RC Circuit and its Time Constant                 

When a resistor, capacitor and battery are connected in series, the current is initially large and has a value V_o/R.  As the capacitor gradually charges, the current decreases.   When the capacitor is fully charged (the capacitor voltage is equal to the battery voltage), the current is zero.  The process of the current going from its maximum value to zero is an exponential decay.  We characterize this process by the "time constant" of the circuit.  At the time t = RC the capacitor will be charged up to approximately 2/3 (or 1-1/e exactly) of its final value.  This time is referred to as the time constant of circuit.  The charging process is illustrated in the figure below showing a graph of capacitor voltage versus time.  A graph of the charge on the capacitor would have the same shape since Q = C V.

Time constants tend to be small, and until recently, it was very difficult and very expensive to capture the charging (or discharging) of a capacitor.  Historically an oscilloscope and a square-wave function generator have been used to measure time constants.  The oscilloscope displays a plot of V vs. t, and the repetitive signal from the function generator continually charges and discharges the capacitor which causes the screen to be refreshed so the display will remain bright.  A square-wave is a wave that has a value of Vm for half of its period and zero for the other half of the cycle.  Not shown in the diagram above is what happens when the signal drops to zero for the second half.  The voltage across the capacitor decreases exponentially to (almost) zero and then the process repeats.

In our lab we want to scale the voltage to be either 3 or 6 blocks tall from zero to Vm so that we will have lines on the screen at 2/3 Vm.  It is very important that the voltage be essentially flat in the time before the function drops to zero.  Otherwise, we will have a systematic error in our calculations that cannot be easily corrected for.  The following graph indicates what you need in terms of flatness before the change.  Notice that the corners of the 'square' wave are rounded off.  The graph shows the square-wave displayed at 2.0 V/div and the voltage on the capacitor at 1.0 V/div.  The  function generator is set for 3.0 Vpp, and the voltage on the capacitor has been shifted until the zero level is 2 divisions below the center of the screen.  The smaller square-wave has been shifted up, so that it is still visible, but does not interfere with the measurement of the time constant.

The time constant is the time that elapses between when the capacitor starts charging and the voltage across the capacitor is 2/3 of the voltage from the function generator.  We can use the vertical segment in the square-wave as our t=0 time.  The Horizontal Position can be used to shift both curves right or left - until the vertical line is on a grid line.  If you look at the red curve, you will see that the time constant is about 0.6 divisions multiplied by the Time/Div.  However, the uncertainty is huge.  We want to expand the horizontal scale so that the rise to 2/3 of the final value takes several divisions.  Doing so will greatly reduce the error bars on the resulting time constant.

PROCEDURE:

1. Begin by measuring the resistance of each of your resistors with a digital multimeter (DMM).  Record the nominal (i.e. marked) value and the measured value for each resistor in your lab notebook.  While the instructions will refer to the nominal value of the resistors given by the color code, you should use your measured values of resistance in the calculations.  The accuracy of the DMM is such that the resistor will add a negligibly small uncertainty to the time constant.  The uncertainty in the capacitance is typically 10% for the capacitor as marked by the manufacturer.
2. Connect your circuit as shown in the following diagram and pictorial.  Although the Function Generator is housed within the DSO, it is an electrically separate device and must be connected (except that the ground side of the generator is connected to the ground of the DS0 as would be the case for any instruments with 3-prong ac plugs).

Since Channel 1 always measures the generator voltage, we will use a 'Tee' on the generator and connect one side to Channel 1and the other end to the circuit as shown in the pictorial.  If the cable connected to Channel 2 has a black (ground) alligator clip, fasten it to the rubber of the cable to keep it from inadvertently shorting out part of the circuit.

3. Turn on your oscilloscope and wait for it to finish booting.  You will begin with a 3.00 volt peak-to-peak square waves at a frequency of 500 Hz.  This frequency is just a convenient place to start.   You will vary the frequency as you make the measurements. Set your function generator using the techniques learned in the previous laboratory.  Since you have limited experience with the DSO, the steps will be given in this lab, but not in future labs.

