How Do You Calculate Sensor Non-Linearity from a Polynomial Equation?

[ilovescience85]

Homework Statement

The input range of a particular sensor is from 0-6 units and it's output is modeled by the equation O = ƒ(I) = 1+2I+0.005I^2-0.00833I^3

Ideally the output should be relate to the input by the straight line equation (ISL) if the form O=kI+a

A) give the law if the ideal straight line for the sensor
B) Plot a graph of the non linearity (N)I, against input I
C) from the graph, determine the maximum non linearity of the sensor, expressed as a percentage if output span
D) Attempt to determine the maximum percentage non linearity by mathematics.

Homework Equations

O = kI+a

The Attempt at a Solution

I not sure where to start here to be honest.
A) O = 1+ 2I + 0.0025I so ideal straight line equation is O= 2.0025I + 1?
B) I know once the above equation is right it's a case of inputting the input range into it to get the graph plot points so in put 0 would give output of 1 and so on but this is all based the correctness of the equation in A.
C) depends on A+B
D) depends on A+B

Any guidance on the above would be greatly appreciated.

 

[elegysix]

I am not familiar with "ideal straight line", but if I had to guess, you should plot the f(I) for 0<I<6, and do regression to find the best fitting line. I would also assume plotting nonlinearity vs input means plot the different between f(I) and the best fitting line at each value of I.

Did your teacher/book explain what is expected for this kind of thing?
I am just guessing here, so I would review what was covered in the course

 

[NascentOxygen]

I suggest that ilovescience85 should ask other students in the class how they are working this problem. Perhaps regression is beyond the course?

Meanwhile, solve the problem assuming the ideal linear characteristic is simply O=1+2I
and you can later speedily rework this if a better-fitting line is intended.

 

[NascentOxygen]

Could you confirm the coefficient of x³.

For your data, Wolframalpha says the straight line of best fit over 0...6 is
O = 0.65 + 2.31 I

Try it: http://m.wolframalpha.com/input/?i=...833x^3},{x,0,6}],a+bx,{a,b},x]&x=0&y=0&js=off

Calculating a line of best fit is probably more complicated than you are expected to use, I'd say.

 

[rezeew]

I would approach this problem by first understanding the given equation and its components. The equation O = ƒ(I) = 1+2I+0.005I^2-0.00833I^3 is a polynomial function that represents the relationship between the input (I) and the output (O) of the sensor. The coefficients 1, 2, 0.005, and -0.00833 represent the contribution of each term to the overall output.

A) The ideal straight line equation for the sensor would be O = kI + a, where k is the slope of the line and a is the y-intercept. In this case, the ideal straight line equation would be O = 2.0025I + 1, as you correctly mentioned.

B) To plot the graph of non-linearity (N)I against input I, you can use the given equation and input values ranging from 0-6 units. This will give you a set of points that can be plotted on a graph, with input I on the x-axis and output N on the y-axis.

C) The maximum non-linearity of the sensor can be determined by finding the maximum deviation of the graph from the ideal straight line. This can be expressed as a percentage of the output span, which is the difference between the maximum and minimum output values. For example, if the maximum output value is 10 and the minimum output value is 2, the output span would be 8. So, if the maximum deviation from the ideal straight line is 1, the maximum non-linearity would be (1/8)*100 = 12.5%.

D) To determine the maximum percentage non-linearity mathematically, you can use the given equation and find the maximum and minimum output values by substituting the input values (0-6 units). Then, you can calculate the difference between the maximum and minimum output values and find the maximum deviation from the ideal straight line. This will give you the maximum non-linearity, which can then be expressed as a percentage of the output span.

Overall, the key to solving this problem is understanding the given equation and using it to find the relationship between the input and output of the sensor. From there, you can plot a graph and use mathematical calculations to determine the maximum non-linearity.