Inductance; and transformers


Sections 21.7 - 21.9

Mutual inductance

Faraday's law tells us that a changing magnetic flux will induce an emf in a coil. The induced emf for a coil with N loops is:

Picture two coils next to each other, end to end. If the first coil has a current going through it,a magnetic field will be produced, and a magnetic flux will pass through the second coil. Changing the current in the first coil changes the flux through the second, inducing an emf in the second coil. This is known as mutual inductance, inducing an emf in one coil by changing the current through another. The induced emf is proportional to the change in flux,which is proportional to the change in current in the first coil. The induced emf can thus be written as:

The constant M is the mutual inductance, which depends on various factors, including the area and number of turns in coil 2, the distance between the two coils (the further apart, the less flux passes through coil 2), the relative orientation of the two coils, the number of turns / unit length in the first coil (because that's what the magnetic field produced by the first coil depends on), and whether the two coils have cores made from ferromagentic material. In other words, M is rather complicated. What's far more important in the equation above is that the emf induced in the second coil is proportional to the change in current in the first.

This effect can be put to practical use. One way to use it is in a transformer, which we'll discuss below. Another is to use it in an ammeter. Conventional ammeters are incorporated directly into circuits, but ammeters don't have to be placed in the current path for alternating current. If a loop connected to a meter is placed around a wire with an AC current in it, an emf will be induced in the loop because of the changing field from the wire, and that will produce a current in the loop, and meter, proportional to the current in the wire.

Self inductance

Coils can also induce emf's in themselves. If a changing current is passed through a coil, a changing magnetic field will be produced, inducing an emf in the coil. Again, this emf is given by:

As with mutual inductance, the induced emf is proportional to the change in current. The induced emf can be written as:

The constant L is known as the inductance of the coil. It depends on the coil geometry, as well as on whether the coil has a core of ferromagnetic material.

We've already discussed resistors and capacitors as circuit elements. Inductors, which are simply wire coils, often with ferromagnetic cores, are another kind of circuit element. One of the main differences between these is what happens to electrical energy in them. Resistors dissipate electrical energy in the form of heat; capacitors store the energy in an electric field between the capacitor plates; and inductors store the energy in the magnetic field in the coil. The energy stored in an inductor is:

In general, the energy density (energy per unit volume) in a magnetic field is:


Electricity is often generated a long way from where it is used, and is transmitted long distances through power lines. Although the resistance of a short length of power line is relatively low, over a long distance the resistance can become substantial. A power line of resistance R causes a power loss of I2R ; this is wasted as heat. By reducing the current, therefore, the I2R losses can be minimized.

At the generating station, the power generated is given by P = VI. To reduce the current while keeping the power constant, the voltage can be increased. Using AC power, and Faraday's law of induction, there is a very simple way to increase voltage and decrease current (or vice versa), and that is to use a transformer. A transformer is made up of two coils, each with a different number of loops, linked by an iron core so the magnetic flux from one passes through the other. When the flux generated by one coil changes (as it does continually if the coil is connected to an AC power source), the flux passing through the other will change, inducing a voltage in the second coil. With AC power, the voltage induced in the second coil will also be AC.

In a standard transformer, the two coils are usually wrapped around the same iron core, ensuring that the magnetic flux is the same through both coils. The coil that provides the flux (i.e., the coil connected to the AC power source) is known as the primary coil, while the coil in which voltage is induced is known as the secondary coil. If the primary coil sets up a changing flux, the voltage in the secondary coil depends on the number of turns in the secondary:

Similarly, the relationship for the primary coil is:

Combining these gives the relationship between the primary and secondary voltage:

Energy (or, equivalently, power) has to be conserved, so:

If a transformer takes a high primary voltage and converts it to a low secondary voltage, the current in the secondary will be higher than that in the primary to compensate (and vice versa). A transformer in which the voltage is higher in the primary than the secondary (i.e., more turns in the primary than the secondary) is known as a step-down transformer. A transformer in which the secondary has more turns (and, therefore, higher voltage) is known as a step-up transformer.

Power companies use step-up transformers to boost the voltage to hundreds of kV before it is transmitted down a power line, reducing the current and minimizing the power lost in transmission lines. Step-down transformers are used at the other end, to decrease the voltage to the 120 or 240 V used in household circuits.

Transformers require a varying flux to work. They are therefore perfect for AC power, but do not work at all for DC power, which would keep the flux constant. The ease with which voltage and current can be tranformed in an AC circuit is a large part of the reason AC power, rather than DC, is distributed by the power companies.

Although transformers dramatically reduce the energy lost to I2R heating in power line, they don't give something for nothing. Transformers will also dissipate some energy, in the form of:

  1. flux leakage - not all the magnetic flux from the primary passes through the secondary
  2. self-induction - the opposition of the coils to a changing flux in them
  3. heating losses in the coils of the transformer
  4. eddy currents

In the iron core of a transformer, electrons would swirl in cross-sectional planes. This current would heat up the transformer, wasting power as heat. To minimize power losses due to eddy currents, the iron core is usually made up of thin laminated slices, rather than one solid piece. Current is then confined within each laminated piece, significantly reducing the swirling tendency as well as the losses by heating.

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