Spectral Analysis
Fundamentals
Signal Theory & Harmonics
Fourier Series Harmonic Decomposition & THD
Sum discrete sinusoidal harmonics to assemble square, triangle, and pulse waveforms.
Direct mathematical demonstration of Fourier analysis. Adjust the count of odd harmonics, inspect Gibbs phenomenon overshoot at step discontinuities, and calculate real-time Total Harmonic Distortion (THD).
Governing Physical Law & Equations
f(t) = \frac{4}{\pi} \sum_{n=1,3,5,\dots}^{N} \frac{1}{n} \sin(n \omega_0 t)
Fourier harmonic expansion of an ideal 50% duty cycle square wave.
Law: Orthogonality of Sinusoidal Basis Functions | Standard Reference: IEEE 519 (Recommended Practice for Harmonic Control in Electric Power)
Adjustable System Parameters
| Parameter |
Nominal Value |
Dynamic Range |
Physical Role |
| Number of Harmonics (N) |
7 |
1 to 31 |
Highest odd harmonic included in the partial sum |
| Fundamental Freq (f_0) |
50 Hz |
10 to 100 Hz |
Base repetition frequency |
| Target Waveform (Type) |
0 |
0 to 2 |
0 = Square, 1 = Triangle, 2 = Sawtooth |
| Signal Noise Ratio (SNR) |
0 % |
0 to 20 % |
Gaussian noise added to the synthesized signal |
Analytical Proof & Derivation
Fourier basis functions are orthogonal over period T. Truncating the square wave expansion at N harmonics produces Gibbs phenomenon overshoot at jump discontinuities converging analytically to (2/π) ∫₀^π (sin(u)/u) du - 1 ≈ 8.9490%.
Verification Benchmark
IEEE 519 benchmark: Fundamental peak is 4/π ≈ 1.2732. Peak overshoot of synthesized square wave for N = 15 is 1.0895 times nominal pulse height (8.95% Gibbs overshoot). Total Harmonic Distortion (THD) conforms to IEEE 519 calculation formula.
Field Engineering Insights
Industrial variable frequency drives generate 5th, 7th, 11th, and 13th current harmonics. IEEE 519 enforces strict limits on Total Demand Distortion (TDD < 5% at Point of Common Coupling) to prevent transformer core overheating.