# Resonant Matching

# Matching WFC to Circuit

For optimal VIC performance, the WFC must be properly matched to the circuit—its capacitance must resonate with the secondary choke at the desired operating frequency. This page covers the matching process and strategies for achieving good resonance.

## The Matching Problem

In a VIC circuit, we have three interdependent parameters:

f₀ = 1 / (2π√(L₂ × C<sub>wfc</sub>))

#### Design Challenge:

<div class="formula-box" id="bkmrk-f%E2%82%80-is-set-by-the-pul" style="background: #f8f9fa; padding: 20px; border-left: 4px solid #007bff; margin: 20px 0;">- **f₀** is set by the pulse generator (typically 1-50 kHz)
- **C<sub>wfc</sub>** is constrained by electrode geometry and water properties
- **L₂** must be designed to complete the resonant match

</div>## Matching Strategies

### Strategy 1: Design L₂ for Given WFC

When WFC geometry is fixed (existing cell):

<div id="bkmrk-measure-cwfc-with-lc" style="background: #e7f3ff; padding: 20px; border-radius: 5px; margin: 20px 0;"><div style="background: #e7f3ff; padding: 20px; border-radius: 5px; margin: 20px 0;">1. Measure C<sub>wfc</sub> with LCR meter
2. Choose target frequency f₀
3. Calculate required L₂:

</div></div>L₂ = 1 / (4π²f₀²C<sub>wfc</sub>)

#### Example:

<div id="bkmrk-cwfc-%3D-10-nf-%28measur" style="background: #e7f3ff; padding: 20px; border-radius: 5px; margin: 20px 0;">- C<sub>wfc</sub> = 10 nF (measured)
- f₀ = 10 kHz (desired)
- L₂ = 1 / (4π² × 10⁴² × 10⁻⁸) = 25.3 mH

</div>### Strategy 2: Design WFC for Given L₂

When using a pre-wound or available choke:

<div id="bkmrk-measure-l%E2%82%82-with-lcr-" style="background: #fff3cd; padding: 20px; border-radius: 5px; margin: 20px 0;"><div style="background: #fff3cd; padding: 20px; border-radius: 5px; margin: 20px 0;">1. Measure L₂ with LCR meter
2. Choose target frequency f₀
3. Calculate required C<sub>wfc</sub>:

</div></div>C<sub>wfc</sub> = 1 / (4π²f₀²L₂)

<div id="bkmrk-design-electrodes-to" style="background: #fff3cd; padding: 20px; border-radius: 5px; margin: 20px 0;">4. Design electrodes to achieve that capacitance

</div>### Strategy 3: Tune with Additional Capacitor

When exact match isn't achievable:

#### If C<sub>wfc</sub> is too low:

Add capacitor in parallel with WFC

C<sub>total</sub> = C<sub>wfc</sub> + C<sub>tune</sub>

#### If C<sub>wfc</sub> is too high:

Add capacitor in series with WFC (less common)

1/C<sub>total</sub> = 1/C<sub>wfc</sub> + 1/C<sub>series</sub>

## Impedance Matching Considerations

Beyond frequency matching, impedance levels affect energy transfer:

#### Secondary Characteristic Impedance:

Z₀ = √(L₂/C<sub>wfc</sub>)

#### Example Comparison:

<div class="formula-box" id="bkmrk-l%E2%82%82-cwfc-f%E2%82%80-z%E2%82%80-10-mh-" style="background: #f8f9fa; padding: 20px; border-left: 4px solid #28a745; margin: 20px 0;"><div class="formula-box" style="background: #f8f9fa; padding: 20px; border-left: 4px solid #28a745; margin: 20px 0;"><table style="width: 100%; border-collapse: collapse; margin-top: 10px;"><thead><tr style="background: #6c757d; color: white;"><th style="padding: 8px; border: 1px solid #ddd;">L₂</th><th style="padding: 8px; border: 1px solid #ddd;">C<sub>wfc</sub></th><th style="padding: 8px; border: 1px solid #ddd;">f₀</th><th style="padding: 8px; border: 1px solid #ddd;">Z₀</th></tr></thead><tbody><tr><td style="padding: 8px; border: 1px solid #ddd;">10 mH</td><td style="padding: 8px; border: 1px solid #ddd;">25 nF</td><td style="padding: 8px; border: 1px solid #ddd;">10 kHz</td><td style="padding: 8px; border: 1px solid #ddd;">632 Ω</td></tr><tr><td style="padding: 8px; border: 1px solid #ddd;">50 mH</td><td style="padding: 8px; border: 1px solid #ddd;">5 nF</td><td style="padding: 8px; border: 1px solid #ddd;">10 kHz</td><td style="padding: 8px; border: 1px solid #ddd;">3162 Ω</td></tr><tr><td style="padding: 8px; border: 1px solid #ddd;">100 mH</td><td style="padding: 8px; border: 1px solid #ddd;">2.5 nF</td><td style="padding: 8px; border: 1px solid #ddd;">10 kHz</td><td style="padding: 8px; border: 1px solid #ddd;">6325 Ω</td></tr></tbody></table>

</div></div>**Higher Z₀ = Higher voltage for same energy**

## Primary-Secondary Matching

For dual-resonant VIC with both L1-C1 and L2-WFC tanks:

