Alternating Current and Alternating Voltage

Alternating electrical quantities alternating current and alternating voltage play a vital role in today’s power system. These quantities improve the versatility and usefulness of the electrical power system. Generating, transmitting, and distributing an alternating power is quite easier than direct power.

Principle of the generation of an alternating electrical quantity
Whenever there is a relative rotational motion between a coil and a magnetic field there will be an alternate in emf induced in the coil. This is the most basic principle of generating an alternating electrical quantity. Whenever a coil rotates inside a static magnetic field the coil produces alternating electricity. Alternatively, if the coil is static and the magnetic field rotates around it, the coil produces the same alternating electricity.

The value of the alternating electricity depends on three factors. Firstly it is directly proportional to the number of conductors in the coil. Secondly, it is directly proportional to the strength of the magnetic field. Thirdly it is directly proportional to the speed at which either the coil or the field is rotating.

Obviously we can generate alternating electricity by rotating either the coil or the magnetic field but practically we use the later one. That means in practical generators alternators the magnetic field rotates around the fixed coil system.

Equation of alternating current and voltage
Let us consider a rectangular coil of N number of turns, rotating in a uniform magnetic field with a constant angular speed of Omega radian per second.

We will measure the time from the vertical axis. Φ is the maximum flux linked with the coil. The magnetic field is along the horizontal axis. Therefore, obviously the maximum flux linkage occurs when the coil is along the vertical axis.

Also, you will consider that at time t the coil rotates anticlockwise with an angle theta in respect of the vertical axis. At that instant, the quantity of flux linked with the coil is given by,

alternating current 1

From the relation between the angular speed and the angular displacement, we get,

alternating current 2
Therefore we can write the equation 1 as,
alternating current 3
Hence, the total flux linkage of the coil at that instant is,

alternating current 4
Now according to Faraday’s law of electromagnetic induction, the induced EMF in the coil is,

alternating current 5
Now if theta = 90°, the value of the induced EMF in the coil is maximum. Let us consider that value as Em. Obviously the expression of Em will be,

alternating current 6
So finally the EMF equation for generating alternating quantities becomes,

alternating current 6
This is the expression of alternating voltage. But in a similar fashion, we can write the expression of alternating current. That expression is,

alternating current 8
Where Im is the amplitude of the current.

The expressions of the alternating voltage and current generated due to relative rotational motion between a magnetic field and a coil tell us that the voltage and current are sinusoidal waves in nature.

Other forms of alternating voltages and alternating currents
If T is the time period of the alternating quantities. Again f is the frequency of the alternating quantities. Then we can write,
alternating current 9
alternating current 10
So by putting the expression of Omega in the alternating voltage and current expressions we get,
alternating current 11

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