## Calibration for a Load Cell

let's have a look at our fairly simple

measurement system for our cantilever

beam load cell it's got four strain

gauges on it which have relatively known

resistances typically a hundred or 120

ohms but they'll have some uncertainties

on them and then when we apply a mass M

on that load cell it'll cause those

resistances to change so if we've got

those resistances hooked up in a

Wheatstone bridge then we'll see that

this measured voltage which is going to

be proportional to this excitation

voltage the power supply voltage this

measured voltage will change as these

resistances change and we know that

that's really close to being a linear

relationship but that measured voltage

is going to be really small so we're

gonna take it out and we're gonna run it

into an AI na 125 instrumentation

amplifier and it's gonna have an offset

some pseudo ground value and some gain

that will set with a resistance and that

will increase the magnitude of this

voltage and offset it to produce an

output voltage that's in the readable

range sort of 0 to 5 volts range that we

can read with our Arduino then we're

going to put that into the Arduino and

get an analog read value out of that so

when we run analog read of say a zero if

that's what we've got hooked up to this

will give us an output value so

following through this chain all of

these are linear operations so this a

should be linearly related to the mass

and if we want to understand how that

works we're going to have to track the

the relationships here and how much

uncertainty that adds so in this

relationship here we're going to start

off with the base resistance of all of

these strain gauges so r1 r2 r3 and r4

and that's going to give us four

constants that we have some uncertainty

about so we just make a list over here

of

how many uncertain constants we've added

so there's four of them there now when

we apply this mass those resistances are

going to change and those Delta R's for

all four of those those are going to be

proportional to the mass so the amount

the resistance changes will depend on

the mass but it's also going to depend

on the exact material properties of this

beam the dimensions of this beam and

where exactly these strain gauges are

located on that beam and where they

actually got put not just where they

were designed so we're gonna wind up

with these Delta R 's equal to some

constant that's gonna pull in all of

those details times the mass and the

constant is going to depend on which

strain gauge we're talking about because

some of them are going to be positive

and some are going to be negative and so

we're introducing four more constants

here that we're uncertain about if we

measured this beam really carefully and

figured out exactly how its going to

bend using our solid mechanics we could

pick some good estimates of what those

constants are so those delta r's change

these resistances we've got a linear

relationship to the resistances and we

can measure the output voltage from our

wheatstonebridge that output voltage is

going to depend on the voltage with no

load on it plus some change in voltage

that's going to depend on a variety of

things some Delta V and that Delta V is

going to be a function of well all the

Delta R's

and all of the original ARS and the

excitation voltage and it's a linear

function so it's going to introduce a

constant another set of constants here

so we've added another four there these

ones here we should be able to get that

directly so it doesn't add any

analyzing our bridge with these

resistances and that will give us our

output measured voltage now then we'll

have a small voltage here that we're

going to have to amplify and that

amplification will it should be pretty

simple we should be able to get the

amplified value being just equal to the

offset plus the gain times the measured

value from the bridge here but that

still introduces two more constants that

we have some uncertainty about because

we know approximately what the offset is

and approximately what the gain is

because of the resistor we chose but

there and finally we've already been

working on figuring out how much

uncertainty we've got in turning a

voltage into an analog read value so

this analog read value is going to be

some function of whatever that output

voltage from the amplifier is and

whatever reference voltage we're using

on our Arduino so we've added that one

we've already kept track of but we've

also got to keep track of the

uncertainty in the reference voltage now

four and four is eight two is ten and

one is eleven that's getting pretty big

eleven constants and all of these are

constants that depends on depend on

things that we don't know very

accurately so if we combine the

uncertainty from all of those then we

should wind up with a total uncertainty

and we're probably going to find that

it's pretty large now if we were very

very careful in our manufacturing we

could make that smaller but it would be

difficult and the other thing to keep in

mind is it could be a little hard to

analyze so in our design of this

measurement system working forward we

can make reasonable estimates of what

we'd like these constants to be and

design accordingly pick the right size

of beam for example so it'll Bend about

the right amount and choose a reasonable

excitation voltage so this will be about

right and we can set the offset and the

gain to give something that'll be

reasonable and we've already found that

we can work with the with the Arduino we

direction and come up with reasonable

values and if we then measured what we

did out we should be able to go back in

a reverse direction and turn our analog

value that we measured this integer

value that came out of analog read go

back that way

and get out of mass so we went forward

at the design stage knowing what mass we

had and figuring out how that was

eventually going to turn into an a value

and we made reasonable choices of

magnitudes along the way and introduced

a lot of uncertain constants that we

need to do detailed analysis to figure

out what they are now this forward

calculation and the inverse calculation

that you could do when you measure that

all works beautifully according to

theory provided you know all of the

details of every single component all

the way along but I'd like to take

another approach and this is the

approach we almost always use in

measurement instead of trying to analyze

every step along the way and get it

exactly right let's assume that we did

get the linear relationships we were

looking for that all of this

understanding of how this measurement

works is correct but that we don't know

what some of these constants are and

that we've got the uncertainty about

those constants we could maybe eliminate

all of these constants by simply

calibrating so if I calibrate I've

already built the system so let's

measure a zero and that'll be just

whatever a value comes out of this end

when I have no mass on here so it's the

a value I get when M is equal to zero

and then let's measure for a reference

mass I can take say a five kilo mass and

hang it on the end and I'll get another

value for a I'll call that a reference

and it's whatever came out of my

measurement system when the mass on the

end was equal to the reference mass so

I've now got two points I can do a

calibration based on those two points I

my best estimate based on those two

points for whatever mass I have an

unknown mass in a later measurement my

best estimate is going to be well it

would be zero if a was

equal to a zero so it'll be a minus a

zero times well it's going to be times

the reference mass and in order to get

that to work out right I'm going to have

to divide by a ref minus a zero so a

minus a zero over a ref minus a zero

will go from zero linearly up to one as

I go from a zero up to a ref and if I

multiply it by M ref that'll translate

that fraction that goes from zero to one

into a mass compared to the reference

mast

I've now by doing these two measurements

I've determined the whole relationship

of all of these eleven constants and

pulled them into just these two

constants that I've actually measured in

my calibration so how do I now know what

the uncertainty is I don't know what's

going on with all of these constants I

can test to find the uncertainty

and that would you simply be a matter of

measuring multiple values of mass and

analyze the variation

and typically we'd be looking for two

standard deviations to give us an

indication of the 95 percent uncertainty

understanding this process is important

to design the measurement system doing

this calibration process is way more

effective in reducing the uncertainty

because it doesn't require that you

would know the exact manufacturing

tolerances on all of the devices in your

system so by all means designed from a

physical understanding of how the system

is working but measure based on a

calibration

you