Design of Experiment

The purpose of this lab is to design and execute an experiment that is assigned by the professor. Our task was to accurately measure the capacitance of an unknown capacitor. We were able to use any components within our parts kits and any equipment available in our lab. With these tools, there are a few methods available to us for measuring capacitance. One would be to create an LC tank circuit and measure the resonance frequency of the circuit. This was originally our plan. We started calculating what the resonance frequency would be for different values of L and C. We found that with our smallest value of inductance and the maximum frequency achievable by the function generator, we would be unable to measure capacitance that was on the order of magnitude of pF or fF. We were also worried that it was more complicated than it was worth. Because of these reasons, we changed our method to measuring the RC time constant of an RC circuit and using this information to calculate the capacitance. With this method, the measurement circuit is very simple and is shown in Figure 1.

A square wave with a low frequency will be used as an instantaneous change in voltage. Since the change in voltage is happening a few times a second, the oscilloscope is actually constantly averaging the experiment, which makes the measurements more accurate.

The first thing that has to be measured is the source resistance of the function generator. We accomplished this by measuring the peak to peak voltage without a load and then adding a 10 Ohm resistor. Using a voltage divider equation, the source resistance could be calculated. Table 1 shows the data that was collected in order to measure the source resistance

The mean value for the source resistance is 46.364 Ohms. This value of source resistance was used for calculation throughout the duration of the experiment.

Now we measured the capacitance of a capacitor that we knew that value of. We wanted to measure a very small capacitance so we could prove the accuracy of our measurements.

We assembled the circuit in figure 1 and started with a 1kOhm resistor. This resistor was then adjusted in order to create a suitable wave form that was accurately measurable with the oscilloscope, depending on the value of the capacitance.

In order to measure the time constant of an RC circuit, you have to know the upper and lower voltage values. We measured these using the oscilloscope. Figure 2 shows this measurement

Figure 2: Measurement of upper and lower voltage values.

Delta V/e was then calculated and this value was added to the lower voltage value. Using voltage cursors, we found where the curve intersected this voltage value. We then centered this value on the oscilloscope and changed to cursors to time cursors. We measured the time difference between the instantaneous voltage change on the input and when the output was equal to delta V/e + V lower. This time difference is the time constant. Figure 3 shows this measurement being made.

Figure 3: Time Constant Measurement

We repeated this measurement three independent times and averaged the time constant value. Using the equation, C = T/(Rs+R) where T is the time constant, the capacitance is calculated.For the capacitor that we knew the value for we got within 1 % of the expected value which was well within the tolerances set forth by the manufacture which were 5%.

At this point we felt confident in our ability to measure capacitance and received the unknown capacitor. We followed the same procedure that's outline above. The following tables report the data that we collected on the unknown capacitor.

The mean value of capacitance was 463.7 nF. This was the value that we reported to the TA. We learned after the measurement that the expected value is 470 nF. This makes our % error 1.34%. This is good considering that this is lower than many manufacture tolerances for capacitors.

Conclusion

In conclusion, Stephen and I successfully wrote and executed an experiment meant to accurately measure the value of an unknown capacitance. The method that we used was measuring the time constant of an RC circuit with a known resistance. One thing that could increase the accuracy of our measurements would be to use an oscilloscope that can make higher accuracy voltage measurements. The voltage cursors on the oscilloscope had a resolution of 20 mV. If this were smaller than the measurement of the capacitance could be more accurate. Due to the low percent error that we had in measuring an unknown capacitance, I am confident in the procedure that we created.