Purpose:

The purpose of this experiment is to investigate voltage variations in the processes of charging and discharging a capacitor. We will demonstrate how variations in the value of resistance and capacitance in an RC circuit effect the charge time of the capacitor. We will also study time constant of a RC circuit. A circuit containing a resistor connected in series with a capacitor is called a RC circuit. When a charged capacitor discharges through a resistance, the potential difference across the capacitor decreases exponentially. The RC time constant is equal to the product of the circuit resistance and the circuit capacitance. It is the time required to charge or discharge the capacitor, through the resistor.

Equipment:

Two resistors
Two capacitors
Power Amplifier
One DMM
Interface box
Computer

Theory:

When a uncharged capacitor is connected across a DC voltage source, the rate at which it charges up decreases as the time passes. At first, the capacitor is easily charged because there is very little charge present on the plates, but as charge accumulates on the plates, the voltage source must do more work in order to move additional charges onto the plates since the plates already have charge of the same sign on them. As a result, the capacitor charges exponentially, quickly at the beginning and more slowly as the capacitor becomes fully charged.

The charge on the plates at any time is given by the equation shown below:

q = q0 (1 - e^ -t/RC) = q0 (1 - e^ -t/τ)

In this equation, q is the maximum charge on the plates and τ is the capacitive time constant. For τ = RC, R is resistance and C is capacitance.

In today’s lab experiment, the charge on the capacitor will be measured indirectly by measuring the voltage across the capacitor being that these two values are directly proportional to each other as seen in the equation: q = CV

At time t = τ = RC, the voltage across the capacitor will have grown to a certain value:

V = V0 (1 - e^-1) = V0 (1 - 1/e) = 0.63 V0

V0 represents the maximum across the capacitor.

When a fully charged capacitor is discharged through a resistor, the voltage across the capacitor decreases with time. This can be seen in the equation:

V = V0 e^-τ/RC = V0 e^-t/τ

After a time t = τ = RC, the voltage across the capacitor has decreased to a value of 0.37 V0. This value is given by:

V = V0 e^-1 = V0/e = 0.37 V0

Computer-Controlled Function Generator:

The power amplifier will be used as a computer-controlled function generator, producing a low-frequency square wave in the lab. The amplitude and frequency of he output signal from the FG to the RC are to be set on the computer screen. To use the output of the power amplifier, connect the load to the banana jacks on the front of it. When it is operating, a green power indicator light on the front panel will light up. It also contains a red current overload light which comes on when ever the maximum current of 1A is exceeded. If it so happens to come on, the output amplitude must be reduced. Always remember to turn off the power supply before making any connections, and do not let the two output terminals of the amplifier touch each other at any time.

Virtual Oscilloscope:

The waveforms from both the FG and the capacitor are displayed on a virtual oscilloscope generated by the computer program.

These are the basic knob functions:

Channel A and B control the position and scale of the green and blue lines.
Volts/div is the scale for the y-axis.
Timebase (ms/div) is the scale for the x-axis.
Move the cursors by clicking the white square.
Move the cursors by dragging them.
Click one of the arrow in the box like icon to make fine adjustments of the cursor positions.

Note: Rp (Rs) is the equivalent resistance of R1 and R2 in parallel (series) connection.
Cp (Cs) is the equivalent resistance of C1 and C2 in parallel (series) connection.

Procedure:

Six measurements are to be done.
Data Set 1: Study the time constant τ (ms) of RC circuit with C1 (~1 µF) and Rp =(~ 5 kΩ) using a square wave with frequency f = 5 Hz and Vo = 4V
Data Set 2: Study the time constant τ (ms) of RC circuit with C1 (~1 µF) and R1 =(~ 10 kΩ) using a square wave with frequency f = 5 Hz and Vo = 4V
Data Set 3: Study the time constant τ (ms) of RC circuit with C1 (~1 µF) and Rs =(~ 20 kΩ) using a square wave with frequency f = 5 Hz and Vo = 4V
Data Set 4: Study the time constant τ (ms) of RC circuit with Cp (~2 µF) and R1 =(~ 10 kΩ) using a square wave with frequency f = 5 Hz and Vo = 4V
Data Set 5: Study the time constant τ (ms) of RC circuit with Cs (~0.5 µF) and R1 =(~ 10 kΩ) using a square wave with frequency f = 5 Hz and Vo = 4V
Data Set 6: Study the time constant τ (ms) of RC circuit with C1 (~1 µF) and R1 =(~ 10 kΩ) at different frequencies (f = 5 Hz, 15 Hz, 30 Hz, 50 Hz) of a square wave with Vo = 4V

Part 1: Preparation
Measure the resistances of R1 and R2 using the DMM and record the values in table 1
Connect R1 and R2 in parallel and measure the equivalent resistance Rp and record it in table 1 to be used for Data set 1
Connect R1 and R2 in series and measure the equivalent resistance Rs and record it in table 1 to be used for Data set 3
Connect C1 and C2 in parallel.
Calculate the capacitance Cp using formula Cp = C1 + C2 and record in table 1.
Connect C1 and C2 in series.
Calculate the equivalent capacitance Cs using formula 1/Cs = 1/C1 + 1/ C2 and record in table 1.
Make connections and shown in the figure in the lab manual.

