CHAPTER 9 INDUCTIVE REACTANCE

9.1       CURRENT IN A PURELY INDUCTIVE CIRCUIT

In the manual ‘Fundamentals of Electricity 1’ the effect of inductance on the behavior of the d.c. ‘magnetising current’, especially when combined with resistance, was examined.  As a d.c. voltage was applied and current started to build up, a ‘back-emf’ was induced in the inductor coil, and it opposed the applied voltage in a decaying manner until the current settled down to a steady value given by Ohm’s Law.

In this chapter we shall examine the behavior of an inductive circuit (initially a ‘purely inductive’ 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. behavior.

FIGURE 9.1

CURRENT IN D.C. AND A.C. CIRCUITS

In the upper part of Figure 9.1 a d.c. voltage is suddenly applied by a switch (a) to a resistive circuit, and (b) to an inductive circuit with both resistance and inductance.

In the case of the resistive circuit (a) the current rises immediately to the value determined by Ohm’s Law, and it stays at that value.  In the case of the inductive circuit (b) the current rises fairly slowly, since the applied voltage is overcoming the ‘back-emf’, or inertia, of the system to make the current grow, as explained in ‘Fundamentals of Electricity 1’.  Eventually the current will settle down at the steady d.c. level determined by Ohm’s Law .  At this point the current ceases to grow and there is no more inertia to overcome.

Consider now the bottom two figures, where an a.c. voltage is suddenly applied by a switch (c) to a purely resistive circuit and (d) to a purely inductive circuit.

In the case of the resistive circuit (c) the voltage has only resistance to overcome, and the current wave follows the voltage wave exactly.  As explained in Chapter 8, the amount at any instant is then determined by Ohm’s Law.  Since the current peaks and valleys coincide with the peaks and valleys of the voltage wave, the current is then ‘in phase’ with the voltage.

In the case of the inductive circuit (d) the process needs to be followed rather more carefully.  Suppose the switch is closed at the instant when the voltage wave is at a positive peak.  Because the load is inductive, the first application of voltage will cause the current to rise sluggishly, as it did in case (b).  It will continue to rise in this manner under the positive drive of the voltage wave up to time ‘A’, by which time the voltage wave has fallen to zero.  There is then no more drive, and the current ceases to rise, as shown at point ‘P’.

After this the voltage becomes increasingly negative, opposing the current flow and causing it to reduce.  At time ‘B’ the voltage is at its negative maximum and the current is reducing at its fastest rate, passing through zero and becoming negative.  Between times ‘B’ and ‘C’ the voltage is still negative, so the current continues to become increasingly negative.  At time ‘C’ the voltage wave has returned to zero and the negative drive has gone, so the current wave levels out at a negative peak ‘Q’.

After time ‘C’ the voltage becomes positive again, now opposing the current’s negative flow.  The current therefore becomes less negative until, at time ‘D’, it has returned to zero.  The voltage however continues positive, so the current continues its rise to become positive again.  The conditions at time ‘D’ are the same as they were at the start time (time ‘O’), and the whole cycle begins again.

It can be seen from (d) that the current wave is ‘late’ compared with the voltage wave by one-quarter of a cycle.  It is said to ‘lag’, and, if one cycle is considered as 360°, it lags by 90°.

In the case shown in (d) the circuit was considered purely inductive (i.e. no resistance).  In practice there will always be some resistance, however small.

To sum up: in purely resistive circuits the current will follow the voltage exactly, whether d.c. or a.c.  In the a.c. case moreover the current will be in phase with the voltage.  In inductive circuits however the current will follow the voltage sluggishly.  In the d.c. case it will rise slowly to its final settled value, whereas in the a.c. case the alternating current wave will not rise slowly but will lag by up to 90° on the voltage wave.

Figure 9.2 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 inductive circuit.

FIGURE 9.2

PURELY RESISTIVE AND PURELY INDUCTIVE CURRENTS

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 inductive circuit however the current wave lags on the voltage wave in time by one-quarter of a cycle, or 90°.  Each current peak occurs 90° after the previous voltage peak.

For the situation where the circuit is not purely inductive but contains also some resistance (the general case) see Chapter 11.

9.2          INDUCTIVE REACTANCE

In the absence of any resistance in the circuit, the applied voltage (V) is at any instant opposed only by the back-emf (E) which is being induced in the inductor by the changing current, so that at all times:

V = - E

The back-emf in an inductor depends on its inductance L (in henrys) and on the rate of change of the current through it, namely:

The inductance L is a property of the inductor itself, but the rate of change of current depends directly on both the amplitude of the current itself (I) and on the speed of change.  In an a.c. system that means on the frequency (f), or number of changes per second.  That is to say:

In fact the actual relationship is:

Rewriting this:

  

by comparison with Ohm’s Law for resistance

the expression 2pfL is the equivalent in an inductive circuit to resistance and, like resistance, is also measured in ohms.  It is called the ‘inductive reactance’ of the circuit (or sometimes just ‘reactance’) and has the symbol ‘X’.  If necessary to distinguish it from the capacitive reactance (see Chapter 10), the symbol ‘X_L’ may be used.

                                

where X_L is in ohms, f in hertz and L in henrys.

See also the end of Chapter 10 for certain notes on the relationship between inductive and capacitive reactance.