Square + triangle = sine (almost)
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From: whit3rd on 8 Mar 2010 16:09 On Mar 8, 8:28 am, "Tim Williams" <tmoran...(a)charter.net> wrote: > I recollect something from Don Lancaster about Magic Sinewaves ... > Looks like it's a PWM tristate thing (requiring an always-on H bridge), but > not really PWM as the edge timings are arbitrary through the cycle. The 'magic sinewaves' approach is a variant on the digital filter theme, using calculated ON/OFF pulses to cancel two or three of the harmonics... but that only buys you a small reprieve from the problem, a low-pass filter to take out the higher harmonics is assumed. Alas, that kills the adjustable- frequency range, unless you make a (expensive) tracking filter. The linear solution of making an accurate triangle wave, then distorting, might get from 5% distortion (which is what a triangle wave is, compared to a sine) down to 1% or less, is terribly limited, too. There's a theorem (the Wiener-Hopf theorem) that says your fit functions work best if they have the same autocorrelation as the thing they fit to... which means a smooth diode response curve is not going to reduce a step-like square wave to sinusoid in a small number of stages, EVER. But, all these 'one percent' solutions don't kill the high harmonics down to the level of a true sinewave oscillator. My old HP 204C was worst-case 0.1% ( - 60 dB) on its distortion right out of the box; compared to the triangle-wave and breakpoint-diodes of an XR2206 at 2.5% before hand-tweaking.
From: Darwin on 8 Mar 2010 16:10 On 7 Mar, 13:47, "Ban" <bans...(a)web.de> wrote: > >http://www.national.com/an/AN/AN-263.pdf > > > (see "Approximation Methods" paragraph beginning at page 8) > > > Hope it helps. > > I downloaded the paper, but what they call *logarithmic* is IMHO *tanh* and > that opamp is not connected very smart either (FIG. 11). I also think it is more a tanh shaping. I used the transistor shaping network for a VCO some years ago and it worked nice, if you do not expect an extremely low distortion rating. A problem is that the output amplitude (before the opamp) is rather small and so there can be noise problems.
From: Don Klipstein on 8 Mar 2010 16:20 In <7f952719-ece0-4a91-bd65-9e981f5c058f(a)v20g2000yqv.googlegroups.com>, whit3rd wrote in part: <SNIP to here> >The linear solution of making an accurate triangle wave, then >distorting, might get from 5% distortion (which is what a triangle >wave is, compared to a sine) A triangle wave has 12.1% distortion. The 3rd harmonic alone has voltage of 11.1% of that of the fundamental. > down to 1% or less, is terribly limited, too. There's a theorem (the >Wiener-Hopf theorem) that says your fit functions work best if they have >the same autocorrelation as the thing they fit to... which means a smooth >diode response curve is not going to reduce a step-like square >wave to sinusoid in a small number of stages, EVER. > >But, all these 'one percent' solutions don't kill the high harmonics >down to the level of a true sinewave oscillator. >My old HP 204C was worst-case 0.1% ( - 60 dB) on its >distortion right out of the box; compared to the triangle-wave >and breakpoint-diodes of an XR2206 at 2.5% before hand-tweaking. - Don Klipstein (don(a)misty.com)
From: Phil Hobbs on 8 Mar 2010 16:26 On 3/8/2010 4:10 PM, Darwin wrote: > On 7 Mar, 13:47, "Ban"<bans...(a)web.de> wrote: > >>> http://www.national.com/an/AN/AN-263.pdf >> >>> (see "Approximation Methods" paragraph beginning at page 8) >> >>> Hope it helps. >> >> I downloaded the paper, but what they call *logarithmic* is IMHO *tanh* and >> that opamp is not connected very smart either (FIG. 11). > > I also think it is more a tanh shaping. I used the transistor shaping > network for a VCO some years ago and it worked nice, if you do not > expect an extremely low distortion rating. A problem is that the > output amplitude (before the opamp) is rather small and so there can > be noise problems. You can do a pretty good job with an LM13700 producing the tanh shape, and then subtracting off a small amount of the original triangle wave to get rid of the cusps at the peaks. Cheers Phil Hobbs -- Dr Philip C D Hobbs Principal ElectroOptical Innovations 55 Orchard Rd Briarcliff Manor NY 10510 845-480-2058 hobbs at electrooptical dot net http://electrooptical.net
From: Jim Thompson on 8 Mar 2010 16:35
On Mon, 08 Mar 2010 16:26:04 -0500, Phil Hobbs <pcdhSpamMeSenseless(a)electrooptical.net> wrote: >On 3/8/2010 4:10 PM, Darwin wrote: >> On 7 Mar, 13:47, "Ban"<bans...(a)web.de> wrote: >> >>>> http://www.national.com/an/AN/AN-263.pdf >>> >>>> (see "Approximation Methods" paragraph beginning at page 8) >>> >>>> Hope it helps. >>> >>> I downloaded the paper, but what they call *logarithmic* is IMHO *tanh* and >>> that opamp is not connected very smart either (FIG. 11). >> >> I also think it is more a tanh shaping. I used the transistor shaping >> network for a VCO some years ago and it worked nice, if you do not >> expect an extremely low distortion rating. A problem is that the >> output amplitude (before the opamp) is rather small and so there can >> be noise problems. > >You can do a pretty good job with an LM13700 producing the tanh shape, >and then subtracting off a small amount of the original triangle wave to >get rid of the cusps at the peaks. > >Cheers > >Phil Hobbs The transfer characteristics of a BIPOLAR diff pair IS a TANH function. The tricky part is containing the signal within the temperature-dependent operating range. ...Jim Thompson -- | James E.Thompson, CTO | mens | | Analog Innovations, Inc. | et | | Analog/Mixed-Signal ASIC's and Discrete Systems | manus | | Phoenix, Arizona 85048 Skype: Contacts Only | | | Voice:(480)460-2350 Fax: Available upon request | Brass Rat | | E-mail Icon at http://www.analog-innovations.com | 1962 | The only thing bipartisan in this country is hypocrisy |