<table id="bkmrk-configuration-condit" style="width: 100%; border-collapse: collapse; margin: 20px 0;"><thead><tr style="background: #17a2b8; color: white;"><th style="padding: 10px; border: 1px solid #ddd;">Configuration</th><th style="padding: 10px; border: 1px solid #ddd;">Condition</th><th style="padding: 10px; border: 1px solid #ddd;">Effect</th></tr></thead><tbody><tr><td style="padding: 10px; border: 1px solid #ddd;">Same frequency</td><td style="padding: 10px; border: 1px solid #ddd;">f₀<sub>pri</sub> = f₀<sub>sec</sub></td><td style="padding: 10px; border: 1px solid #ddd;">Maximum voltage magnification</td></tr><tr><td style="padding: 10px; border: 1px solid #ddd;">Slight offset</td><td style="padding: 10px; border: 1px solid #ddd;">f₀<sub>sec</sub> ≈ 0.95-1.05 × f₀<sub>pri</sub></td><td style="padding: 10px; border: 1px solid #ddd;">Broader response, easier tuning</td></tr><tr><td style="padding: 10px; border: 1px solid #ddd;">Harmonic</td><td style="padding: 10px; border: 1px solid #ddd;">f₀<sub>sec</sub> = 2× or 3× f₀<sub>pri</sub></td><td style="padding: 10px; border: 1px solid #ddd;">Secondary resonates on harmonic</td></tr></tbody></table>

## Finding Resonance

### Method 1: Frequency Sweep

1. Connect oscilloscope across WFC
2. Sweep generator frequency slowly
3. Watch for voltage peak
4. Note frequency of maximum amplitude

### Method 2: Phase Measurement

1. Monitor current and voltage simultaneously
2. At resonance, current and voltage are in phase (phase = 0°)
3. Below resonance: capacitive (current leads)
4. Above resonance: inductive (current lags)

### Method 3: Minimum Current

For a series resonant circuit driven from a voltage source:

- Current is minimum at anti-resonance (parallel resonance)
- May need to reconfigure measurement

## Troubleshooting Mismatch

<table id="bkmrk-symptom-likely-cause" style="width: 100%; border-collapse: collapse; margin: 20px 0;"><thead><tr style="background: #dc3545; color: white;"><th style="padding: 10px; border: 1px solid #ddd;">Symptom</th><th style="padding: 10px; border: 1px solid #ddd;">Likely Cause</th><th style="padding: 10px; border: 1px solid #ddd;">Solution</th></tr></thead><tbody><tr><td style="padding: 10px; border: 1px solid #ddd;">No clear resonance peak</td><td style="padding: 10px; border: 1px solid #ddd;">Very low Q (high losses)</td><td style="padding: 10px; border: 1px solid #ddd;">Reduce water conductivity, lower DCR</td></tr><tr><td style="padding: 10px; border: 1px solid #ddd;">Resonance far from expected</td><td style="padding: 10px; border: 1px solid #ddd;">Wrong L or C values</td><td style="padding: 10px; border: 1px solid #ddd;">Measure components, recalculate</td></tr><tr><td style="padding: 10px; border: 1px solid #ddd;">Resonance drifts during operation</td><td style="padding: 10px; border: 1px solid #ddd;">Temperature change, bubbles</td><td style="padding: 10px; border: 1px solid #ddd;">Allow warmup, improve gas venting</td></tr><tr><td style="padding: 10px; border: 1px solid #ddd;">Multiple resonance peaks</td><td style="padding: 10px; border: 1px solid #ddd;">Coupled modes, parasitics</td><td style="padding: 10px; border: 1px solid #ddd;">Check for stray coupling</td></tr></tbody></table>

## Fine Tuning Tips

#### For L₂ Adjustment:

<div id="bkmrk-add%2Fremove-turns-%28la" style="background: #d4edda; padding: 20px; border-radius: 5px; margin: 20px 0;"><div style="background: #d4edda; padding: 20px; border-radius: 5px; margin: 20px 0;">- Add/remove turns (large adjustment)
- Adjust core gap if gapped (medium)
- Use adjustable ferrite slug (fine)

</div></div>#### For C<sub>wfc</sub> Adjustment:

<div id="bkmrk-add-parallel-capacit" style="background: #d4edda; padding: 20px; border-radius: 5px; margin: 20px 0;"><div style="background: #d4edda; padding: 20px; border-radius: 5px; margin: 20px 0;">- Add parallel capacitor (increases C)
- Change water level (changes effective area)
- Adjust electrode spacing (if possible)

</div></div>#### For Frequency Adjustment:

<div id="bkmrk-pll-feedback-to-trac" style="background: #d4edda; padding: 20px; border-radius: 5px; margin: 20px 0;">- PLL feedback to track resonance
- Variable frequency oscillator
- Multiple operating modes

</div>## Complete Matching Checklist

<div id="bkmrk-%E2%98%90-measure-or-calcula" style="background: #f8f9fa; padding: 20px; border-radius: 5px; margin: 20px 0;">1. ☐ Measure or calculate C<sub>wfc</sub>
2. ☐ Measure or calculate L₂
3. ☐ Calculate expected f₀ = 1/(2π√(L₂C))
4. ☐ Verify f₀ is within driver frequency range
5. ☐ Calculate Z₀ = √(L₂/C)
6. ☐ Estimate R<sub>total</sub> (DCR + solution R)
7. ☐ Calculate Q = Z₀/R
8. ☐ Build circuit and measure actual resonance
9. ☐ Fine-tune as needed
10. ☐ Verify Q meets design goals

</div>**VIC Matrix Calculator:** The Simulation tab performs complete matching analysis. Enter your choke and WFC parameters, and it calculates resonant frequency, Q factor, voltage magnification, and shows warnings if components are mismatched.

*Chapter 6 Complete. Next: The VIC Matrix Calculator →*