# LC Circuits

# LC Circuit Fundamentals

An LC circuit consists of an inductor (L) and a capacitor (C) connected together. These circuits form the foundation of resonant systems and are central to understanding how the VIC operates.

## Components of an LC Circuit

### The Inductor (L)

An inductor stores energy in its magnetic field when current flows through it. Key properties:

- **Inductance (L):** Measured in Henries (H), represents the inductor's ability to store magnetic energy
- **Inductive Reactance:** X<sub>L</sub> = 2πfL (increases with frequency)
- **Current lags voltage by 90°** in a pure inductor

### The Capacitor (C)

A capacitor stores energy in its electric field between two conductive plates. Key properties:

- **Capacitance (C):** Measured in Farads (F), represents the capacitor's ability to store electric charge
- **Capacitive Reactance:** X<sub>C</sub> = 1/(2πfC) (decreases with frequency)
- **Current leads voltage by 90°** in a pure capacitor

## Series LC Circuit

**Circuit Configuration:** L and C connected in series with the source

#### Total Impedance:

Z = √(R² + (X<sub>L</sub> - X<sub>C</sub>)²)

#### At Resonance (X<sub>L</sub> = X<sub>C</sub>):

<div id="bkmrk-z-%3D-r-%28minimum-imped" style="background: #f8f9fa; padding: 20px; border-radius: 5px; margin: 20px 0;">- Z = R (minimum impedance)
- Current = V/R (maximum current)
- Voltage across L = Voltage across C = Q × V<sub>source</sub>

</div>### Series LC Behavior

<table id="bkmrk-frequency-condition-" style="width: 100%; border-collapse: collapse; margin: 20px 0;"><thead><tr style="background: #007bff; color: white;"><th style="padding: 10px; border: 1px solid #ddd;">Frequency</th><th style="padding: 10px; border: 1px solid #ddd;">Condition</th><th style="padding: 10px; border: 1px solid #ddd;">Circuit Behavior</th></tr></thead><tbody><tr><td style="padding: 10px; border: 1px solid #ddd;">f &lt; f₀</td><td style="padding: 10px; border: 1px solid #ddd;">X<sub>C</sub> &gt; X<sub>L</sub></td><td style="padding: 10px; border: 1px solid #ddd;">Capacitive (current leads voltage)</td></tr><tr style="background: #e7f3ff;"><td style="padding: 10px; border: 1px solid #ddd;">f = f₀</td><td style="padding: 10px; border: 1px solid #ddd;">X<sub>C</sub> = X<sub>L</sub></td><td style="padding: 10px; border: 1px solid #ddd;">Resistive (current in phase with voltage)</td></tr><tr><td style="padding: 10px; border: 1px solid #ddd;">f &gt; f₀</td><td style="padding: 10px; border: 1px solid #ddd;">X<sub>L</sub> &gt; X<sub>C</sub></td><td style="padding: 10px; border: 1px solid #ddd;">Inductive (current lags voltage)</td></tr></tbody></table>

## Parallel LC Circuit

**Circuit Configuration:** L and C connected in parallel

#### At Resonance:

<div id="bkmrk-impedance-approaches" style="background: #f8f9fa; padding: 20px; border-radius: 5px; margin: 20px 0;"><div style="background: #f8f9fa; padding: 20px; border-radius: 5px; margin: 20px 0;">- Impedance approaches infinity (in ideal case)
- Current from source is minimum
- Large circulating current flows between L and C

</div></div>**Also called:** Tank circuit, because it "tanks" or stores energy

## Characteristic Impedance (Z₀)

The characteristic impedance is a fundamental property of any LC circuit:

Z₀ = √(L/C)

This value represents:

<div class="formula-box" id="bkmrk-the-impedance-at-res" style="background: #f8f9fa; padding: 20px; border-left: 4px solid #28a745; margin: 20px 0;">- The impedance at resonance for a parallel LC circuit
- The ratio of voltage to current in a traveling wave
- A design parameter for matching circuits

</div>## Energy Transfer in LC Circuits

In an ideal LC circuit (no resistance), energy oscillates perpetually between the inductor and capacitor:

1. **Capacitor fully charged:** All energy stored in electric field (E = ½CV²)
2. **Current building:** Energy transferring to inductor
3. **Maximum current:** All energy stored in magnetic field (E = ½LI²)
4. **Current decreasing:** Energy transferring back to capacitor
5. **Cycle repeats** at the resonant frequency

## LC Circuits in the VIC

The VIC uses LC circuits in two critical locations:

### Primary Side (L1-C1)

- L1 = Primary choke inductance
- C1 = Tuning capacitor
- Tuned to the driving frequency from the pulse generator
- Develops the initial voltage magnification

### Secondary Side (L2-WFC)

- L2 = Secondary choke inductance
- WFC = Water Fuel Cell capacitance
- May be tuned to the same or a harmonic frequency
- Delivers magnified voltage to the water

**Design Principle:** The relationship between L and C values determines not only the resonant frequency but also the characteristic impedance, which affects how much voltage magnification is achievable.

## Practical Considerations

- **Component tolerances:** Real components have tolerances that affect the actual resonant frequency
- **Parasitic elements:** Inductors have parasitic capacitance, capacitors have parasitic inductance
- **Temperature effects:** Component values can drift with temperature
- **Losses:** Real circuits have resistance that dampens oscillations

*Next: Quality Factor (Q) Explained →*