We have seen that whenever an electric current flows through a conductor a magnetic field is formed (section 4.3). The opposite can also occur and under certain conditions and a magnetic filed can be responsible for the flow of an electrical current. This phenomena can occur in two ways:

*Dynamic Induction*– the conductor moves through a magnetic field.*Static Induction*– the magnetic field around a conductor changes.

In both cases the magnetic field changes relative to the conductor.

### 6.1 Dynamic Induction

If the ends of a metal wire are connected to a sensitive indicating instrument the needle will jump when the wire is suddenly moved through the poles of a magnet (figure 6.1a). The movement of the needle is due to an EMF being induced in the wire as it moves through the magnetic field, this EMF causes a current to flow if the wire forms a closed circuit. When the wire’s movement stops the needle returns to zero as the EMF is no longer being induced (figure 6.1b). If the wire is pulled back across the magnetic field the needs will be seen to jump in the opposite direction (figure 6.1c).

It can be found that the magnitude of the induced EMF is proportional to the strength of the magnet, the length of the conductor in the magnetic field and velocity of the wire, thus;

where:

*e* = instantaneous induced EMF (V)

B = flux density of the magnetic file (T)

*l* = length of conductor in the field (m)

*v* = velocity of the conductor (m/s) (note that velocity is speed in a given direction).

Therefore *e* will be a steady value if the conductor moves with a constant velocity through a uniform field. However, in a non-uniform field or if the velocity of the conductor is not constant the induced EMF will change with time and *e* is used to indicate an instantaneous EMF, that is the EMF at a given instant.

The induced EMF can also be defined in terms of the rate at which magnetic lines of flux are ‘cut’ by the wire. If a conductor cuts one weber of flux in one second the EMF induced will be one volt. Thus:

where:

Φ = total magnetic flux ‘cut’ (Wb)

*t* = time, (s)

The direction of the induced EMF (obviously the direction of the current in a close circuit relies on the direction of the induced EMF) depends on the direction of the movement and the direction of the magnetic field (remember that the direction of the field is taken to be north to south). The direction of the EMF, movement or field can be found using *Fleming’s right-hand (generator) rule* (figure 6.2).

### 6.2 Simple Generator

Consider a loop of wire rotating between the poles of a magnet, such as that shown in figure 9a. Connections with the ends of the wire are made using brushes and slip ring’s so that the loop can rotate freely. Figure 6.3b shows a view down a section of the loop, AB, at different positions as the loop rotates through the magnetic field. When AB is in either position 1 and 5 it is moving along the lines of flux but not cutting them, therefore no EMF is induced. In positions 2,4,6 and 8 AB cuts the flux lines at an angle so that an EMF is induced. At positions 3 and 7 the conductor moves directly across the flux lines, cutting it at a maximum rate and the maximum EMF is induced. Fleming’s right-hand rule can be used to find the direction of the induced EMF and therefore the direction in which the current flows (figure 6.3b). Notice that the direction of the induced EMF (and current) when the coil moves from left to right across the field is in the opposite direction to the EMF induced when the coil moves from right to left. The magnitude and direction of the induced EMF is plotted against time in figure 6.3c, this is an alternating EMF that alternately acts in different directions.

Note that the current flowing in the side of the loop AB at any point in the cycle will be in the opposite direction to the current flowing in the side CD. Therefore, the current will flow down one side of the loop and back up the other, reducing to zero when the loop is horizontal before flowing back around the loop in the opposite direction.

### 6.3 Static Indication

If the coil of wire remains stationary an EMF can still be induced within it by varying a magnetic field that passes through it. Therefore the lines of magnetic flux ‘cut’ the conductor rather that the conductor ‘cutting’ the lines of flux. When the change in magnetic flux occurs it is often referred to as a change in ‘flux linkage’ indicating that there has been a change in the amount of magnetic flux ‘linking’ with or passing through a conductor or coil.

