CHAPTER 9 CAPACITANCE

9.1       THE CAPACITOR

A capacitor, or ‘condenser’ as it used to be called, is a device for storing electrostatic energy.  In the early days, when the only sources of electricity were the electrostatic machines such as the Van der Graaf and Wimshurst, electrical energy so created was stored in ‘Leyden Jars’ for future use (Figure 9.1).  For many years the property of capacitance was measured in the unit ‘jar’.

FIGURE 9.1

LEYDEN JAR

The modern condenser or capacitor is the direct descendent of the Leyden jar, and the modern unit of measurement is the ‘farad’ (F).  The farad is however an extremely large unit much too large for practical use, so the unit one-millionth of a farad, or one microfarad (mF), is in general use.  One jar is equal to about one-thousandth of a microfarad (0.001mF or 10^-9 F).

Care is needed to distinguish between the following:

Capacitor:                  the actual device for storing the energy

Capacitance:             the ability to store the energy, measured in mF

The word ‘capacity’ should not be used in this connection.

The ability to store energy is best explained by looking at the construction of a capacitor.  It consists of several parallel metal plates, in flat or cylindrical form, separated from a similar set of metal plates by a thin insulating substance such as glass, mica or paraffin-waxed paper.  These insulating layers are called the ‘dielectric’.  When a potential difference (or voltage) is applied across each set of plates, a strong electric field is set up between them through the thin dielectric.  The closer the plates are together, the stronger the electric field.  This field causes electric strain in the material of the dielectric, causing it to behave like a spring which has been squeezed in a vice.  When an external circuit is provided between the two sets of plates - say by a wire connecting them - this electric spring is released and gives up its energy.

Although the property of capacitance is the principal reason for providing a capacitor, capacitance is found in many other places, often where it is not wanted.  When it exists in this way it is called ‘self-capacitance’.  It is particularly noticeable in cables and overhead power lines, but it is also to be found in machine windings and transformers - in fact anywhere where conductors are arranged close to one another with a thin layer of insulation between.  Self-capacitance exists not only between adjacent conductors but also between conductors and earth or cable-sheath.

9.2       CHARGE AND DISCHARGE OF CAPACITOR

The mechanism of charge can best be understood by considering the mechanical analogy

of Figure 9.2.

FIGURE 9.2

CAPACITANCE - AN ANALOGY

A large, rigid tank is completely full of water which is regarded as incompressible.  Down the middle is a flexible elastic membrane.  One side of the tank is connected through a valve to a water supply under pressure, and the other to suction.

Initially the valve is closed.  Both sides of the membrane are at equal pressure and the membrane is undistorted.  If now the valve is opened and water admitted under pressure, it will flow into the right side of the tank and out from the left side.  The water movement through the tank itself, being over a wide cross-section, will be small compared with the movement of water in the pipes.  As the water in the tank is displaced from right to left, so the membrane becomes distorted to the left and stretches, imposing increasing pressure on the right-hand side.  Eventually, when the distortion is such as to produce a pressure equal to that of the incoming water, the water flow will cease.  A definite volume of water will have entered the tank on the right-hand side, and an equal amount will have departed from the left.  The stretched membrane will be in a state of elastic strain.

The valve can now be closed, leaving the membrane in the position shown.  The right side of the tank is under pressure and the static energy is stored in the stretched elastic membrane.  Although the water can move in either direction through the external piping, in considerable quantity in the case of a large tank, there is no transfer of water within the tank across the membrane.

Dielectric

A (Charge)

FIGURE 9.3

CAPACITOR

An electric capacitor (Figure 9.3) behaves in much the same way.  A d.c. voltage is applied across the two plates of a capacitor by closing battery switch ‘A’, so that one plate is at a higher potential than the other.  The dielectric, which can be regarded as an ‘electrically compressible’ substance, is subject to a strong electric field which puts it into a state of electric strain, just as the stretched membrane was in a state of mechanical strain.

