Square + triangle = sine (almost)

"Harold Larsen"

A triangle wave contains only odd harmonics too.

A "sawtooth" wave contains all integer harmonics.

.... Phil

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

"Ron Tanner" "Phil Allison"

Ever noticed how sine waves are flat topped and pass through zero at a 45 degree angle ?

Not much like your hut with pitched roof wave.......

..... Phil

This reminds of the XR2206 chip that makes square, triangle and sine using analog technology.

Why would you need to simulate a sine wave? It is well characterized in the literature and there are lots of extra ones lying around unused.

You can approximate a sine wave by putting a triangle wave through a circuit that has a hyperbolic tangent shaped transfer function. The following circuit (from "Musical Applications of Microprocessors " by Hal Chamberlin)approximates that function by using the conduction characteristics of two back to back diodes at low currents:

Triangle in 14V pk-pk

o-------o-------------------- .-. | .-. | | - | | 1M | | ^ | | 150 '-' | '-' | | | | | | | | |-+ | | | -o--------o------>|-+ Sine out 1V pk-pk | | | | o---------o .-. | .-. 1M | | | | | | | V | | 150 '-' - '-' | - | ---------|----------- === GND

(created by AACircuit v1.28.6 beta 04/19/05

Though I haven't tried to do it the author claims that with precision components and adjustment the circuit can be adjusted to under 1% harmonic distortion. You could do a similar thing with a differential amplifier or an OTA.

Heh, no that doesn't work.

The usual way to approximate a sine wave is to blunt the sharp tips off a triangle wave with diodes. With enough tweaking it gets very close.

If this circuit is really published the way you drew it, it shows how little a uP guy knows about analogue. The distortion may be even higher than of the triangle wave at the input, and

As pointed out by other participants, you can obtain a sine wave from a triangle wave thanks to a nonlinear transform of the signal. The National Semiconductor application note 263 is worth reading and contains a paragraph dedicated to those techniques:

(see "Approximation Methods" paragraph beginning at page 8)

Hope it helps.

Not in this world it doesn't. Both contain only the odd harmonics but in varying amounts. You get from square to triangle by integrating it. _ _ _ _| |_| |_| |_

A square wave is sum (-1)^(2n+1).sin((2n+1)wt)/(2n+1) n=0 .. inf

When you integrate a square wave you get a triangle wave - usually available off the timing capacitor with a bit of buffering.

/\ /\ /\ \/ \/ \/

The expression for the square wave can be integrated to give:

A triangle wave is sum sin((2n+1)wt)/(2n+1)^2 n=0 .. inf

You could take the linear combination of triangle + square/3 to null out the third harmonic but the waveform would look nothing like a sine wave because of all the other uncancelled higher harmonics.

And the zero crossing would be perpendicular which is not right.

None at all.

A much better way ISTR originally poineered by HP is to take a triangle wave and apply diode shaping to it. First order is to just clip the top off and the next order chamfers the rough edges then a low pass filter.

Neater methods by varying gain with amplitude exist too. Although the neatest of all is probably based on log shaping. Almost all of these tricks have been displaced by direct digital synthesis now.

Natsemi has an app note that reviews sine generation methods that you might find interesting:

And venerable Intersil ICL8038 part that first embodied square, triangle and a sinewave shaper on one chip is still online at

Regards, Martin Brown

Use no filtering and only the triangle waveform: pass thru what 40-50 years ago was called a DFG (diode function generator) at least 16 segments for each polarity; THD result can be quite low (less than 0.5% aka 46dB THD.

Correction: SIX diodes for each polarity, NOT 16 (remembered incorrectly).

You have a crappy "eye"; no cigar - in fact no tobacco!

...and one does not need a movie star to SINE the autograph!

Yep, the circuit is exactly the way it's drawn in the book - the FET is listed as a 2N3819. When I built the circuit I think it gave me a THD of like 10%, so I was wondering what black magic the author was using to get it below 1%. Looking at it more carefully I can understand why, it's an approximation to an approximation - the curve of the hyperbolic tangent function (e^2x -1)/(e^2x + 1) which approximates a sine is itself approximated by an ordinary diode law exponential. I think the reason it was included is that it's cheap: at the time the book was published (1980) OTAs probably cost the equivalent of $10 each and if someone were assembling a "voice per board" type synthesizer with a lot of voices the cost of an OTA and assorted components to make a sine wave for each voice might become prohibitive. In a synthesizer perhaps they figure the signal is just going to be stuffed through a low pass VCF anyhow so the THD is not such a big deal.

I like the idea of using a Taylor series to generate a sine transfer function; what kind of multiplier would you use to raise the input to the 3rd and 5th powers? Some kind of translinear network?

I recall reading about an extension of fT multiplier and analog multiplier (Gilbert cell) type circuits, where you basically join more B-E's and collectors together in the right pattern and ratio of areas to create a sine (or cosine) function approximation directly, by adding up subsequent terms of the Taylor expansion. Cool.

Tim

.-------\ .--| | \ .------\ o--o---+ | X | >----| | \ .---------. Vin | '--| | / | X | >----|-0.543 | | '-------/ .--| | / | |---o | | '------/ .--|+1.543 | | | | '---------' | | | '------------------o-------------' (created by AACircuit v1.28.6 beta 04/19/05

You need only the 3rd power for my above stated accuracy. With the shown coefficients for the subtraction you get 0.25% THD.

I downloaded the paper, but what they call *logarithmic* is IMHO *tanh* and that opamp is not connected very smart either (FIG. 11).

Maybe the news reader is confused, but 'Larsen' just posted as 'Tanner'.

RL