Square + triangle = sine (almost)
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From: MooseFET on 10 Mar 2010 21:29 On Mar 10, 1:47 pm, whit3rd <whit...(a)gmail.com> wrote: > On Mar 9, 9:38 pm, "JosephKK"<quiettechb...(a)yahoo.com> wrote: > > > Just to be off the wall, what is the integral of a triangle wave? > > How about the second and third integrals? > > To close the loop and make an oscillator, using a square wave > from a Schmitt trigger, one can use one stage of integration > (to a triangle wave) or three, but not two. Two stages > gets to a double-parabola, three stages to a double-cubic. > The double-cubic has its small pointy defect at the crest > as does the triangle wave. > > The three-stage thing is called a phase shift oscillator if you make > it without the Schmitt trigger part. > > Alas, those all require multiple-gang variable resistors instead of a > single knob to adjust. Have you ever priced a good three-gang > pot? Or capacitor, for that matter? You can use just one pot with the wiper grounded to make two stages where the gain varies proportionally to the pot setting. The circuit below does this. Since a two integrator oscillator design requires two gain stages that vary proportional to the frequency desired, this will do the trick over a reasonable range. Version 4 SHEET 1 1404 680 WIRE 208 -176 32 -176 WIRE 464 -176 288 -176 WIRE 1072 -144 1008 -144 WIRE 1216 -144 1152 -144 WIRE 464 -80 464 -176 WIRE 544 -80 464 -80 WIRE 320 -48 320 -64 WIRE 1216 -48 1216 -144 WIRE 1296 -48 1216 -48 WIRE -448 -32 -688 -32 WIRE -144 -32 -448 -32 WIRE 32 -32 32 -176 WIRE 32 -32 -64 -32 WIRE 288 -32 240 -32 WIRE 464 -16 464 -80 WIRE 464 -16 352 -16 WIRE 32 0 32 -32 WIRE 288 0 32 0 WIRE -688 16 -688 -32 WIRE 1088 32 1088 16 WIRE 320 48 320 16 WIRE 464 48 464 -16 WIRE 880 48 464 48 WIRE 1008 48 1008 -144 WIRE 1008 48 960 48 WIRE 1056 48 1008 48 WIRE 32 64 32 0 WIRE 1216 64 1216 -48 WIRE 1216 64 1120 64 WIRE 1056 80 1008 80 WIRE 240 112 240 -32 WIRE 384 112 240 112 WIRE 464 112 464 48 WIRE -688 128 -688 96 WIRE 1088 128 1088 96 WIRE 32 176 32 144 WIRE 240 192 240 112 WIRE 240 304 240 272 WIRE -448 352 -448 -32 WIRE 1008 352 1008 80 WIRE 1008 352 -448 352 WIRE -32 432 -32 416 WIRE 96 432 96 416 WIRE -32 528 -32 512 WIRE 96 528 96 512 FLAG 320 -64 vcc FLAG -32 416 vcc FLAG 320 48 vee FLAG 96 416 vee FLAG 96 528 0 FLAG -32 528 0 FLAG -688 128 0 FLAG 32 176 0 FLAG 240 304 0 FLAG 1088 16 vcc FLAG 1088 128 vee FLAG 544 -80 ver1out FLAG 1296 -48 ver2out SYMBOL voltage 96 416 R0 WINDOW 123 0 0 Left 0 WINDOW 39 0 0 Left 0 SYMATTR InstName V1 SYMATTR Value -12 SYMBOL voltage -32 416 R0 WINDOW 123 0 0 Left 0 WINDOW 39 0 0 Left 0 SYMATTR InstName V2 SYMATTR Value 12 SYMBOL voltage -688 0 R0 WINDOW 123 0 0 Left 0 WINDOW 39 0 0 Left 0 SYMATTR InstName V3 SYMATTR Value 1 SYMBOL res -48 -48 R90 WINDOW 0 0 56 VBottom 0 WINDOW 3 32 56 VTop 0 SYMATTR InstName R1 SYMATTR Value 10k SYMBOL res 48 160 R180 WINDOW 0 36 76 Left 0 WINDOW 3 36 40 Left 0 SYMATTR InstName R2 SYMATTR Value {10E3*PotSetting} SYMBOL res 192 -160 R270 WINDOW 0 32 56 VTop 0 WINDOW 3 0 56 VBottom 0 SYMATTR InstName R3 SYMATTR Value 10k SYMBOL res 224 176 R0 SYMATTR InstName R4 SYMATTR Value 10k SYMBOL res 480 96 R90 WINDOW 0 0 56 VBottom 0 WINDOW 3 32 56 VTop 0 SYMATTR InstName R5 SYMATTR Value 10k SYMBOL Opamps\\LT1056 320 -80 R0 SYMATTR InstName U1 SYMBOL Opamps\\LT1056 1088 0 R0 SYMATTR InstName U2 SYMBOL res 1056 -128 R270 WINDOW 0 32 56 VTop 0 WINDOW 3 0 56 VBottom 0 SYMATTR InstName R6 SYMATTR Value 10k SYMBOL res 864 64 R270 WINDOW 0 32 56 VTop 0 WINDOW 3 0 56 VBottom 0 SYMATTR InstName R7 SYMATTR Value 10k TEXT -392 -176 Left 0 !.step param PotSetting=0.001 1 0.2 TEXT -392 552 Left 0 !.op