If the battery switch ‘A’ is now opened, the capacitor will be left in that state of strain - it is said to be ‘charged’ - and it will remain so until discharged or until it discharges itself by internal leakage.  Some large oil-filled capacitors have been known to hold their full charge for many months.

The current entering one side and leaving the other side of the capacitor is the ‘charging current’, exactly akin to the water entering one side and leaving the other side of the tank.

If switch ‘B’ is now closed, the two plates are short-circuited together, and the charge on the positive plate is conveyed back to the negative, driven by the dielectric ‘spring’ unwinding.  The stored energy has been released, and the dielectric has relaxed.

The amount of energy that could be stored in the water tank analogy depended on the volume of water and the elastic properties of the membrane.  In the case of a capacitor the amount of energy that can be stored depends on the total plate area, on their distance apart and on the electrical properties of the dielectric used.  Taking these into account, the ability of any given capacitor to store electric energy is called its ‘capacitance’, symbol C.

If the given capacitor has a capacitance of C farads, and a voltage V is applied across it, the amount of electrostatic energy stored is

½CV²   joules

(This should be compared with the magnetic energy ½LI² stored in an inductance (see Chapter 8), or with the kinetic energy ½mv² stored in a moving mass.)

The water analogy showed that passing the water in and out through pipes and under pressure caused the tank to store energy, and on reversal to allow the stretched membrane to relax.  Similarly passing a current into a capacitor ‘charges’ it and causes it to store electric energy; reversing that current discharges it and recovers the energy (Figure 9.4).

FIGURE 9.4

POSITIVE AND NEGATIVE CHARGING

The analogy can be taken a little further.  If the water pump were reversed so that water enters the tank from the left and leaves it from the right, the membrane would simply stretch to the right until its pressure balanced the incoming pressure from the left.  Similarly, if the charging current is reversed (Figure 9.4), the left plate of the capacitor would become positive and the right negative.  The electric field across the dielectric would still be present but reversed in direction and would still be in a state of electric strain.

So it is possible, in general, to charge a capacitor in either direction, and it will store the same amount of energy (depending on the voltage) in either case: this energy is ½CV².

An exception to this statement applies to electrolytic capacitors which, for chemical reasons, may not be reverse-charged.  These capacitors consist of a single spiral aluminium foil coated wth a very thin film of aluminium oxide which acts as the dielectric.  The electrolyte itself (ammonium borate) acts as the second ‘plate’ and makes contact with the metal case.  The very thin dielectric film allows the ‘plates’ to come very close to each other and so to increase the capacitance greatly.  In fact the electrolytic capacitor has a far greater capacitance, size for size, than the conventional type and is now widely used, especially in electronic circuits.

However, any attempt to reverse the polarity will destroy the oxide film, and their application is therefore limited.  The polarity of such capacitors is clearly marked on them to prevent their reverse connection.  One suitable use for them is for smoothing a rectified a.c. circuit, where the d.c. polarity is always maintained.

9.3       TIME CONSTANT

Figure 9.5 shows a capacitor, capacitance C, charging from a d.c. source V through a resistance R.

FIGURE 9.5

VOLTAGE / TIME CURVE

When the switch is closed (Figure 9.5(a)) the capacitor is at first without charge, which means that there is no potential difference, or voltage, across its plates.  At the instant of closing therefore, the full applied voltage V appears across the resistance only, and, by Ohm’s Law, the current I through it is   this flows round the loop and into the capacitor and is therefore the initial charging current of the capacitor.

After a short time (Figure 9.5(b)) the current has produced some charge in the capacitor - suppose it has acquired a voltage E.  This must be in a direction to oppose the applied voltage V and to reduce its effectiveness; it is very similar to the back-emf in an inductance when the current is rising (see Chapter 8).

