Phase Difference and Phase Shift

Phase Variance is used to define the difference in degrees or radians when two or more alternative quantities reach maximum or zero values.

Previously, we found that a Sinusoidal waveform is an alternate magnitude that can be presented graphically in the time field along a horizontal zero axis. In addition, as an alternative magnitude, we found that sinus waves have a positive maximum value at π/2 time, a negative maximum value at 3π/2 time, and zero values occur along the baseline of 0, π and 2π.

However, not all sinusoidal waveforms pass exactly through the zero axis point at the same time, but can be "shifted" to the right or left of 0ocompared to another sinus wave.

For example, comparing a voltage waveform to that of a current waveform.This then produces an angular shift or Phase Difference between two sinusoidal waveforms. Any sine wave that does not exceed zero at t = 0 has a phase shift.

The phase difference or phase shift, also called the sinusoidal waveform, is the angle of Φ (Greek letter Phi) in radians or degrees in radians, where the waveform shifts from a specific reference point along the horizontal zero axis. In other words, phase shift is the lateral difference between two or more waveforms along a common axis, and sinusoidal waveforms of the same frequency can have a phase difference.

The phase difference of an alternative waveform can vary between Φ 0 and the maximum time period, during a full cycle the T of the waveform and anywhere along this horizontal axis, depending on the angular units used Φ = 0 to 2π (radians) or Φ = 0 to 360o.

Phase difference can also be expressed as time shift in seconds, representing part of the time period, for example +10ms or – 50uS, but it is usually more common to express the phase difference as an angular measurement.

Then, the equation of the instantaneous value of a sinusoidal voltage or current waveform that we developed in the previous sinusoidal waveform will need to be changed to take into account the phase angle of the waveform, and this becomes the new general expression.

phase difference
Phase Difference Equation
  • Here:
  • A m – the amplitude of the waveform.
  • ωt – the angular frequency of the waveform in radians/sec.
  • Φ (phi) – the phase angle in degrees or radiians that the waveform shifts left or right from the reference point.

If the positive slope of the sinusoidal waveform passes through the horizontal axis "before" t = 0, then the waveform shifts to the left with Φ >0 and the phase angle is positive, +Φ will give a leading phase angle.In other words,0o appears earlier than the vector produces a counterclockwise reversal.

Similarly, if the positive slope of the sinusoidal waveform passes through the horizontal x axis after a period of t = 0 "after", then the waveform shifts to the right, that is, <0 ve faz açısı doğrudan negatif olacaktır -Φ üreten vektörün saat yönünde dönmesini sağlayan 0  there is a delayed phase angle, as it occurs after Φ o.Both situations are shown below.

Phase Relationship of a Sinusoidal WaveForm

phase difference

First, let's say that two alternative amounts, such as a voltage, v and a current, have the same frequency at hertz. Since the frequency of the two quantities is the same as the angular velocity, the ω must also be the same. Therefore, at any time, we can say that the phase of the voltage will be the same as the phase of the current of V.

Then the angle of rotation in a given time period will always be the same, and therefore the phase difference between the two amounts of V and I will be zero and Φ = 0.Since the frequency of the voltage, V and current, I are the same, they must both reach their maximum positive, negative and zero values at the same time during a full cycle (although their amplitude is different).Then it is said that two alternative quantities, V and I, are in "the same phase".

Two Sinusoidal WaveForms – "intra-phase"

phase difference

Now let's consider that voltage, V and current, I have a 30 o phase difference between them, that is ( Φ = 30 o or π /6 radian).Both alternate quantities have the same frequency to rotate at the same speed, that is, this phase difference remains constant at all moments in the time, then, the phase difference is represented by phi, between the two quantities of 30.

Phase Difference of a Sinusoidal WaveForm

phase difference

The above voltage waveform starts from scratch along the horizontal reference axis, but at the same time the value of the current waveform is still negative and does not exceed this reference axis until after 30 o.Then there is a Phase difference between the two waveforms as the current passes the horizontal reference axis, which reaches the maximum peak and zero values after the voltage waveform.

Since the two waveforms are no longer "in-phase", therefore they must be an amount of "out of phase" determined by phi, Φ, and in our example this is 30 o.So we can say that the two waveforms are now 30 out ofphase.It can also be said that the current waveform lags behind the voltage waveform with phase angle Φ.Then in our example above there is a Delayed Phase Difference of two waveforms, so this is how the above will be given as an expression for both voltage and current:

phase difference

Where there is a phase angle Φ and current, I delays voltage",

Similarly, the current has a positive value of I, and the reference voltage will be a "pioneer" voltage up to a phase angle of the current waveform, to an axis field that reached the maximum peak and zero values some time ago.It is then said that the two waveforms have a Leading Phase Difference, and this expression for both voltage and current will be as follows:

phase difference

Where there is current, it "directs" the V voltage with Phase angle Φ(leading)"

The phase angle of a sinus wave can be used to describe the relationship of one sinus wave to another using the terms "Precursor" and "Delayed" to indicate the relationship between two sinusoidal waveforms of the same frequency drawn on the same reference axis.The two waves are out of phase with 30o in the example above.

The relationship between the two waveforms and the resulting phase angle can be measured anywhere along the horizontal zero axis, where each waveform passes in the direction of positive or negative "same slope".

In AC power circuits, the ability to define the relationship between a voltage and a current sinus wave within the same circuit is very important and forms the basis of AC circuit analysis.

Cosine WaveForm

Now we know that if one waveform is "shifted" to the right or left of 0 ocompared to another sinus wave, the expression of this waveform is A m sin(ωt ± Φ ).However, if the waveform passes the horizontal zero axis with a positive slope 90o or π/2 radians before the reference waveform, the waveform is called the Cosine WaveForm, and the expression is:

phase difference
Cosine Expression

The cosine wave is called only "cos", in electrical engineering the sinus wave is important as well as an important place in the cosine wave. It has the same shape as the sine wave equivalent, that is, it is a sinusoidal function, but +90 is shifted in front ofit or a full quarter period.

Phase Difference Between Sinus Wave and Cosine Wave

phase difference

Alternatively, we can say that the sinus wave is a cosine wave that has shifted in the other direction with -90 o. In both cases, the following rules will always apply when dealing with sinus waves or angled cosine waves.

Sinus and Cosine Wave Relations

phase difference

When comparing two forms of sinusoidal wave, it is more common to express their relationship as sinus or cosine with amplitudes that go positive, and this is achieved using the following mathematical identifications.

phase difference

Using the above relationships, we can transform any sinusoidal waveform from sinus wave to a cosine wave or vice versa, with or without angular or phase difference from sinus wave.

In the next tutorial on phasers, we will use a graphical method to represent or compare the phase difference between two sinusoids by looking at the phaser representation of a single-phase AC amount, together with some phaser algebras related to the mathematical addition of two or more phasers.