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_pk/π)[sin(ωt) + (1/3)sin(3ωt) + (1/5)sin(5ωt) + ...]

Harmonic	Frequency	Relative Amplitude
1st (fundamental)	f	100%
3rd	3f	33.3%
5th	5f	20%
7th	7f	14.3%

Pulse Train (Variable Duty Cycle)

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

a_n = (2V_pk/nπ) × sin(nπD)

Effect of Duty Cycle:

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

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):

At 10 kHz (1st): Response = 100%
At 30 kHz (3rd): Response ≈ 0.6%
At 50 kHz (5th): Response ≈ 0.2%

Harmonic Resonance

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

Scenario	Effect	Action
Drive at f₀	Fundamental resonates	Normal operation
Drive at f₀/2	2nd harmonic resonates	May be useful or problematic
Drive at f₀/3	3rd harmonic resonates	Subharmonic driving
Harmonic hits SRF	Choke self-resonates	Avoid—causes problems

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

Circuit resonance: f₀ = 30 kHz
Drive frequency: f_drive = 10 kHz
3rd harmonic of drive (30 kHz) excites resonance

Advantages:

Lower switching frequency (easier on semiconductors)
Different pulse characteristics
May interact differently with WFC

Disadvantages:

Harmonic has lower amplitude than fundamental
Reduced efficiency (energy in unused harmonics)
More complex analysis

Pulse Shaping for Harmonic Control

Adjusting pulse shape can control harmonic content:

Technique	Effect
Slower edges (rise/fall time)	Reduces high-order harmonics
Duty cycle = 1/n	Eliminates nth harmonic
Trapezoidal waveform	Controlled harmonic rolloff
Sine wave drive	No harmonics (pure fundamental)

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.