Capacitors

Unit 9: Capacitors

9.1.1 Capacitor

Conductive plates

Lead

A capacitor is an electrical device which is used to store electrical energy in the same way as the bucket is used for storing water or a tank for storing gas. Each of these devices has fixed capacity which does not depend on the quantity to be stored Capacitors are electronic components designed to store electric charge. Figure 9.1 shows the schematic diagram of parallel plate capacitors. They consist of two conductive plates separated by an insulating material called a dielectric. Who voltage is applied across the plates, one plate accumulates a positive charge while the other accumulates an equal and opposite negative charge, resulting in the storage of electric charge in the capacitor.

The ability of a capacitor to store charge is determined by its capacitance (C), which is measured in farads (F). The capacitance depends on the physical characteristics of the capacitor, such as the area of the plates, the distance between them, and the properties of the dielectric material.

The quantity of charge Q on a capacitor is directly proportional to the potential difference between the plates; that is, Qα n

Where:

Q = CV

(9.1)

Electric Field

Electric Charge

Fie: 9.1

A Palle Plate Capacitor.

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Capacitors are very useful to reduce voltage fluctuations cicctronic power supplies to transmit signals, to detect electro-magnetic oscillations at radio frequencies and to other imporant electronic circuits.

Q = Electric charge stored in the capacitor (measured in coulombs. C)

C = Capacitance of the capacitor (measured in farads, F)

V = Voltage applied across the capacitor (measured in volts, V)

The proportionality constant depends on the shape and separation of Plates. The capacitance of a capacitor is defined as

"The amount of charge required to create a unit potential difference between Parallel plates.

From equation (9.1), the capacitance C of a capacitor can be measured as

Hence, capacitance is the charge stored per unit potential

dee if the charge stored per uni difference.

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Capacitor also used as

sensor

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9.1.2 Types of Capacitor

Capacitors play a very important role in various electrical circuits, especially radio circuit. Here we classify some common types of capacitors.

Variable Capacitors:

This type of capacitor is used to control the capacitance continuously for tuning transmitters and receivers signals.

Film Capacitors:

Film Capacitors consisting of a relatively large family of capacitor with the difference being in Meir dielectric properties. They are available in almost any value and stages as high as 1500 volts. Film capacitors are used in power electronics devices, phase shifters, X-ray flashes and pulsed lasers.

Ceramic capacitor

These capacitors are used in high-frequency circuits ranging from audio to radio frequency. These capacitors are also called disc capacitors.

Electrolytic Capacitors:

These capacitors are generally used when very large capacitance values are required. The majority of electrolytic capacitors are polarized. Electrolytic Capacitors are generally used in DC power supply circuits due to their large capacitance's and small size which help to reduce the ripple voltage.

9.1.3 Capacitance of a parallel plate Capacitor

It consist of two parallel metallic plates of equal area A which are separated by a distance das shown in figure 9.2. Both parallel plates carry charges of magnitude +Q and -Q respectively. The surface charge density (o) is defined as the charge stored (2) per unit area (4).

Yariable capacitor

Filmcapacitor

eramic capacitor

Electroivre Capacitor

Murton's lan

Therefore, the electric field in this case will be

Αερ Το

.... (9.2)

the plates is uniform, the

As the field between the plates is uniform, therefore. the magnitude of the potential difference between the plates equals Ed. i.e., V-Ed, Substituting the value of V from equation (9.1), we get,

Area - A

Centre

A parallel-plate capacitor consists of two parallel conducting plates, each of area A, separated by a distance d.

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Unit 9: Capacitors

V Q

이

E=a= Cd

.. (9.3)

By comparing the equations (9.2) and (9.3), we get relation for capacitance as given below

(9.4)

Equation (9.4) shows that the capacitance of a parallel-plate capacitor is proportional to the area of plates and inversely proportional to the plate separation

Effects of dielectric materials

Experimentally it is found that capacitance of a capacitor can be creased by filling the space between the plates with dielectrics. Dielectrics are electrically insulating materials that in rease the ability of storing charges on the plates of the capacitor. Polythene and axed paper are the examples of dielectrics.

Microscopically the molecules of the dielectric materials become polarized by the charged plates of the capacitor as shown in figure 9.3. This polarization is responsible to increase the capacitance of the capacitor. When the dielectric is inserted between the

plates of a capacitor, then permittivity of Negative charge such capacitor will increase by the amount of or where e, is the relative permittivity of a substance and it is called the dielectric

constant.

Polarized Smolecule

Fig: 9.

Positive charge

A capacitor filled with electric material.

