# Generation of a Sine Wave

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A fundamental concept behind the operation of alternating current systems is that voltage and current waveforms will be sinusoidal – a Sine Wave. This is best explained by considering how a coil of wire behaves when rotated in a magnetic field.

Faraday's Induction Law states that any change in the magnetic environment of a wire will cause a voltage to be "induced" in the wire. This can be expressed mathematically as:

where:

V - induced voltage, V (volts)

B - magnetic flux density, T (tesla)

l - length of conductor, m (metre)

v - velocity between conductor and magnetic field m.s-1 (metre per second)

θ - angle between conductor and magnetic field, radians (or degrees)

If the wire is moved parallel to the magnetic field, the resulting angle is zero and the induced voltage with be zero. If the wire is moved perpendicular to the magnetic field (sin θ = 1) then the maximum voltage will be induced (for a given field and velocity). At any other angle the voltage will be proportional to the sine of the angle.

## Generation of Voltage

In practice voltage is generated by rotating a coil of wire through a magnetic field in a generator. The illustration shows this process. Initially the coil is perpendicular to the magnetic field generating maximum voltage. As the coil rotates the voltage decreases according to the sine of the angle until the conductor is parallel to the magnetic field. Further rotation then increases the voltage until once again it is at a maximum (but in the opposite direction).

For each revolution a complete sine wave is generated. The number of sine wave cycles generated per second (the frequency) depends on how quickly the generator is rotating.

In practice each generator coil will have several turns of wire. For 'n' turns, the total voltage will be 'n' times that given by the above equation. Real generator winding are often more complex than that of a single coil, however the basic goal of constructing machines to generate a sine wave still apply.

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#### View 3 Comments (old system)

1. drdimitri says:
11/19/2012 10:00 PM

Actually, in practice the wires will be embedded in a ferromagnetic rotor, thus the voltage produced will be a square-wave and not sinusoidal. Also, in practice the rotor will be the magnet and the stator will have the windings (synchronous machine).

• Steven says:
11/20/2012 12:26 PM

Thanks for the comments drdimitri. They bring out a valid point.

The theory described above is correct and useful in explaining how a sinusoidal emf is generated, which was the intent of the page. As an additional reference, readers can refer to Hughes, Electrical Technology Seven Edition, page 192, which describes much the same analysis.

You are correct that practical machines are more complicated than the simple coil type shown and that the magnetic flux does not behave the same. The high reluctance of the air gap (compared to the rotor and stator) forces the magnetic flux to be perpendicular between the rotor and stator. To obtain a sine wave the turns are distributed in closely spaced slots around the rotor and varied in a sinusoidal manor.

• drdimitri says:
11/20/2012 12:45 PM

Exactly! This is one of the questions I always ask my students and at first they always forget that the air gap introduces a very high reluctance, so they think that "by default" a generator will produce a pure sine wave. It actually takes a lot of effort to produce a sine wave, as you have pointed out (distributed windings)...

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