curve fitting the calibration data of a thermistor voltage...

Curve Fitting the Calibration Dataof a Thermistor Voltage Divider

Gerald Recktenwald

Portland State University

Department of Mechanical Engineering

[email protected]

March 5, 2013

EAS 199B: Engineering Problem Solving

Thermistor Calibration Measurements

To make measurements with an Arduino, the

thermistor is located in the upper leg of a

voltage divider.thermistor

10 kΩ

5V

Analog input

For the calibration measurements, the

thermistor is placed in an insulated thermos,

along with a thermometer and water.

Voltage divider forArduino input

+5V or digital output

analog inputInsulatedCoffee mug

Thermistorprobe

ReferenceThermometer

EAS 199B: Engineering Problem Solving page 1

Calibration Data Set

Data storage

• Repeated readings of the voltage divider are

averaged. The averaged values are displayed

in the Arduino Serial Monitor.

• The displayed data is copied and pasted into

columns of an Excel spreadsheet.

• The first row of the spreadsheet is the

temperature for the data set.

• Columns under the first row are the averaged

raw analog input values.

• After the measurements are complete, the data

in the Excel worksheet is saved as a tab-

delimited text file.

Temperaturein first row

Analog input valuesin other rows

1 2 3 nc...


Analysis of Calibration Data with Matlab

The analysis of the calibration data involves these steps

1. Read the data into Matlab.

2. Plot histograms of the raw readings to determine the variability of the calibration

readings.

3. Compute the mean of the raw readings

4. Curve fit the raw voltage divider readings as a function of temperature

5. Swap the roles of the data to curve fit the temperature as a function of voltage divider

readings.

The quality of the curve fits is assessed with plots of the residuals.


Read the Calibration Data into Matlab

The built-in load function reads an array of plain text data into a matrix

D = load(’thermistor_data.txt’)

D is a matrix. The first row of D is the temperature.

T = D(1,:);

The remaining rows are voltage divider readings

V = D(2:end,:);

Most of the work involves analyzing the data in the new

matrix V.

Temperaturein first row

Analog input valuesin other rows

1 2 3 nc...


Read the Calibration Data into Matlab

Compute the average of each of the columns in V

Vave = mean(V);

Vave is a row vector with the same number of elements as T.

The length (number of elements) in both T and Vave is equal to the number of columns

in the data set.


Plot the Calibration Data

It is easy to plot the mean data.

plot(T,Vave,’o’);

xlabel(’Temperature (C)’);

ylabel(’Analog reading’);

But we are getting ahead of

ourselves. We should first look at

the quality of the data by making

histograms of the readings for each

calibration temperature.0 10 20 30 40 50 60

300

350

400

450

500

550

600

650

700

750

Temperature (C)A

nalo

g re

adin

g

For convenience, the following page lists a Matlab function to load and plot the data.


Plot Calibration Data

function plot_calibration_data(fname)

% plot_calibration_data Plot raw calibration data for the thermistor

% -- Supply a default file name

if nargin<1, fname=’thermistor_data.txt’; end

% -- Load the data and compute summary statistics for each column

D = load(fname);

T = D(1,:); % Temperature values in first row

V = D(2:end,:); % Voltage data in columns from second row until end

Vave = mean(V); % Mean of data in each column

% -- Plot the averaged reading versus the calibration temperature

plot(T,Vave,’o’);

xlabel(’Temperature (C)’); ylabel(’Analog reading (10-bit scale)’);


Histogram of the Calibration Data (1)

It is easy to plot a histogram of the data in

the first column.

hist(V(:,1))

xlabel(’Analog reading (10-bit scale)’)

ylabel(’Number of readings’)

V(:,1) is the vector of data in the first

column of V. The : is a wildcard meaning

all of the row indices.

There appears to be some spread in the

data, but look at the range!739.1 739.2 739.3 739.4 739.5 739.60

2

4

6

8

10

12

14

16

18

20

Analog reading (10−bit scale)

Num

ber

of r

eadi

ngs

The analog readings are integer values. The entire set of readings at this temperature fall

between 739.1 and 739.6. The non-integer values occur because each of the readings in

the data set is an average of 20 readings.



There is no significant variability for the

data in the first column.

Let’s check the other columns (other

temperatures).

