output hardware

PWM frequency restrictions

When using pulse width modulation to control the power of an output, restrictions of the output system must be considered. While the ESP32 is capable of producing PWM signals in the range of several $\mathrm{kHz}$, and passive components like resistors are able to follow those with ease, active components like transistors, MOSFETs or optocouplers are more limited.

Two parameters are typically used to characterize a such output components:

1. The time $t_\mathrm{ON}$ between activation of a control signal and full activation of the corresponding output.
2. The time $t_\mathrm{OFF}$ between deactivation of a contral signal and full deactivation of the corresponding output.

For most optocouplers and solid state relays those are in the range of $t_\mathrm{ON} + t_\mathrm{OFF} \approx 1 \, \mathrm{ms}$, which corresponds to the maximum frequency

$$f_\mathrm{max} = \frac{1}{t_\mathrm{ON} + t_\mathrm{OFF}} \approx 1 \, \mathrm{kHz}.$$

However, when the output power is to be controled by PWM, the range of usable frequencies is even more restricted. Using an $8$ bit control value and a PWM frequency of $f$, this results in $255$ different power output levels (excluding $0$ or "off"). To use the full range, a full on/off cycle of the output component must therefore be smaller than $\frac{1}{255}$ of the control signal period or $f_\mathrm{max} \leq 255 f$.

Whenever the output is unable to follow this higher frequency, a reduced number of effective output states $n_\mathrm{eff}$ is the result. Before a minimum control value $n_\mathrm{min}$ is chosen, the output is unable to fully switch on before the control signal is turned off again:

$$n_\mathrm{min} = 255 \cdot f \cdot \left( t_\mathrm{ON} + t_\mathrm{OFF} \right)$$

On the other hand, whenever the the control value is higher than $n_\mathrm{max}$, the output will be unable to turn off completely:

$$n_\mathrm{max} = 255 - n_\mathrm{min}.$$

In both cases the actual power output might be nonlinear and the control value does not accurately represent it. The following table shows the approximate control value ranges for an optocoupler with $f_\mathrm{max} = 1 \, \mathrm{kHz}$.

$f / (\mathrm{Hz})$	$n_\mathrm{eff}$	$n_\mathrm{min}$	$n_\mathrm{max}$
0.5	255	0	255
10	250	3	252
20	245	5	250
30	240	8	247
40	235	10	245
50	230	13	242
100	204	26	230
150	179	38	217
200	153	51	204
250	128	64	191
300	102	77	179
350	77	89	166
400	51	102	153
450	26	115	140
500	0	128	128