Lab 4 - Charge and Discharge of a Capacitor

Introduction

Capacitors are devices that can store electric charge and energy. Capacitors have several uses, such as filters in DC power supplies and as energy storage banks for 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 physics 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 capacitor. You will study this circuit and ways to change its effective capacitance by combining capacitors in series and parallel arrangements.

Discussion of Principles

A capacitor consists of two conductors 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. This is written algebraically as The charge C is measured in units of coulomb (C), the voltage

ΔV 

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 circuit, a resistor and a capacitor are connected in series with a battery and a switch. See Fig. 1. When the switch is in position 1 as shown in Fig. 1(a), charge on the conductors 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 The solution to is Eq. (2)

ΔV −

Q

C

− R

dQ

dt

= 0. 

is where

Q_f 

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 The solution to Eq. (4)

Q

C

− R

dQ

dt

= 0. 

is where

Q₀ 

represents the initial charge on the capacitor at the beginning of the discharge, i.e., at

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).

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, 

and 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. We can use the definition

(I = dQ/dt) 

of current through the resistor and Eq. (3)

Q = Q_f1 − e^{(−t / RC)} 

and Eq. (5)

Q = Q₀e^{(−t / RC)} 

to get an expression for the current during the charging and discharging processes. where

I₀ =

ΔV0

R

 

in Eq. (8)

charging: I = +I₀e^−t/RC 

and Eq. (9)

discharging: I = −I₀e^−t/RC 

is the maximum current in the circuit at time t = 0. Then the potential difference across the resistor will be given by the following. Note that during the discharging process the current will flow through the resistor in the opposite direction. Hence I and

ΔV 

in Eq. (9)

discharging: I = −I₀e^−t/RC 

and Eq. (11)

discharging: ΔV = − ΔV₀e^−t/RC 

are negative. This voltage as a function of time is shown in Fig. 3. It is useful to describe charging and discharging in terms of the potential difference between the conductors (i.e., "the voltage across the capacitor"), since the voltage across a capacitor can be measured directly in the lab. By using the relationship

Q = C ΔV, 

Eq. (3)

Q = Q_f1 − e^{(−t / RC)} 

and Eq. (5)

Q = Q₀e^{(−t / RC)} 

that describe the charging and discharging of a capacitor can be rewritten in terms of the voltage. Merely divide both equations by

C, 

and the relationships become the following. Note that these two equations are similar in form to Eq. (3)

Q = Q_f1 − e^{(−t / RC)} 

and Eq. 5

Q = Q₀e^{(−t / RC)} 

. The graph of the voltage across the capacitor versus time is shown in Fig. 4 below. By rearranging Eq. (12)

charging: ΔV = ΔV_f1 − e^(−t/RC) 

we get Take the natural log (ln) of both sides of this expression and multiply by –1 to obtain A plot of

−ln((ΔV_f − ΔV)/ΔV_f) 

versus time will produce a straight line graph with a slope of 1/RC. Similarly, for the discharging process, Eq. 13

discharging: ΔV = ΔV₀e^(−t/RC) 

can be rewritten to get Take the natural log (ln) of both sides of this expression and multiply by –1 to obtain A plot of

−ln(ΔV)/ΔV₀) 

versus time will produce a straight line graph with a slope of 1/RC.

Using a Square Wave to Simulate the Role of a Switch

In this experiment, rather than using a switch, we will be using a signal generator that can generate periodic wave forms of varying shapes, like a sine wave, triangular wave, and square wave. Both the frequencies and amplitudes of the wave forms can also be adjusted. Here we will use the signal generator to produce a time-varying voltage, with a square wave form, across the capacitor similar to the one shown in Fig. 5. The output voltage from the signal generator changes back and forth from a constant positive value to a constant zero volts in equal intervals of time t. The time

T = 2t 

is the period of the square wave. During the first half of the cycle, when the voltage is positive, it is similar to the switch being in position 1. During the second half of the cycle, when the voltage is zero, it is the same as the switch being in position 2. So the square wave, which is a DC voltage that is turned on and off periodically, serves as both battery and switch in the setup in Fig. 1 The signal generator allows this switching to be done repeatedly, and it is possible to optimize the data collection by adjusting the frequency of the repetition. This frequency will depend on the time constant of the RC circuit. When the time t is larger than the time constant τ of the RC circuit, the capacitor will have enough time to charge and discharge, and the voltage across the capacitor will be as shown in Fig. 4.

Objective

In this experiment a (computer-emulated) oscilloscope will be used to monitor the potential difference, and thus, indirectly, the charge on a capacitor. The voltage measurements will be used in two different ways to compute the time constant of the circuit. Finally, capacitors will be connected in parallel to examine their equivalent circuit capacitance.

Equipment

Procedure

Please print the worksheet for this lab. You will need this sheet to record your data.

Setting Up the RC Circuit

The RLC circuit board that you will be using consists of three resistors and two capacitors among other elements. See Fig. 6 below. In theory you can, therefore, have different combinations of resistors and capacitors. In this experiment you will use the 33-Ω and 100-Ω resistors and the two capacitors.

Procedure A: Time Constant of the Circuit

We will use the computer to emulate the oscilloscope in this experiment.

Procedure B: Calculation of Capacitance by Graphical Methods

Procedure C: Measuring Effective Capacitance

Capacitance adds directly when capacitors are connected in parallel and inversely when connected in series. This is opposite of the rule for resistors. For capacitors in parallel, the effective capacitance is given by and for capacitors in series, the effective capacitance is

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