Editing Pulse-width modulation (section)

==Principle==
=== Periodic pulse wave ===
[[Image:Duty cycle general.svg|thumb|350px|Fig. 1: a periodic [[pulse wave]], showing <math>y_\text{min}</math>, <math>y_\text{max}</math> and constant duty cycle D.]]

If we consider a periodic [[pulse wave]] <math>f(t)</math> with period <math>T</math>, low value <math>y_\text{min}</math>, a high value <math>y_\text{max}</math> and a constant duty cycle D (Figure 1), the average value of the waveform is given by:

<math display="block">\bar{y} = \frac{1}{T}\int^T_0f(t)\,dt</math>

As <math>f(t)</math> is a pulse wave, its value is <math>y_\text{max}</math> for <math>0 < t < D \cdot T</math> and <math>y_\text{min}</math> for <math>D \cdot T < t < T</math>. The above expression then becomes:

<math display="block">\begin{align}
  \bar{y} &= \frac{1}{T} \left(\int_0^{DT} y_\text{max}\,dt + \int_{DT}^T y_\text{min}\,dt\right)\\
          &= \frac{1}{T} \left(D \cdot T \cdot y_\text{max} + T\left(1 - D\right) y_\text{min}\right)\\
          &= D\cdot y_\text{max} + \left(1 - D\right) y_\text{min}
\end{align}</math>

This latter expression can be fairly simplified in many cases where <math>y_\text{min} = 0</math> as <math>\bar{y} = D \cdot y_\text{max}</math>. From this, the average value of the signal (<math>\bar{y}</math>) is directly dependent on the duty cycle D.

However, by varying (i.e. modulating) the duty cycle (and possibly also the period), the following more advanced pulse-width modulated waves allow variation of the [[average]] value of the waveform.

=== Intersective method PWM ===
[[File:Pwm.svg|frameless|400x400px]]

The intersective method is a simple way to generate a PWM output signal (magenta in above figure) with fixed period and varying duty cycle is by using a [[comparator]] to switch the PWM output state when the input waveform (red) intersects with a [[Sawtooth wave|sawtooth]] or a [[Triangle wave|triangle]] waveform (blue).

Depending on the type of sawtooth or triangle waveform (green in below figure), intersective PWM signals (blue in the below figure) can be aligned in three manners:

[[File:Three_PWM_types.svg|frameless|400x400px]]

* Leading [[Signal edge|edge]] modulation (top plot) uses a reverse sawtooth wave to generate the PWM. The PWM's leading edge is held at the leading edge of the window and the trailing edge is modulated.
* Trailing edge modulation (middle plot) uses a normal sawtooth wave to generate the PWM. The PWM's trailing edge is fixed and the leading edge is modulated.
* Centered pulses (bottom) uses a triangle waveform to generate the PWM. The pulse center is fixed in the center of the time window and both edges of the pulse are moved to compress or expand the width.

==== Time proportioning ====
Many digital circuits can generate PWM signals (e.g., many [[microcontrollers]] have PWM outputs). They normally use a [[counter (digital)|counter]] that increments periodically (it is connected directly or indirectly to the [[clock signal|clock]] of the circuit) and is reset at the end of every period of the PWM. When the counter value is more than the reference value, the PWM output changes state from high to low (or low to high).<ref>{{cite web |last=Barr |first=Michael |date=1 September 2001 |title=Introduction to Pulse Width Modulation (PWM) |url=https://barrgroup.com/Embedded-Systems/How-To/PWM-Pulse-Width-Modulation |website=Barr Group}}</ref> This technique is referred to as '''time proportioning,''' particularly as '''time-proportioning control'''<ref>''Fundamentals of HVAC Control Systems,'' by Robert McDowall, [https://books.google.com/books?id=UMk1EUp-W-UC&pg=PA21&dq=%22time+proportioning%22 p. 21]</ref> – which ''proportion'' of a fixed cycle time is spent in the high state.

The incremented and periodically reset counter is the discrete version of the intersecting method's sawtooth. The analog comparator of the intersecting method becomes a simple integer comparison between the current counter value and the digital (possibly digitized) reference value. The duty cycle can only be varied in discrete steps, as a function of the counter resolution. However, a high-resolution counter can provide quite satisfactory performance.