If a detector is attached to a coil it will indicate an EMF has been induced when an externally produced magnetic field increases or decreases through the coil (figure 6.4b). When the field is switched off or remains constant the inductor will show that no EMF is induced (figure 6.4a). The magnitude of the induced EMF is dependent on the rate at which the magnetic flux changes, not the amount by which the flux changes: a small amount of magnetic flux changing rapidly can induce a greater EMF than a large change of flux taking place slowly.

An EMF of one volt will be induced in a coil by a flux changing at a rate of one weber per second. Hence:

where:

*e* = average EMF induced (V)

Φ_{1} = final value of magnetic flux (Wb)

Φ_{2} = initial value of magnetic flux (Wb)

*t* = time taken for flux to change from Φ_{1} to Φ_{2 }(s).

N = number of turns on the coil.

To avoid calculus this equation assumes that the change in flux occurs at a steady rate, or at least any fluctuations in the change are ignored, hence *e* is the average EMF induced over the period of flux change.

The above equation is for a coil responding to an externally produced magnetic field. If the coil is not in an externally produced magnetic field but is connected through a switch to a DC power supply it will produce its own magnetic field since it will act as a solenoid (section 4.4). As soon as the switch is closed the coil will begin to set up a magnetic field. The magnetic flux will pass through the coil itself and since this flux is increasing in the instant the switch is thrown it will induce an EMF within the coil. If this induced EMF had the same direction as the power supply the current would increase out of control until the coil melted. Therefore in this case the induced EMF must act against the applied EMF since we know that coils don’t spontaneously melt. This phenomena is known as Lenz’s law:

*The direction of an induced EMF is always such as to oppose the effect producing it.*

Thus, the EMF induced by magnetic flux change, in turn caused by an increasing current, will oppose the current and try to prevent it flowing. Similarly the EMF induced by a reducing current will try to maintain the current. The induced EMF can not stop the change in current since the current will eventually reach a steady value and the magnetic flux will also become steady, it does however slow it down for as long as there are no other changes.

### 6.4 Mutual Inductance

Mutual induction occurs between two coil where one is creating a magnetic flux as current flows through it and the other has an EMF induced in it by this flux.

Consider two coils placed next to each other (figure 6.5a). The left-hand coil is connected to a DC power supply and the right-hand coil is connected to a centre-zero detector. While no current flows in the left-hand coil there will be no effect on the right-hand coil and the detector remains at zero (figure 6.5a). When the switch is closed the current begins to flow through the left-hand coil, however it will not reach a steady value immediately but increases gradually because of Lenz’s law. While the current increases in the left-hand coil the magnetic flux created by it also increases, some of this flux will pass through the right-hand coil. Since this flux is increasing it will induce an EMF in the right-hand coil and the detector’s needle will move from zero (figure 6.5b).

Once the current in the left-hand coil has reached a steady value the flux it produces also becomes steady and therefore the flux passing through the right-hand coil is no longer changing. Therefore no EMF is induced in the right-hand coil and the detector returns to zero (figure 6.5c).

If the switch is opened the current in the left-hand coil will reduce to zero but because of Lenz’s law the current will reduce gradually, thus the flux passing through the right-hand coil also reduces. This changing flux induces an EMF in the right-hand coil and the detector moves from zero (figure 6.5d). Note that the deflection of the detector’s needle will be in the opposite direction to that which occurred when the flux was increasing.

Once the current in the right-hand coil has reached zero the coil no longer produces a flux and so there is no flux passing through the right-hand coil and the detector returns to zero.

This phenomena of an EMF being induced in one coil when a current in another coil changes is called *mutual induction* and a *mutual inductance* (M) is said to exist between the coils. Note that induction takes place as the current increases or decreases but an amount of inductance exists between the two coils even if the current is steady. The unit of mutual inductance is the henry (H) and it can be defined in two ways –

(a) The mutual inductance between two coils is one henry when a current changing in the first coil at a rate of one ampere per second induces an EMF of one volt in the second coil. Thus:

where:

e = average EMF induced in the second coil (V)

M = mutual inductance between the coils (H)

I1 = initial current in the first current (A)

I2 = final current in the final current (A)

t = the time for the change to occur (s)

Note that again e is the average EMF induced to avoid calculus.

