Calculating Phase Shift and Phase Difference

In this post we comprehensively discuss how to calculate and solve phase shift and phase difference in AC sinusoidal waveforms, using appropriate formulas.

What are Phasors?

Phasors analyze AC circuit behavior when frequencies are the same. Combining phasors depends on their relative phase.

Sinusoidal waveforms reach max values at π/2 and 3π/2, and zero values at 0, π, and 2π. Some waveforms may be shifted compared to others, resulting in a phase difference. Any waveform not intersecting zero at t = 0 has a phase shift.

What is Phase Shift?

The displacement of a sinusoidal waveform along the horizontal axis from a reference point is called the phase shift, denoted by Φ. Both degrees and radians can be used to measure it.

There might be a phase discrepancy between sinusoidal waveforms at the same frequency. Phase difference (Φ) can vary from 0 to the waveform’s maximum time period (T).

It can be written as a time shift, τ, in seconds, or as an angular measurement. The phase angle must be taken into account when adjusting the equation for a sinusoidal waveform’s instantaneous value.

How to Calculate Phase Difference

Phase difference can be calculated using the following formula:

A_t = A_max x (sinωt ⁺_– Φ)

In this formula, the variables are defined as follows:

• Am represents the amplitude of the waveform.
• ωt represents the angular frequency of the waveform in radian/sec.
• Φ (phi) represents the phase angle in degrees or radians, indicating the shift of the waveform either to the left or right from the reference point.

A sinusoidal waveform shifts leftward and has a positive phase angle (+Φ) and leading phase angle if its positive slope hits the horizontal axis before t = 0.

To put it another way, the vector rotates counterclockwise when the waveform comes before 0^o.

Likewise, if the sinusoidal waveform’s positive slope crosses the horizontal axis after t = 0, it has moved to the right, producing a lagging phase angle and a negative phase angle (-Φ). In this instance, the vector rotates clockwise since the waveform occurs later than 0^o. The two scenarios are displayed below.

Voltage Current in-phase

Let’s start by considering voltage (v), current (i), and angular velocity (ω) at the same frequency (ƒ).

This implies that there will never be a phase gap between them (Φ = 0), as their phases will always coincide.

Even if their amplitudes differ, both voltage (v) and current (i) will reach their greatest and zero levels simultaneously over a whole cycle. We refer to this alignment as being “in-phase.”

Voltage Current Out of Phase

Let us now imagine that there is a 30^o phase difference between the voltage, (v), and the current, (i) i.e., (Φ = 30^o or π/6 radians).

Since the two alternating values have the same frequency and rotate at the exact same rate, the phase difference between them will always be identical.

As a result, the phase difference of 30 degrees between the two variables can be illustrated by phi (Φ).

The voltage waveform starts at zero on the reference axis, while the current waveform remains negative and doesn’t intersect the reference axis until 30^o later.

This creates a phase difference between the two waveforms, with the current waveform lagging behind the voltage waveform by 30^o.

Therefore, the two waveforms are considered to be out-of-phase by 30^o, with the current waveform lagging.

The formula for calculating out of phase voltage and current in AC waveform is given below:

Voltage, (V_t) = V_msinωt

Current, (i_t)= I_msin(ωt – Φ)

The phase angle Φ causes a lag between the current i and the voltage v.

Understanding Phase Angle

The phase angle represents the relative position of two sine waves, either “leading” or “lagging.” This relationship is identified by graphing the waveforms along the same reference axis.

In our example, the waveforms are 30^o out of phase, indicating that either current lags voltage or voltage leads current by 30^o.

This relation may be assessed at any position on the horizontal zero axis where the waveforms have the same slope direction.

The ability to define the relationship between voltage and current sine waves is critical in AC power circuits, and it serves as the foundation for AC circuit analysis.

What is Cosine Waveform

When a waveform is shifted to the right or left of 0^o in relation to another sine wave, its representation changes to A_m sin(ωt ± Φ).

If the waveform intersects the horizontal zero axis with a positive slope of 90^o or π/2 radians before the reference waveform, it is called a Cosine Waveform and the expression transforms into the following equation:

sin(ωt + 90^o ) = sin(ωt + π/2) = cosωt

When it comes to electrical engineering, the cosine wave, or just “cos,” is just as significant as the sine wave.

The cosine wave is a sinusoidal function that is displaced by +90^o, or one full quarter of a period prior to the sine wave but has the same structure as its sine wave cousin.

Relationship Between a Sine wave and a Cosine wave

On the contrary, a sine wave may be defined as a cosine wave that has been displaced by -90 degrees in the opposite direction. In either case, the following guidelines always hold true when working with sine or cosine waves that have an angle.

• cos(ωt + Φ ) = sin(ωt + Φ + 90^o)
• sin(ωt + Φ ) = cos(ωt + Φ + 90^o)

In the comparison of two sinusoidal waveforms, it is customary to represent their relationship as either a sine or cosine function with positive amplitudes. This can be achieved by employing specific mathematical identities.

• sin(A ± B) = sin(A)cos(B) ± cos(A)sin(B)
• cos(A ± B) = cos(A)cos(B) ± sin(A)sin(B)

Therefore we get:

• -sin(ωt) = sin(ωt ± 180^o)
• -cos(ωt) = cos(ωt ± 180^o)
• ±sin(ωt) = cos(ωt ± 90^o)
• ±cos(ωt) = sin(ωt ± 90^o)

With the help of the formulas mentioned above, it becomes possible for us to convert sinusoidal waveforms, for example from sine wave to a cosine wave, or vice versa. This conversion is regardless of whether or not they exhibit an angular or phase difference.