Project 5

# Linear Regression: Internal Resistance of a Test Cell

Chapter 28, Physics for Scientists and Engineers by Fishbane, Gasiorowicz, and Thornton

## INTRODUCTION

A one-meter slide wire potentiometer has been calibrated using a standard cell. This potentiometer has been used to measure the emf of a test cell. The terminal voltage of the same test cell is then measured as different load resistors are connected across the test cell. These data are used to determine the internal resistance of the test cell.

Three sets of data were obtained. For each set, the following values of load resistors, R, were:

• 150, 100, 60, 30, 15, 10, 8, 6, and 4 ohm.

For the three sets of data, the measured voltages, V, were as follows:

• 1.467, 1.466, 1.465, 1.457, 1.442, 1.428, 1.416, 1.401, and 1.369 V.

• 1.261, 1.273, 1.255, 1.251, 1.228, 1.203, 1.183, 1.150, and 1.105 V.

• 1.525, 1.523, 1.517, 1.498, 1.466, 1.446, 1.418, 1.403, and 1.340 V.
A linear least squares analysis using the MATLAB program, p23best1, will yield information about the degree to which these data fit a straight line. The procedure for measuring the voltage across the load resistor and the corresponding circuit diagram is shown in the manual for the experiment P23 The Potentiometer: Internal Reistance of a Test Cell.

## EXERCISES

1. Using the three sets of data, plot graphs with terminal voltage V on the vertical axis and current, I on the horizontal axis.

2. Draw the best fit straight line through each set of data points.

3. Run the program, p23best1, and report the results for the three sets of data shown above.

4. For each case, determine the internal resistance of the test cell from the best fit parameters.

5. Modify the program, p23best1, to calculate the chi-square function for each case. How do these values of chi-square compare with N - M, where N is the number of data points and M is the number of parameters used in the fit.