Pulse-width modulation

Periodic pulse wave

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

$${\displaystyle {\bar {y}}={\frac {1}{T}}\int _{0}^{T}f(t)\,dt}$$

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

$${\displaystyle {\begin{aligned}{\bar {y}}&={\frac {1}{T}}\left(\int _{0}^{DT}y_{\text{max}}\,dt+\int _{DT}^{T}y_{\text{min}}\,dt\right)\\&={\frac {1}{T}}\left(D\cdot T\cdot y_{\text{max}}+T\left(1-D\right)y_{\text{min}}\right)\\&=D\cdot y_{\text{max}}+\left(1-D\right)y_{\text{min}}\end{aligned}}}$$

This latter expression can be fairly simplified in many cases where ${\displaystyle y_{\text{min}}=0}$ as ${\displaystyle {\bar {y}}=D\cdot y_{\text{max}}}$. From this, the average value of the signal (${\displaystyle {\bar {y}}}$) 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

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 or a triangle waveform (blue).

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

• Leading 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.

Delta modulation

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

Asynchronous delta-sigma PWM

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 the middle plot). When the integral of the error exceeds the limits (the upper and lower grey lines in the middle plot), the PWM output changes state. By integrating the difference of the error with the input signal, delta-sigma modulation shapes noise of the resulting spectrum is more concentrated in higher frequencies above the input signal's band.

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)

Direct torque control is a method used to control AC motors. It is closely related to 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.

Servo and motors

PWM is used to control servomechanisms; see servo control.

The PWM output of a microcontroller is almost never used to directly drive a motor. Instead, this signal is interpreted by a motor controller, which converts it into a more accurate PWM signal for the motor. This kind of "PWM controller" used in robotics is derived from PWM controllers used in hobby radio control vehicles.^[21]

Telecommunications

In telecommunications, PWM is a form of signal modulation where the widths of the pulses correspond to specific data values encoded at one end and decoded at the other.

Pulses of various lengths (the information itself) will be sent at regular intervals (the carrier frequency of the modulation).

          _      _      _      _      _      _      _      _     
Clock    | |    | |    | |    | |    | |    | |    | |    | |    
       __| |____| |____| |____| |____| |____| |____| |____| |____

                 _      __     ____          ____   _
PWM signal      | |    |  |   |    |        |    | | |
       _________| |____|  |___|    |________|    |_| |___________

Data       0     1       2      4      0      4     1      0

The inclusion of a clock signal is not necessary, as the leading edge of the data signal can be used as the clock if a small offset is added to each data value to avoid a data value with a zero length pulse.

                _      __     ___    _____   _      _____   __     _   
PWM signal     | |    |  |   |   |  |     | | |    |     | |  |   | |  
             __| |____|  |___|   |__|     |_| |____|     |_|  |___| |_____

Data            0       1      2       4     0        4      1     0

Power delivery

PWM can be used to control the amount of power delivered to a load without incurring the losses that would result from linear power delivery by resistive means. Drawbacks to this technique are that the power drawn by the load is not constant but rather discontinuous (see Buck converter), and energy delivered to the load is not continuous either. However, the load may be inductive, and with a sufficiently high frequency and when necessary using additional passive electronic filters, the pulse train can be smoothed and an average analog waveform recovered. Power flow into the load can be continuous. Power flow from the supply is not constant and will require energy storage on the supply side in most cases. (In the case of an electrical circuit, a capacitor to absorb energy stored in (often parasitic) supply side inductance.)

High frequency PWM power control systems are easily realisable with semiconductor switches. As explained above, almost no power is dissipated by the switch in either on or off state. However, during the transitions between on and off states, both voltage and current are nonzero and thus power is dissipated in the switches. By quickly changing the state between fully on and fully off (typically less than 100 nanoseconds), the power dissipation in the switches can be quite low compared to the power being delivered to the load.

Modern semiconductor switches such as MOSFETs or insulated-gate bipolar transistors (IGBTs) are well-suited components for high-efficiency controllers. Frequency converters used to control AC motors may have efficiencies exceeding 98%. Switching power supplies have lower efficiency due to low output voltage levels (often even less than 2 V for microprocessors are needed), but still more than 70–80% efficiency can be achieved.