Therefore, the capacitance of such capacitor is given as

E,C

The above relation shows that the charge storing capacity of a capacitor is enhanced by the dielectric which permits it to store &, times more charge for the same potential difference.

NOW SON

...(9.5)

The relative permittivity e, of the material is the ratio of the capacitance of a capacitor with a given materials filling the space between the plates to the capacitance of the same capacitor when space is evacuated. Typical values of &, are presented in table 9.1. ■

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Unit : Capacitors

193 of 327 D

Table No. 9.1: Dielectric Constant (E)

Insulating

Dielectric

S/No.

material

constant

Vacuum

Air

1.0006

Teflon

Mineral Oil

Polyethylene

Waxed paper

Epoxy

PVC

3.7

Nylon

Bakelite

Mica

Water

Worked Example 9.1

A parallel plate capacitor consists of two plates with an area of 0.01 m2 each separated by a distance of 0.002 m. The capacitor is filled with a dielectric material having a relative permittivity (er) of 4. Calculate the capacitance of this capacitor.

Solution:

Step 1: Formula:

C = 4A/d

where:

C is the capacitance,

E is the vacuum permittivity constant (approximately 8.854 x 10-12 Fm), Er is the relative permittiyty (given as 4).

A is the area of one plate (0.01 m2).

d is the separation distance between the plates (0.007 m).

Step 2:

Now, substitute the given values into the formula:

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C = (8.854 x 10-12 F/m) x 4 x (0.01 m2) / (0.002 m)

8.854 x 10

× 4 × (0.01 / 0.002) F

Coesions Entre

C 3.5416 x 10-11 F

So, the capacitance of the parallel plate capacitor is approximately 3.5416 x 10-11 Farads (F).

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Self-Assessment Questions:

1. The capacitance of a capacitor formed by two parallel metal plates each 200 cm2 in area separated by a dielectric 4 mm thick is 0.0004 microfarads. A potential difference of 20,000 V is applied. Calculate the total chappen de plates?

9.1.4 Combinations of Capacitors

Two or more capacitors often are combined in electric circuits. Capacitor

We can calculate the equivalent capacitance of certain combinations with the help circuit diagrama The circuit symbols for capacitors, bateries, and switches are showire figure 9.4.

Parallel Combing on of capacitors

In a parallel comination of capacitors, two or more capacitors are connected side by side with their positive terminals connected together and their negative terminals connected together. This arrangement allows the capacitors to share the same voltage across their terminals while increasing the total capacitance of the combination.

When capacitor are connected in parallel, the total capacitance (C) of the combination is the sum of the individual capacitances

C. C. C), and so on) of the capacitors connected:

C-C+C2+++C

11mbel

Rattery

symbol

Open

Closed

Fig:9.4

Circuit symbols for capacitors.batteries, and switches.

Consider three capacitors are connected in parallel combination as shown figure 9.5 and all three diagrams are equivalent. we know that potential difference cross each capacitors connected in parallel are the same and equal to the potential difference applied across the combination. i.c..

V = = =

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(a) a pictorial representation of three capacitors connected in parallel to a battery

9.5

(b) circuit diagram showing the three capacitors connected in parallel to a battery

(e) circuit diagram showing the equivalent capacitance of the capacitors connected in parallel.

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V = V1 + V2 + V2

To find an expression for equivalent capacitance C use the eq. (9.1) to find the potential difference across cach actual capacitor and equivalent capacitor.

V1 = 2. V2 = 2. V2 = 2 and V = therefore,

we have

Cea GG G3

1 1 1

Cea C1 C2 G

--- (9.7)

This expression shows that the reciprocal of the equivalent capaciance is the algebraic sum of the reciprocal of the individual capacitance. Thus, the equivalent capacitance of a series combination is always less than any individual capacitance.

Self-Assessment Questions:

1. Find the equivalent capacitance of the circuit shown in figure and all capacitors have the

same capacitance of 15pF

2. Two capacitors X and Y are connected in series across a 100 V supply and it is observed that the potential difference across them are 60 V and 40 V respectively. A capacitor of 2 uF capacitance is now connected in parallel with X and the potential difference across Y rises to 90 volts. Calculate the capacitance of X and Y.

9.2.1 Charging and discharging of a capacitor through a resistor

Consider a circuit having a capacitance C and a resistance R which are connected in series with a battery of potential diffrence V through switch as shown in the figure 9.7. This circuit is called RC circuit,

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(a)

(b)

Fig: 9.7 Resistor Capacitator (RC) circuit.

