Tutorial 9 - Inductors  (A-level Extension)

The content of this tutorial is not on the AQA syllabus (or, for that matter, the OCR or EDEXCEL).  It is on the SQA Advanced Higher syllabus.  It is part of the option A (Alternating Currents) on the Welsh Board (WJEC).  Other Option A content can be found in Tutorials 9 and 10.

I have included it here because students studying the AQA Electronics Option need to know something about the concepts of AC Theory to understand the ideas of filters.  Also students studying a syllabus that is not AQA will find these notes helpful.

 Contents

Before studying this tutorial, you may wish to revise Electromagnetic Induction

Inductors are simpler components than capacitors.  At its simplest an inductor is a coil of wire. In the picture above, the wire is wrapped around a magnetic core.  The coils for a demountable transformer can be used as inductors. Capacitors are very sensitive to changes in temperature, or exceeding the working voltage. There are no such problems with inductors.

There are different kinds of inductors as shown as symbols in the picture below: 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.  A perfect inductor has zero resistance.

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.

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

But we see that the bulb is slightly dimmer.  There seems to be some extra "resistance".  This "resistance" is caused by the reactance of the inductor.

### 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)

You may remember the magnetic field of a bar magnet looking like this: 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.  The rise in voltage is exponential, but further study is not needed at this level.

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).  Consider a simple inductor, which is a coil of wire wrapped around a cylindrical piece of core material, which might not even be magnetic: The equation that links all these terms together is: 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 hollow square former 2.0 cm × 2.0 cm. The length of the solenoid is 5.0 cm. Calculate the inductance.

Inductance and Current
The self-inductance of a coil can be worked out from the equation: From this, we can say that the induced EMF is the product of the inductance and the rate of change of the current.

• The dI/dt term is the rate of change of the current.

• Units for dI/dt are amps per second (A s-1).

• The minus sign tells us that the EMF is opposing the applied voltage.

The graph below shows the idea: The dI/dt term is obtained by taking the tangent from the graph and working 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.  Notice that the maximum rate of change of the current is just when the inductor is turned on.

 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?

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.

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 = ½ × 0.5 H× 4. 0 (A)2 = 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.

 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?