# Capacitors and Capacitance

### Electrostatic Field

Figure 44.1 represents two parallel metal plates, A and B, charged to different potentials. If an electron that has a negative charge is placed between the plates, a force will act on the electron tending to push it away from the negative plate B towards the positive plate, A. Similarly, a positive charge would be acted on by a force tending to move it toward the negative plate. Any region such as that shown between the plates in Figure 44.1, in which an electric charge experiences a force, is called an electrostatic field. The direction of the field is defined as that of the force acting on a positive charge placed in the field. In Figure 44.1, the direction of the force is from the positive plate to the negative plate. Such a field may be represented in magnitude and direction by lines of electric force drawn between the charged surfaces. The closeness of the lines is an indication of the field strength. Whenever a p.d. is established between two points, an electric field will always exist.

Figure 44.2(a) shows a typical field pattern for an isolated point charge, and Figure 44.2(b) shows the field pattern for adjacent charges of opposite polarity. Electric lines of force (often called electric flux lines) are continuous and start and finish on point charges; also, the lines cannot cross each other. When a charged body is placed close to an uncharged body, an induced charge of opposite sign appears on the surface of the uncharged body. This is because lines of force from the charged body terminate on its surface.

The concept of field lines or lines of force is used to illustrate the properties of an electric field. However, it should be remembered that they are only aids to the imagination.

The force of attraction or repulsion between two electrically charged bodies is proportional to the magnitude of their charges and inversely proportional to the square of the distance separating them,

This is known as Coulomb’s law.

Hence the force between two charged spheres in air with their centres 16 mm apart and each carrying a charge of C1.6 µC is given by:

Electric Field Strength

Figure 44.3 shows two parallel conducting plates separated from each other by air. They are connected to opposite terminals of a battery of voltage V volts. There is therefore an electric field in the space between the plates. If the plates are close together, the electric lines of force will be straight and parallel and equally spaced, except near the edge where fringing will occur (see Figure 44.1). Over the area in which there is negligible fringing,

where d is the distance between the plates. Electric field strength is also calledpotential gradient.

Capacitance

Static electric fields arise from electric charges, electric field lines beginning and ending on electric charges. Thus the presence of the field indicates the presence of equal positive and negative electric charges on the two plates of Figure 44.3. Let the charge be CQ coulombs on one plate and šQ coulombs on the other. The property of this pair of plates which determines how much charge corresponds to a given p.d. between the plates is called their :

The unit of capacitance is the farad F (or more usually f.LF D 10š6 F or pF D 10š12 F), which is defined as the capacitance when a p.d. of one volt appears across the plates when charged with one coulomb.

For example, the p.d. across a 4 µF capacitor when charged with 5 mC is determined as follows:

Capacitors

Every system of electrical conductors possesses capacitance. For example, there is capacitance between the conductors of overhead transmission lines and also between the wires of a telephone cable. In these examples the capacitance is undesirable but has to be accepted, minimised or compensated for. There are other situations where capacitance is a desirable property.

Devices specially constructed to possess capacitance are called capacitors (or condensers, as they used to be called). In its simplest form a capacitor consists of two plates that are separated by an insulating material known as a dielectric. A capacitor has the ability to store a quantity of static electricity.

The symbols for a fixed capacitor and a variable capacitor used in electrical circuit diagrams are shown in Figure 44.4 The charge Q stored in a capacitor is given by:

Electric Flux Density

Unit flux is defined as emanating from a positive charge of 1 coulomb. Thus electric flux is measured in coulombs, and for a charge of Q coulombs, the flux D Q coulombs.

Electric flux density D is the amount of flux passing through a defined area A that is perpendicular to the direction of the flux:

Permittivity

At any point in an electric field, the electric field strength E maintains the electric flux and produces a particular value of electric flux density D at that point. For a field established in vacuum (or for practical purposes in air), the

where εo is called the permittivity of free space or the free space constant. The value of εo is 8.85 × 1012 F/m.

When an insulating medium, such as mica, paper, plastic or ceramic, is introduced into the region of an electric field the ratio of D/E is modified:

where εr , the relative permittivity of the insulating material, indicates its insulating power compared with that of vacuum:

The insulating medium separating charged surfaces is called a dielectric. Compared with conductors, dielectric materials have very high resistivities. They are therefore used to separate conductors at different potentials, such as capacitor plates or electric power lines.

For example, if two parallel plates having a p.d. of 200 V between them are spaced 0.8 mm apart, then

#### The Parallel Plate Capacitor

For a parallel-plate capacitor, as shown in Figure 44.5(a),

where εo D 8.85 x 10š12 F/m (constant), εr = relative permittivity, A = area of one of the plates, in m2, and d D thickness of dielectric in m.

Another method used to increase the capacitance is to interleave several plates as shown in Figure 44.5(b). Ten plates are shown, forming nine capacitors with a capacitance nine times that of one pair of plates.

If such an arrangement has n plates then capacitance C / (n - 1).

For example, a parallel plate capacitor has nineteen interleaved plates each 75 mm by 75 mm separated by mica sheets 0.2 mm thick. Assuming the relative permittivity of the mica is 5, the capacitance of the capacitor is

given by:

#### Capacitors Connected in Parallel and Series

##### (a) Capacitors connected in parallel

Figure 44.6 shows three capacitors, C1, C2 and C3, connected in parallel with a supply voltage V applied across the arrangement.

When the charging current I reaches point A it divides, some flowing

into C1 , some flowing into C2 and some into C3 . Hence the total charge QT(D I ð t) is divided between the three capacitors. The capacitors each store a charge and these are shown as Q1, Q2 and Q3 respectively.

