 nF</td><td style="padding: 10px; border: 1px solid #ddd;">4.72 nF</td><td style="padding: 10px; border: 1px solid #ddd;">3.54 nF</td></tr><tr><td style="padding: 10px; border: 1px solid #ddd;">200</td><td style="padding: 10px; border: 1px solid #ddd;">28.3 nF</td><td style="padding: 10px; border: 1px solid #ddd;">14.2 nF</td><td style="padding: 10px; border: 1px solid #ddd;">9.44 nF</td><td style="padding: 10px; border: 1px solid #ddd;">7.08 nF</td></tr><tr><td style="padding: 10px; border: 1px solid #ddd;">500</td><td style="padding: 10px; border: 1px solid #ddd;">70.8 nF</td><td style="padding: 10px; border: 1px solid #ddd;">35.4 nF</td><td style="padding: 10px; border: 1px solid #ddd;">23.6 nF</td><td style="padding: 10px; border: 1px solid #ddd;">17.7 nF</td></tr></tbody></table>

## Including EDL Effects

For more accurate modeling at lower frequencies or smaller gaps:

#### EDL Capacitance per Electrode:

C<sub>edl</sub> = c<sub>dl</sub> × A

Where c<sub>dl</sub> ≈ 20-40 µF/cm² for stainless steel in water.

#### Total with EDL:

1/C<sub>total</sub> = 1/C<sub>geo</sub> + 2/C<sub>edl</sub>

(Factor of 2 because both electrodes have EDL)

#### Example:

<div class="formula-box" id="bkmrk-a-%3D-100-cm%C2%B2%2C-d-%3D-1-m" style="background: #fff3cd; padding: 20px; border-left: 4px solid #ffc107; margin: 20px 0;">- A = 100 cm², d = 1 mm, c<sub>dl</sub> = 25 µF/cm²
- C<sub>geo</sub> = 7.08 nF
- C<sub>edl</sub> = 25 µF/cm² × 100 cm² = 2500 µF = 2.5 mF
- 1/C = 1/7.08nF + 2/2.5mF ≈ 1/7.08nF
- C<sub>total</sub> ≈ 7.08 nF (EDL negligible)

</div>## 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:

<table id="bkmrk-factor-effect-on-c-t" style="width: 100%; border-collapse: collapse; margin: 20px 0;"><thead><tr style="background: #dc3545; color: white;"><th style="padding: 10px; border: 1px solid #ddd;">Factor</th><th style="padding: 10px; border: 1px solid #ddd;">Effect on C</th><th style="padding: 10px; border: 1px solid #ddd;">Typical Change</th></tr></thead><tbody><tr><td style="padding: 10px; border: 1px solid #ddd;">Temperature increase</td><td style="padding: 10px; border: 1px solid #ddd;">C decreases (ε<sub>r</sub> drops)</td><td style="padding: 10px; border: 1px solid #ddd;">-0.4%/°C</td></tr><tr><td style="padding: 10px; border: 1px solid #ddd;">Gas bubble formation</td><td style="padding: 10px; border: 1px solid #ddd;">C decreases (less water)</td><td style="padding: 10px; border: 1px solid #ddd;">-5% to -30%</td></tr><tr><td style="padding: 10px; border: 1px solid #ddd;">Water level drop</td><td style="padding: 10px; border: 1px solid #ddd;">C decreases</td><td style="padding: 10px; border: 1px solid #ddd;">Proportional</td></tr><tr><td style="padding: 10px; border: 1px solid #ddd;">Electrode coating</td><td style="padding: 10px; border: 1px solid #ddd;">C may decrease</td><td style="padding: 10px; border: 1px solid #ddd;">Variable</td></tr><tr><td style="padding: 10px; border: 1px solid #ddd;">Applied voltage</td><td style="padding: 10px; border: 1px solid #ddd;">Minor change</td><td style="padding: 10px; border: 1px solid #ddd;">±5%</td></tr></tbody></table>

## Design Workflow

#### 1. Determine Required C

C<sub>wfc</sub> = 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<sub>r</sub>) = 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 →*