Phase and Phase Difference of Currents and Voltages

The phase means the angular advancement of a waveform from the zero-reference within a cycle. To understand the basic concept of phase let us draw a simple sinusoidal wave.

For example, let us consider point P on the x-axis. Suppose the wave propagates up to that point from zero within a time t. The angular velocity of propagation of the wave is omega. Therefore we can write, the expression of the angular advancement as
1
We can show the position of the coil in the magnetic field producing this alternating electrical quantity as follows.
2
We call this angle as the phase angle.

Phase Difference
For practical purposes, the absolute phase is not a very useful parameter. Rather the comparison between the phases of two different alternating electrical quantities is much useful. At a certain instant, the phase of two different electrical signals may be different. And the angular difference between these two phases is the phase difference between these two signals.

Suppose there are two coils of different turns. We have fitted them in the same alignment. Then we rotate these two coils with the same angular speed in a common magnetic field. Each time the plane of the coils coincides with the axis of the field, the induced EMF in the both of the coils becomes maximum. On the other hand, when both the coils become perpendicular to the axis of the field, the induced EMF in both of the coils becomes zero. In this way, the sinusoidal wave of the induced EMF in both of the coils alters together in the same fashion. Therefore the angular velocity, frequency, and shape of the waveform are the same for both emfs. But the maximum value, as well as any other instantaneous value of the signals, are different because the numbers of turns in the coils are different. Let us write the expression of these two EMFs as
3
Now we will fit these two coils in two different alignments as shown in the figure below.

The angular difference between the alignments of the coils is theta.

Now we will rotate these coils in a common magnetic field. Obviously the speed of the rotation of these two coils is the same. But here are you observe when the first coil comes to its horizontal position, the second coil remains theta angle behind it. So the maximum EMF occurs in the second coil after theta angle of the same occurrence in the first coil. Similarly, the first coil reaches its vertical position theta angle before the second coil reaches the vertical position. Therefore, the EMF becomes zero at the first coil theta phase before the second coil. In this way, there will be a phase difference between the waveforms of emf of two coils. In this case, we represent the EMFs as
4
In that case, the second coil follows the first coil, so EMF of the second coil lags that of the first coil. But if the second coil rotates ahead of the first coil then the EMF of the second coil leads that of the first coil. In that case, we represent the EMFs as,
5
Therefore, we have observed that positive theta represents the leading and negative theta represents the legging phase difference.

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