(b) The mutual inductance between two coils is one henry when a current of one ampere in the first coil produces a flux linkage of one weber turn in the second coil. A weber turn is the flux passing through a coil multiplied by the number of turns in the coil. Thus:

where:

M = mutual inductance between the coils (H)

Φ = the amount of magnetic flux set up by the first coil which passes through the second (Wb)

N2 = the number of turns in the second coil

I1 = the current on the second coil (A)

In practice, a number of factors affect the mutual inductance between coils. There are:

- the number of turns in the coils – more turns give higher mutual inductance;

the distance between the coils – the greater the distance, the lower the mutual inductance; - the position of one coil relative to the other – if the coils are at right angles, not much of the magnetic flux set up by the first coil will pass through the second, and the mutual inductance will be low.
- the presence of a magnetic circuit – if both coils are wound on an iron-core this will not only increase the magnetic flux set up but will channel almost all of it through the second coil, considerably increasing the mutual inductance.

The symbols used for mutual inductors are shown in figure 6.6.

Example

Two air-cored coils have a mutual inductance of 0.4H and one coil of 1200 turns has a flux linkage of 5nWb when a steady current flows in the other, calculate the value of the steady current.

When the current in the coil system is increased to 25A, an average EMF of 80V is induced in the 1200 turn coil. Calculate the time taken for the current to change.

For iron-cored inductors the amount of flux set up will not be directly proportional to the magnetising current because of a phenomena called saturation. This will be described latter.

### 6.5 Self Inductance

When we were considering mutual induction we saw that a change in the current flowing in one coil creates a changing magnetic flux that induces an EMF in the second coil. However, the first coil also experiences the changing magnetic flux because the flux passes through both coils (figures 11b,c and d), therefore an EMF must be induced in the first coil as well.

The induction of an EMF in an isolated coil, due to a change of current within itself that results in a change of the magnetic flux linkage, is called *self inductance* (L). The unit of self inductance, like that of mutual inductance, is the henry (H). Self inductance can be defined in two ways –

(a) The self inductance of a coil is one henry when a rate of change of current of one ampere per second in the coil induces an EMF of one volt in it. Thus:

where:

*e* = average induced EMF (V)

L = self inductance of system (H)

I_{1} = initial current in coil (A)

I_{2} = final current in coil (A)

*t* = time taken for current from I_{1} to I_{2}, (s)

(b) The self inductance of a coil is one henry if a current of one ampere in the coil produces within it a flux linkage of one weber turn. A weber turn is the flux passing through a coil multiplied by the number of turns in the coil. Thus:

where:

L = self inductance of coil (H)

Φ = magnetic flux linkage of the coil (Wb)

N = number of turns

I = current carried by the system (A)

The factors affecting the self inductance of a conductor are:

- the number of turns – more turns gives higher self inductance;
- the way the turns are arranged – a short thick coil will have higher self inductance than a long, thin one;
- the presence of a magnetic circuit – if the coil is wound on an iron core, the same current will set up a greater magnetic flux and the self inductance will be higher.

The symbol for an inductor is shown in figure 6.7.

Example

A coil has a self inductance of 3H and a resistance of 8Ω. Calculate the current when the coil is connected to a 12V DC supply. The supply is switched off and the current falls to zero in 0.2s, calculate the average EMF induced.

