# Harmonic Analysis

# Harmonic Analysis

VIC circuits are typically driven by non-sinusoidal waveforms (pulses, square waves), which contain harmonics. Understanding how these harmonics interact with the resonant circuit is important for predicting actual performance and potential interference effects.

## Fourier Analysis Basics

Any periodic waveform can be decomposed into a sum of sinusoids:

#### Fourier Series:

f(t) = a₀ + Σ\[aₙcos(nωt) + bₙsin(nωt)\]

Where n = 1, 2, 3... are the harmonic numbers (n=1 is fundamental)

## Harmonic Content of Common Waveforms

### Square Wave

50% duty cycle square wave contains only odd harmonics:

V(t) = (4V<sub>pk</sub>/π)\[sin(ωt) + (1/3)sin(3ωt) + (1/5)sin(5ωt) + ...\]

<div id="bkmrk-harmonic-frequency-r" style="background: #e7f3ff; padding: 20px; border-radius: 5px; margin: 20px 0;"><table style="width: 100%; border-collapse: collapse; margin-top: 15px;"><thead><tr style="background: #6c757d; color: white;"><th style="padding: 8px; border: 1px solid #ddd;">Harmonic</th><th style="padding: 8px; border: 1px solid #ddd;">Frequency</th><th style="padding: 8px; border: 1px solid #ddd;">Relative Amplitude</th></tr></thead><tbody><tr><td style="padding: 8px; border: 1px solid #ddd;">1st (fundamental)</td><td style="padding: 8px; border: 1px solid #ddd;">f</td><td style="padding: 8px; border: 1px solid #ddd;">100%</td></tr><tr><td style="padding: 8px; border: 1px solid #ddd;">3rd</td><td style="padding: 8px; border: 1px solid #ddd;">3f</td><td style="padding: 8px; border: 1px solid #ddd;">33.3%</td></tr><tr><td style="padding: 8px; border: 1px solid #ddd;">5th</td><td style="padding: 8px; border: 1px solid #ddd;">5f</td><td style="padding: 8px; border: 1px solid #ddd;">20%</td></tr><tr><td style="padding: 8px; border: 1px solid #ddd;">7th</td><td style="padding: 8px; border: 1px solid #ddd;">7f</td><td style="padding: 8px; border: 1px solid #ddd;">14.3%</td></tr></tbody></table>

</div>### Pulse Train (Variable Duty Cycle)

Pulse train with duty cycle D contains both odd and even harmonics:

a<sub>n</sub> = (2V<sub>pk</sub>/nπ) × sin(nπD)

#### Effect of Duty Cycle:

<div id="bkmrk-d-%3D-50%25%3A-only-odd-ha" style="background: #fff3cd; padding: 20px; border-radius: 5px; margin: 20px 0;">- **D = 50%:** Only odd harmonics (even harmonics cancel)
- **D = 25%:** Strong 2nd harmonic, weak 4th
- **D = 33%:** No 3rd harmonic (3rd harmonic null)
- **Narrow pulse:** Wide harmonic spectrum, many significant harmonics

</div>## Resonant Circuit Response to Harmonics

A resonant circuit acts as a bandpass filter. It responds most strongly to frequencies near f₀:

```
Response
    │
    │          Fundamental
    │              ↓
    │             ╱╲
    │            ╱  ╲              3rd harmonic
    │           ╱    ╲                 ↓
    │          ╱      ╲              (small response)
    │         ╱        ╲           ┌─┐
    │        ╱          ╲          │ │
    └───────────────────────────────────────→ f
             f₀          3f₀
```

#### Response at Harmonic Frequencies:

H(nf) = 1 / √\[1 + Q²(n - 1/n)²\]

For high Q circuits, harmonics far from f₀ are strongly attenuated.

#### Example (Q=50, f₀=10 kHz):

<div class="formula-box" id="bkmrk-at-10-khz-%281st%29%3A-res" style="background: #f8f9fa; padding: 20px; border-left: 4px solid #28a745; margin: 20px 0;">- At 10 kHz (1st): Response = 100%
- At 30 kHz (3rd): Response ≈ 0.6%
- At 50 kHz (5th): Response ≈ 0.2%

</div>## Harmonic Resonance

If a harmonic happens to fall near f₀, it can cause problems or opportunities:

