# Voltage Magnification

# Voltage Magnification at Resonance

Voltage magnification is the cornerstone of VIC circuit operation. At resonance, the voltage across reactive components (inductors and capacitors) can be many times greater than the input voltage. This is how the VIC develops high voltages across the water fuel cell while drawing modest current from the source.

## The Principle of Voltage Magnification

In a series resonant circuit, even though the total impedance is at minimum (just resistance), the individual voltages across L and C can be much larger than the source voltage. This isn't "free energy"—it's the result of energy continuously cycling between the inductor and capacitor.

#### Key Insight:

At resonance, V<sub>L</sub> and V<sub>C</sub> are equal in magnitude but opposite in phase. They cancel each other in the circuit loop, but individually each represents a real voltage that can do work.

## Voltage Magnification Formula

#### Q-Based Magnification:

V<sub>output</sub> = Q × V<sub>input</sub>

#### Impedance-Based Magnification:

Magnification = Z₀ / R = (1/R) × √(L/C)

Both formulas give the same result since Q = Z₀/R for a series circuit.

## Practical Examples

<table id="bkmrk-input-voltage-q-fact" style="width: 100%; border-collapse: collapse; margin: 20px 0;"><thead><tr style="background: #28a745; color: white;"><th style="padding: 10px; border: 1px solid #ddd;">Input Voltage</th><th style="padding: 10px; border: 1px solid #ddd;">Q Factor</th><th style="padding: 10px; border: 1px solid #ddd;">Output Voltage</th><th style="padding: 10px; border: 1px solid #ddd;">Application</th></tr></thead><tbody><tr><td style="padding: 10px; border: 1px solid #ddd;">12V</td><td style="padding: 10px; border: 1px solid #ddd;">10</td><td style="padding: 10px; border: 1px solid #ddd;">120V</td><td style="padding: 10px; border: 1px solid #ddd;">Low-Q experimental setup</td></tr><tr><td style="padding: 10px; border: 1px solid #ddd;">12V</td><td style="padding: 10px; border: 1px solid #ddd;">50</td><td style="padding: 10px; border: 1px solid #ddd;">600V</td><td style="padding: 10px; border: 1px solid #ddd;">Typical VIC circuit</td></tr><tr><td style="padding: 10px; border: 1px solid #ddd;">12V</td><td style="padding: 10px; border: 1px solid #ddd;">100</td><td style="padding: 10px; border: 1px solid #ddd;">1200V</td><td style="padding: 10px; border: 1px solid #ddd;">High-Q optimized circuit</td></tr><tr><td style="padding: 10px; border: 1px solid #ddd;">24V</td><td style="padding: 10px; border: 1px solid #ddd;">50</td><td style="padding: 10px; border: 1px solid #ddd;">1200V</td><td style="padding: 10px; border: 1px solid #ddd;">Higher input voltage approach</td></tr></tbody></table>

## Where the Magnified Voltage Appears

### In a Series LC Circuit

- **Across the inductor:** V<sub>L</sub> = Q × V<sub>source</sub> (leads current by 90°)
- **Across the capacitor:** V<sub>C</sub> = Q × V<sub>source</sub> (lags current by 90°)
- **Across resistance:** V<sub>R</sub> = V<sub>source</sub> (in phase with current)

### In the VIC Circuit

The water fuel cell acts as the capacitor, so the magnified voltage appears directly across the water:

**VIC Voltage Path:**

Source → L1 → C1 (series resonance for initial magnification)

Transformed via coupling to → L2 → WFC (secondary resonance)

Result: High voltage across water fuel cell electrodes

## Two Approaches to Magnification

### Method 1: Maximize Q

Increase Q by reducing resistance:

- Use copper wire instead of resistance wire
- Use larger gauge wire
- Minimize connection resistances
- Use low-ESR capacitors

### Method 2: Optimize Z₀/R Ratio

Increase characteristic impedance relative to resistance:

- Increase inductance (more turns, larger core)
- Decrease capacitance (for same resonant frequency, requires more inductance)
- The ratio √(L/C) determines Z₀

#### Design Trade-off:

For a given resonant frequency f₀ = 1/(2π√LC):

<div class="formula-box" id="bkmrk-higher-l-with-lower-" style="background: #f8f9fa; padding: 20px; border-left: 4px solid #17a2b8; margin: 20px 0;"><div class="formula-box" style="background: #f8f9fa; padding: 20px; border-left: 4px solid #17a2b8; margin: 20px 0;">- Higher L with lower C → Higher Z₀ → Higher magnification (but more wire, more DCR)
- Lower L with higher C → Lower Z₀ → Lower magnification (but less wire, less DCR)

</div></div>The optimal design balances these factors.

