Analog realization of trignometric function

1. Analog realization of trignometric function

After a frantic search to realize sine/cosine functions in analog, I came across the patent reference https://patentimages.storage.googlea.../US4164729.pdf. Can anybody explain , how the linear approximation was arrived? (refer page 10)

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2. Re: Analog realization of trignometric function

Here's a way to approximate the sine function using an analog multiplier (from this paper.).

3. Re: Analog realization of trignometric function

Can you please explain the circuit function

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4. Re: Analog realization of trignometric function

I have a hunch that patent writers tend to omit certain details about an invention, so as not to 'give away the store'.

The article tells how sines and cosines are derived "by means of an approximation which forms the "heart' of the converter." As far as I can tell it's 'almost' but not totally explained in the diagrams and text.

There is a network of weighted resistors but without values labelled.

It's possible the author discovered a mathematical relationship between sines and radians, resulting in a particular waveform which he could apply to achieve more accurate results. Etc.

And perhaps since then the inventor (or another person) found another method to do the same task, or a simpler method. Etc.

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5. Re: Analog realization of trignometric function

Resistor values are listed in the patent article, column 11. One wonders if such precision is needed (example, 8.1523k ohms, 1.9576k ohms).

Anyway these seem to make up the ladder network. Just for speculation I suppose it could shape a ramp wave (linear) into a sinewave. The difference between the two is portrayed below. Resistor values could be determined by calculation and/or experimentation.

6. Re: Analog realization of trignometric function

Few more searches let me to circuit diagram and equation as shown. I checked this in microcap. While it works perfectly for static digital inputs , but on dynamic inputs from counter, the circuit fails to converge. I think that the feedback system needs to be analyzed properly. Expert opinion would be higly appreciated. In the circuit, R1 is 1000 ohms and R2 is 562 ohms( Also checked with 555 ohms)

7. Re: Analog realization of trignometric function

I don't see how your circuit is related to the given rational approximation function. There must be two variable terms (D/A circuits controlled by the angle argument) and a circuit implementing a quotient. If I understand right, in the original tracking converter, the quotient operation is realized implicitly by comparing a numerator and denominator term and varying the angle argument, that's not feasible for direct implementation of the approximation.

8. Re: Analog realization of trignometric function

Consider 10 bit DAC. If Ei is set to 1V DC. Then by varying the digital inputs from 0-1023, Eo varies from from sin(0 to pi/2). By smaller manipulations, it is possible to get the entire sine wave.

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