739.1 739.2 739.3 739.4 739.5 739.60

2

4

6

8

10

12

14

16

18

20


Num

ber

of r

eadi

ngs

The histogram of the second column is

obtained with

hist(V(:,2))

which produces the plot to the right.

Again, the readings fall in a very narrow of

analog input values.665 665.1 665.2 665.3 665.4 665.50

5

10

15

20

25

30


Num

ber

of r

eadi

ngs



We can automate the creation of histograms with a loop over all columns in V.

First, extract the number of columns for the data set with the built-in size command

nc = size(V,2)

This is nice: our data analysis automatically adjusts to the size – number of columns and

number of rows – of the data set. No mouse clicks involved!

With nc known, the following loop computes histograms for each of the columns in V

nc = size(V,2)

for j=1:nc

figure % Open a new plot window

hist(V(:,j)) % Histogram of data in column j

xlabel(’Analog reading (10-bit scale)’)

ylabel(’Number of readings’)

end



We can combine the histograms in a single window by using the subplot command to

create an array of plots.. The syntax is

subplot(nrow,ncol,p)

where nrow is the number of rows in the array, ncol is the number of columns in the

array, and p is a counter for the plot number in the array.

The diagram to the right shows

three possible layouts of subplots.

All practical combinations of

nrow and ncol are allowed.1

2

subplot(2,1,p)p = 1,2

1

3

5

2

4

6

subplot(3,2,p)p = 1,2,...6

1 2

3 4

nc

nprow

subplot(nprow,2,p)p = 1,2,...nc

••••••



The following code snippet uses the subplot command to layout the histograms for the

columns of V in an nprow × 2 array

nc = size(V,2); % number of columns

nprow = ceil(nc/2); % number of rows for array of subplots

for j=1:nc

subplot(nprow,2,j); % Move to the next subplot

hist(V(:,j)); % Bin the data and plot the histogram

ylim( [0, 40] ); % Set the same y-axis limits in all plots

end

The plot_thermistor_histograms.m function (on the next slide) mass-produces

histograms: one for each column in the data set.



function plot_thermistor_histograms(fname)

% plot_thermistor_histograms Plot histograms of calibration data for thermistor

if nargin<1, fname=’thermistor_data.txt’; end % Default name of data file

% -- Load the data and compute summary statistics for each column

D = load(fname);

T = D(1,:); % Temperature values in first row

V = D(2:end,:); % Voltage data in columns from second row until end

Vave = mean(V); % Mean of data in each column

Vmed = median(V); % Median of data in each column

Vstd = std(V); % Standard deviation of each column

nc = size(V,2); % Number of columns in V

nprow = ceil(nc/2); % Number of rows in the array of histograms

fprintf(’\n T(C) mean(V) median(V) std_dev(V)\n’);

for j=1:nc

subplot(nprow,2,j); % Move to the next subplot

hist(V(:,j)); % Bin the data and plot the histogram

ylim( [0, 40] ); % Set the same y-axis limits in all plots

text(min(V(:,j))+0.1, 33, sprintf(’T = %-6.1f’,T(j))); % label the subplot

fprintf(’%8.1f %8.1f %8.1f %8.2f\n’,T(j),Vave(j),Vmed(j),Vstd(j));

end



Variability at all temperatures the data sets is negligible.

739 739.2 739.4 739.6 739.8 7400

20

40T = 51.9

665 665.2 665.4 665.6 665.8 6660

20

40T = 42.4

578 578.2 578.4 578.6 578.8 5790

20

40T = 32.4

528 528.2 528.4 528.6 528.8 5290

20

40T = 27.1

493 493.5 494 494.5 4950

20

40T = 23.7

470 470.2 470.4 470.6 470.8 4710

20

40T = 21.3

447 447.5 448 448.5 4490

20

40T = 19.0

429 429.2 429.4 429.6 429.8 4300

20

40T = 17.3

386 386.5 387 387.5 3880

20

40T = 13.2

325 325.5 326 326.5 3270

20

40T = 6.4

314 314.5 315 315.5 3160

20

40T = 6.0


Summary data

The plot_thermistor_histograms.m function also prints the mean, median and

standard deviation of each column of data. Remember: each column contains the analog

readings for a given calibration temperature.

plot_thermistor_histograms

T(C) mean(V) median(V) std_dev(V)

51.9 739.3 739.3 0.08

42.4 665.3 665.3 0.08

32.4 578.4 578.4 0.07

27.1 528.7 528.7 0.06

23.7 494.1 494.1 0.07

21.3 470.5 470.5 0.07

19.0 448.1 448.1 0.07

17.3 429.6 429.6 0.08

13.2 386.9 386.9 0.07

6.4 325.9 325.9 0.15

6.0 315.0 315.0 0.07

The next step is to create a curve fit of the analog readings versus the temperature.