Variable-speed computer fan controllers usually use PWM, as it is far more efficient when compared to a potentiometer or rheostat. (Neither of the latter is practical to operate electronically; they would require a small drive motor.)

Light dimmers for home use employ a specific type of PWM control. Home-use light dimmers typically include electronic circuitry that suppresses current flow during defined portions of each cycle of the AC line voltage. Adjusting the brightness of light emitted by a light source is then merely a matter of setting at what voltage (or phase) in the AC half-cycle the dimmer begins to provide electric current to the light source (e.g. by using an electronic switch such as a triac). In this case, the PWM duty cycle is the ratio of the conduction time to the duration of the half AC cycle defined by the frequency of the AC line voltage (50 or 60 Hz, depending on the country).

These rather simple types of dimmers can be effectively used with inert (or relatively slow reacting) light sources such as incandescent lamps, for example, for which the additional modulation in supplied electrical energy, which is caused by the dimmer, causes only negligible additional fluctuations in the emitted light. Some other types of light sources, such as light-emitting diodes (LEDs), however, turn on and off extremely rapidly and would perceivably flicker if supplied with low-frequency drive voltages. Perceivable flicker effects from such rapid-response light sources can be reduced by increasing the PWM frequency. If the light fluctuations are sufficiently rapid (faster than the flicker fusion threshold), the human visual system can no longer resolve them, and the eye perceives the time average intensity without flicker.

In electric cookers, continuously variable power is applied to the heating elements, such as the hob or the grill, using a device known as a simmerstat. This consists of a thermal oscillator running at approximately two cycles per minute, and the mechanism varies the duty cycle according to the knob setting. The thermal time constant of the heating elements is several minutes, so that the temperature fluctuations are too small to matter in practice.

Audio effects and amplification

Varying the duty cycle of a pulse waveform in a synthesis instrument creates useful timbral variations. Some synthesizers have a duty-cycle trimmer for their square-wave outputs, and that trimmer can be set by ear; the 50% point (true square wave) is distinctive because even-numbered harmonics essentially disappear at 50%. Pulse waves, usually 50%, 25%, and 12.5%, make up the soundtracks of classic video games. The term PWM as used in sound (music) synthesis refers to the ratio between the high and low levels being secondarily modulated with a low-frequency oscillator. This gives a sound effect similar to chorus or slightly detuned oscillators played together. (In fact, PWM is equivalent to the sum of two sawtooth waves with one of them inverted.)^[22]

Class-D amplifiers produce a PWM equivalent of a lower-frequency input signal that can be sent to a loudspeaker via a suitable filter network to block the carrier and recover the original lower-frequency signal. Since they switch power directly from the high supply rail and low supply rail, these amplifiers have efficiency above 90% and can be relatively compact and light, even for large power outputs. For a few decades, industrial and military PWM amplifiers have been in common use, often for driving servomotors. Field-gradient coils in MRI machines are driven by relatively high-power PWM amplifiers.

Historically, a crude form of PWM has been used to play back PCM digital sound on the PC speaker, which is driven by only two voltage levels, typically 0 and 5 V. By carefully timing the duration of the pulses, and by relying on the speaker's physical filtering properties (limited frequency response, self-inductance, etc.) it was possible to obtain an approximate playback of mono PCM samples, although at a very low quality, and with greatly varying results between implementations. The Sega 32X uses PWM to play sample-based sound in its games.

In more recent times, the Direct Stream Digital sound encoding method was introduced, which uses a generalized form of pulse-width modulation called pulse-density modulation, at a high enough sampling rate (typically in the order of MHz) to cover the whole acoustic frequency range with sufficient fidelity. This method is used in the SACD format, and reproduction of the encoded audio signal is essentially similar to the method used in class-D amplifiers.

Electrical

SPWM (sine–triangle pulse-width modulation) signals are used in solar inverter design. These switching signals are fed to the FETs that are used in the device. The device's efficiency depends on the harmonic content of the PWM signal. There is much research on eliminating unwanted harmonics and improving the fundamental strength, some of which involves using a modified carrier signal instead of a classic sawtooth signal^[23][24][25] in order to decrease power losses and improve efficiency. Another common application is in robotics where PWM signals are used to control the speed of the robot by controlling the motors.