# Electrode Geometry

# Electrode Geometry & Spacing

The physical design of WFC electrodes directly determines its electrical characteristics—capacitance, resistance, and field distribution. Proper geometry is essential for achieving target resonant frequencies and efficient operation.

## Parallel Plate Electrodes

The simplest configuration with straightforward calculations:

#### Capacitance:

C = ε₀ε<sub>r</sub>A / d

#### For Water (ε<sub>r</sub> ≈ 80):

C (pF) ≈ 708 × A(cm²) / d(mm)

#### Example:

<div class="formula-box" id="bkmrk-10-cm-%C3%97-10-cm-plates" style="background: #f8f9fa; padding: 20px; border-left: 4px solid #007bff; margin: 20px 0;">- 10 cm × 10 cm plates = 100 cm²
- 2 mm gap
- C = 708 × 100 / 2 = 35,400 pF = 35.4 nF

</div>## Concentric Tube Electrodes

Cylindrical geometry provides more surface area:

#### Capacitance:

C = 2πε₀ε<sub>r</sub>L / ln(r<sub>outer</sub>/r<sub>inner</sub>)

#### Simplified (for small gap relative to radius):

C ≈ ε₀ε<sub>r</sub> × 2πr<sub>avg</sub>L / d

Where d = r<sub>outer</sub> - r<sub>inner</sub>

#### Example:

<div class="formula-box" id="bkmrk-inner-tube%3A-20-mm-od" style="background: #e7f3ff; padding: 20px; border-left: 4px solid #17a2b8; margin: 20px 0;">- Inner tube: 20 mm OD
- Outer tube: 22 mm ID
- Length: 100 mm
- Gap: 1 mm
- C ≈ 708 × π × 2.1 × 10 / 1 = 46.7 nF

</div>## Tube Array Configurations

Multiple tubes in parallel increase total capacitance:

```
    Top View of 9-Tube Array:

           ┌───┐
         ┌─┤   ├─┐
       ┌─┤ └───┘ ├─┐
     ┌─┤ └───────┘ ├─┐
   ┌─┤ └───────────┘ ├─┐
   │ └───────────────┘ │
   │   Alternating     │
   │   + and − tubes   │
   └───────────────────┘

    Each concentric pair adds to total capacitance.
    C_total = C₁ + C₂ + C₃ + ... (tubes in parallel)
```

## Electrode Spacing Trade-offs

<table id="bkmrk-gap-size-capacitance" style="width: 100%; border-collapse: collapse; margin: 20px 0;"><thead><tr style="background: #28a745; color: white;"><th style="padding: 10px; border: 1px solid #ddd;">Gap Size</th><th style="padding: 10px; border: 1px solid #ddd;">Capacitance</th><th style="padding: 10px; border: 1px solid #ddd;">Resistance</th><th style="padding: 10px; border: 1px solid #ddd;">Field Strength</th><th style="padding: 10px; border: 1px solid #ddd;">Practical Issues</th></tr></thead><tbody><tr><td style="padding: 10px; border: 1px solid #ddd;">Very small (&lt;0.5 mm)</td><td style="padding: 10px; border: 1px solid #ddd;">Very high</td><td style="padding: 10px; border: 1px solid #ddd;">Low</td><td style="padding: 10px; border: 1px solid #ddd;">Very high</td><td style="padding: 10px; border: 1px solid #ddd;">Bubble blocking, arcing risk</td></tr><tr style="background: #d4edda;"><td style="padding: 10px; border: 1px solid #ddd;">Small (0.5-1.5 mm)</td><td style="padding: 10px; border: 1px solid #ddd;">High</td><td style="padding: 10px; border: 1px solid #ddd;">Medium-low</td><td style="padding: 10px; border: 1px solid #ddd;">High</td><td style="padding: 10px; border: 1px solid #ddd;">**Sweet spot**</td></tr><tr><td style="padding: 10px; border: 1px solid #ddd;">Medium (1.5-3 mm)</td><td style="padding: 10px; border: 1px solid #ddd;">Medium</td><td style="padding: 10px; border: 1px solid #ddd;">Medium</td><td style="padding: 10px; border: 1px solid #ddd;">Medium</td><td style="padding: 10px; border: 1px solid #ddd;">Easy to build</td></tr><tr><td style="padding: 10px; border: 1px solid #ddd;">Large (&gt;3 mm)</td><td style="padding: 10px; border: 1px solid #ddd;">Low</td><td style="padding: 10px; border: 1px solid #ddd;">High</td><td style="padding: 10px; border: 1px solid #ddd;">Low</td><td style="padding: 10px; border: 1px solid #ddd;">Needs more voltage</td></tr></tbody></table>

