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Magnetism and electricity have been proven to be fundamentally linked. A magnetic flux will create a current in a wire just as a current in a wire will create a magnetic flux. This property is called Lenz's law and is the basis for how an inductor functions. The reason why this happens can be explained through conservation of energy, when a force (magnetic field) acts upon a wire, there is no way for that wire to store the energy, so that force turns into electric current and we get an EMF. Lenz's law can be explained by Faraday's law of induction:
Faraday's law of induction states that the EMF produced is the result of a change in magnetic flux with respect to time.
An inductor is nothing more than a piece of copper wire wound as a coil which makes it more permeable to magnetic flux.
An inductors job in a circuit is very similar to a capacitor it stores energy for later use. As current flows through an inductor, the current builds a magnetic flux around the inductor. This magnetic flux was the product of induction and will try and resist any change in current due to Lenz's law.
An inductor is a very simple and crude device, it is used in very similar applications to a capacitor.
LRC Circuits
a very popular application to an inductor is the creation of an oscillator using a capacitor and an inductor. Putting these two components in series with one another will create an oscillator at a certain frequency, this was primarily used in radio tuning as you had an inductor hooked up to a variable capacitor to turn into certain frequencies. LRC circuits can get much more complicated, but that is for another time.
Transformers
The principle role for an inductor is the transformer, a device that is used to modulate voltage across two circuits.
A Transformer is simply two inductors coupled via a piece of iron.
A transformer only works with an alternating current as a constant DC current will not produce a changing magnetic flux and thus wont create an EMF.
The voltage change across the transformer is proportional to the number of turns difference across the transformer.
N = Number of turns
Note about Inductors
Now, just as a capacitor is any break in your circuit, an inductor is ANY connection in your circuit. This is another problem that plagues analog circuits. Every wire is going to be creating a magnetic flux and every wire is going to be inductively coupled some how. This wont destroy your circuits, but it is something to be keen of when designing a circuit.
Formula Sheet
Faraday's Law
Transformers