### Author Topic: About 0V, 1V, +, *, identity element and why a multiplier requires calibration  (Read 360 times)

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#### RoGeorge

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##### About 0V, 1V, +, *, identity element and why a multiplier requires calibration
« on: October 13, 2019, 11:43:56 am »
By definition, to measure is to compare with an etalon (AKA standard) unit.

Now, the tale, it all started with a stupid idea, how to get a 1V reference with a multiplier and a comparator:
- take an analog multiplier, and tie together both inputs, as a result, Vout = V2in
- note that for Vin = 1V, Vout would be 1V, too, because 1*1=1
- now, hook a voltage comparator between the input and output of the analog multiplier
- for any input less than 1V (we are talking only positive voltages for now) the output of the multiplier will be less than 1V
- for any input bigger than 1V the output of the multiplier will be bigger than 1V
- if we browse the Vin around the 1V point (e.g. swipe a voltage with a potentiometer), the comparator will switch when we pass over the 1V input point
- in conclusion, we can detect where the 1V is, and ideally, we can get exactly 1V without having a voltage reference.

Obviously, this doesn't work in real life, because there is no such thing like an ideal multiplicator.

A multiplier must to be calibrated, so in a way, the multiplier itself contains a voltage reference inside.  It's even more clear if we look at the measuring units, because Volt*Volt should give Volt2, which is not the same as Volt.  The output of an analog multiplier produces Volts, not square Volts.

Going a little into the math, for why 1*1=1, this property is called the identity element, let's call it e.
By definition, depending on the type of the operation (natural numbers addition, matrices multiplication, etc.) the identity element can be:
- valid only as a left side identity
(e some_operation a) = a
- valid only as a right side identity
(a some_operation e) = a
- valid on both sides, sometimes called neutral element
(e some_operation a) = (a some_operation e) = a

Let's take some examples
- Zero is the identity element for addition:  0 + a = a + 0 = a
- One is the identity element for multiplication:  1 * a = a * 1 = a

If we want to find out a zero, it's simple, and it does not need us to pick an arbitrary ethalon unit.  We can simply use a comparator and symmetry, so by switching the comparator's inputs between them, we can tell if something is "0" without having a Zero Etalon unit.

However, we can not use the same techniques (with a multiplier) in order to identify a "1", a One Etalon Unit, because there is no such thing as a multiplier that does not require calibration.

Why a multiplier requires calibration, while a comparator doesn't?

Other said, why 1 unit (e.g. a Volt, an Ampere, or whatever other "1" unit) requires calibration (so an etalon/standard unit), while 0 doesn't require calibration, and doesn't requires a "0" etalon?

It's some bamboozling between math and physics here, which I can not pinpoint yet.
« Last Edit: October 13, 2019, 11:45:56 am by RoGeorge »

#### dietert1

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##### Re: About 0V, 1V, +, *, identity element and why a multiplier requires calibration
« Reply #1 on: October 13, 2019, 02:18:15 pm »
You already wrote the answer: 0 = - 0 and 1 != -1.
In a so-called analog multiplier the multiplication is achieved by some nonlinear physical process, like current in a pn transition as a function of voltage. That has a built in scale which defines the size of the 1. This scale depends on temperature and the making of the semiconductor. Now, if you put a PT temperature sensor onto the same chip, that is pretty linear in temperature and represents your identity funtion to compare with, you are roughly were you want to go.

When i ask myself what existing device may be implementing your idea: Chip designers use two different nonlinear functions to do it. The usual name is "band-gap voltage reference" and they are used everywhere nowadays.

Regards, Dieter

Smf