Measurement of the Time Constant in an RC Circuit

photo of all of the equipment, a function generator, an oscilloscope and a breadboard with resistors and capacitors

In this lab experiment we will measure the time constant τ of an RC circuit via three different methods. In figure 1 we've sketched a series RC circuit.

sketch of the standard RC circuit. A resistor is in series with a capacitor, and a battery can be switched in and out of the circuit
Figure 1 - Diagram of an RC Circuit

When the switch is in position 1, the voltage source supplies a current to the resistor and the capacitor. Charge is deposited on the plates of the capacitor. At first there is very little charge on the plates, however, as time goes on the charge on the plates builds up and the increased voltage across the capacitor will reduce the flow of current through the circuit. We can see this in the following loop equation:

Vo + Vr + Vc = 0
or
Vo - iR - q/C = 0

As q gets larger, i must get smaller to compensate. As time goes on, the current will eventually approach zero. When the switch is moved to position 2, the battery is removed from the circuit, and the charge that has built up in the capacitor flows through the resistor. In this case the equation is:

iR + q/C = 0
or
dq/dt R + q/C = 0

This first order differential equation has a solution in the form of an exponential:
q(t) = qo e(- t / τ)

Where τ = RC. This decaying function is plotted in figure 2:

Sketch of an exponential function, which decays down to zero smoothly
Figure 2 - Exponential Decay

Given the values of R and C in most circuits, it is very hard to "watch" the decay. In this lab we will cheat a little bit, we will connect our RC circuit not to a voltage supply with a switch, but to a function generator that is outputting a square wave. This will act as an "on" and "off" voltage supply hundreds or thousands of times per second. We can then observe the voltage across the circuit on an oscilloscope, and measure τ from there.

Procedure:
Using your oscilloscope, measure the square-wave output from the function generator. Set the peak-to-peak voltage to at least ten volts and position the waveform on the oscilloscope screen in a way that it is easy to measure voltages. As you do this, also verify that the period measured on your oscilloscope is what you would expect from the frequency from the function generator. A common mistake in this lab is to use an uncalibrated timescale.

On your breadboard connect a capacitor and resistor in series. Pick a pair with an RC of 10-4 seconds or smaller. Note however that if you pick a capacitor with a very small capacitance, then the capacitance of the rest of the circuit will dominate your measurement of τ. If your results for method number #1 disagree strongly with methods #2 and #3, then you have ignored the warning about picking a very small capacitance.

Method #1
The first way to measure the τ is to read R & C directly off of the components themselves. Since capacitors typically have uncertainties of ± 20%, what is the uncertainty associated with this measurement?

Method #2
Connect your oscilloscope to measure the voltage across the capacitor. See figure 3. Note that the capacitor should connect to ground, not the resistor. Think about this detail when you make your measurements. If you measure incorrectly, you can ground both sides of the capacitor, in effect removing it from the circuit.

photo of the circuit. Power comes in from the function generator, flows through the resistor, and then the capacitor, back to the function generator. The voltage is measured across the capacitor.
Figure 3 - Photo of RC Circuit

On your scope inspect the voltage across the capacitor. It should look something like Figure 4. Note that you may have to adjust the triggering on your oscilloscope.

sketch of proper waveform, which increases in an approach to a maximum, and then undergoes smooth decay in an exponential fashion.

Same waveform as above, but actual data are plotted.
Figure 4 - Top: Diagram of Voltage Response Bottom: Voltage across the Capacitor on the Oscilloscope

Extend the period of the function generator so that it looks like the capacitor is fully discharging. "Blow up" this section of the graph by changing the time scale, this way you can inspect it more closely. See Figure 5.

Closeup of oscilloscope showing exponential decay
Figure 5 - "Blow Up" of the decaying part of the waveform

Our second method of measuring the time constant will be a "one point" measurement. Since e-1 = 0.368, take the difference between the highest and lowest voltages, multiply this by 0.368, and add it to the lowest voltage. That will be the voltage across the capacitor after one τ. Find this voltage level on your screen and measure how long it took for the voltage across the capacitor to decay to this value. As voltage corresponds to one τ worth of decay, it is a direct measurement of τ. Remember to include an estimate of error in your notebook. Estimate the error the same way you would estimate the measurement error made when using a ruler.

Method #3
The final way of measuring τ is to take data at many points. Make use of the fact that we have digital oscilloscopes by saving the data on to a USB memory stick and importing the data into Excel. If you then calculate a column that is the natural log (ln) of your voltage, you can graph these data against time and get an estimate for τ (actually, -1/τ). Use the computers in lab to get the slope and the error on the slope. Be aware that some of your data are more valuable than others, and delete points that are going to throw off your curve (think about the relative error of some points compared to others).

In your conclusion discuss the RC circuit, compare your values for τ with their uncertainties, and include your thoughts on the best way to measure τ. Note that you conclusion should be decently longer than ones you have written so far for this class, and that if your discussion does not include comments upon the uncertainties in your measurements, it will not be considered complete.