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Switching-Waveform Harmonic Source Spectrum Estimator

Expand a periodic trapezoidal voltage with linear rise/fall edges into discrete Fourier lines and a strict amplitude bound; this is not a measured EMI spectrum.

FORMULAv1.0.0
INPUT PARAMETERS

Enter a periodic trapezoidal switching waveform

Calculate exact discrete Fourier lines for linear rise/fall edges; the maximum frequency automatically determines the harmonic order.

CALCULATION RESULT

Calculation result

Calculated locally
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Enter the parameters and run the calculation to see results and model assumptions here.

FORMULA & TRACEABILITY

Formulas and applicability

Formulas, variables, and model boundaries are published. References last reviewed on 2026-09-01.

5 formulas · 2 references
CORE EQUATIONS

Core equations

These equations directly produce the primary results and define the tool’s core model.

2
F01

Periodic Trapezoidal Switching Waveform

T = 1/f₀；tfall,start = tᵣ/2 + DT − t꜀/2；v(t) 为分段线性上升、平台、线性下降与低电平

Define one period using voltage swing ΔV, period T, 50%-crossing duty cycle D, rise time tᵣ, and fall time t꜀; this definition gives a DC average of ΔV·D.

f₀
Switching fundamental frequencyHz
T
Switching periods
ΔV
Voltage swing between high and low levelsV
D
Duty cycle defined by 50% level crossings1
tᵣ
Full duration of the linear rising transition used by the models
t꜀
User-defined full duration of the linear falling transitions
Applicability
  • Waveform period, swing, duty cycle, and edge times remain stable
  • (tᵣ+t꜀)/2 does not exceed either the high-level or low-level duration
  • Edges are linear ramps; ringing, overshoot, jitter, and spread-spectrum modulation are outside the model
F02

Discrete Fourier Harmonic Coefficients

Cₙ = (1/T)∫₀ᵀv(t)e^(−jnω₀t)dt；Vₙ,pk = 2|Cₙ|，n ≥ 1

Analytically integrate the piecewise-linear trapezoid. The implementation derives Cₙ from the two rectangular edge-derivative pulses instead of approximating line amplitudes with discrete samples.

Cₙ
Complex Fourier coefficient of harmonic nV
ω₀
Fundamental angular frequency 2πf₀rad/s
Vₙ,pk
Peak amplitude of sinusoidal line nV
n
Positive integer harmonic order1
Applicability
  • Periodic steady-state waveform
  • Discrete lines exist only at n·f₀
  • The time origin changes phase but not magnitude
DERIVED & CONVERSION

Derived and conversion equations

Derive units, levels, and supporting engineering quantities from the core values.

1
F04

Harmonic RMS and Level Conversion

Vₙ,rms = Vₙ,pk/√2；Lₙ = 20log₁₀(Vₙ,rms/1 μV)

Express each non-zero discrete line as an equivalent sinusoidal RMS value and convert it to dBμV. A theoretical null has no finite logarithmic value and remains a spectral null.

Vₙ,rms
RMS value of sinusoidal harmonic component nV RMS
Lₙ
Voltage level referenced to 1 μV RMSdBμV
Applicability
  • Convert non-zero harmonic lines only
  • dBμV is the source-node voltage level, not a measuring-receiver port reading
BOUNDARIES & RULES

Boundary and rule equations

Check model applicability, measurement conditions, and regulatory rules.

2
F03

Equal-Edge Closed-Form Check

tᵣ=t꜀=tₑ 时：Vₙ,pk = 2ΔV·D·|sinc(nD)·sinc(nf₀tₑ)|

When rise and fall times are equal, the general analytic integral reduces to a two-sinc closed form used for independent regression checks, with sinc(x)=sin(πx)/(πx).

tₑ
Equal linear rise and fall times
sinc
Normalized sinc function1
Applicability
  • Rise and fall times are equal
  • Duty cycle uses the same 50% level definition as this page
F05

Strict Amplitude Bound for Discrete Lines

V̂ₙ,pk = min{2ΔV·min(D,1−D), ΔV/(πn)·[min(1,1/(πnf₀tᵣ))+min(1,1/(πnf₀t꜀))]}

Derived from the mean-absolute-value bound on Fourier coefficients and the triangle inequality for the two edge-derivative integrals. It shows −20/−40 dB/dec trends without implying continuous spectrum between harmonics.

V̂ₙ,pk
Mathematical upper bound on peak amplitude at order nV
n
Continuous or integer harmonic order1
Applicability
  • The bound is no smaller than the exact discrete line for the same input
  • Used only for trends and display scale, not standards-limit decisions
REFERENCES

References

01Texas Instruments · Application Report SCAA082A, Section 1.2 Clock SignalsHigh-Speed Layout Guidelines2017 revision · The official application report explains that a real clock is a finite-edge trapezoid, can be decomposed by Fourier series, and has higher harmonics governed by rise and fall time.
02Texas Instruments · Application Report SCAA031, Higher Harmonic ComponentsEMI Prevention in Clock-Distribution Circuits1997 · The official report relates digital waveforms, Fourier harmonics, and spectrum envelopes while treating source spectrum separately from transmission-line and antenna effects.
Engineering use notice

Results use the models and assumptions shown on this page for design estimates and pre-compliance risk review. Complex structures, dispersion, near-field coupling, and test setup can cause significant deviation.

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