==== Spectrum ====
The resulting [[spectrum|spectra]] (of the three alignments) are similar. Each contains a [[DC component]], a base sideband containing the modulating signal, and phase modulated [[Carrier signal|carriers]] at each [[harmonic]] of the frequency of the pulse. The amplitudes of the harmonic groups are restricted by a <math>\sin x / x</math> envelope ([[sinc function]]) and extend to infinity. The infinite bandwidth is caused by the nonlinear operation of the pulse-width modulator. In consequence, a digital PWM suffers from [[aliasing]] distortion that significantly reduce its applicability for modern [[communication system]]s. By limiting the bandwidth of the PWM kernel, aliasing effects can be avoided.<ref>{{cite journal |last=Hausmair |first=Katharina |author2=Shuli Chi |author3=Peter Singerl |author4=Christian Vogel |date=February 2013 |title=Aliasing-Free Digital Pulse-Width Modulation for Burst-Mode RF Transmitters |journal=IEEE Transactions on Circuits and Systems I: Regular Papers |volume=60 |issue=2 |pages=415–427 |citeseerx=10.1.1.454.9157 |doi=10.1109/TCSI.2012.2215776 |s2cid=21795841}}</ref>

On the contrary, [[delta modulation]] and [[delta-sigma modulation]] are random processes{{Clarification needed|reason=What exactly is meant by "random" here?|date=February 2024}} that produces a continuous spectrum without distinct harmonics. While intersective PWM uses a fixed period but a varying duty cycle, the period of delta and delta-sigma modulated PWMs varies in addition to their duty cycle.

===Delta modulation===
{{Main|Delta modulation}}[[File:Delta_PWM.svg|frameless|400x400px]]

Delta modulation produces a PWM signal (magenta in above figure) which changes state whenever its integral (blue) hits the limits (green) surrounding the input (red). 

=== Asynchronous delta-sigma PWM ===
{{Main|Delta-sigma modulation#Asynchronous delta-sigma modulation}}

[[File:Sigma-delta_PWM.svg|frameless|400x400px]]

[[Asynchronous circuit|Asynchronous]] (i.e. unclocked) delta-sigma modulation produces a PWM output (blue in bottom plot) which is subtracted from the input signal (green in top plot) to form an error signal (blue in top plot). This error is integrated (magenta in middle plot). When the integral of the error exceeds the limits (the upper and lower grey lines in middle plot), the PWM output changes state. By integrating the difference of the error with the input signal, delta-sigma modulation [[Noise shaping|shapes noise]] of the resulting spectrum to be more in higher frequencies above the input signal's band.

===Space vector modulation===

{{Main|Space vector modulation}}

Space vector modulation is a PWM control algorithm for multi-phase AC generation, in which the reference signal is sampled regularly; after each sample, non-zero active switching vectors adjacent to the reference vector and one or more of the zero switching vectors are selected for the appropriate fraction of the sampling period in order to synthesize the reference signal as the average of the used vectors.

===Direct torque control (DTC)===

{{Main|Direct torque control}}

Direct torque control is a method used to control AC motors. It is closely related with the delta modulation (see above). Motor torque and magnetic flux are estimated and these are controlled to stay within their hysteresis bands by turning on a new combination of the device's semiconductor switches each time either signal tries to deviate out of its band.

===PWM sampling theorem===
The process of PWM conversion is non-linear and it is generally supposed that low pass filter signal recovery is imperfect for PWM. The PWM sampling theorem<ref>J. Huang, K. Padmanabhan, and O. M. Collins, “The sampling theorem with constant amplitude variable width pulses”, IEEE transactions on Circuits and Systems, vol. 58, pp. 1178 - 1190, June 2011.</ref> shows that PWM conversion can be perfect:

<blockquote>Any [[bandlimited]] [[baseband]] signal whose amplitude is within ±0.637 can be represented by a PWM waveform of unit amplitude (±1). The number of pulses in the waveform is equal to the number of [[Nyquist rate|Nyquist samples]] and the peak constraint is independent of whether the waveform is two-level or three-level.</blockquote>

For comparison, the [[Nyquist–Shannon sampling theorem]] can be summarized as:

<blockquote>If you have a signal that is bandlimited to a bandwidth of f<sub>0</sub> then you can collect all the information there is in that signal by sampling it at discrete times, as long as your sample rate is greater than 2f<sub>0</sub>.<ref>{{cite web |last=Wescott |first=Tim |date=August 14, 2018 |title=Sampling: What Nyquist Didn't Say, and What to Do About It |url=http://www.wescottdesign.com/articles/Sampling/sampling.pdf |publisher=Wescott Design Services |quote=The Nyquist-Shannon sampling theorem is useful, but often misused when engineers establish sampling rates or design anti-aliasing filters.}}</ref></blockquote>