## Electric Field Calculation

#### Field Strength (uniform field approximation):

E = V / d

#### Example:

<div class="formula-box" id="bkmrk-v-%3D-1000-v-%28from-vic" style="background: #fff3cd; padding: 20px; border-left: 4px solid #ffc107; margin: 20px 0;"><div class="formula-box" style="background: #fff3cd; padding: 20px; border-left: 4px solid #ffc107; margin: 20px 0;">- V = 1000 V (from VIC magnification)
- d = 1 mm = 0.001 m
- E = 1000 / 0.001 = 1,000,000 V/m = **1 MV/m**

</div></div>**Note:** Water breakdown occurs at ~30-70 MV/m, so typical VIC fields are well below breakdown.

## Surface Area Considerations

Larger electrode area provides:

- Higher capacitance (more energy storage)
- Lower current density (longer electrode life)
- More sites for gas evolution
- Better heat dissipation

But requires:

- Larger choke inductance (to maintain resonant frequency)
- More water volume
- Larger enclosure

## Dimensional Design Process

#### Step 1: Determine Target Capacitance

From resonant frequency and available inductance:

C<sub>target</sub> = 1 / (4π²f₀²L₂)

#### Step 2: Choose Geometry Type

Plates, tubes, or array based on available materials and space.

#### Step 3: Select Gap Distance

Balance capacitance needs with practical concerns (1-2 mm typical).

#### Step 4: Calculate Required Area

A = C × d / (ε₀ε<sub>r</sub>)

#### Step 5: Dimension the Electrodes

For plates: Choose L × W. For tubes: Choose radius and length.

## Practical Design Example

#### Target: f₀ = 10 kHz, L₂ = 50 mH available

**Required capacitance:**

C = 1/(4π² × 10000² × 0.05) = 5.07 nF

**Using parallel plates with 1.5 mm gap:**

A = 5.07 × 10⁻⁹ × 0.0015 / (8.854×10⁻¹² × 80) = 10.7 cm²

**Electrode size:** ~3.3 cm × 3.3 cm plates (quite small!)

**For more practical size, use 1 mm gap:**

A = 7.1 cm² → 2.7 × 2.7 cm plates

*Note: Very small WFC! May need to increase L₂ for practical electrode sizes.*

## Edge Effects

Real electrodes have fringing fields at edges that increase effective capacitance:

- For parallel plates, add ~0.9d to each edge dimension
- For tubes, end effects can add 5-10% to capacitance
- Guard rings can reduce edge effects in precision applications

## Electrode Alignment

#### Critical Requirements:

<div id="bkmrk-parallelism%3A-plates-" style="background: #f8d7da; padding: 15px; border-radius: 5px; margin: 20px 0;">- **Parallelism:** Plates must be parallel for uniform field
- **Concentricity:** Tubes must be truly concentric
- **Uniform gap:** Variations cause hot spots and non-uniform current
- **Insulating spacers:** Use non-conductive materials (PTFE, ceramic)

</div>## Gas Evolution Considerations

When gas is produced, it affects the electrical characteristics:

- Bubbles displace water, reducing effective capacitance
- Bubble layer increases resistance
- Vertical orientation helps bubbles rise and escape
- Perforated electrodes allow better bubble release

**VIC Matrix Calculator:** The Water Profile section calculates WFC capacitance from your electrode dimensions. Enter geometry type, dimensions, and spacing to get accurate capacitance values for circuit design.

*Next: Water Conductivity &amp; Dielectric Properties →*