# Do analog circuits exist that are essentially mathematical functions in DC?

I am curious if a class of analog circuits exist that would take say a 0-5 V input signal and output a mathematical function like a sine function mapping 0-5 V to 0-2pi, or a log, exponent, polynomial? I could see the associated coefficients also being DC analog inputs, say

V_out = V1 * V_in ^ 2 + V2 * V_in + V3


I know op-amps can be configured as differentiators, integrators, adders, multipliers, so one way might be to have a high frequency oscillator working with these components to do a DC-AC-DC type conversion, but this seems overcomplicated.

Analog Devices have papers on the subject, e.g. this one on log amps.

The AD538 is a monolithic real-time computational circuit which provides precision analog multiplication, division, and exponentiation. The combination of low input and output offset voltages and excellent linearity results in accurate computation over an unusually wide input dynamic range. Laser wafer trimming makes multiplication and division with errors as low as 0.25% of reading possible, while typical output offsets of 100 microV or less add to the overall off-the-shelf performance level. Real-time analog signal processing is further enhanced by the device's 400 kHz bandwidth.

It's important to note those limitations. 400kHz bandwidth is not very high compared to doing the same arithmetic in the digital domain. That's one of the reasons why almost all control systems computation is done in digital; the others are better linearity, noise tolerance, thermal stability and power consumption.

• That would be an obvious next concern, though my interest was in combining a microprocessor with these self-contained calculating units that would not use its clock cycles to perform all of these calculations and introduce timing issues. Additionally I am looking at using them for feedback control and was interested in whether the non-discrete nature would always be superior to the discrete stepping of a digital system. – J Collins Feb 15 '17 at 11:42
• You have to analyse these things in terms of noise. Discrete stepping manifests as "quantisation noise". Continuous signals have noise levels. Those errors of 0.25% are almost exactly the same as the quantisation error of an 8-bit system. A 16-bit system could be far better. – pjc50 Feb 15 '17 at 15:15
• @pjc50: That "0.25%" was quoted for multiplication (so there's two logarithms and an antilogarithm performed); minimum 10-bit accuracy. Also, the error bars shrink when the signals do, so that analog accuracy might hold for 4 orders of magnitude different base signal levels. Digitized, that'd take about 23 bits. – Whit3rd Feb 15 '17 at 23:00

Yes, quite a few elementary functions are available in analog circuits. There are accurate amplifiers and summing circuits, which covers linear functions of any available input. Heating of a resistor produces a thermal signal proportional to current-squared, and voltage bias of a semiconductor diode (or transistor) produces an exponential current, and both these effects can have high precision (better than 1%) application. square-law current meter Logarithm converter

It is also possible to use voltage-clipping techniques, with Zener or rectifier diodes, or comparators, to modify a linear response with arbitrarily many 'breakpoints'. The resulting networks are awkward to design, but have, historically, been employed to make sine and other functions. Many compensation schemes use (or used) such tricks to linearize a thermocouple temperature scale.

Do analog circuits exist that are essentially mathematical functions in DC?