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Unit 9: Capacitors

When the swtich is at position I as shown in Fig. 9.7(a), the capacitor begins to store charge. If at any time during charging, the charge Q flowing through the circuit and Q is the charge on the capacitor, then sum of potential difference between the plates of the capacitor and across the resistor is equal to the potential difference supply-from-the battery. As the current stops flowing when the capacitor is fully charged,When Q-Q (the maximum value of the charge on the capacitor), then

Q-CV

Experiments shows that the charging process of a capacitor exhibits the exponential behaviour therefore we can write its equations

9 Ro(1-e-R)

(8)

where Q, represents the final charge on the capacitor that accumulates. Where the quantity r RC is called the tim constant or the capacitive time constant of the circuit apd.it,has dimensions of time. Farther.

if RC <<1, Q will tain its final value rapidly

if RC 1, it will do so slowly.

Thus, RC determines the rate at which the

capacitor charges (or discharges) itself through a resistance.

If rt, the from equation (9.8)

Q = Q(1-1) = Q(1--)

Q = Q(1

Qo (1-2718) 96 (1-0.368)

Q = 0.6320......(9.9)

Therefore the time constant is the duration of time for the capacitor in which 63.2% of its maximum value charge is deposited on the plates during the charging of the capacitor The equation (9.8) shows that charge builds up exponentially during the charging process (See Fig. 9.8 a).

Charpia C

BA120

0.3680.

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When the switch is moved to position 2, for the circuit which shows the discharging of the charged capacitor (see Fig. 9.8(b). The battery is now out of the circuit and the capacitor will discharge itself through resistor R. This discharging process of the capacitor follows the following equation.

Time (s)

Cocooning Centro

Q = Qoe (9.10)

(9.10)

where Qo represents the initial charge on the capacitor at the beginning of the discharge, i.e., at 1-0. The above expression shows that the charge

Capacitator (a) charging and (b) discharging verses time.

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Unit 9: Capacitors

decays exponentially when the capacitor discharges, and that it takes an infinite amount of time to fully discharge.

If r = RC = t, the equation (9.9) becomes

Q = Qoe-1

2.713

Q = 0.368 Q......(9.1)

Time constant of a RC circuit is thus also the time during which the charge on the capacitor falls from its maximum value to 0.368 of its maximum value.

9.2.2 Energy Stored in Capacitor

Charging of a capacitor always involves some expenditure of engy by the charging agency. This energy is stored in the electrostatic field set up in the dielectric mediu. On discharging the capacitor, the field collapses and the stored energy is released. Many of those who work with electronic equipment have at some time verified that a capacitor can store energy. If the plates of a charged capacitor are connected by a conductor such as a wire, charge moves between each plate and its connecting wire until the capacitor is uncharged.

The discharge can often be observed as a visible spark. If you accidentally ouch the opposite plates of a charged capacitor, your fingers act as a pathway for discharge and the result is an electric shock. The degree of shock you receive depends on the capacitance and the voltage applied to the capacitor. Such a shock could be dangerous if high voltages are present as in the power supply of a home theater system. Because the charges can be stored in a capacitor even when the system is turned off, unplugging the system does not make it safe to open the case and touch the components inside.

Let us consider a capacitor connected to sourge of potential difference Initially, when the capacitor is uncharged, the potential difference between the plates is zero. Finally when charge +Q and Q deposited on the plates, the potential difference between the plates becomes V. The average voltage on the capacitor during the charging process is V/2. See chapter 08 section 8.5,2. The equation (8.7) can be modified for this case

2 Q

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The th which com the

camera when we take photographs is due the energy released from the capacitor.

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During cardiac arrest, heart defibrillator is used to give a sudden surge of a large amount of electrical energy to the patient's chest to retrieve the

normal heart function.

==QV -τον

Q = CV

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Unit 9: Capacitors

U =

CV2 2

.. (9.12)

For a given capacitance, the stored energy increases as the charge and the potential difference increase. In practice, there is a limit to the maximum energy (or charge) that can be stored because, at a sufficiently large value of V, discharge ultimately occurs between the plates. For this reason, capacitors are usually labeled with a maximum operating voltage.

Self-Assessment Questions:

1. A 2200 uF capacitor is charged to a potential difference of 9.0 V then discharged through a 100 k resistor. Calculate (a) the initial charge stored by the capacitor, (b) the time constant of the circuit. (c) calculate the potential difference after a time of 300s equal to the time constant.

An air-capacitor of capacitance 0.005 μF is connected to a direct voltage of 500 V, is disconnected and then immersed in oil with a relative permittivity of 2.5. Find the energy stored in the capacitor before and after immersion.

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32°C Haze

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