From: Phil Hobbs on 10 Mar 2010 22:32 On 3/10/2010 9:18 PM, MooseFET wrote: > On Mar 10, 1:50 am, Bitrex<bit...(a)de.lete.earthlink.net> wrote: >> MooseFET wrote: >>> On Mar 9, 8:36 pm, Bitrex<bit...(a)de.lete.earthlink.net> wrote: >>>> Jim Thompson wrote: >>>>> On Tue, 09 Mar 2010 10:10:26 -0500, Phil Hobbs >>>>> <pcdhSpamMeSensel...(a)electrooptical.net> wrote: >>>>>> On 3/9/2010 9:59 AM, Jim Thompson wrote: >>>>>>> On Mon, 08 Mar 2010 21:42:22 -0800, >>>>>>> "JosephKK"<quiettechb...(a)yahoo.com> wrote: >>>>>>>> On Mon, 08 Mar 2010 09:25:15 -0700, Jim Thompson<To-Email-Use-The-Envelope-I...(a)My-Web-Site.com> wrote: >>>>>>>>> On Mon, 08 Mar 2010 00:01:44 -0800, Muzaffer Kal<k...(a)dspia.com> >>>>>>>>> wrote: >>>>>>>>>> On Sun, 07 Mar 2010 23:11:20 -0800, >>>>>>>>>> "JosephKK"<quiettechb...(a)yahoo.com> wrote: >>>>>>>>>>> On Sat, 6 Mar 2010 20:21:10 -0800, D from BC<myrealaddr...(a)comic.com> wrote: >>>>>>>>>>>> In article<4b9324ee.4432...(a)news.tpg.com.au>, rontan...(a)esterbrook.com >>>>>>>>>>>> says... >>>>>>>>>>>>> On Sun, 7 Mar 2010 14:31:48 +1100, "Phil Allison"<phi...(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 athttp://www.intersil.com/data/FN/FN2864.pdfhasa >>>>>>>>>> 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. >>>>>>> The only place I can remember using it in an actual product was for >>>>>>> linearizing a flat-face CRT sweep (RADAR)... and there it was >>>>>>> piecewise _curve_ fitting. >>>>>>> ...Jim Thompson >>>>>> Breakpoint amps are nearly always a crutch. One poor guy I tried to >>>>>> help (15 years back) ignored my advice and wound up with a multi-diode >>>>>> breakpoint amp stuck inside a crystal oven to keep the breakpoints from >>>>>> going all over the place with temperature. Blech. (It was in a fancy >>>>>> measurement system, too. Got all sorts of industry awards.) >>>>>> The Widlar approach (National AN4, Figure 8) uses BJT saturation to make >>>>>> nice sharp breakpoints that don't drift much. Of course you have to >>>>>> wait for the transistor to come out of saturation. >>>>>> About the only good use of breakpoint amps I've seen is inside >>>>>> complicated FB loops, e.g. to approximately correct for the nonlinearity >>>>>> of VCOs and heaters. This reduces the variation of loop gain and so >>>>>> makes frequency compensation easier. Drift and inaccuracy are not a big >>>>>> problem in those sorts of applications. >>>>>> Cheers >>>>>> Phil Hobbs >>>>> Ah, yes! Thanks for the reminder! I also linearized a frequency >>>>> hopping VCO for OmniSpectra _many_ years ago... for jumping close to >>>>> desired frequency, so the PLL lock was faster... a cavity beast :-) >>>>> I would never use _just_ diodes, rather use them with OpAmps or >>>>> comparators, such as... >>>>> http://analog-innovations.com/SED/ClampForLarkin.pdf >>>>> (A Christmas gift, 2007. But he remains a cranky old git :-) >>>>> http://analog-innovations.com/SED/LevelDetectAndFollow-LM339.pdf >>>>> http://analog-innovations.com/SED/LevelDetectAndFollow-TL431.pdf >>>>> http://analog-innovations.com/SED/PerfectDiodeForChargerIsolation.pdf >>>>> ...Jim Thompson >>>> The first schematic looks like the start of a decent guitar fuzzbox >>>> pedal! I think one could set more breakpoints with different slopes by >>>> using more comparators with the breakpoint voltage on the non inverting >>>> inputs and putting resistors in series with the diodes, right? >> >>>> Back before guitar practice amps with DSP became commodity hardware, >>>> Peavey had a patented technology called "TransTube" that purported to >>>> make a solid state amp have a tone more like a tube amp. I wonder if >>>> they used a similar piecewise linear technique to make the amp have a >>>> softer clipping characteristic. >> >>> At lowish frequencies, you can do this: >> >>> ---------------------------------------/\/\---+----Out >>> ! ! >>> +-----------------/\/\----+-/\/\---+---/\/\----+ >>> ! ! ! ! >>> ! --!-\ ! ! >>> In ---+--------!+\ !>-- ! >>> !>---+--/\/\--+---!+/ ! >>> --!-/ ! ! ! >>> ! ! ---/\/\--GND ! >>> GND--/\/\--+--/\/\---+------------------------/\/\-- >> >>> With rail to rail op-amps, you can get a total of 6 knees from Vee to >>> Vcc in the output >>> swing. >> >> I'm having trouble following that circuit - it looks clever, but how >> does it work? > > Start with Vin = 0 > Both op-amps are working normally. > A small change in Vin is given gain as it goes towards output > > Increase Vin and at some point the 2nd op-amp hits the rail. > Now the gain is less. > > Increase Vin and the 1st opamp hits the rail. > Now the gain is even lower. > > Increase Vin and the (-) input of the 2nd opamp is going up > to where the 2nd opamp comes off the rail and swings downwards > The gain is further reduced. > > The same works in the other direction. Plessey used to sell packaged RF amps that worked a bit like that--they were meant to be cascaded to form logarithmic IF strips. They had a gain of 10 dB for small signals, dropping to 0 dB when they saturated. With a string of them in series, upping the signal input by 10 dB railed another amplifier, and so reduced the overall gain by 10 dB. That way, the differential gain was dVout/dVin ~ 1/Vin, which makes it approximately logarithmic. These were different from the usual successive-detection DLVAs, because they actually performed logarithmic compression on the RF, not just the detected signal. 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: MooseFET on 11 Mar 2010 09:10 On Mar 10, 7:32 pm, Phil Hobbs <pcdhSpamMeSensel...(a)electrooptical.net> wrote: [....] > >>> At lowish frequencies, you can do this: > > >>> ---------------------------------------/\/\---+----Out > >>> ! ! > >>> +-----------------/\/\----+-/\/\---+---/\/\----+ > >>> ! ! ! ! > >>> ! --!-\ ! ! > >>> In ---+--------!+\ ! >-- ! > >>> ! >---+--/\/\--+---!+/ ! > >>> --!-/ ! ! ! > >>> ! ! ---/\/\--GND ! > >>> GND--/\/\--+--/\/\---+------------------------/\/\-- > > >>> With rail to rail op-amps, you can get a total of 6 knees from Vee to > >>> Vcc in the output > >>> swing. [....] > Plessey used to sell packaged RF amps that worked a bit like that--they > were meant to be cascaded to form logarithmic IF strips. They had a > gain of 10 dB for small signals, dropping to 0 dB when they saturated. > With a string of them in series, upping the signal input by 10 dB railed > another amplifier, and so reduced the overall