RC Time Constant

Purpose:
The purpose of this experiment is to show how a capacitor charges and discharges exponentially. It is also to determine the time constant of an RC circuit.

Equipment: 2 resistors, 2 capacitors, power amplifier, Digital multmeter, interface box, computer

Theory: A capacitor charges exponentially, that is that initially charge accumulates easily on the plates but then it becomes increasingly difficult to keep adding charge to the capacitor. This is because more work is required to add more charge when there is already charges with the same sign already there. The charge on the plates is modelled by the equation q = q0 (1-e -t/RC) = q0 (1-e -t/τ). In the experiment, instead of measuring charge, voltage is measured and then compared to charge with the equation q = CV. Since τ = RC at time t, the voltage across capacitor is a value of 1 – 1/e of V0 which is 0.63 of V0. The discharge of the capacitor is calculated with the equation of V = V0 e-t/RC = V0 e-t/τ. After a certain time, V = V0 e-1 = V0/e = 0.37 V0.

Procedure:
In the first part of the experiment, the first step is to measure the resistance of Resistors 1 and 2, as well as the resistance of the two resistors connected in parallel and in series. Then, the capacitance of capacitors 1 and 2 is measured, as well as the equivalent capacitance of them connected in series and parallel.
In the second part of the experiment, the time constant of an RC circuit is measured in Data set 1-5, and the charging and discharging process in Data set 6. In order to make the RC circuit, connect the power, interface, resistors, and capacitors as is shown in Figure 2. In steps 1-5, maximum voltage (V0) is always 4V and frequency (f) is always 5 Hz. In Data Sets 2,4, and 5, only Resistor 1 (R1) is used. In Data Set 1, the resistors are connected in parallel while in Data Set 3, the resistors are connected in series. In Data Sets 1-3, only capacitor 1 (C1) is used. In Data Set 4, both capacitors are connected in parallel and in Data Set 5 they are connected in series. All of these measurements are measured using a computer and a digital multmeter (DMM). The computer program automatically records and saves each data set. For each Data Set, when the circuit is ready, you click “Start” and then “Read” and then move the cursors so that the value measured is 0.63V0 so that the time constant is being measured.
In Data Set 6, the charging and discharging of the RC circuit is studied by doing four different measurements with different frequencies (5, 15, 30, 50 Hz). This time, instead of measuring the value from 0 to 0.63V0, the whole range is measured as the whole charge and discharge of the capacitor is being measured.

Table 1: R1 | R2 | Rs | Rp | C1 | C2 | Cs | Cp | 10.63 Ω | 10.63 Ω | 21.20 Ω | 5.31 Ω | 1.32 μF | 1.01 μF | 0.57 μF | 2.33 μF |

Table 2: Data Set | f (Hz) | T (ms) | R (kΩ) | C (μF) | τth (ms) | x1 (ms) | x2 (ms) | τexp (ms) | y1 (V) | y2 (V) | y2 – y1 | 1 | 5.0 | 200 | 5.31 | 1.32 | 7.01 | 46.78 | 52.66 | 6.38 | 6.01 | 2.51 | 2.50 | 2 | | | 10.63 | 1.32 | 14.03 | 151.06 | 165.43 | 14.37 | 0.04 | 2.51 | 2.47 | 3 | | | 21.20 | 1.32 | 27.98 | 67.02 | 94.68 | 27.66 | 0.18 | 2.51 | 2.33 | 4 | | | 10.63 | 2.33 | 24.77 | 53.72 | 79.79 | 26.07 | 0.01 | 2.51 | 2.50 | 5 | | | 10.63 | 0.57 | 6.06 | 64.89 | 71.81 | 6.92 | 0.04 | 2.51 | 2.47 |

Table 3 (Data Set 6): Data | V0 (V) | R (kΩ) | C (μF) | τth (ms) | f (Hz) | T (ms) | T/τth | y2 – y1 (V) | Shape of wave across C | Does V0 = y2 – y1? | 1 | 4.0 | 10.00 | 1.00 | 10 | 5 | 88.30 | 8.83 | 3.97 | | Yes | 2 | | | | | 15 | 32.98 | 3.30 | 3.16 | | No | 3 | | | | | 30 | 17.62 | 1.70 | 1.98 | | No | 4 | | | | | 50 | 9.84 | 0.98 | 1.29 | | No |