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I have been reading about analog computing yet have not been able to find material about a certain issue; in reference to arithmetic, it is said that op-amps are used to perform these tasks, for example addition with a summing amplifier:

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What about when one wants to sum larger numbers? (>10^5) I have though about say using smaller input voltages and then multiplying them with a high gain but that would require the op-amp to have a higher rail voltage to avoid saturation, as well as potentially loosing accuracy if one wanted to keep the voltages bounded between the rails.

In analog computing, is there are formal method of summing (or any form of arithmetic) large numbers (without working with thousands and thousands of volts) whilst not sacrificing accuracy in the process?

NBoss
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    Just use a different scale factor: 1 V = 1000 units, or whatever. Of course you'll have trouble adding large numbers to small numbers (say 1,000,000 + 1=?), but that's a fundamental issue with analog computing. – The Photon Aug 23 '18 at 02:16
  • Image the scaling needed for solving a differential equation, on an analog computer? Twas a useful exercise. – analogsystemsrf Aug 23 '18 at 03:02
  • @ThePhoton So even the best computers used by NASA and some of the CMOS chips can't perform these operations with accuracy (such as 1,000,000 + 1)? Is there no way in which these devices manage such tasks without needing infinitesimally accurate power supplies? – NBoss Aug 23 '18 at 05:29
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    @NBoss, a digital computer can easily compute 1,000,000 + 1. An analog computer would have trouble doing it. AFAIK, NASA doesn't use analog computers, though they may have done back in the 1960's. AFAIK, nobody is building CMOS chips for the purpose of analog computing on any scale larger than a quad op-amp. – The Photon Aug 23 '18 at 05:31
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    For example, in your summing circuit, the scaling factor of the four inputs are inversely proportional to the resistor values R1, R2, R3. Now look on digikey and see what's the tightest tolerance resistor you can reasonably buy. If one input is 1 uV and another input is 1,000,000 uV, but the resistor values are only +/- 100 ppm, will the output usefully perform the addition you've asked it to? – The Photon Aug 23 '18 at 05:42
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    @NB: "*... without needing infinitesimally accurate power supplies?*" The negative feedback means that the output is independent of the supply to a point. – Transistor Aug 23 '18 at 06:08
  • This is slightly related to [my question](https://electronics.stackexchange.com/questions/373492/analog-analog-multiplication-part-of-a-hybrid-cpu-for-fun) which will show you how troublesome it will become for you. – Harry Svensson Aug 23 '18 at 08:57

2 Answers2

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The high voltage needed problem can be solved by choosing the correct scaling. If you used 1 uV to represent a value of one you can work at scales of 10^6. You would however have problems generating and measuring such fine graduations of voltage.

This assumes that your circuit is ideal. In practice the resistors and opamp have tolerances that will severely impact the accuracy of your results if you are trying to work to this precision.

If you need accurate arithmetic at scales of 10^5 go with a digital representation and adders.

RoyC
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Floating point maths eg a 16 bit FP number, would also have exactly the same issue.

Mathematicians rearrange and reorder equations to move and remove identities and constants, so that you simply do not have to do this kind of computation. With some effort, the big numbers can be separated from the little numbers, and handled separately to be reinserted later, or reduced to constants, that are pre-calculated.

Whereas double-precision floating point has allowed extreme laziness in calculation, analog computing requires the same mathematical effort that one would put in before starting calculation, if you had to do the calculations by hand. Manual calculation was the default for the people who developed analog computing.

Analog computing was not used for financial problems that require that sort of exactitude. Financial calculations are primarily additions, and multiple precision addition was already trivial for human, mechanical, and early electronic computors.

It was to solve engineering problems with real valued numbers, and as we know, an engineer is man who knows that one plus one is close enough to two.

Henry Crun
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