What is Phase Difference : Formula & Its Equation

In the mechanism of vibrations, a phase is a portion of a period where a point finishes after the entire passage across the zero or the reference position. This concept is even applicable for simple harmonic movements where the phase experienced by vibrating bodies and waves. While coming to a waveform, the position of the wave-particle in a periodic signal is termed as “Phase”. The entire phase of the signal is 360 degrees. The term crucially comes into action when either two or more waves interfere with each other or they travel in a similar medium. Corresponding to this approach, the term “Phase difference” was evolved. A detailed explanation of phase difference, its equation, waveforms, and formula will be explained in this article

What is Phase Difference?

Considering the sinusoidal waveform, the phase difference is explained as the time gap where the wave either falls behind or leads in correspondence to another wave. This term is just a characteristic of a single wave and it is the relative characteristic of either two or more waves. This is even termed as either offset or angle. This is generally symbolized by the letter Ф. The entire phase of the signal is shown as 2∏.

As it is discussed the phase of a wave either lags or leads with that of another wave, leading corresponds that a wave is at the forefront of the other having the same amount of frequency. Whereas lagging corresponds that a wave is at the back of the other wave with the having amount of frequency.

In order to know completely about phase difference, it is more important to also know about the terms “Phase quadrature” and “Phase opposition”. This term corresponds to when a waveform is either leads or lags by 90⁰. While phase opposition corresponds when a waveform is either leads or lags by 180⁰.

To have a clear and detailed analysis of phase difference, looking at the below-shown phase difference waveform figure can be clearly understood.

The time period and phase factors of a wave are in inverse proportion with another wave which signifies that

t in degrees = (1/360 f) in degrees

t in radians = (1/6.28 f) in radians

Here, ‘f’ represents the signal frequency and ‘t’ represents the time interval.

Equation

The equation of two sinusoidal waves can be shown as

A_(t) = A_max × sin (ωt±Ф)

This is the phase difference equation.

Here, ‘A_max’ is the maximum amplitude level of the wave

‘ωt’ represents the angular frequency of the wave measured in radians/second

And ‘Ф’ represents the angle calculated in degrees/radians where the wave undergoes shift left or right making one position as a reference point. When the positive incline of the sine wave moves through the horizontal axis before the time period is ‘0’, then there will be a shift to the left position and then Ф > 0, which means that the angle has positive characteristics. + Ф gives a leading angle. In a clear way, this means that + Ф appears before the time of 0⁰ thus generating an anticlockwise vector direction.

In the same way, when the positive incline of the sine wave moves through the horizontal axis after some time period of t = ‘0’, then there will be a shift to the right position and then Ф < 0, which means that phase angle has negative characteristics. – Ф gives a lagging angle. In a clear way, this means that – Ф appears after the time of 0⁰ thus generating a clockwise vector direction.

In order to clearly depict these leading and lagging conditions, let us consider the pictorial descriptions that show the phase relationship of the sine wave.

Phase Relationship

This section explains the phase relationship of a sinusoidal waveform.

Initially, let us consider that two varying quantities which are voltage and current where these quantities possess a similar frequency of ‘f’ (Hz). When the frequency is the same for these two varying quantities, the angular velocity (ω) will also be similar. With this, it can be known that at any period of time, the phase voltage (v) = phase current (i).

So, the degree of rotation within the specific time period will be the same, and phase variation that exists between these two quantities will be null which is Ф = 0. This shows that at the time of one full cycle, the values of the current ad voltage will reach maximum values, then the two varying quantities will be in a similar phase.

Now, let us consider the other scenario that voltage and current quantities have a variation of 30⁰ where (Ф = 30⁰ or ∏/6 rad). As these quantities have the same rotational speed, they also have similar frequency levels and this phase variation will be constant at any time period.

In the above picture, the voltage wave initiates at zero position across the horizontal axis, whereas the current wave is in negative value and will not move across the reference position till Ф > 30⁰. With this, there creates a phase difference.

Variation of Waveforms

In order to consider the difference of waveforms, we can clearly know in three different conditions those are

• Out of phase
• In Phase

Out of Phase

In the scenario that two varying waveforms possess similar frequency but not the same phase, then those waves are stated to be in “Not in Phase”. Here, the phase variation will not be zero. The below image shows the out-of-phase condition for two sinusoidal waves. Whereas in the case of the In-Phase wave, the retardation exists in the format of 1/3, 2/4, 2/5… more.

In the out of phase condition, there exists another two scenarios which are leading and lagging phases.

Leading Phase

The leading wave corresponds that a wave is at the forefront of the other with the same amount of frequency. For a leading wave, the voltage and current equations are given by

Voltage (Vt) = Vm × sin ωt

Current (it) = Im × sin (ωt – Φ)

Here, ‘Ф’ corresponds to the angle difference of the leading wave.

Lagging Phase

The lagging wave corresponds that a wave is at the back of the other with the same amount of frequency. For a leading wave, the voltage and current equations are given by

Voltage (Vt) = Vm × sin ωt

Current (it) = Im × sin (ωt + Φ)

Here, ‘Ф’ corresponds to the angle difference of the lagging wave.

In Phase Sine Waveforms

In phase, waveforms are those where the phase difference between the two sinusoidal waves is zero. This happens only when the two waves have similar frequency and phase levels. For these in-phase waves, the retardation appears as an entire number of wavelengths such as 3, 4, 5…. The below picture shows the in-phase waveforms.

The waveforms have similar frequency levels but possess various amplitude levels (increased voltage). This is all about the phase difference of waveforms.

Voltage and Current Phase Relationships to R, L, C

The R, L, and C circuits are termed with the name Resonance circuit. This section explains the concept of voltage and current performance of the inductor, resistor, and capacitor circuits in correspondence to the phase.

In the resistor device, the voltage and the current will be in a similar phase. Because of this, the phase variation between the waves is ‘0’. In the capacitor device, the voltage and currents are out of phase and the current value precedes voltage by 90⁰. Whereas in the inductor device, the voltage and currents are out of phase and the voltage value precedes current by 90⁰. The operation of the inductor is completely contrary to the behavior of the capacitor.

And this is all about the concept of phase difference. This article gives a clear analysis of what is phase difference, its equations, formula, waveforms, and phase relationship. It is also more important to know what is the relation between path difference and phase difference?