gain by 10 dB. > > That way, the differential gain was dVout/dVin ~ 1/Vin, which makes it > approximately logarithmic. These were different from the usual > successive-detection DLVAs, because they actually performed logarithmic > compression on the RF, not just the detected signal. I have often thought that the same sort of thing would be good in an FM IF strip. In that case, the 3rd power part of the series is what does the most to remove the noise. The odd numbered terms are less effective. As a result, it would be very nice to be able to make the limiting stage have the right amount of 3rd power and less of all the others. > > 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 nethttp://electrooptical.net
From: Phil Hobbs on 12 Mar 2010 09:29 On 3/11/2010 9:10 AM, MooseFET wrote: > On Mar 10, 7:32 pm, Phil Hobbs > <pcdhSpamMeSensel...(a)electrooptical.net> wrote: > [....] >>>>> At lowish frequencies, you can do this: >> >>>>> ---------------------------------------/\/\---+----Out >>>>> ! ! >>>>> +-----------------/\/\----+-/\/\---+---/\/\----+ >>>>> ! ! ! ! >>>>> ! --!-\ ! ! >>>>> In ---+--------!+\ !>-- ! >>>>> !>---+--/\/\--+---!+/ ! >>>>> --!-/ ! ! ! >>>>> ! ! ---/\/\--GND ! >>>>> GND--/\/\--+--/\/\---+------------------------/\/\-- >> >>>>> With rail to rail op-amps, you can get a total of 6 knees from Vee to >>>>> Vcc in the output >>>>> swing. > > [....] >> Plessey used to sell packaged RF amps that worked a bit like that--they >> were meant to be cascaded to form logarithmic IF strips. They had a >> gain of 10 dB for small signals, dropping to 0 dB when they saturated. >> With a string of them in series, upping the signal input by 10 dB railed >> another amplifier, and so reduced the overall gain by 10 dB. >> >> That way, the differential gain was dVout/dVin ~ 1/Vin, which makes it >> approximately logarithmic. These were different from the usual >> successive-detection DLVAs, because they actually performed logarithmic >> compression on the RF, not just the detected signal. > > I have often thought that the same sort of thing would be good in an > FM IF > strip. In that case, the 3rd power part of the series is what does > the most > to remove the noise. The odd numbered terms are less effective. As a > result, > it would be very nice to be able to make the limiting stage have the > right > amount of 3rd power and less of all the others. > If you use an FM detector that's insensitive to AM, e.g. a wideband PLL, it'll work better than a limiter at low SNR. Essentially all reasonable demodulation schemes work equivalently at high SNR, where the AM and PM fluctuations are linear functions of the additive noise amplitude. At low SNR, limiters start suppressing the signal in favour of the noise. That leads to dropouts--periods during which the signal disappears and only the noise is left--which really set the threshold for detection. PLLs using wideband AGC instead of limiting don't exhibit threshold, although the detected signal does get noisier as the SNR declines. (You need the AGC to keep the loop bandwidth reasonably constant.) The Plessey scheme can be thought of as a kind of ultrawideband AGC, I suppose--as long as the compression comes after the narrowest IF filter, it should be nearly equivalent. The only other problem there would be AM-PM conversion. Most kinds of amplifiers slow down when you rail them, which leads to phase errors. (The good news for FM is that this needs only a 1-D calibration, instead of 2-D as in I/Q schemes.) 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 12 Mar 2010 09:39