The effective voltage trying to charge the capacitor is now reduced to (V - E), with E growing all the time.  This is the voltage appearing across the resistor, so the charging current I is falling steadily.  This is once again the classic case of the rate of charge depending on the amount present, which, as in the inductive case, gives an exponential voltage/time curve, as shown in Figure 9.5(c).

The voltage-charge (E) on the capacitor rises exponentially until it eventually equals the applied voltage, at which point E = V, the charging current stops and the charge voltage remains steady at the value V.

The rate of rise of this curve is determined by its ‘time constant’, the meaning of which was explained for an inductive curve in Chapter 8. In the case of a capacitive curve such as that of Figure 9.5, the time constant has the value CR.

FIGURE 9.6

TIME CONSTANT OF A CAPACITIVE CIRCUIT

Figure 9.6 is given the same treatment as Figure 8.4 in the preceding chapter.  A tangent is drawn at the origin, cutting the final steady-voltage line at P.  A perpendicular PN is drawn; ON then represents the time constant of this particular capacitor, and the rate of rise (gradient) of the voltage at the start is  volts per second.

It should be noted that, the higher the resistance R or the capacitance C, the longer the time constant, and this is extensively used to provide time-delay circuits.

FIGURE 9.7

DISCHARGING CAPACITOR

If a capacitor is charged, and is then switched to discharge through a resistance, the voltage would decay exponentially towards zero, and the discharge current would follow the same pattern (Figure 9.7).  The time constant would still be CR and could be arrived at as before by drawing a tangent at t = 0.  The amount of decay at the time equal to the time constant (time P) is again  or about 63%, leaving 37% still to decay.

 

FIGURE 9.8

R-C TIME-DELAY CIRCUIT

Figure 9.8 shows a simple delay circuit for operating some device.  When the switch is closed, the R-C circuit begins to charge at a rate determined by its time constant, the capacitor’s voltage rising all the time.  When it reaches a certain predetermined level, it becomes sufficient to operate a relay which only then switches on the device to be controlled.

If R is made variable, the time constant can be altered at will and so the time delay made adjustable.  This type of delay device is widely used because it consists of simple and cheap elements (a capacitor and a resistor) and no moving parts.

9.4       CAPACITORS IN PARALLEL AND SERIES

The behaviour of capacitors when placed in parallel or series is best explained by considering how the energy is disposed between them.

Parallel

If a voltage V is applied to, say, three parallel capacitors each of capacitance C, then the full voltage is applied to each, and the energy stored by each is ½CV² (Figure 9.9).  The total energy stored by the three is therefore three times this, namely  

	FIGURE 9.9 CAPACITORS IN PARALLEL If C’ is the capacitance of the equivalent capacitor which stores the same total energy, then this energy will be ½C’V2 \ ½C’V2 = 3 CV2 2 or C’ = 3C To generalise, the capacitance of a single capacitor equivalent to n in parallel is n times that of each individual capacitor, assuming that they are of equal value. Series If a voltage V is applied to, say, three series capacitors each of capacitance C, then one-third of the applied voltage will appear across each (Figure 9.10).  Therefore the energy stored by each is The total energy of the three is therefore three times, or  FIGURE 9.10 CAPACITORS IN SERIES If C’ is the capacitance of the equivalent capacitor which stores the same total energy, then this energy will be ½C’V2 \ ½C’V2 = 1 CV2 6 or C’ = 1 C 3 or 1 = 3 1 C’ C To generalise, the reciprocal capacitance of a single capacitor equivalent to n in series is n times the reciprocal capacitance of each individual capacitor, assuming that they are of equal value.  It should be noted that the equivalent capacitance of series capacitors is smaller than that of the individual elements. Summary The total capacitance of capacitors in parallel add directly:                                                 e.g.      C’ =     C1  +    C2    +      C3       +      …. but that of capacitors in series add inversely: e.g. 1 = 1 + 1 + 1 + …. C’ C1 C2 C3 (Compare the sum of series and parallel resistances in Chapter 5. The formulae for series and parallel are interchanged from those above.)