Tutorial 10 A  Mutual Inductance and Transformers 

Learning Objectives 
To learn about mutual inductance. To use magnetic circuit theory to explain transformer equations. To consider the structure of a transformer and different types of transformer. To identify and quantify losses in a transformer. To learn about the induction coil 

Key Questions 
What is mutual inductance? How do we derive the transformer equations? How are transformers made? Why are transformers not 100 % efficient? How can the transformer effect be made to work with DC? 

Mutual Inductance In previous tutorials, we have seen the effect of self inductance in a single coil. We pass a current through the coil, we get a magnetic field. If we make that magnetic field pass by a second coil, we get an induced voltage in that second coil.
We actually don't have to have any movement at all. If we change the current, there is a change in the magnetic field, and that induces a voltage in a second unconnected coil. This is called the transformer effect or mutual induction.
If a changing current I_{1} A is passed through Coil 1, there will be a changing voltage E_{2} V in Coil 2. This can be worked out with the formula:
It's the same pattern as the formula for selfinductance. The term M is the mutual inductance which has units of Henry (H).
Mutual Inductance and Reluctance Let us have both coils wound around a closed iron ring like this:
Coil 1 has N_{1 }turns and has a supplied current of I_{1} A flowing in it. Coil 2 with N_{2} turns is connected to a complete circuit which has an induced current I_{2} A flowing. The iron ring has a reluctance S H^{1}. Within the ring, there is a flux F Wb. Since this is a magnetic circuit, we can say that F is the same throughout. From magnetic circuits (Tutorial 7) we know that:
So we can write:
Since the flux is produced by the current in Coil 1, we can write an expression for the mutual inductance (which affects Coil 2):
We now untidy things by multiplying by (N_{1}/N_{1}). Yes I know it is 1, but it's essential for the argument:
From the equation above:
So substituting in, we get:
And cancelling gives us:
The Current Equation We know from above that:
The flux is the same, so the F term cancels out to leave: I_{1}N_{1} = I_{2}N_{2}
This rearranges to:
This assumes that the transformer is perfect, i.e. there are no losses.
The Voltage Equation We know that:
Coil 1 has N_{1} turns and Coil 2 has N_{2} turns. The EMF for Coil 1 is E_{1} V and the EMF for Coil 2 is E_{2 }V. The rate of change in flux is the same. So we can write:
Since EMF is a voltage we can rewrite the equation as:
We can then rearrange the equation to give:


Transformers Transformers use the principle of mutual induction to change the value of alternating voltages and currents, as we have seen in the example above. The devices are generally very efficient with low losses and life expectancy is long. The picture below shows a variety of different transformers that I have at home:
The transformer is a machine that is simplicity itself. It consists of:
The transformer in the picture above is referred to as a core transformer. This is the sort that schools and colleges have for demonstration purposes, the demountable transformer.
Most practical transformers have the secondary wrapped around the primary. This saves space and reduces flux leakage. The construction is referred to as a shell transformer. The picture below shows the inside of the labpack, which has a shell transformer.
This transformer has several wires coming from its coils. They are called taps and allow for a range of voltages that are selected by switches. The switches are sometimes called tapchangers. The thermal switch switches off if excessive current is taken from the transformer, and prevents it getting too hot. The rectifier turns the AC to DC (strictly speaking, rectified AC). In this particular example, the rectifier does not work after a reverse voltage ruined it.
The two coils are electrically completely different circuits. Either of the coils can act as a primary. The laminated core is made up of layers of soft iron separated by an insulating layer of varnish or glue. The laminated soft iron core is shown in the picture below:
This reduces losses from eddy currents. Eddy currents can have a significant heating effect. If there aren't any laminations, the core can get very hot indeed.
Soft iron loses its magnetism immediately the current is turned off. Therefore the magnetic field can change forwards to backwards as the current changes.
The ratio of the input voltage to the output voltage is the same as the ratio of the number of turns on the primary to the number of turns on the secondary. We have seen the equation before:
If N_{1} is greater than N_{2}, we have a stepdown transformer, because the voltage is reduced. A stepup transformer increases the voltage.
If a transformer is 100 % efficient (and it nearly is) we can say that: power in = power out Equation: V_{1}I_{1} = V_{2}I_{2}
Therefore we can say that when the voltage is lower, the current is bigger. We have seen the transformer equation in terms of current:
In practice, the transformer is about 97 % efficient. When a large transformer is transferring a lot of energy, even 3 % losses produce a fair amount of heat. Therefore the transformer is cooled with oil, which is pumped to heat exchangers.
The giant transformers here step the output voltage of a power station from 25 000 V to 132 000 V for transmission.
Transformers can only work with alternating current; they cannot work with direct current.
Losses in a Transformer In schools and colleges, we often teach that the power into a transformer is the same as the power out. This is summed up in the equation:
P = V_{1}I_{1} = V_{2}I_{2}
This is alright as a first approximation. If it were completely correct, this tutorial would stop here. There is no such thing as a 100 % efficient device. Even in the best transformer, there are losses, which we will explore further. What are the sources of inefficiency in a transformer?
This is called hysteresis, shown on the graph below:
A few points to note:
The efficiency is given by the equation:
The strange looking symbol, h, is "eta", a Greek letter long 'ē'. It used as the physics code for efficiency. The equation can be rewritten as:
Efficiency is a fraction, and is often expressed as a percentage.
Power Factor Another important quantity for transformers is the power factor. Since a transformer is reactive device connected to an alternating supply, some energy is lost in making the alternating magnetic field. Therefore the power that can be usefully got out of the transformer is not simply the product of the voltage and the current. For more details, see my Electrical Principles notes, Tutorial 5 and Tutorial 6.
The product of voltage and current is called the apparent power, and is given the physics code S. Its units are voltampère (VA). The equations for this is:
S = VI
The true power, physics code P, is the product of the apparent power and the power factor. It is measured in watts (W) and is given by:
True Power (W) = Apparent Power (VA) × power factor
In Physics code, this is written:
The term cos f is the power factor and we can see this in the power phase vector diagram below.
The reactive power, Q, is the power taken by the changing magnetic field. The units are reactive voltamp (var). Work has to be done to make the magnetic field. Some energy is got out as the magnetic field collapses as the voltage falls, but it's not the same as the work done to make the magnetic field. Some is lost as shown on the hysteresis graph above. The formula is:
For many transformers, the power factor works out at about 0.8. Let's do a worked example:
Iron losses are constant, since they depend on the frequency of the alternating current. Most transformers are connected to a 50 Hz AC supply. Copper losses stem from the resistance of the wire. So they will vary with the current. If the load is reduced to half, the current in both the primary and secondary will be reduced to half. Therefore the losses will be reduced to a quarter, since losses depend on the square of the current.
The maximum efficiency occurs when the copper losses are equal to the iron losses. Working out the maximum efficiency needs to be approached carefully. Follow through the worked example to see how.


