Experiments with a capacitor

Introduction

A storage oscilloscope enables the voltage/time graph for a capacitor charging through a resistor to be displayed and, from the print-out, a value of the time constant for the circuit to be calculated. This provides all the information required for calculating the value of the capacitor.

Equipment required

• A PC oscilloscope such as the 2205 with PicoScope Education software.
• A capacitor of unknown value (approximately 1 µF for the values used here).
• A battery (a 9 V PP3 was used but any type or value can be used)
• Suitable leads and connectors.

Background theory

When an uncharged capacitor is connected to a DC supply through a resistor, the potential difference across its plates (Vc) increases rapidly at first but then slows down until the value is equal to that of the DC supply (Vs). When Vc = Vs then there is no current flowing through the resistor and the capacitor is fully charged. The time constant of the circuit, t, is defined as the time taken for the value of Vc to reach 63.2% of the final value which can be taken as Vs. We also have that t = RC where R is the value of the resistor in ohms and C is the capacitor value in farads. For a full treatment of the mathematics behind this definition see the Appendix at the end of this note.

Practical considerations

The PicoScope 2205 oscilloscope can give a time axis of almost any value and so the values of R and C can be any convenient value. It should be noted however that the input impedance of the PicoScope 2205 is approximately 1 MΩ and will be connected in parallel with the capacitor. If this input impedance is not to significantly change the maximum value of Vc then the charging resistor should have a value < 1 MΩ by a factor of about 100.

Experiment setup

Arrange the components to give a series RC circuit as shown. Some means of discharging the capacitor between measurements is required and a flying lead will suffice. In our experiment a 1 µF capacitor was used with a 3900 Ω resistor and a 9 V battery.

The PC oscilloscope settings are typically: Timebase 2 ms/div, Channel A input +10 V DC, Trigger Single shot, rising to capture a pulse when capture is started. Use a sensitivity of 150 mV or so at a pre-trigger value of about 5%.

Carrying out the experiment

Discharge the capacitor by shorting across its terminals with the flying lead. Press the space bar on the PC and close the switch. A trace similar to that shown should be obtained. Adjust the timebase settings as required and repeat the run. To repeat a run, open the switch, discharge the capacitor, press the space bar on the PC and close the switch again. Save the screen image.

Calculations

Measurements can either be made from the computer printout or the screen cursors available in PicoScope can be used to determine the maximum value of Vc. Once the time constant value of Vc has been determined the cursors can be repositioned so that the time can be read from the display. Measure the maximum value of Vc reached on the printout. Call this value Vs. Calculate 63.2% of Vs and draw a dotted line across the graph parallel to the x–axis. From the point where the 63.2% line crosses the trace, drop a line down to the x-axis (see diagram above). Calculate the value of t from the graph.

If t = RC then C = t/R.

From the example trace Vs = 9 V and 63.2% is 5.7 V giving a t value of 4.1 ms

Since R = 3900 we get C = 4.1 x 10^–3 / 3900.

Which gives C = 1.05 µF

Questions and discussion of results

1. How does this result compare with the expected value of the capacitor? Is it within the normal 10% tolerance?

Further study

The time constant can be derived from the discharge characteristic as well as the charging one except that the time is calculated from the time taken for Vc to fall to 36.8% of Vs. This is less easy to set up since the triggering is more difficult. It could be used with C values of the order of 1000 μF where the times are several seconds so that manual triggering could be used. The circuit to be used is changed to that shown. The capacitor is first charged by switching to the battery and then discharged using the switch in the other position.

The circuit can be adapted to small C values if the switch and battery are replaced by a square wave generator. This alternately charges and discharges the capacitor giving both characteristics. Adjust the generator frequency and the timebase controls to get the best trace.

If you have any comments or suggestions for improvements please e-mail experiments@picotech.com.

Appendix

Suppose the cell has EMF Vs and the voltages across the resistor and capacitor at any instant are Vc and Vr. As the capacitor charges, the value of Vc increases and is given by Vc = q/C where q is the instantaneous charge on the plates. At this instant (time t) there will be a current I flowing in the circuit.

We know that current is given by the rate of change of charge. So:

We also know that Vs = Vc + Vr and Vc = q/C.

From Ohm’s Law we have that Vr = iR:

Since Vs = Vc + Vr we get:

Rearranged…

This equation can be shown to have the solution:

where Q is the maximum charge i.e. C x Vs.

When t = RC the equation becomes:

q = Q(1 – e^–1)

But:

1 – e^–1 = 1 – 0.368
= 0.632

i.e. the value of q is 63.2% of the maximum value of Q when t = RC.