Tutorial 5 A - Inductor Properties

Learning Objective

Recall that an inductor produces a magnetic field.

To understand that inductors have an inductance.

To explain that inductors have a reactance due to their inductance.

Key Questions

What is inductance?

How do we demonstrate the idea of reactance?

How can we measure inductance?

## Properties of Inductors

Inductors are simpler components than capacitors.  At its simplest an inductor is a coil of wire.

Photos from Wikipedia

Capacitors are very sensitive to changes in temperature, or exceeding the working voltage.   There are no such problems with inductors.

The circuit symbols of different kinds of inductors as shown in the picture below.

The diagram shows the features in an inductor that we will use in the discussion of the Physics of inductance.

We will consider single inductors which have a self-inductance, for which the Physics code is L and the units are Henrys (H).  The unit is named after Joseph Henry (1797 – 1878), an American physicist, who did a lot of pioneering work with electromagnetism.

As far as electric currents are concerned, an inductor is simply a piece of coiled wire and should not affect the flow of charge at all.

Suppose we connect an inductor in series with a bulb.  The circuit is below:

The inductor has zero resistance.  If we connect the circuit to a DC battery, the bulb lights up to full brightness, just as we would expect.

Now let’s connect it to an AC supply of exactly the same voltage.  We would expect the bulb to light up to exactly the same brightness.   But what we see is the bulb is slightly dimmer.

 Explain which AC voltage we should use: peak or RMS.

### Why does an inductor have a reactance?

Electric currents always produce a magnetic field.  That is an unchangeable law of Physics.  A direct current produces a steady magnetic field, while an alternating current produces a magnetic field that is changing all the time.

You can make a voltage across the ends of a wire by moving a magnet past a wire, or by moving a wire past a magnet.  You can even make a voltage by having a magnet sitting next to a wire that is stationary, but you have got to change the magnetic field.  If the magnetic field remains the same, no voltage will be induced, however strong the magnetic field.

A current-carrying coil of wire will act as an electromagnet, even though the coil itself is made of a non-magnetic material like copper.  Let’s think about what would happen when we switch on a current that passes through a coil of wire.  When we switch on the current, a magnetic field is made as the current flows.  As the field being made, a reverse voltage is made to oppose the increase in voltage across the inductor.

We can use Faraday’s and Lenz’s laws to help us to model the situation.  There are some terms that we need to know to help us think it through:

 Quantity Physics Code Units Flux density (magnetic field strength) B Tesla (T) Magnetic flux F Weber (Wb) Electromotive Force (voltage) E (Curly ‘E’) Volts (V)

Note that in some browsers the font for curly 'E' isn't supported.  I have not always been consistent in using it either.

You may remember the magnetic field of a bar magnet looking like this:

Picture from Wikipedia

Flux is simply the total number of field lines.  There are 9 field lines in this picture.

Flux density is high when the field lines are close together.  That means that the magnetic field is strong.  So the magnetic field strength is shown by the concentration of field lines.  If the field line a spread out, the field is weaker.

The magnetic flux is the product between the field strength and the area.  In physics code it is written:

Faraday’s and Lenz’s Laws tell us that a change in a magnetic field will produce an electromotive force (EMF) that will act to oppose the change.  This can be summed up by the equation:

The term N is the number of turns in the coil, and the dF/dt is the rate of change of flux, i.e. how much the flux changes in a small time interval.

If the dt term is very small, then the reverse EMF is very large.  This has important implications for electronic circuits with inductive components.

If we plot a graph of the voltage across a conductor when it is switched on, we get something like this:

When the voltage is not changing when DC flows, then there is zero reverse EMF, and the current flows normally as if the inductor were simply a wire.  However, some work per coulomb is done to build up the magnetic field when the inductor is switched on.

However, in AC, as the current is changing direction all the time, there is a reverse EMF being induced all the time to prevent the change from happening.  The higher the frequency, the bigger the reverse EMF.

### Inductance

The inductance of an inductor is the property by which the inductor induces a voltage in itself in response to a changing current.  It has the Physics code L and is measured in Henrys (H).

The simplest inductor is a solenoid and the inductance of a solenoid is worked out using the following equation:

The terms involved are:

 Quantity Physics Code Units Inductance L Henry (H) Permeability of free space m0 (= 4p × 10-7) H m-1 Number of turns N Area of the solenoid A m2 Length l m

The permeability of free space is a constant that is common in electromagnetism.  Its Physics code is m0, (pronounced “mu-nought”).  The symbol m is “mu”, a Greek lower-case letter ‘m’.

 An inductor is made of a solenoid of 1200 turns of copper wire around a square former 2 cm × 2 cm.  The length of the solenoid is 5 cm.  Calculate the inductance.

### Inductance and Current

The self-inductance of a coil can be worked out from the equation:

The dI/dt term is the rate of change of the current:

We can determine this by measuring the gradient at any point, as shown in the diagram above.  We need to take a tangent and work out its rise and run.  This is not a particularly easy way of measuring L.  But if we know L, we can easily work out the reverse EMF.

 A solenoid of 2.3 × 10-2 H and negligible resistance is connected across a 12 V battery.  What is the rate of increase of current in the solenoid as it’s turned on?

## Energy in an inductor

If we have an inductor in a DC circuit, we find that it makes little difference.  However, if we turn the current off, the magnetic field collapses.  A large reverse EMF is produced that can give you a shock as the energy is released.  The reverse voltage spike will wreck electronic components, but there is a simple way to get round this, which we will look at later.

The energy held in an inductor is given by the equation:

 Worked example What is the energy stored in a inductor of 0.5 H when a current of 4.0 amps is flowing through it? Answer  E = ˝ × 0.5 × 4.02 = 4 J

If it takes 0.1 s for the magnetic field to collapse, the 4.0 J is dissipated in 0.1 s, i.e. at a rate of 40 W.

To learn how this equation is derived, click HERE to go to the extension page, or click the button at the bottom of this page.

 Question 4 A current of 5.0 A is flowing through an inductor of 10 H.        (a) What is the energy contained in the inductor.  (b) What is the reverse voltage if it takes 0.05 s for the magnetic field to collapse?

### Reverse voltage spikes

The reverse EMF can be reduced considerably by reducing the voltage slowly, over a period of several seconds.  Another way is to have a reverse biased diode to protect rectifier packs and other electronic components.  This conducts the reverse voltage spike harmlessly away.  The diode is wired in parallel with the inductive component.  The arrangement is shown in the diagram below:

Large DC motors have many advantages compared to AC motors.  They can be easily reversed, and their speed can be controlled easily.  However they have large inductances because they have lots of coils of wire and big lumps of magnetic material.  So switching on and off by simply closing and opening a switch is not an option unless you like electrical sparks flying.  This is why you need control gear such as this to increase and decrease the current slowly.

 Explain why the reverse biased diode is needed and how it protects the transistor.