R.C Time Constant
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Abstract
This experiment was split up into two separate investigations. The aim of the first investigation was to calculate the time interval for a capacitor to be discharged to 37% of its original value through a resistor. This result is known as the time constant of the decay, given the Greek letter (t).
From my results I plotted a graph of Exponential decay of voltage against time, the value I obtained in this investigation was
t = 10.10 ± 0.21 seconds
The actual value of t was 10.28s. I obtained this value from the equation t = RC (1). With (R) being the resistance of the resistor and (C) being the capacitance of the capacitor.
The aim of the second investigation was similar to the first investigation, but this time I discharged the capacitor through a multimeter. The results were plotted as a graph of Voltage decay against time and I obtained a value of t = 200 ± 0.21
Introduction
RC circuits are among the most useful, simple and robust passive electric circuits and play integral roles in everyday equipment such as traffic lights, pacemakers and audio equipment. While their applications are numerous and varied, they are mostly used for their signal filtering capabilities and surprisingly precise timing abilities.[1]
The R.C circuits used in this experiment are similar in concept. It is important to know how these circuits work if we are going to improve on their design and efficiency in years to come.
Theory
A capacitor acts as a store for electrical charge. If the capacitor has a capacitance (C) and a voltage (V) is applied across the capacitor, the amount of charge (Q) stored by the capacitor is given by
Q=CV (2)
The current (I) is a measure of the rate of change of charge Q(t) as a function of time as shown in equation (3)
I(t) = Q(t)/RC (3)
After further calculations, it is found that dV(t)/dt = -V(t)/RC (4)
With V(t) being the voltage across the capacitor.
This differential equation has the solution
V(t) = Vo.exp(-t/RC) (5)
(5) can be recast as lnV(t)=ln Vo(-t/RC)
Methodology
The capacitor C is charged to a voltage V by depressing and holding the tapping key. The discharge of the capacitor can be monitored by measuring the voltage across the RC combination.
For investigation 1, I charged the capacitor to its full value. The time taken to charge the capacitor to its full value was noted, and repeated 2 more times to get an average time. I then monitored the D.C voltage across the resistor when the tapping key is closed and the voltage across the capacitor at 1 second time intervals, from one second to 20 seconds inclusive, when the tapping key is released. I repeated this step 2 more times to get an average value of voltage. I also noted the time for the voltage across the capacitor to fall to 10% of its original value. This would give me an estimate of the time scale of the delay. Notes of results were taken and calculations were made.
For investigation 2 I disconnected the resistor and charged the capacitor to its full value again, but this time, I discharged the capacitor through the multimeter. Notes of voltage across the capacitor were taken at 15 second time intervals. This procedure was carried out a further 2 times to get average values of voltage at 15 second time intervals. Notes on how long it took the voltage across the capacitor to fall to 10% of its original value were taken to estimate the time scale of the delay.
Results and Discussions
The capacitor was fully charged at 1.527volts. The time taken for the voltage across capacitor to fall to 10% of its original value is given at 22.37 seconds. For investigation 1, I obtained an average value of resistance of the resistor to be 51400 ohms and the capacitor had the value 20 micro Farads. This would give us the time constant at a value of
t = RC
t = 51400 x 0.00002
t = 10.28 seconds
My own calculation, gave me a value of
t = 10.00 ± 0.22 seconds
This value was obtained from drawing a graph of my results, which is called graph 1.
With Vo being the initial voltage, and Vo being 37% of the initial voltage.
For investigation 2, the resistor was disconnected and the capacitor was discharged through the multimeter. The values I obtained for voltage decay are shown in graph 2, and from the graph, the time constant (t) was given at a value of
t = 200 ± 0.42 seconds
This result could then be used to calculate the