Fourier Series Harmonic Amplitude, RMS & Gibbs Phenomenon Overshoot Calculator

Compute Fourier harmonic coefficients, Parseval energy %, and Total Harmonic Distortion (THD).

Fundamental Sine Amplitude b₁ (4A/π)
6.366 units
7-th Harmonic Amplitude b_7 0.91 units
Parseval Energy Captured (%) 94.96 %
Gibbs Overshoot Peak (+17.9% of Step) 5.895 units
Inputs
⚡ Scenarios:
V / units
Peak amplitude A of the symmetric square wave (±A).
Hz
Fundamental frequency f₁ in Hz.
harmonic N
Highest odd harmonic order N included in the partial sum (1, 3, 5, 7, ...).
Results & Graph
Fundamental Sine Amplitude b₁ (4A/π)
6.366 units
7-th Harmonic Amplitude b_7
0.91 units
Parseval Energy Captured (%)
94.96 %
Gibbs Overshoot Peak (+17.9% of Step)
5.895 units

Fundamental Sine Amplitude b₁ (4A/π)

Graph 1 • Fourier Series Harmonics • Domain Architecture & Live Telemetry Maps Square Wave Peak Amplitude A, Fundamental Frequency Hz, Max Harmonic Order N through Fourier Series Harmonics Formula.
Fundamental Sine Amplitude b₁ : 6.366 units Fourier Series Harmonics • Invariants & Metrics Fundamental Sine Amplitude b 6.366 units 7-th Harmonic Amplitude b_7 0.91 units Parseval Energy Captured (%) 94.96 % Gibbs Overshoot Peak (+17.9% 5.895 units
Graph 2 • Fundamental Sine Amplitude b₁ (4A/π) Across Square Wave Peak Amplitude A Sweep (40% → 160%) Evaluates b_n = 4A / (n·π) for odd n; ParsevalPowerFraction = Σ_{odd k≤N} (8 / ( across 7 operating points (current input highlighted).
0 units 5.60 units 11.20 units 2.55 2 3.82 3 5.09 4 6.37 5 7.64 6 8.91 7 10.19 8
Graph 3 • Joint Sensitivity: Square Wave Peak Amplitude A × Fundamental Frequency Hz Compares Fundamental Sine Amplitude b₁ (4A/π) when Fundamental Frequency Hz is shifted +25% (solid) vs. −25% (dashed).
0 units 4.55 units 9.10 units --- Fundamental Frequency Hz @ 75% 4.46 -30% 5.41 -15% 6.37 +0% 7.32 +15% 8.28 +30%
Summary: Fundamental (f₁=60 Hz) amplitude is 6.366, 7-th harmonic (420 Hz) amplitude is 0.909, and the first 4 odd harmonics capture 94.96% of total square-wave power.
Summary: Fundamental (f₁=60 Hz) amplitude is 6.366, 7-th harmonic (420 Hz) amplitude is 0.909, and the first 4 odd harmonics capture 94.96% of total square-wave power.
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Formula & Mathematical Methodology: Fourier Series Harmonics Formula

bn = 4A / (n·π) for odd n
ParsevalPowerFraction = Σodd k≤N (8 / (k²·π²))
THD = 1 / (8/π² − 1)
b_n = 4A / (n·π) for odd n; ParsevalPowerFraction = Σ_{odd k≤N} (8 / (k²·π²)); THD = √(1 / (8/π²) − 1)
  • squareWavePeakAmplitudeA: Square Wave Peak Amplitude A (V / units)
  • FundamentalFrequencyHz: Fundamental Frequency Hz (Hz)
  • maxHarmonicOrderN: Max Harmonic Order N (harmonic N)

Rounding: Internal floating-point calculations run in 64-bit IEEE-754 double precision; display values round to 2–4 decimal places.

Worked Example: Worked Mathematical Example: Fourier Series Harmonics

Compute Fourier harmonic coefficients, Parseval energy %, and Total Harmonic Distortion (THD).

  1. Specify the function parameters, matrix entries, or confusion matrix counts.
  2. Apply the closed-form or iterative formula: b_n = 4A / (n·π) for odd n; ParsevalPowerFraction = Σ_{odd k≤N} (8 / (k²·π²)); THD = √(1 / (8/π²) − 1).
  3. Examine the primary result, error bounds, and 3 interactive SVG plots.
Result: Computes Fourier Series Harmonics via b_n = 4A / (n·π) for odd n; ParsevalPowerFraction = Σ_{odd k≤N} (8 / (k²·π²)); THD = √(1 / (8/π²) − 1).

Frequently Asked Questions

How does the Fourier Series Harmonic Amplitude, RMS & Gibbs Phenomenon Overshoot Calculator compute the solution?

It evaluates b_n = 4A / (n·π) for odd n; ParsevalPowerFraction = Σ_{odd k≤N} (8 / (k²·π²)); THD = √(1 / (8/π²) − 1) directly in your browser and updates three custom SVG visualizations in real time.

Can I solve backwards for an input parameter using Goal Seek?

Yes. Click "🎯 Goal Seek" in the results toolbar to solve backwards for any numeric input field.

Does this calculator run locally offline?

Yes. All calculations run 100% locally in your browser and work offline via the CalculatorGraph PWA.

Mathematics & Statistics · Numerical Analysis & Calculus

Fourier Series Harmonic Amplitude, RMS & Gibbs Phenomenon Overshoot Calculator: inputs, method, and how to read the result

Compute Fourier harmonic coefficients, Parseval energy %, and Total Harmonic Distortion (THD).

Choose inputs for the case you are checking

The defaults are illustrative. Replace them with values for your case and keep the units shown beside each field.

  • Square Wave Peak Amplitude A: Peak amplitude A of the symmetric square wave (±A).
  • Fundamental Frequency Hz: Fundamental frequency f₁ in Hz.
  • Max Harmonic Order N: Highest odd harmonic order N included in the partial sum (1, 3, 5, 7, ...).

What the calculation includes

With the default values shown, Fundamental Sine Amplitude b₁ (4A/π) is 6.366 units. The calculation follows Fourier Series Harmonics Formula; the formula and variable definitions above show how the inputs combine.

One-input check: With the other defaults held constant, changing Square Wave Peak Amplitude A from 5 V / units to 6 V / units changes Fundamental Sine Amplitude b₁ (4A/π) from 6.366 units to 7.639 units. This is a comparison inside the calculator's stated model, not a prediction.

Where to use caution

  • Uses exact analytical or standard numerical analysis formulations from NIST DLMF.
  • Double-precision floating-point arithmetic has ~15–17 significant decimal digits of precision.

Scope: Global Mathematical & Statistical Standard Check the assumptions and limitations against your situation before relying on the estimate.

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