RC Circuit and its Time Constant

When a resistor, uncharged capacitor and battery are connected in series, the current is initially large and has a value Vo/R.  As the capacitor gradually charges, the current decreases.   The uncharged capacitor has a voltage of zero.  When it is fully charged (the capacitor voltage is equal to the battery voltage), the current is zero.  As the capacitor discharges, the process of the voltage 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 (really 1-1/e or about 63%) of its final value.  This time is referred to as the time constant of circuit.  The charging process is illustrated in Figure 1 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.  Notice how the thick line indicating the time constant rises up almost to  2/3 of the battery voltage.

 Figure 1. Voltage of a charging capacitor showing the time constant.

The discharge of the capacitor is shown in Figure 2.  In this case we are going to approximate the time constant by the amount of time that it takes the voltage to drop to 1/3 of its maximum value. As you can see, this is a slightly longer time, but it offers an advantage of quickly measuring the time constant using the DSO.  It turns out that the approximate time constant can be easily corrected as we will see later.

 Figure 2. Voltage of a discharging capacitor showing the approximate time constant.

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.  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.  The oscilloscope displays a plot of V vs. t, and the repetitive signal from the function generator (changing from zero to a maximum with a constant period) continually charges and discharges the capacitor which causes the screen to be refreshed so the display will remain bright.  Figure 1 is the charging portion of the square wave and Figure 2 is the discharge.  Together they would show one complete cycle.

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 1 divisions below the center of the screen.  The smaller square-wave has been shifted down, 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 in the resulting time constant.

 Figure 3. Necessary shapes to obtain accurate data.

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.
2. Use the capacitance meter that you used earlier in the term to measure the capacitance.  Be sure to adjust the range so that you have 2 or 3 nonzero digits displayed.  Record your values in your notebook.  Be sure to turn the meter off when you are finished.
3. 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).
• Use the 2.7 x 105 ohm resistor and the 0.001 µF  (also known as 1 nF) capacitor for R and C, respectively.
• Connect a BNC to the Gen Out and wire the circuit, making sure the black lead is connected to the capacitor and the red lead to the reisistor.  Use an alligator clip lead to connect the resistor to the capacitor.
• Place the BNC cable that has only one alligator clip on 'Ch 2' and connect the clip to the junction of the resistor and capacitor as shown.  This will be your experimental value channel.
• To measure the voltage being output by the generator (which may not necessarily be the voltage that you set due to possible overloading), connect the short cable with BNC connectors at each end to 'Ch 1' and to the 'Tee' on the generator output.

1. 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 50 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.
 Set up 'Ch 1' and 'Ch 2' and the Triggering Press 'Ch 1' and then 'Coupling'.  If it is not on 'DC' set it to 'DC' and turn the soft menu off. Press 'Ch 2' and then 'Coupling'.  If it is not on 'DC' set it to 'DC' and turn the soft menu off. Set the 'Ch 1' sensitivity to 2 V/div and the 'Ch 2' sensitivity to 1 V/div. Set the timebase to 5ms/division. Press the 'Trigger' button, make sure the 'Source' is 'Ch 1' and 'Mode' is 'Normal' then turn off the soft menu. Set the Function Generator to output to a square wave Press 'Menu' on the function generator Press the 'Soft Key' beside 'Output Type' Rotate the knob to select 'Square' Press and Release the knob. Set the frequency to 50 Hz Press the 'Soft Key' beside frequency Press the arrows to get to the most significant digit Rotate the knob to select 5 press the arrow to the right and set the digit to zero repeat the previous step until the first three digits are 500 (don't worry if there is a decimal point and ignore the other digits for now) Move the arrow to the right until the units below the number are red Rotate the knob until the frequency shown is 50 Hz or 0.0500 kHz Press and Release the knob. Next we will set the amplitude to 3 Vpp Press the 'Soft Key' beside 'Ampl' Follow the same steps as you did for the frequency until the setting shows 300 for the most significant digits.  (Ignore the decimal and any trailing digits) Move the arrow to the right until the units below the number are red Rotate the knob until the display shows 3.00 Vpp Press the 'On/Off' button just above the 'Gen' cable to turn the output on.  You should hear a 'click' and the button will be green.
1.  If the 'Run/Stop' button is Red, press it to make it green.  You should see traces on your screen.  If not, adjust the 'Trigger Level' until you see a display on the screen.
2. 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.  The time base setting of 50 ms/div will show about 3 cycles with the menu turned off and two cycles with the menu on.
3. You should see a yellow trace on the screen.  Check that your V/div for 'Ch 1' is set to 2.00V/div.  You should see a square wave that is 1.5 divisions from peak-to-peak.  The 'Ch 1'  position should be adjusted until the square wave is in the lower 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. Adjust the vertical position of 'Ch 2'  until the bottom line is 1 block below the center.  If the upper level portion is not 2 blocks 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.
4. 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.
5.  In order to get an accurate time constant, you need to spread the signal out more on the screen. You want to keep the onset of discharging the capacitor near the left hand edge of the screen while spreading the graph out.   To do this alternately reduce your Time/div and adjust your horizontal position.
6. Adjust your horizontal position until the vertical line of 'Ch 1' lies on top of one of the grid lines on the screen.
7. Carefully measure the time constant as the number of horizontal divisions from the onset of discharging 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.  Do your best to estimate the location of the center of the trace line to the nearest 0.1 division.
8. 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.
9. Record your values in a table in your notebook like the one below.
 Trial Marked R Marked C Freq # Div Time/Div Approx TC
1. Repeat the time constant measurements and calculations for the test of the trials in the table below. 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.
 Trial Resistor  (ohms) Capacitor Value 1 3.3 x 104 0.001 µF 2 2.7 x 105 100 nF 3 1.5 x 104 0.1 µF 4 2.7 x 103 0.05 µF

The approximation where we used 1/3 of the maximum voltage to find the time constant introduced a systematic error.  It tends to overestimate the time constant - by the ratio of 36.8% to 33.3% for all measurements as shown in the figure below.

To correct for overestimating the time constant we need to multiply the approximate time constant we measured by 0.948.   This is then the best value we can obtain experimentally for the time constant of the circuit.

1. Make a table in your notebook with the following columns:
 Trial Marked R Marked C R meas C meas Theory TC Approx TC Exp TC
1. Complete the table and calculate the theoretical time constants as the product (Rmeas)(Cmeas) with both in SI units and the experimental time constant as 0.948 times the approximate (i.e. measured) time constant.

1. 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.

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.