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 r A / d 

 For Water (ε r ≈ 80): 

 C (pF) ≈ 708 × A(cm²) / d(mm) 

 Example: 

 

 10 cm × 10 cm plates = 100 cm² 

 2 mm gap 

 C = 708 × 100 / 2 = 35,400 pF = 35.4 nF 

 

 Concentric Tube Electrodes 

 Cylindrical geometry provides more surface area: 

 Capacitance: 

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

 Simplified (for small gap relative to radius): 

 C ≈ ε₀ε r × 2πr avg L / d 

 Where d = r outer - r inner 

 Example: 

 

 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 

 

 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 

 Gap Size 

 Capacitance 

 Resistance 

 Field Strength 

 Practical Issues 

 Very small (<0.5 mm) 

 Very high 

 Low 

 Very high 

 Bubble blocking, arcing risk 

 Small (0.5-1.5 mm) 

 High 

 Medium-low 

 High 

 Sweet spot 

 Medium (1.5-3 mm) 

 Medium 

 Medium 

 Medium 

 Easy to build 

 Large (>3 mm) 

 Low 

 High 

 Low 

 Needs more voltage 

 Electric Field Calculation 

 Field Strength (uniform field approximation): 

 E = V / d 

 Example: 

 

 

 V = 1000 V (from VIC magnification) 

 d = 1 mm = 0.001 m 

 E = 1000 / 0.001 = 1,000,000 V/m = 1 MV/m 

 

 

 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 target = 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 r ) 

 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: 

 

 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) 

 

 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. 

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