Square + triangle = sine (almost)
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From: George Herold on 8 Mar 2010 21:48 On Mar 8, 8:57 am, MooseFET <kensm...(a)rahul.net> wrote: > On Mar 7, 4:10 pm, whit3rd <whit...(a)gmail.com> wrote: > > > > > > > On Mar 6, 8:04 pm, rontan...(a)esterbrook.com (Ron Tanner) wrote: > > > > On Sun, 7 Mar 2010 14:31:48 +1100, "Phil Allison" <phi...(a)tpg.com.au> > > > wrote: > > > >> If a squarewave contains all odd harmonics of the fundamental > > > >> frequency, and a triangle all even, > > > > > ** Sorry - that is WRONG . > > > > > A triangle wave contains only odd harmonics too. > > > OK thanks for the pull-up, but how about using a triangle-square wave > > > mix, in place of a filter, to simulate a sinewave . > > > To generate a square wave and triangle wave together, just hook a > > Schmitt trigger to an op amp configured as integrator (resistor from > > input to (-) node, feedback capacitor from op amp output to (-) node. > > The output of the op amp then is fed to the Schmitt trigger (a '555 > > does this without the op amp, but its triangle waves are curvey > > because of that). > > > The result is a square wave and a triangle wave, both of constant > > amplitude, which changes frequency with a single variable resistor > > (the integrator input resistor). > > > SO, now you want to mix a square and triangle? If you generate them > > by the method above, fundamentals ARE OUT OF PHASE by ninety degrees, > > and so are all of the harmonics. You get neither cancellation nor > > reinforcement > > by any simple summing. > > If you use a quad comparator, you can do some interesting stuff. With > just 2 more comparators, you can make this: > > ------ ------ > --- --- --- --- > ------ > > It will be in phase with the triangle wave. It can be made to have no > 3rd harmonic fairly easily. By trading off the 3rd you can have a > reduced > 5th. > > It is too early in the morning for me to be sure, but I think that if > you > fiddle it just right and add in some of the triangle wave, you can get > low > values for both the 3rd and 5th. > > It also seems to me that there should be a way to make a very > nonlinear > PWMing action that when low pass filtered leaves a moderately good > estimate > of a sine wave. This could allow you to hold the sine wave shape over > perhaps > a couple of decades.- Hide quoted text - > > - Show quoted text - That's neat, are you making a 'few' bit DAC? George H.
From: George Herold on 8 Mar 2010 21:59 On Mar 8, 4:09 pm, whit3rd <whit...(a)gmail.com> wrote: > 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. That's Great! Thanks, I tried to filter a square wave into a sine and was deeply disappointed. (If you filtered it hard enough there was just not enough left.) George H. > > 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: JosephKK on 9 Mar 2010 00:42 On Mon, 08 Mar 2010 09:25:15 -0700, Jim Thompson <To-Email-Use-The-Envelope-Icon(a)My-Web-Site.com> wrote: >On Mon, 08 Mar 2010 00:01:44 -0800, Muzaffer Kal <kal(a)dspia.com> >wrote: > >>On Sun, 07 Mar 2010 23:11:20 -0800, >>"JosephKK"<quiettechblue(a)yahoo.com> wrote: >> >>>On Sat, 6 Mar 2010 20:21:10 -0800, D from BC <myrealaddress(a)comic.com> wrote: >>> >>>>In article <4b9324ee.4432562(a)news.tpg.com.au>, rontanner(a)esterbrook.com >>>>says... >>>>> >>>>> On Sun, 7 Mar 2010 14:31:48 +1100, "Phil Allison" <phil_a(a)tpg.com.au> >>>>> wrote: >>>>> >>>>> > >>>>> >"Harold Larsen" >>>>> >> >>>>> >> If a squarewave contains all odd harmonics of the fundamental >>>>> >> frequency, and a triangle all even, >>>>> > >>>>> > >>>>> > ** Sorry - that is WRONG . >>>>> > >>>>> > A triangle wave contains only odd harmonics too. >>>>> > >>>>> >http://en.wikipedia.org/wiki/Triangle_wave >>>>> > >>>>> >A "sawtooth" wave contains all integer harmonics. >>>>> > >>>>> >>>>> OK thanks for the pull-up, but how about using a triangle-square wave >>>>> mix, in place of a filter, to simulate a sinewave . >>>>> >>>>> I have not seen that method applied or described anywhere, but it >>>>> makes a fair approximation, at least to my eye. >>>>> >>>>> Harold Larsen >>>> >>>> >>>>This reminds of the XR2206 chip that makes square, triangle and sine >>>>using analog technology. >>> >>>Sure enough, as does the ICL8038. Part of the question is how it is done. >> >>The datasheet at http://www.intersil.com/data/FN/FN2864.pdf has a >>pretty good schematic and explanation which shows how it's done. > >Yep. "Piecewise-Linear", aka break-point analysis... taught in better >engineering schools ;-) > > ...Jim Thompson I first saw it in a synchro to digital converter about 1973. I had to think hard for a while before i "got" it.