On Fri, 12 Mar 2010 09:29:27 -0500, Phil Hobbs <pcdhSpamMeSenseless(a)electrooptical.net> wrote: >On 3/11/2010 9:10 AM, MooseFET wrote: >> On Mar 10, 7:32 pm, Phil Hobbs >> <pcdhSpamMeSensel...(a)electrooptical.net> wrote: >> [....] >>>>>> At lowish frequencies, you can do this: >>> >>>>>> ---------------------------------------/\/\---+----Out >>>>>> ! ! >>>>>> +-----------------/\/\----+-/\/\---+---/\/\----+ >>>>>> ! ! ! ! >>>>>> ! --!-\ ! ! >>>>>> In ---+--------!+\ !>-- ! >>>>>> !>---+--/\/\--+---!+/ ! >>>>>> --!-/ ! ! ! >>>>>> ! ! ---/\/\--GND ! >>>>>> GND--/\/\--+--/\/\---+------------------------/\/\-- >>> >>>>>> With rail to rail op-amps, you can get a total of 6 knees from Vee to >>>>>> Vcc in the output >>>>>> swing. >> >> [....] >>> Plessey used to sell packaged RF amps that worked a bit like that--they >>> were meant to be cascaded to form logarithmic IF strips. They had a >>> gain of 10 dB for small signals, dropping to 0 dB when they saturated. >>> With a string of them in series, upping the signal input by 10 dB railed >>> another amplifier, and so reduced the overall gain by 10 dB. >>> >>> That way, the differential gain was dVout/dVin ~ 1/Vin, which makes it >>> approximately logarithmic. These were different from the usual >>> successive-detection DLVAs, because they actually performed logarithmic >>> compression on the RF, not just the detected signal. >> >> I have often thought that the same sort of thing would be good in an >> FM IF >> strip. In that case, the 3rd power part of the series is what does >> the most >> to remove the noise. The odd numbered terms are less effective. As a >> result, >> it would be very nice to be able to make the limiting stage have the >> right >> amount of 3rd power and less of all the others. >> > >If you use an FM detector that's insensitive to AM, e.g. a wideband PLL, >it'll work better than a limiter at low SNR. Essentially all reasonable >demodulation schemes work equivalently at high SNR, where the AM and PM >fluctuations are linear functions of the additive noise amplitude. > >At low SNR, limiters start suppressing the signal in favour of the >noise. That leads to dropouts--periods during which the signal >disappears and only the noise is left--which really set the threshold >for detection. PLLs using wideband AGC instead of limiting don't >exhibit threshold, although the detected signal does get noisier as the >SNR declines. (You need the AGC to keep the loop bandwidth reasonably >constant.) > >The Plessey scheme can be thought of as a kind of ultrawideband AGC, I >suppose--as long as the compression comes after the narrowest IF filter, >it should be nearly equivalent. > >The only other problem there would be AM-PM conversion. Most kinds of >amplifiers slow down when you rail them, which leads to phase errors. >(The good news for FM is that this needs only a 1-D calibration, instead >of 2-D as in I/Q schemes.) > >Cheers > >Phil Hobbs For FM it's hard to beat the old analog way... discriminator. I used an active filter version on my very first modem design (300 Baud, "muff" coupled to a telephone :-) ...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 |