Part 2: Data Recording
Turn on the computer and the power amplifier
Open the RC time constant lab icon and click run to begin data recording.
Click Data set 1 and follow the instructions on the screen.
Choose a square wave with Vo = 4V, f = 5Hz and START.
Change volts/div and timebase by dragging their pointers so that at least one complete cycle is displayed on the screen.
Click “Read” and move the cursors to read X1 (ms) and Y1 (V) at V = 0 and X2 and Y2 at V = 0.63 Vo
Save data and continue to next measurement.
If you are satisfied with Date Set 1 turn off the power amplifier and make connections for Data Set 2 repeating the above procedure.
If you are satisfied with Date Set 2 turn off the power amplifier and make connections for Data Set 3 repeating the above procedure.
If you are satisfied with Date Set 3 turn off the power amplifier and make connections for Data Set 4 repeating the above procedure.
If you are satisfied with Date Set 4 turn off the power amplifier and make connections for Data Set 5 repeating the above procedure.
If you are satisfied with Date Set 5 turn off the power amplifier and make connections for Data Set 6 repeating the above procedure.
Click Data set 6 and follow the instructions on the screen.
Change frequency to 15 Hz, 30 Hz and 50 Hz to retrieve the rest of the measurements.
Print out a copy for each group member and close down program and power supplies.

Data & Analysis:

Table 1:
R1 |R2 |Rs |Rp |C1 (µF) |C2 (µF) |Cs (µF) |Cp (µF) | |10.51 kΩ |10.09 kΩ |20.60 kΩ |5.15 kΩ |1.01 µF |0.99 µF |2.00 µF |2.00 µF | |
Table 2:

Data Set |f (Hz) |T (ms) |R (kΩ) |C (µF) | τthe (ms) |X1 (ms) |X2 (ms) |τexp (ms) |Y1 (V) |Y2 (V) |Y2 - Y1 | |1 |5.0 |200 |5.15 |1.01 |5.20 |56.38 |61.70 |5.32 |0.01 |2.51 |2.50 | |2 | | |10.51 |1.01 |10.61 |65.43 |76.06 |10.63 |0.01 |2.51 |2.50 | |3 | | |20.60 |1.01 |20.81 |46.81 |67.55 |20.74 |0.01 |2.51 |2.50 | |4 | | |10.51 |2.00 |21.02 |59.04 |79.26 |20.22 |0.01 |2.51 |2.50 | |5 | | |10.51 |0.50 |5.25 |63.83 |68.62 |4.79 |0.01 |2.51 |2.50 | |
Table 3:

Data |V0 (V) |R (kΩ) |C (µF) | τthe (ms) |f (Hz) |T (ms) |T/τthe |y1 - y2 (V) |Wave shape across C |Does V0 = y1 - y2 | |1 |4.0 |10.00 |1.00 |10.00 |5 |0.200 |0.020 |3.93 |Broad & Long |~V0 | |2 | | | | |15 |0.067 |0.007 |3.62 |Broad,Short |No | |3 | | | | |30 |0.033 |0.003 |2.54 |Short, Narrow |No | |4 | | | | |50 |0.020 |0.002 |1.63 |Very Sharp |No | |Questions & Exercises:

The shapes of the waveforms from the capacitor change with the frequency of the input square wave because the actual time required to fully charge or discharge the capacitor can only be changed by changing the value of either the capacitor itself or the resistor in the circuit. The voltage drop across the capacitor alternates between charging up to Vc and discharging down to zero according to the input voltage.
The voltage across a capacitor after 2τ in a charging process would be twice the value of V therefore V would equal 1.26. The voltage across a capacitor after 2τ in a discharging process would be decreased by 2 therefore V =0.63.
With V=V0 e^-t/RC, it mathematically takes infinite time for a capacitor in a circuit to be discharged. In order for a capacitor to be discharged to less than 1% of its initial voltage, It would take 4.6 RC
A 2.0 μf capacitor in a circuit in series with a resistor of 1.0 MΩ is charged with a 6.0 V battery. In order for it to charge the capacitor to 3/4 of its maximum voltage, it would take about 1.4 seconds.
The conclusion that I can draw from the comparison of data set 2 with data set 4 is that the time constant of data set 4 is twice the amount of set 2. This is because the capacitance of data set 4 was set to 2 μf whereas in data set 4 it was set to 1 μf. The resistance of R1 was still kept the same though.
The conclusion that I can draw from the comparison of data set 2 with date set 5 is that dat set 5 was half than that of data set 2. This time, the resistance was still kept the same, but with the circuit being connected in series this time, the capacitance is halved to 0.5 μf make the RC = 5.00.

Conclusion:

Overall this lab experiment ended very successfully. As always we did embark upon some complications when it came to taking the measurements of the data that we needed to collect. Partially because for the data pertaining to data set 6, we did not place the cursors on the tip or the base, therefore we had to repeat that part of the procedure in order to get a more accurate reading. Other than that we did everything else correctly when it came to changing the amount of current and voltage that needed to be inputted into the computer program to read our data. Although it is evident that we did make some minor errors as stated previously we accomplished the original purpose of this lab being that the RC time constant of an RC circuit is indeed equal to the product of the circuit resistance and the circuit capacitance.