10.1 CURRENT IN A PURELY CAPACITIVE CIRCUIT
In the manual ‘Fundamentals of
Electricity 1’ the effect of capacitance on the behaviour of the d.c. ‘charging
current’, especially when combined with resistance, was examined. As a d.c. voltage was applied and a charging
current set up, the growing charge on the capacitor increasingly opposed the
applied voltage until the charging current decayed and ceased.
This chapter examines the behaviour
of a capacitive circuit (initially a ‘purely capacitive’ one without
resistance) when an a.c. voltage is applied to it. It will be seen to be apparently very
different from its d.c. behaviour.
This can best be explained, without
resorting to mathematics, in the following way.
FIGURE 10.1
PURELY CAPACITIVE CIRCUIT
Suppose an a.c. charging current,
instead of being the classical ‘sine-wave’ shape, is square shaped - that is,
as the heavy lines in Figure 10.1(a). It
is constant and positive during the first quarter cycle (from A to B); constant
and negative during the second and third quarter cycle (B to C and C to D); and
constant and positive again during the fourth quarter cycle (from D to E).
Then during the period A to B the
capacitor is charging at a constant rate, its charge voltage growing steadily
and opposing the applied voltage (V). Its charge voltage (EC) is
therefore represented by the red
line, increasing uniformly and negatively between A and B to point P.
During the next quarter cycle (B to
C) the charging current has reversed and is constant; it will steadily
discharge the capacitor until at instant C it is again without charge. The red line rises uniformly from point P to
zero.
During the next quarter cycle (C to
D) the charging current is still uniform and negative, so the capacitor will
charge up negatively. This assists the applied voltage, and its
charge voltage is therefore represented by the red line above the axis,
increasing uniformly and positively between C and D to point Q.
Finally, during the fourth quarter
cycle (D to E) the charging current becomes positive again and, being uniform,
discharges the negatively charged capacitor steadily. The capacitor voltage EC (red line) therefore falls uniformly from Q to zero
during this quarter cycle.
The applied voltage is opposed only
by the capacitor’s charge voltage and is therefore equal and opposite to the
red EC curve - it is the
full-line curve V. (Compare the purely inductive case (Chapter
9) where it is only the ‘back-emf’ E
of the inductor which opposes the applied voltage V.)
For this explanation the charging
current wave was assumed to be square-topped.
This is of course not the typical case; normally the current wave would
be a sine curve. If the square-topped
and angled lines of Figure 10.1(a) are ‘rounded off’, implying a gradual rather
than a sudden change, then we approach the sine-wave curves of Figure
10.1(b). The charging current curve I is in heavy line, and the capacitor
voltage curve EC is as
shown in red.
The applied voltage is opposed only
by the capacitor’s charge voltage and is therefore equal and opposite to the
red EC curve - it is the
light full-line curve V as in Figure
10.1(a).
If Figure 10.1(b) is examined
closely, it will be seen that the heavy current curve I is ahead in time of the applied voltage curve V; the current is said to lead the applied voltage (as distinct from
the inductive case where it lags). In
the case described it leads by one-quarter of a cycle, or 90°.
To sum up: if an alternating voltage
is applied to a purely capacitive circuit, the alternating current which will
flow will lead the applied voltage by 90°
Figure 10.2 shows, for comparison,
the currents (dotted curves) which would be caused by applying an alternating
voltage (V) as in curve (a)
(b) to a purely resistive
circuit
(c) to a purely capacitive
circuit.
It is shown in Chapter 8 that the
current in the purely resistive circuit follows the applied voltage exactly -
its peaks, valleys and zeros occur always at the same instants. The current is then ‘in phase’ with the
voltage.
With a purely capacitive circuit
however the current wave leads the
voltage wave in time by one-quarter of a cycle, or 90°. Each current peak occurs 90° before the next voltage
peak. (Compare the purely inductive case
of Chapter 9 where the current wave lags by 90°. Particularly compare Figure
10.2 with Figure 9.2.)
For the situation where the circuit
is not purely capacitive but contains also some resistance, see Chapter
11.
FIGURE 10.2
PURE RESISTIVE AND PURE
CAPACITIVE CURRENTS
10.2 CAPACITIVE REACTANCE
From Figure 10.1(b) it is seen that
the charge EC on the
capacitor is at all times equal and opposite to the value of the applied
voltage V at any given instant. But the charging current I of a capacitor depends on its capacitance C and on the applied voltage V. (Think of a tank filling with water under
pressure. The rate of flow into the tank
will depend on the depth of water already there (its contents at the moment),
and the pressure of the water entering.)
But the value of current depends
also on the frequency with which the alterations take place; for obviously if
the frequency is doubled, the same charge or discharge must take place in half the time, so the value of current
throughout the cycle is doubled.
Therefore I is proportional to
the capacitance (C), to the applied
voltage (V) and to the frequency (f) -that is I µ f x C x V. In fact the actual relationship is:
If it is turned and written:
by comparison with Ohm’s Law for
resistance
the expression is the equivalent in a
capacitive circuit to resistance and, like resistance, is also measured in
ohms. It is called the ‘capacitive
reactance’ of the circuit and has the symbol ‘X’. If necessary to
distinguish it from the inductive reactance (see Chapter 9), the symbol ‘XC’ may be used.
\ Capacitive reactance
where XC is in ohms, f in hertz and C in farads.
10.3 RELATIONSHIP BETWEEN CAPACITIVE AND INDUCTIVE REACTANCE
Chapters 9 and 10 developed the
concept of two kinds of reactance - inductive (XL) and capacitive (XC). Both are measured in ohms and both act to
limit the current, when an a.c. voltage is applied, as if they were resistances
behaving as the resistance in Ohm’s Law.
FIGURE 10.3
CAPACITIVE AND INDUCTIVE
CURRENTS
Their essential difference is that
an inductive reactance draws a current which lags 90° on the applied a.c. voltage, whereas a capacitive reactance draws a
current which leads 90° on the voltage. The two, IL
(red) and IC (blue)
respectively, of different magnitudes, are shown in Figure 10.3 related to the
applied voltage V.
It will be seen that, since each
current wave is displaced 90° either side of the voltage wave, there
is 180° between them - that is, the inductive and capacitive current waves
are exactly opposite to each other in phase - they are said to be
‘anti-phase’. One can be regarded merely
as the negative of the other, or, put slightly differently, XL and XC can be regarded as having opposite signs.
Convention regards inductive reactance
(being far the more common in power engineering) as positive, which makes
capacitive reactance negative. If it
were stated that two circuits had ‘20 ohms’ and ‘-15 ohms’ reactance, it would
be inferred that the former was inductive and the latter capacitive.
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