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
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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.
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.
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.
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
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
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:
(Compare the sum of series and
parallel resistances in Chapter 5. The
formulae for series and parallel are interchanged from those above.)
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