For example, capacitance’s of 1 µF, 3 µF, 5 µF and 6 µF are connected in parallel to a direct voltage supply of 100 V.

The equivalent capacitance C = C1 + C2 + C3 + C4 = 1 + 3 + 5 + 6 = 15 µF

(b) Capacitors connected in series

Figure 44.7 shows three capacitors, C1, C2 and C3, connected in series across a supply voltage V. Let the p.d. across the individual capacitors be V1, V2 and V3 respectively as shown.

Let the charge on plate ‘a’ of capacitor C1 be CQ coulombs. This induces an equal but opposite charge of šQ coulombs on plate ‘b’. The conductor between plates ‘b’ and ‘c’ is electrically isolated from the rest of the circuit so that an equal but opposite charge of CQ coulombs must appear on plate ‘c’, which, in turn, induces an equal and opposite charge of šQ coulombs on plate ‘d’, and so on. Hence when capacitors are connected in series the charge on each is the same.

(Note that this formula is similar to that used for resistors connected in parallel).

For example, capacitance’s of 3 µF, 6 µF and 12 µF are connected in series across a 350 V supply. The circuit diagram is shown in Figure 44.8.

In practice, capacitors are rarely connected in series unless they are of the same capacitance. The reason for this can be seen from above where the lowest valued capacitor (i.e. 3 µF) has the highest p.d. across it (i.e. 200 V) which means that if all the capacitors have an identical construction they must all be rated at the highest voltage.

For the special case of two capacitors in series:

Dielectric Strength

The maximum amount of field strength that a dielectric can withstand is called the dielectric strength of the material.

Energy Stored in Capacitors

The energy, W, stored by a capacitor is given by:

Practical Types of Capacitor

Practical types of capacitor are characterised by the material used for their dielectric. The main types include: variable air, mica, paper, ceramic, plastic, titanium oxide and electrolytic.

1. Variable air capacitors. These usually consist of two sets of metal plates (such as aluminium), one fixed, the other variable. The set of moving plates rotate on a spindle as shown by the end view of Figure 44.9.

As the moving plates are rotated through half a revolution, the meshing, and therefore the capacitance, varies from a minimum to a maximum value. Variable air capacitors are used in radio and electronic circuits where very low losses are required, or where a variable capacitance is needed. The maximum value of such capacitors is between 500 pF and 1000 pF.

2. Mica capacitors. A typical older type construction is shown in Figure 44.10.

Usually the whole capacitor is impregnated with wax and placed in a bake- lite case. Mica is easily obtained in thin sheets and is a good insulator.

However, mica is expensive and is not used in capacitors above about 0.2 µF. A modified form of mica capacitor is the silvered mica type. The mica is coated on both sides with a thin layer of silver that forms the plates.

Capacitance is stable and less likely to change with age. Such capacitors have a constant capacitance with change of temperature, a high working voltage rating and a long service life and are used in high frequency circuits with fixed values of capacitance up to about 1000 pF.

3. Paper capacitors. A typical paper capacitor is shown in Figure 44.11 where the length of the roll corresponds to the capacitance required. The whole is usually impregnated with oil or wax to exclude moisture, and then placed in a plastic or aluminium container for protection. Paper capacitors are made in various working voltages up to about 150 kV and are used where loss is not very important. The maximum value of this type of capacitor is between 500 pF and 10 µF. Disadvantages of paper capacitors include variation in capacitance with temperature change and a shorter service life than most other types of capacitor.

4. Ceramic capacitors. These are made in various forms, each type of construction depending on the value of capacitance required. For high values, a tube of ceramic material is used as shown in the cross section of Figure 44.12. For smaller values the cup construction is used as shown in Figure 44.13, and for still smaller values the disc construction shown in Figure 44.14 is used. Certain ceramic materials have a very high permittivity and this enables capacitors of high capacitance to be made which are of small physical size with a high working voltage rating. Ceramic capacitors are available in the range 1 pF to 0.1 µF and may be used in high frequency electronic circuits subject to a wide range of temperatures.

5. Plastic capacitors. Some plastic materials such as polystyrene and Teflon can be used as dielectrics. Construction is similar to the paper capacitor but using a plastic film instead of paper. Plastic capacitors operate well under conditions of high temperature, provide a precise value of capacitance, a very long service life and high reliability.

6. Titanium oxide capacitors have a very high capacitance with a small physical size when used at a low temperature.

7. Electrolytic capacitors. Construction is similar to the paper capacitor with aluminium foil used for the plates and with a thick absorbent material, such as paper, impregnated with an electrolyte (ammonium borate), separating the plates. The finished capacitor is usually assembled in an aluminium container and hermetically sealed. Its operation depends on the formation of a thin aluminium oxide layer on the positive plate by electrolytic action when a suitable direct potential is maintained between the plates. This oxide layer is very thin and forms the dielectric. (The absorbent paper between the plates is a conductor and does not act as a dielectric.) Such capacitors must always be used on d.c. and must be connected with the correct polarity; if this is not done the capacitor will be destroyed since the oxide layer will be destroyed. Electrolytic capacitors are manufactured with working voltage from 6 V to 600 V, although accuracy is generally not very high. These capacitors possess a much larger capacitance than other types of capacitors of similar dimensions due to the oxide film being only a few microns thick.

The fact that they can be used only on d.c. supplies limit their usefulness.

#### Discharging Capacitors

When a capacitor has been disconnected from the supply it may still be charged and it may retain this charge for some considerable time. Thus precautions must be taken to ensure that the capacitor is automatically discharged after the supply is switched off. Connecting a high value resistor across the capacitor terminals does this.