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 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. 

 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. 

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