4. Connect one end of a 2.7 x 10⁵ ohm resistor to the Red lead of the cable attached to 'Gen'.  Connect one end of a 0.001 µF  (also known as 1 nF) capacitor to the black lead of that cable.  Place both of the free ends together and hold in place with the Red lead from Channel 2.  You should see traces on your screen.
5. Next you will need to set your timebase (sec/div).  A signal of frequency 500 Hz has a period of 0.002 s or 2 m/s.  If we wish to see several periods displayed, we will need a time/div that is approximately two to three times the period.  A time base setting of 500 us/div will show about 3 cycles with the menu turned off and two cycles with the menu on. 
6. Next we will set the vertical sensitivity and adjust the position of our traces. If the 'CH1' button is not already illuminated, turn on 'Channel 1' by pressing the 'CH1' button.  You should see a yellow trace on the screen.  Adjust your V/div until it is set to 2.00V/div.  You should see a square wave that is 1.5 divisions from peak-to-peak.  The coupling will need to be set to 'DC' and the position adjusted until the square wave is in the upper third of the grid.  If your wave is substantially less than 1.5 divisions, check your wiring for problems.  If you can't find the cause, ask your instructor. Once the square-wave has been completed, press the Ch2 button and set the sensitivity to 1.0 V/div.  Adjust the vertical position until the bottom line is 2 blocks below the center.  If the upper level portion is not 1 block above the center, adjust the Amplitude of the function generator to make the entire waveform 3 divisions tall.  As you change the Amplitude, you will likely have to touch up the vertical position.
7. If you look carefully at the screen it has a slight flicker.  Press the 'Run/Stop' button and the last measurement will be held on the screen. Alternately reduce your Time/div and adjust your horizontal position - keeping the onset of charging the capacitor near the left hand edge of he screen while spreading the graph out.  Carefully measure the time constant as the number of horizontal divisions from the onset of charging until the line 'crosses' the finely divided line across the middle of the screen.  You should note that the trace on the screen is relatively thick and that expanding the graph too much makes it hard to determine where the trace crosses the x-axis.  Record your measurement for the time constant in terms of divisions (along with the sec/div) and the resulting approximate time constant in seconds.  Notice the word approximate.  Using the time to charge to 2/3 of the maximum value makes for better experimental measurements, but causes a systematic error that we will compensate for later.
8. Record your values of R and C, the generator frequency, the number of divisions needed to reach the 2/3 level, and the number of seconds/div in a table in your notebook. Calculate the theoretical time constant as the product RC (with both in SI units).  This number also has an error associated with it due to the inaccuracy of our knowledge of the values of the resistance and capacitance.  The manufacturer of the capacitors guarantees the values to within 10%.   The likely error in the resistances you measured is much smaller than that so we will ignore it.  Thus, the percent error in your theoretical time constant is also 10%.  Calculate the absolute error in your theoretical time constant.  Record your theoretical time constant and %error in your lab notebook. Do your theoretical and experimental values agree to within your errors?
9. Repeat the time constant measurements and calculations for the following combinations of resistors and capacitors: Make sure that you see a shape like that shown in Figure 2- almost flat just before changing level before you take data values. Change the frequency on the generator if necessary.

10. When complete, turn off the function generator output, remove the resistor and capacitor and turn any instruments that you have used (i.e. DMM and DSO) off. 

Correction of a systematic error

The approximation where we used 2/3 of the maximum voltage to find the time constant has a systematic error.  It tends to overestimate the time constant - by the same factor for all measurements as shown in the figure below.

Calculate the quantity 3(1-1/e) [or the quantity 3(1-e^-1) if that is easier to enter into your calculator] and record it in your notebook.  The value should be less than 2.0.  Divide that number by 2.  This is the correction factor that will reduce your approximate experimental time constant to the best experimental value.  Make a table in your notebook with the following columns: R, C, t_approx, t_exp, and t_theory. The theoretical time constant is the product of the measured resistance and the printed capacitance.

When completing your write-up, you may wish to make a graph of Experimental Time Constants versus Theoretical Time Constants.  Ideally, the graph would be a line of slope 1 that passes through the origin.  Any errors due to miscalculation or mistakes in reading would prominently show on the plot. 

Last Modified November, 2014