From: Ban on 9 Mar 2010 01:20 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. Here is a dimensioned citcuit using this principle, just a differential amp like in the app-note. .------o--------o--------------------- +2.5V | | | .-. .-. .-. 13.5k | |/ | |10k | | 10k |/| | | 10k| | ___ \ /'-' '-' '-' .--|___|--. \ | | | | | | | | | |\ | Sineshaper | | o------o---|-\ | Out 5Vpp | | | | >--o--- | o--------)-------o--|+/ | | | | |/ OPA365 | | | .-. | | | | | Triangle | | | | |10k In ___ | |/ \| '-' o-|___|-)--o-| Array |-. | 5Vpp 10k | | |> <| | === | | | | | GND | .-. .-. .-. .-. | | | | |47 | | | | 390| | | | | 47| | | |390 | '-' '-' '-' '-' | | | | | | === '---o----' === | GND | GND | | o-----. | | | | \| | |/ |---o--| Array <| |> CA3046 | | 1k 1k | | '----------o-----------------------o -2.5V (created by AACircuit v1.28.6 beta 04/19/05 www.tech-chat.de) sorry for the size. I can not promise 0.1% THD like Phil Hobbs, but around 0.3% is achievable. The opamp goes slightly into saturation, so it cuts the remaining corners of the triangle wave, much like subtracting them. I do not throw away dynamic range like in the app.note. The amplitude on the base of the diff. pair should be around */- 4Vt or 180mVpp for minimum distortion. ciao Ban
From: Fred Bartoli on 9 Mar 2010 03:22
MooseFET a �crit : > On Mar 8, 8:28 am, "Tim Williams" <tmoran...(a)charter.net> wrote: >> "MooseFET" <kensm...(a)rahul.net> wrote in message >> >> news:c968f0a3-64bc-46e1-8a14-7b36a8e75d0f(a)b9g2000pri.googlegroups.com... >> >>> If you use a quad comparator, you can do some interesting stuff. With >>> just 2 more comparators, you can make this: >>> ------ ------ >>> --- --- --- --- >>> ------ >> I recollect something from Don Lancaster about Magic Sinewaves and how you >> can get arbitrarily low harmonics from certain optimal patterns of on and >> off, given sufficiently accurate timing, and I suppose some sort of >> filtering. I never did figure out if it's supposed to be a tristate >> waveform (as above) > > The waveform I drew can be made by simply adding two pulse trains > with > different duty cycles. The fact that 3 time 60 degrees is 180 degrees > is how you can get the 3rd harmonic to go away. > > If you use more steps, you can get the first N harmonics to drop to > zero. The same is true for line segments instead of steps. > Which is nothing more than the analog variant of a transversal filter that you can build from a divider, a shift register and a few weighted summing resistors. -- Thanks, Fred. |