## Energy Considerations

Voltage magnification doesn't violate energy conservation:

#### Power In = Power Dissipated

At steady-state resonance:

<div id="bkmrk-current-through-circ" style="background: #e7f3ff; padding: 20px; border-radius: 5px; margin: 20px 0;"><div style="background: #e7f3ff; padding: 20px; border-radius: 5px; margin: 20px 0;">- Current through circuit: I = V<sub>source</sub>/R
- Power from source: P = V<sub>source</sub> × I = V<sub>source</sub>²/R
- Power dissipated in R: P = I²R = V<sub>source</sub>²/R (same!)

</div></div>The high voltage across L and C represents *reactive power*—energy that sloshes back and forth but isn't consumed.

## Real Power vs. Reactive Power

<table id="bkmrk-type-symbol-unit-des" style="width: 100%; border-collapse: collapse; margin: 20px 0;"><thead><tr style="background: #6c757d; color: white;"><th style="padding: 10px; border: 1px solid #ddd;">Type</th><th style="padding: 10px; border: 1px solid #ddd;">Symbol</th><th style="padding: 10px; border: 1px solid #ddd;">Unit</th><th style="padding: 10px; border: 1px solid #ddd;">Description</th></tr></thead><tbody><tr><td style="padding: 10px; border: 1px solid #ddd;">Real Power</td><td style="padding: 10px; border: 1px solid #ddd;">P</td><td style="padding: 10px; border: 1px solid #ddd;">Watts (W)</td><td style="padding: 10px; border: 1px solid #ddd;">Actually consumed, heats resistors</td></tr><tr><td style="padding: 10px; border: 1px solid #ddd;">Reactive Power</td><td style="padding: 10px; border: 1px solid #ddd;">Q (or VAR)</td><td style="padding: 10px; border: 1px solid #ddd;">Volt-Amperes Reactive</td><td style="padding: 10px; border: 1px solid #ddd;">Oscillates, stored in L and C</td></tr><tr><td style="padding: 10px; border: 1px solid #ddd;">Apparent Power</td><td style="padding: 10px; border: 1px solid #ddd;">S</td><td style="padding: 10px; border: 1px solid #ddd;">Volt-Amperes (VA)</td><td style="padding: 10px; border: 1px solid #ddd;">Total power flow</td></tr></tbody></table>

## Magnification in the VIC Matrix Calculator

The VIC Matrix Calculator displays voltage magnification in several ways:

### In Choke Designs

- **Q Factor:** Calculated from inductance and DCR
- **Voltage Magnification:** Equals Q for series resonance
- **Z₀/R Magnification:** Alternative calculation method
- **Example Output:** Shows actual voltage with 12V input

### In Circuit Profiles

- **Q\_L1C:** Q factor of primary side (L1 with C1)
- **Q\_L2:** Q factor of secondary side (L2 with WFC)
- **Voltage Magnification:** Expected magnification at resonance

**Practical Note:** Real circuits achieve somewhat less than theoretical magnification due to losses not accounted for in simple models (core losses, radiation, dielectric losses in capacitors, etc.). Expect 70-90% of calculated values in practice.

## Safety Warning

#### ⚠️ High Voltage Hazard

Resonant circuits can develop dangerous voltages even from low-voltage sources:

<div id="bkmrk-a-12v-source-with-q%3D" style="background: #f8d7da; padding: 15px; border-radius: 5px; border-left: 4px solid #dc3545; margin: 20px 0;">- A 12V source with Q=50 produces 600V peaks
- These voltages can cause electric shock or burns
- Energy stored in capacitors remains after power is removed
- Always discharge capacitors before handling circuits
- Use appropriate insulation and safety equipment

</div>*Chapter 1 Complete. Next: The Electric Double Layer (EDL) →*