ELEC2134 Chap.5 Pulse Trains and Rectifier Spectra
Pulse Trains and Rectifier Spectra
Pulse trains make the connection between waveform geometry and spectral structure especially clear. Amplitude and duty ratio set the average value, while pulse width controls the zeros of the sinc envelope sampled at harmonic locations.
The chapter also applies period reasoning to rectifiers: full-wave rectification halves the input period, so a 50 Hz sinusoid produces DC and harmonics at multiples of 100 Hz.
Two time scales perform different jobs. The repetition period sets the spacing of allowed harmonic lines, while the pulse width sets the width of the continuous sinc envelope those lines sample.
Changing one without the other therefore changes a different visual feature. Keeping this distinction explicit prevents the common error of using pulse width as the repetition period.
Duty ratio also predicts useful landmarks before integration. The zero-index coefficient is the time average AD, and envelope zeros occur when the harmonic index times D reaches a nonzero integer.
A time shift of the pulse changes coefficient phase but leaves these magnitudes and zero locations unchanged. Rectifier spectra follow the same principle: find the output's smallest period first, then build its frequency grid.
What this chapter covers
- 01
Duty ratio as pulse width divided by period
- 02
Centred-pulse exponential Fourier coefficients
- 03
Sinc envelope height and zero locations
- 04
Narrow-pulse and wide-spectrum tradeoff
- 05
Full-wave rectifier period halving
- 06
DC and harmonic locations in rectified signals
Worked example · free
Spectrum of a centred pulse train
- +1D = tau/T = 1/4 and f0 = 1/T = 250 Hz.
- +2For a centred pulse, Cn = A D sinc(nD) = 2 sinc(n/4).
- +1At n = 0, the limiting sinc value is one, so C0 = A D = 2.
- +2The first positive sinc zero satisfies nD = 1, giving n = 4 and frequency 4f0 = 1000 Hz.
Key terms
- Pulse train
- A periodic sequence of pulses with fixed amplitude, width and repetition period.
- Duty ratio
- The fraction of each period during which a unipolar pulse is active.
- Sinc envelope
- The continuous curve whose samples determine rectangular-pulse harmonic magnitudes.
- Spectral zero
- A frequency where the transform envelope or sampled coefficient magnitude is zero.
- Full-wave rectification
- An operation that makes both input half-cycles have the same output polarity.
Pulse Trains and Rectifier Spectra FAQ
Must a pulse-train duty ratio be below 50%?
No. Duty ratio is simply tau divided by T and may lie anywhere from zero to one for a standard unipolar pulse train. Values above 50% mean the pulse is active for most of each period. The same coefficient expression applies when the waveform definition remains valid.
What controls pulse-spectrum line spacing?
The repetition period controls the spacing: lines occur at integer multiples of f0 = 1/T. Pulse width controls the sinc envelope and its zero locations. Changing T while holding pulse width fixed changes the sampling density, while changing width reshapes the envelope itself.
Why is there no 50 Hz line after ideal full-wave rectification?
A full-wave-rectified 50 Hz sinusoid repeats every half input cycle, so its new period is 0.01 s and its fundamental frequency is 100 Hz. Its spectrum contains DC plus 100, 200, 300 Hz and higher multiples. The original 50 Hz is not an allowed harmonic location.
Does shifting a pulse change its spectrum?
A time shift changes each complex coefficient's phase but leaves coefficient magnitude, DC level, harmonic spacing and sinc-zero indices unchanged. Centred and edge-aligned pulses with the same amplitude, width and period therefore share a magnitude spectrum. Their phase spectra record the difference in time position.
How can I check the first sinc zero?
Set nD equal to the first nonzero integer and solve for the harmonic index. The result must be an integer harmonic to appear as an exact missing line in the sampled series. If it is not an integer, the continuous envelope has a zero between harmonic samples rather than at a spectral line.
Exam move
Draw one pulse with its amplitude, width and period labelled before using a coefficient formula. Compute D and f0 separately, then predict DC and the first envelope zero. Practise explaining why narrow pulses spread spectral energy more widely. For rectifiers, ignore the input label initially and find the output's smallest repeating interval directly; that prevents the common 50 Hz error.
Check each spectrum against three landmarks: line spacing, DC height and first missing harmonic. When coefficient signs alternate, keep magnitude nonnegative and carry the sign through phase rather than drawing a negative amplitude line.
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