Depth Study - Charge and Discharge of A Capacitor

Depth Study - Charge and Discharge of a Capacitor
INTRODUCTION
Capacitors 1 are parallel plate devices that can store electric charge and energy. Capacitors have
several uses, such as filters in DC power supplies and as energy storage banks f or pulsed lasers.
Capacitors pass AC current, but not DC current, so they are used to block the DC component of
a signal so that the AC component can be measured. Plasma physics2 makes use of the energy
storing ability of capacitors. In plasma physics short pulses of energy at extremely high voltages
and currents are frequently needed. A capacitor can be slowly charged to the necessary voltage
and then discharged quickly to provide the energy needed. It is even possible to charge several
capacitors to a certain voltage and then discharge them in such a way as to get more voltage
(but not more energy) out of the system than was put in.
This experiment features an RC circuit, which is one of the simplest circuits that uses a capaci-
tor. You will study this circuit and determine the charging and discharging features of a capacitor.
DISCUSSION OF PRINCIPLES
A capacitor consists of two conductive plates separated by a small distance. When the conductors
are connected to a charging device (for example, a battery), charge is transferred from one conductor
to the other until the difference in potential between the conductors due to their equal but opposite
charge becomes equal to the potential difference between the terminals of the charging device. The
amount of charge stored on either conductor is directly proportional to the voltage, and the constant
of proportionality is known as the capacitance 3. This is written algebraically as
Q = C∆V. (1)
The charge C is measured in units of coulomb (C), the voltage ∆V in volts (V), and the capacitance
C in units of farads (F). Capacitors are physical devices; capacitance is a property of devices.
Charging and Discharging
In a simple RC circuit4 , a resistor and a capacitor are connected in series with a battery and a
switch. See Fig. 1.
1
http://en.wikipedia.org/wiki/Capacitor
2
http://en.wikipedia.org/wiki/Plasma (physics)
3
http://en.wikipedia.org/wiki/Capacitance
4
http://en.wikipedia.org/wiki/RC circuit
1
Figure 1: A simple RC circuit
When the switch is in position 1 as shown in Fig. 1(a), charge on the plates builds to a
maximum value after some time. When the switch is thrown to position 2 as in Fig. 1(b),
the battery is no longer part of the circuit and, therefore, the charge on the capacitor cannot be
replenished. As a result the capacitor discharges through the resistor. If we wish to examine the
charging and discharging of the capacitor, we are interested in what happens immediately after the
switch is moved to position 1 or position 2, not the later behavior of the circuit in its steady state.
For the circuit shown in Fig. 1(a), Kirchhoff’s loop equation can be written as
Q dQ
∆V − −R = 0. (2)
C dt
The solution to is Eq. (2) is
h i
Q = Qf 1 − e(−t/RC) (3)
where Qf represents the final charge on the capacitor that accumulates after an infinite length of
time, R is the circuit resistance, and C is the capacitance of the capacitor. From this expression
you can see that charge builds up exponentially during the charging process. See Fig. 2(a).
When the switch is moved to position 2, for the circuit shown in Fig. 1(b), Kirchhoff’s loop
equation is now given by
Q dQ
−R = 0. (4)
C dt
The solution to Eq. (4) is
Q = Q0 e(−t/RC) (5)
where Q0 represents the initial charge on the capacitor at the beginning of the discharge, i.e., at
2
t = 0. You can see from this expression that the charge decays exponentially when the capacitor
discharges, and that it takes an infinite amount of time to fully discharge. See Fig. 2(b).
Figure 2: Change versus time graphs
Time Constant τ
The product RC (having units of time) has a special significance; it is called the time constant
of the circuit. The time constant is the amount of time required for the charge on a charging
capacitor to rise to 63% of its final value. In other words, when t = RC,
Q = Qf 1 − e−1

(6)
and
1 − e−1 = 0.632. (7)
Another way to describe the time constant is to say that it is the number of seconds required for the
charge on a discharging capacitor to fall to 36.8% (e−1 = 0.368) of its initial value.
3
AIM
Conduct an experiment to initially charge a given capacitor and then discharge
the same capacitor in order to be able to plot a graph of voltage vs time and charge vs
time for a charging capacitor and a graph of voltage vs time and charge vs time for the
discharging capacitor. You can determine the charge of the capacitor through the
voltage and charge relationship. In addition, you must calculate the time constant from your
graphs, compare them and describe its significance.
EQUIPMENT
1. Capacitor (4700 uF or equivalent)
2. Resistor (5100 ohms or equivalent)
3. Connecting wires
4. Multimeter (to measure voltage, range 0 - 20 V)
5. Timer/stopwatch (use increments of 10 seconds)
6. DC Power supply (8 or 10 V)
PROCEDURE
RC Charging Circuit
When a voltage source is applied

to an RC circuit, the capacitor, C
charges up through the
resistance, R
RC Discharging Circuit
When a voltage source is

removed from a fully charged RC
circuit, the capacitor, C will
discharge back through the
resistance, R
4
ADDITIONAL REQUIREMENTS
In addition to the requirements of the experiment above, conduct second hand research on
the following regarding capacitors and include it in your final report. You research much show significant
depth of knowledge and application:
1. Function and purpose of capacitors

2. Types of capacitors and their limitations
3. Energy stored in capacitors
4. Dielectrics and capacitance
5. Capacitors in series and parallel
6. Applications of RC circuits
REPORT REQUIREMENTS
1. Introduction / Theory /Aim

2. Risk Assessment
3. Procedure(s)
4. Results
5. Graphs
6. Calculations
7. Discussion and conclusion
8. Additional requirements
9. References (Harvard)
USEFUL LINKS
1. https://youtu.be/X4EUwTwZ110
2. https://youtu.be/ucEiEic-kZ4
3. https://youtu.be/oJ0TsR9g_vk
4. https://youtu.be/ikZkzdFLkKA
5. https://youtu.be/8ZfFrAhraSA
6. https://youtu.be/3IQ-TO6zJgk

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Depth Study - Charge and Discharge of a Capacitor

Charging and Discharging

The solution to is Eq. (2) is

The solution to Eq. (4) is

Figure 2: Change versus time graphs

1 − e−1 = 0.632. (7)

When a voltage source is applied

When a voltage source is

1. Function and purpose of capacitors

1. Introduction / Theory /Aim

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