About 0V, 1V, +, *, identity element and why a multiplier requires calibration - Page 1
Forum section: electronics / general

author: RoGeorge
author_rank: Super Contributor
timestamp: October 13, 2019, 11:43:56 am
content: 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 multipliermust 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 theidentity 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) = aLet'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 = aIf 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.

author: dietert1
author_rank: Super Contributor
timestamp: October 13, 2019, 02:18:15 pm
content: 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