On Mon, 6 Mar 2000, "Grif" w. keith griffith wrote:

> At 10:44 AM 3/5/00 -0600, you wrote..
> >It finally dawned on me yesterday how magic sine waves can be sythesized.
>
> Major math snip:::
>
> Scott:  doesn't your code produce an analog output?  I was of the opinion
> the whole reason for the Magic part of the synthesis was using saturated
> switching at a much higher freq than the final output and stuffing it thru
> a low pass filter, getting a somewhat usable, higher power sine wave
> output?  Or did I just learn something this early in the week???  ( that
> would be good,,, then I wouldn't have to work as hard the rest of the week!)


True, the ultimate objective is to produce an analog output. However, the
goal with the so called 'magic sinewaves' is to find the MINIMUM number of
switch transitions. The resulting bit stream is filtered such that a sine
wave is produced. The reason for minimizing the transitions is because
some switches, particularly those associated with high power applications
like in AC inverters, require a significant amount of energy to switch or
cannot be efficiently switched at high frequencies. As an example, you may
have a sine wave synthesis circuit where you want to drive the gate of a
MOSFET directly from a pic I/O pin. Pic's have fairly beefy outputs,
however they're very weak when compared to real MOSFET drivers. But given
enough time, the PIC I/O can switch a MOSFET (assuming of course that Voh
is above Vgs threshold - i.e. logic level Vgs MOSFETs may be required). If
you attempted to drive the MOSFET at too high of frequency with this pic
I/O pin, then chances are that it will never switch!

In addition, if you can use harmonic cancellation, then the constraints on
the analog filter may be relaxed. PWM generated waveforms relax the filter
constraints by maximizing the switching frequency. If the PWM carrier is
100 times the sinewave you're trying to generate, then you're analog
filter is much easier to design. There aren't any harmonics (to an
approximation) between the sine wave frequency you're generating and the
carrier frequency of the PWM.

Magic sine waves explicitly remove these harmonics by strategically
placing the pulses (or edges). The pulses, when viewed individually,
contain harmonics at the fundamental (of the sine wave being generated)
and at all integer multiples above. But when the pulses are taken
together, the 2nd, 3rd and so on harmonics are cancelled. The number of
harmonics cancelled depends on the quantization of time for the edges. For
example, in the previous post, I had a graph of a 12-bit pulse stream
010110100000. The second and third harmonics were exactly cancelled, but
there is a 4th and 5th harmonic present. Doubling the number of pulses to
24 allows a pulse stream to be created that will cancel the 4th harmonic.
In general, if you have a pulse stream containing n! pulses, then it's
possible to exactly cancel the first n harmonics. However, n! grows rather
quickly. It's simply not practical to attempt to cancel the first 10
harmonics (after the fundamental) by creating a pulse stream containing
10! or 3,628,800 pulses.

Consequently, if you wish to cancel the first 10 harmonics but with a
pulse stream containing only a 1000 pulses, you're not going to achieve
your goal. However, you may discover a particular stream that exactly
cancels the 2nd, 3rd, and 4th, and attenuates the 5th through 10th
harmonics by 20dB. From that you may design an appropriate analog filter.

AFAIK, there is no way to apriori determine the appropriate length of the
pulse stream to achieve some required harmonic profile. I think
Lancaster's contribution to this technology is to discover certain lengths
that perform well - or at least better than pulses streams that are
slightly shorter or longer. What I'm playing around with is a way you can
study the characteristics of one stream versus another.

Scott