CHAPTER 8 INDUCTANCE

8.1       WHAT IS INDUCTANCE?

Wherever a magnetic field is produced by an electric current passing through a circuit, that circuit displays the phenomenon of ‘inductance’.

Before looking at the effects of inductance on a d.c. circuit, it will be useful to see what is its nature by looking at a mechanical analogy.

FIGURE 8.1

GRINDSTONE ANALOGY

Suppose there is a large grindstone with a turning handle (Figure 8.1).  It is old, and its bearings are stiff and rusty, giving a lot of friction.  If we try to turn the handle, even slowly, we must overcome this friction, causing heat and loss of energy at the bearings and making ourselves hot with the effort expended.

But there is another type of opposition to our attempts to turn the wheel - its inertia.  It is heavy, and in order to accelerate it we must not only overcome friction but also provide it with an accelerating force in order that it shall gather speed.  The greater the weight or inertia, the greater the force needed to accelerate.  Also, the greater the acceleration desired, the greater the force we must apply. (This is Newton’s Second Law of Motion.)

An electric circuit exhibits the same effects.  It has resistance, and, in order for a current to flow, a pressure in the form of a voltage is needed to overcome it.

But an electrical circuit has inertia too.  It opposes, like the grindstone, any attempt to speed up the current or to cause it to grow.  And the faster it has to grow, the greater the voltage needed to be applied, quite apart from that needed to overcome resistance.  This inertia in an electrical circuit is called ‘inductance’ and is due to the fact that any electric current causes magnetisation, and that effect is greatly increased by the presence of iron (which magnetises easily).

Some circuits, especially those without coils and without iron, have resistance but very little inductance.  They are referred to as ‘resistive circuits’.  Others which have coils, and especially those with iron such as generators, motors and transformers, have both resistance and considerable inductance.  They are referred to as ‘inductive circuits’.  In the fairly rare cases where the resistance is so small that it can be neglected compared with the inductance (say the grindstone with ball bearings) the circuit is called ‘purely inductive’.

How inductance arises in a circuit due to its magnetisation and causes it to display electrical inertia, or ‘sluggishness’, is explained in the following paragraphs.

8.2       THE INDUCTOR (OR ‘CHOKE’)

Faraday’s Law of Electromagnetic Induction, as explained in Chapter 6, states that, if a conductor moves in a magnetic field, an emf (or voltage) is induced in it.  Such movement need only be relative; it is equally true if the magnetic field moves past a stationary conductor.

Movement implies change - that is to say, Faraday’s Law applies also to any conductor around which the field is changing, that is, growing or decreasing.

FIGURE 8.2

INDUCTANCE AND BACK-EMF

Suppose there were a coil of wire through which a current is flowing, as in Figure 8.2.  Then, by Oersted’s principle, there is a magnetic field concentrated along its axis.  If now the current started to change - say to increase - the magnetic field through all the turns of the coil would also be increasing.  This is then a changing field which, by Faraday’s Law, induces in each turn an emf (or voltage), and its direction would be such as to oppose the change - that is, to try to prevent the current in this case increasing.

What happens is shown diagrammatically in Figure 8.2.  A voltage V is applied through a variable resistance R to the coil.  For any given setting of R the current I through the coil (assumed to have no resistance of its own) is given by Ohm’s Law:

I	=	V
		R

If now R is decreased to R’ with a view to increasing the current in the coil, the increasing current gives rise to an induced voltage E in the coil in a direction opposed to V.  This induced voltage E is called the ‘back-emf’ of the coil.  Consequently the net voltage appearing across R’ is no longer V but is now (V - E), and, by Ohm’s Law:

I	=	V – E
		R’

Although R has been reduced to R’, I is not proportionately higher because E reduces the effective voltage.  In other words, Ohm’s Law does not seem to apply in this case.

The back-emf E depends on the rate of change of current  through the coil and on the physical construction, including the number of turns, of the coil.  It is written:

E	=	-L	di
			dt

As stated  . is the rate of change of current (positive if increasing), and L is a property of the coil.  The minus sign indicates that its direction opposes the increasing current, so that E is then negative.

L is called the ‘inductance’ of the coil; for any given coil it is a fixed quantity, but it differs from coil to coil.  The presence of iron in the core increases L considerably.  Inductance is measured in the unit ‘henry’ (H).

A coil carrying electric current, especially one with an iron core, becomes thereby magnetised, and an electromagnet is a store of energy.  The energy stored in a coil of inductance L (in henrys) and carrying a current I (in amperes) is:

½LI² (joules)

(Compare the kinetic energy of a mass m moving with velocity v; it is ½mv².)  If ever the current in the coil is stopped, this energy has to be given up, in one form or another.

8.3       SWITCHING AN INDUCTIVE CIRCUIT WITH RESISTANCE

A special case arises when a voltage is suddenly switched on to a circuit containing resistance R and an inductance L (assumed to have no resistance of its own).  Before the switching no current at all was flowing.  When the switch is closed the current starts to flow and tries to build up, but this change is opposed by a back-emf proportional to the rate of build-up and which reduces the effective voltage to (V - E).