The negative sign indicates that Lenz’s law applies and the induced EMF will oppose the change producing it. In this case the linking flux is reducing because the current is reduced, therefore the induced EMF tries to keep the current flowing. The greater the self inductance of the coil, the longer it will take for the current to reduce to zero. Note that the self inductance does not affect the steady current that flows in a DC system and I = U/R can still be used to find the steady current (i.e. when the current is steady and not changing the coil acts as a plane resistor). Because the inductance EMF opposes any changes in current it is called a *back EMF.* It will always oppose change but never prevent it. Back EMFs cause current changes in high inductance DC systems to be slow. Iron-cored coils with high self inductance are thus called chokes because they smooth and stifle current fluctuations.

### 6.6 Energy Stored In A Magnetic Field

A perfect inductor would be a coil of wire that has inductance but no resistance. In practice this is impossible since inductors are made from coiled wire and wires have resistance. However, ignoring resistance can help us examine the energy changes that occur in inductance. Energy is required to set up a magnetic field and this energy is not lost as heat but stored in the magnetic field. When the current causing a magnetic field is switched off the field collapses and the stored energy is released to the circuit in terms of an induced back EMF.

The energy stored in a magnetic field set up by a current flowing in a coil is given by:

where:

W = energy stored in the magnetic field (J)

L = self inductance (H)

I = final current flowing through the coil (A)

In practice, an inductor will consist of a coil of wire that will have resistance. This resistance can be shown separately from the inductance as shown in figure 6.8.

### 6.7 Growth And Decay Curves

For the circuit shown in figure 6.8, as soon as the switch is close a DC current will start flowing through the coil. The initial rate of the rise in current will be high, resulting in a large induced EMF. Obeying Lenz’s law, this EMF will oppose the current and the rate at which the current increases will decline until a steady current is reached, from this point the coil acts as a resistor. Therefore, the current increase plotted against time (figure 6.9) will be a curve which levels out at a value of I = ER; where E is the supply voltage and R is the resistance of the coil.

When the switch in figure 6.9 is opened a decaying current occurs, fast at first and then slower as the induced EMF opposes the change. The instantaneous current against time curve (figure 6.10) is the inverse of the previous curve.

The ratio of L/R is called the time constant (τ) which can be defined as the time taken for the voltage or current to reach its final value if it continued to change at its initial rate and:

When the power supply is switched on the instantaneous current (i.e. the current at any given time, *t*) is given by:

When the power supply is switched off the instantaneous current is given by:

where:

E = supply EMF (V)

R = resistance of the coil (Ω)

*t *= time (s)

The previous two equations are included for completeness but don’t worry about the maths if you are not comfortable with exponential functions.

### 6.8 Switching Inductive Circuits

When the power supply connected to an inductance is switched the current that maintains the magnetic field stops and the magnetic flux fades. The energy in the magnetic field can be stored no longer and must be released back into the circuit as a current driven by the induced EMF. Thus if a power supply is simply switched off, the sudden release of energy can cause an large induced EMF that can spark across a switch or melt the insulation on the coil wire. This means that the circuit shown in figure 6.8 could be quite dangerous. To avoid damage, circuits are designed so as to allow the induced EMF to driven a current around a discharge circuit. Figure 6.11 shows three possible methods that can be used to dissipate the energy stored in the magnetic field of an inductor.

In figure 6.11a when the switch is opened the induced EMF drives the current through R_{D} rather than arcing across the switch. However, the value of R_{D} must be high enough so that current does not flow through it when the switch is closed, but low enough so that with the switch open a significant current will flow through it rather than arcing across the switch.

Figure 6.11b show an improved circuit that contains a voltage-sensitive resistor. At normal supply voltages these resistors have very high resistances and little current will flow through them. When the switch is opened however, the induced EMF causes the resistance of the special resistor to drop and provides a discharge path for the current.

A third option, shown in figure 6.11c, is to include a diode in place of the discharge resistor. Diodes are semi conducting devises that have very low resistances to currents that flow in the direction of the arrow and block current flowing in the other direction. Hence, with the switch closed no current can flow through the discharge circuit. With the switch opened the induced EMF pushes current around the circuit in the opposite direction and the diode allows it to flow through the discharge circuit.