Different Transformer Types We have seen above the core type transformer that is used in the demountable transformer in physics labs. The coils are made of enamelled copper wire. The enamel acts as an electrical insulator. A picture is shown below:
There is a range of accessories that allow the tutor to demonstrate induction heating and welding. Students like to see the demonstration that shows nails being welded together.
The more common type of transformer is the shell type in which the secondary is wrapped around the primary as in the picture below:
In power transformers the core is generally made of silicon steel, a material in which hysteresis and eddy currents are minimised. Even so there can be a marked effect for very large power transformers, so cooling systems are needed. In very large transformers, insulated aluminium wire used to keep the costs down.
Audio frequency transformers are rated from a few mVA to about 20 VA and operate at frequencies in the range of human hearing, 20 Hz to 20 kHz. They are similar in construction to a shell transformer, but in small audio frequency transformers, the core is made of ferrite. In larger ones the core is made of silicon steel. They are used for impedance matching, which we will look at in Tutorial 10 C.
In radio frequency transformers the core is of air, ferrite, or dust. Ferrite is a ceramic material that has a high resistivity, but similar magnetic properties to silicon steel. Dust cores consist of fine particles of iron, each particle of which is insulated from its neighbour.
Radio transmission uses the transformer effect with an air (or even vacuum) core. The transmitter acts as the primary of a very inefficient transformer, while the aerial on your radio set is the secondary. Tuned filters consisting of inductors and capacitors can be made to resonate at the desired frequency to select the right frequency.


DC Transformers The sorts of transformers we have seen above will not work with direct current. A secondary voltage will only be induced when the transformer is turned on or off. While the DC is flowing there will be a constant level of flux in the core. There will be heat losses due to the resistance in the copper windings. Otherwise all we have is a physics curiosity.
The earliest experiments with transformers used induction coils. These devices use DC, and have a make and break circuit (See Tutorial 4B). The diagram below shows the idea:
Image from Wikimedia Commons By Original work: PieterJanR. This version: Chetvorno
This formed the basis of induction coils in school and college physics labs:
Image from Wikimedia Commons By Harry Winfield Secor  Downloaded from Harry Winfield Secor (1920) The How and Why of Radio Apparatus, 1st Ed., Experimenter Publishing Co., New York, p.5 on Google Books, Public Domain,
These devices could easily generate 100 kV at the terminals. They were used in early radio transmission equipment and early Xray tubes.
One of the earliest physicists who did work on the induction coil was Father Nicholas Callan (1799  1864) who was Professor of Natural Philosophy at Maynooth College in Ireland. He found that when a secondary coil was wrapped around a primary, hefty electric shocks could be experienced if the primary circuit was broken. His coil could easily generate 60 000 V.
Image from Wikimedia Commons  Auguste Blanqui It is said that his voltmeter consisted of a number of Catholic seminary students in series. Presumably precise calibration of such an instrument was not easy and there was always a high degree of uncertainty. He measured the voltage on how high they jumped when he turned the contraption on. This could be an explanation for the generally crusty nature of Catholic priests in Ireland at this time...
The secondary voltages were considered highly dangerous in later years, and the use of induction coils in schools and colleges was banned.
Induction coils are used in car ignition systems, although nowadays the switching is done electronically.