As the current increases, its rate of rise slows down; so therefore does the back-emf E, and the net voltage (V - E) approaches nearer and nearer to V.  Eventually the current levels off, and, since there is now no change, there is no back-emf, and the full voltage V appears across the resistance R, giving the steady current by Ohm’s Law

I	=	V
		R

FIGURE 8.3

SWITCHING ON AN INDUCTIVE CIRCUIT

This type of current rise, shown in Figure 8.3, is known as ‘exponential’ and is found in all branches of physics where the rate of change depends on the amount already present.  In this case, where a voltage is suddenly applied to a circuit containing inductance and resistance, the current rises, not suddenly, but at a reduced rate, or ‘sluggishly’, the rate falling off ‘exponentially’ until it finally settles down at a value given by Ohm’s Law, namely

I	=	V
		R

In the discussion so far the coil has been assumed to be inductive but to have no resistance of its own (L but no R).  In practice of course, all coils must have some resistance, but it is convenient to regard that resistance as separate from the purely inductive coil.

One aspect of this treatment should be realised.  Since the back-emf depends on the rate of change of current   any attempt to stop the current suddenly by opening the switch causes the rate to rise steeply towards infinity, and therefore a very large back-emf would be induced to oppose the change - it would be many times greater than the applied voltage V.  This greatly increased voltage would appear across the open switch contacts (which could be regarded as a resistance of very high ohmic value) and would cause severe sparking or arcing at the switch contacts and possibly voltages dangerous to personnel.

Therefore a d.c. inductive circuit of any size must never be simply broken by a switch. Special precautions must be taken, one of which is shown in Figure 8.4.

FIGURE 8.4

DISCHARGING INDUCTANCE

The inductive coil, which in practice has some resistance of its own (R₁), is shunted by another resistance (R₂).  In normal use the switch is closed and current flows in parallel through both the coil and the shunt resistance, the I²R₂ energy in the latter being wasted as heat.

When the switch is opened, the current already flowing in the coil, instead of being stopped, finds a backward path through R₂ and continues to circulate round the coil and the shunt resistance.  Eventually the stored energy in the coil will be dissipated in heat loss in both R₂ and R₁ (= I²R₂ + I²R₁), and the current will fall exponentially to zero.  The rate of

change, even at the beginning, is therefore quite slow, so the back-emf is also low, and the voltage appearing across the switch contacts is quite small and causes little sparking - it is in fact only equal to the volt-drop IR₂ across the shunt resistance at the start.  The slow decay of current in the coil may however delay the release of whatever mechanism the coil is driving, such as the opening of a solenoid-operated valve.

A shunt resistance used in this way is often called a ‘discharge resistance’ because it discharges and dissipates the energy stored in the coil and reduces contact sparking.  The greater its ohmic value the quicker the discharge of energy (I²R₂), but the greater the ‘spark voltage’ (IR₂) appearing across the switch contacts.  It is always necessary to make a compromise, taking into account also the time delay for the coil’s discharge.

8.4       TIME CONSTANT

It has been shown that, when an inductive circuit is switched on, the rate of build-up of current is relatively slow, and it slows down still more in an exponential curve as the steady-state condition, given by Ohm’s Law, is approached.  It is necessary to quantify such exponential curves so as to distinguish between fast and slow changes.  For this purpose a ‘time constant’ is used.

There is a general, but wrong, idea that the time constant is a measure of the time taken for the current (or other quantity) to settle down, but a moment’s thought will show that, in theory, it never quite reaches its steady state, and it is therefore not possible to pin-point any instant in time at which it has done so.

FIGURE 8.5

TIME CONSTANT OF AN INDUCTIVE CIRCUIT

Figure 8.5 shows a typical exponential rising curve of, for example, a current after switch-on.  If the circuit consists of an inductance L (henrys) and a total resistance R (ohms), it can be shown mathematically that the initial rate of rise (i.e. the slope at time t = 0) when a voltage V is applied is equal to .amperes per second.

If the tangent is drawn at this point and extended to cut the final steady-state line at a point P, and if a perpendicular PN is dropped from P to the time axis, then the time represented by ON is called the ‘time constant’ of that curve. It is expressed in seconds (or milliseconds).

It can also be shown mathematically that for any exponential curve at all the value of current at point Q - that is, after a time equal to the time constant - is  of its ultimate value.  ‘e’ is the ‘exponential number’ equal to 2.718, so that at Q the current has risen to 63% of its ultimate steady value (not, it should be noted, 100%), no matter how fast or slow it is rising.

The time constant (in seconds) of a circuit such as that shown in Figure 8.3, and whose current/time curve is repeated in Figure 8.5, is equal to , where L is in henrys and R in ohms.  From this it can readily be seen that a highly inductive circuit (large L) is likely to have a long time constant, and also that any inductive circuit with low resistance (i.e. a purely inductive’ circuit) will also have a long time constant.  To reduce the constant and so to speed up the response it is necessary either to reduce L, or to increase R, or both. The time constant depends on both.

Increasing V, and so the slope at the start, would also reduce the response time (but not the time constant), but this is not always possible, since V is usually the unalterable system voltage.  This is however done in the ‘field forcing’ of generators - see manual ‘Electrical Generation Equipment’.

Similarly the current of an inductance discharging through a shunt resistance as shown in Figure 8.4 will decay exponentially as shown there, and its time constant will again be .  In this case however R will be the total resistance (R₁ + R₂) of the discharge loop and will include both the shunt discharge resistance and the internal resistance of the inductance itself.  Since (R₁ + R₂) on discharging will be far greater than R₁ alone on closing, the decay time constant of a discharging inductance fitted with a discharge resistor, such as a machine’s field, is in general much shorter than the build-up time - that is to say, it discharges more rapidly than it builds.

The correct symbol for time constant is the Greek letter ‘tau’.