CHAPTER 10 CAPACITIVE REACTANCE

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 (E_C) 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 E_C (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 E_C 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 E_C 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 E_C 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 E_C 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 ‘X_C’ may be used.

                    \ Capacitive reactance

where X_C 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 (X_L) and capacitive (X_C).  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, I_L (red) and I_C (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, X_L and X_C 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.

Taking this concept a little further, since the inductive or capacitive loads here considered are regarded as ‘drawing’ currents - that is, taking currents in - a capacitive load (which draws a leading current) could be equally regarded as supplying - that is, giving out - a lagging current.  This idea will not be considered further here, but it is used in the application of power-factor correction, which is dealt with in the manual ‘Electric Motors’.