<table id="bkmrk-scenario-effect-acti" style="width: 100%; border-collapse: collapse; margin: 20px 0;"><thead><tr style="background: #dc3545; color: white;"><th style="padding: 10px; border: 1px solid #ddd;">Scenario</th><th style="padding: 10px; border: 1px solid #ddd;">Effect</th><th style="padding: 10px; border: 1px solid #ddd;">Action</th></tr></thead><tbody><tr><td style="padding: 10px; border: 1px solid #ddd;">Drive at f₀</td><td style="padding: 10px; border: 1px solid #ddd;">Fundamental resonates</td><td style="padding: 10px; border: 1px solid #ddd;">Normal operation</td></tr><tr><td style="padding: 10px; border: 1px solid #ddd;">Drive at f₀/2</td><td style="padding: 10px; border: 1px solid #ddd;">2nd harmonic resonates</td><td style="padding: 10px; border: 1px solid #ddd;">May be useful or problematic</td></tr><tr><td style="padding: 10px; border: 1px solid #ddd;">Drive at f₀/3</td><td style="padding: 10px; border: 1px solid #ddd;">3rd harmonic resonates</td><td style="padding: 10px; border: 1px solid #ddd;">Subharmonic driving</td></tr><tr><td style="padding: 10px; border: 1px solid #ddd;">Harmonic hits SRF</td><td style="padding: 10px; border: 1px solid #ddd;">Choke self-resonates</td><td style="padding: 10px; border: 1px solid #ddd;">Avoid—causes problems</td></tr></tbody></table>

## Sub-Harmonic Driving

It's possible to drive the circuit at a sub-multiple of f₀ and let a harmonic excite resonance:

#### Example: 3rd Harmonic Drive

<div id="bkmrk-circuit-resonance%3A-f" style="background: #d4edda; padding: 20px; border-radius: 5px; margin: 20px 0;"><div style="background: #d4edda; padding: 20px; border-radius: 5px; margin: 20px 0;">- Circuit resonance: f₀ = 30 kHz
- Drive frequency: f<sub>drive</sub> = 10 kHz
- 3rd harmonic of drive (30 kHz) excites resonance

</div></div>#### Advantages:

<div id="bkmrk-lower-switching-freq" style="background: #d4edda; padding: 20px; border-radius: 5px; margin: 20px 0;"><div style="background: #d4edda; padding: 20px; border-radius: 5px; margin: 20px 0;">- Lower switching frequency (easier on semiconductors)
- Different pulse characteristics
- May interact differently with WFC

</div></div>#### Disadvantages:

<div id="bkmrk-harmonic-has-lower-a" style="background: #d4edda; padding: 20px; border-radius: 5px; margin: 20px 0;">- Harmonic has lower amplitude than fundamental
- Reduced efficiency (energy in unused harmonics)
- More complex analysis

</div>## Pulse Shaping for Harmonic Control

Adjusting pulse shape can control harmonic content:

<table id="bkmrk-technique-effect-slo" style="width: 100%; border-collapse: collapse; margin: 20px 0;"><thead><tr style="background: #007bff; color: white;"><th style="padding: 10px; border: 1px solid #ddd;">Technique</th><th style="padding: 10px; border: 1px solid #ddd;">Effect</th></tr></thead><tbody><tr><td style="padding: 10px; border: 1px solid #ddd;">Slower edges (rise/fall time)</td><td style="padding: 10px; border: 1px solid #ddd;">Reduces high-order harmonics</td></tr><tr><td style="padding: 10px; border: 1px solid #ddd;">Duty cycle = 1/n</td><td style="padding: 10px; border: 1px solid #ddd;">Eliminates nth harmonic</td></tr><tr><td style="padding: 10px; border: 1px solid #ddd;">Trapezoidal waveform</td><td style="padding: 10px; border: 1px solid #ddd;">Controlled harmonic rolloff</td></tr><tr><td style="padding: 10px; border: 1px solid #ddd;">Sine wave drive</td><td style="padding: 10px; border: 1px solid #ddd;">No harmonics (pure fundamental)</td></tr></tbody></table>

## Harmonic Interaction with Multiple Resonances

In dual-resonant VIC (primary + secondary), harmonics may interact with both:

```
Response
    │
    │     Primary        Secondary
    │     resonance      resonance
    │         ↓              ↓
    │        ╱╲            ╱╲
    │       ╱  ╲          ╱  ╲
    │      ╱    ╲        ╱    ╲
    │     ╱      ╲      ╱      ╲
    │    ╱        ╲    ╱        ╲
    │   ╱          ╲  ╱          ╲
    └──────────────────────────────────→ f
          f₀,pri       f₀,sec
```

If f₀,sec = 3 × f₀,pri, then:

- Fundamental drives primary resonance
- 3rd harmonic drives secondary resonance
- This is sometimes called "harmonic matching"

## Practical Harmonic Considerations

#### EMI Concerns:

Harmonics can cause electromagnetic interference. Shield appropriately and consider filtering if needed.

#### Measurement:

Use an oscilloscope with FFT function or spectrum analyzer to view harmonic content of your signals.

#### Design Rule:

For clean resonance, ensure no significant harmonics fall within the passband (f₀ ± f₀/Q) of unintended resonances.

## Harmonic Analysis in VIC Matrix Calculator

**Calculator Feature:** The simulation can show frequency response across a range that includes harmonics. When analyzing a design, check whether any harmonics of your drive frequency fall near the circuit's resonant points or SRF values.

*Next: Transformer Coupling Effects →*