Comb filtering PWM

Kragen Javier Sitaker, 2019-10-28 (4 minutes)

A phase-correct PWM signal has a lot of its energy at the fundamental PWM frequency and, typically, more at its harmonics --- though which harmonics depends on the duty cycle it's producing at the moment. A unity-gain feedforward comb filter has nulls at a fundamental frequency and its harmonics --- if the gain is negative, then it has a null at DC and at all multiples of the fundamental, while if the gain is positive, then it has a null at only odd multiples of the fundamental (and a gain of 2 at DC).

Typically the PWM harmonics are thought of as undesirable noise on the PWM signal. The good news is that, at a constant PWM output level, all of the PWM noise is at these harmonics; so, if we could notch them out, PWM could perfectly reproduce our desired output voltage or current.

There are many ways to do this, but in particular I wanted to explore using transmission-line comb filters. A transmission line leading to an open circuit will feed a delayed copy of its input back to that input; this is a unity-gain feedforward comb filter. A 10-meter coaxial transmission line with a typical propagation speed of 0.5c will add a signal delayed by about 130 ns to its input.

(I'm a little bit vague on exactly how this needs to be hooked up in a circuit to get the desired effect, in particular for a high-power signal where efficiency is important.)

Suppose that your PWM frequency is 7.5 MHz, carefully locked to this delay. Then the transmission-line comb filter will cancel the first, third, fifth, seventh, and higher odd harmonics from the PWM signal, leaving only the even harmonics.

If this filter can be cascaded with a second similar filter made with a 5-meter coaxial transmission line, that will cancel the second, sixth, tenth, and higher 2*odd harmonics from the PWM signal, leaving only the harmonics divisible by 4.

A third such filter made of 2.5 meters of such line will cancel the remaining harmonics not divisible by 8: 4, 12, 20, 28, and so on.

A fourth such filter made of 1.25 meters will cancel the remaining harmonics not divisible by 16: 8, 24, 40, and so on.

At this point the first 15 harmonics of the PWM signal have been perfectly reactively canceled; the first unfiltered harmonic is the 16th, at 120 MHz.

By adding a delayed copy of the PWM signal to itself four times, we can make a stairstep approximation of the desired signal, with I think any of 34 = 243 different voltage levels; there is a small remaining amount of quantization noise remaining at 16 and more times the PWM carrier frequency.

Ten-meter-long delay lines might sound impractically large, but Horowitz & Hill tell us that old oscilloscopes commonly used a dual-core coax with two signal lines in a double helix inside the outer shield; the signal propagation along the helix was what you would normally expect for signal propagation along a normal coaxial cable center. If this helix were 20 mm in diameter and the coiled wire were 1 mm wide (including insulation), each 2 mm of delay line would provide 31.4 mm of delay, so 10 m of delay would fit into a length of only 640 mm.

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