Calibration Curve Fit Equation

The calibration process suggests a curve fit of the form V = f(T ): the water

temperature was chosen by mixing warm and cold water supplies, and the voltage output

was recorded with the Arduino.

We could use a polynomial for V (T ).

V = c1Tn

+ c2Tn−1

+ · · · + cn−1T + cn. (1)

However, to control the temperature of the fish tank, we will need T = f(V ).

T = a1Vn

+ a2Vn−1

+ · · · + an−1V + an. (2)


Polynomial Curve Fit in MATLAB

Assume the data is in the T and

Vave vectors.

c = polyfit(T,Vave,2);

Tfit = linspace(min(T),max(T));

Vfit = polyval(c,Tfit);

plot(T,Vave,’o’,Tfit,Vfit,’r--’);

xlabel(’Temperature (C)’);

ylabel(’Analog output’);

0 10 20 30 40 50 60300

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Temperature (C)

Ana

log

outp

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The fit looks good, but we should dig deeper


Residual of the Curve Fit (1)

We often use R2 as an indicator of the fit. There is more information in the residuals

ri = yi − yfit(xi)

where (xi, yi) are the measured data and yfit(x) is the fitting function, e.g., a

polynomial.

ri = yi - yfit(xi)


Residual of the Curve Fit (1)

Given the curve fit evaluated with polyfit, we can compute the residuals in one line

c = polyfit(T,Vave,2); % coefficients of polynomial curve fit: V = f(T)

rfit = Vave - polyval(c,T); % residuals of the fit: r(xi) = yi - f(xi)

Vave is a row vector of the average for each column

polyval(c,T) is a row vector of the curve fit evaluated at each of the measured T values.

Vave - polyval(c,T) is a row vector of residuals: the difference between the measured

(average) V values and the V values predicted by the curve fit.

The fit_thermistor_calibration function puts the curve fit and residual computation

and plot together in one m-file.


Matlab Function for Curve Fit

function fit_thermistor_calibration(npoly,fname)

% fit_thermistor_calibration Polynomial curve fits of calibration data for thermistor

% ... See fit_thermistor_calibration.m for missing code to read data and compute Vave

% -- Perform curve fit

c = polyfit(T,Vave,npoly);

fprintf(’\nCurve fit coefficients\n’); fprintf(’\t%12.7e\n’,c);

% -- Plot the curve fit

Tfit = linspace(min(T),max(T));

vfit = polyval(c,Tfit);

subplot(2,1,1)

plot(T,Vave,’o’,Tfit,vfit,’r--’);

xlabel(’Temperature (C)’); ylabel(’Analog output’);

legend(’Data’,sprintf(’Degree %d poly fit’,npoly),’Location’,’Northwest’);

% -- Evaluate residuals at the known (measured) input values

rfit = Vave - polyval(c,T);

subplot(2,1,2)

plot(T,rfit,’o’);

xlabel(’Temperature (C)’); ylabel(’Residual of the fit’);


Matlab Quadratic Curve Fit

fit_thermistor_calibration(2) produces a quadratic fit and a residual plot

0 10 20 30 40 50 60300

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Temperature (C)

Ana

log

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DataDegree 2 poly fit

0 10 20 30 40 50 60−5

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Temperature (C)

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he fi

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Matlab Quadratic Curve fit

Notice that the residuals appear to have a pattern.

The quadratic fit is missing information in the data.

0 10 20 30 40 50 60300

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Temperature (C)

Ana

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0 10 20 30 40 50 60−5

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Temperature (C)

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Matlab Cubic Curve Fit

fit_thermistor_calibration(3) produces a cubic fit. The residuals appear to be

random When the residuals are random, stop increasing the order of the polynomial.

0 10 20 30 40 50 60300

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Temperature (C)

Ana

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0 10 20 30 40 50 60−4

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Temperature (C)

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curve fitting the calibration data of a thermistor voltage...

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