Home >> Electric Circuits >> AC Circuits >> Phasor Diagram and Phasor Addition

Phasor Diagram and Phasor Addition

by | Last updated Jul 5, 2021 | AC Circuits

Phasor Diagram is a graphical representation of the relation between two or more alternating quantities in terms of magnitude and direction. In other words, it depicts the phase relationship between two or more sinusoidal waveforms having the same frequency.

Phasor is a straight line with an arrow at one end, rotating in an anticlockwise direction. The length of the line represents the magnitude of the sinusoidal quantity and the arrow represents the direction. Learn how to represent a phasor.

Phasor diagram

Let us consider two sinusoidal AC waveforms as shown below, in which the current(i) lag behind the voltage(v) by an angle φ.

The phasor diagram is drawn corresponding to time zero on the horizontal axis as the reference. The length of the phasor is proportional to the voltage or current that is considered at any instant.

Sinusoidal waveform

In this waveform, assume the phase difference or phase shift be φ = 300. While drawing the phasor diagram, one phasor is designated as a reference phasor. The other phasors are drawn either by lagging or leading with respect to the reference axis.

If you observe the above waveform, the voltage waveform(v = Vm sin ωt) starts at time zero on the horizontal axis. Hence it is designated as the reference phasor and is drawn along the horizontal reference axis as shown below in blue color.

The current waveform(i = Im sin ωt) reaches the horizontal axis with a delay by an angle of φ = 300, called the lagging phase difference. As the phasors are rotating in an anticlockwise direction, the current phasor is drawn behind the voltage phasor by 300 as shown below in red color.

Phasor diagram - lagging

On the other hand, if the current waveform is ahead of the voltage by the same angle φ = 300, it is called the leading phase difference. In this case, the current phasor is drawn ahead of the voltage phasor in the anticlockwise direction by angle 300.

Phasor diagram - leading

Phasor Addition

Most of the time, we deal with two or more alternating quantities in AC circuit analysis. In such cases, it is necessary to add or subtract two or more sinusoidal waveforms having the same frequency. It can be done either by graphical method or by analytical method.

Let us take a look at those two methods in detail.

Graphical Method

In this method, the addition or subtraction of two waveforms is done by plotting the phasors on the scale. If two waveforms are in-phase with each other, they can be added similar to the addition of two DC values.

For example, let us consider two AC voltages V1 = 25 V and V2 = 30 V, which are in-phase with each other. They can be added together to get the total voltage of VT = V1 + V2 = 25 + 30 = 55 V. Its phasor diagram is shown below.

Phasor diagram for addition

How to add two phasors?

To add two sinusoidal waveforms that are out of phase with each other, their phase angle must be considered.

Choose a phasor as a reference and draw it along the X-axis. Now, draw the other phasors at that instant, one after another. To get the resultant phasor, join the origin of the first phasor and the endpoint of the final phasor.

Now, the length of the phasor from origin to the last point represents the magnitude of the resultant phasor. The angle formed by the resultant phasor with respect to the reference axis represents the phase angle.

For example, let us consider two voltages V1 = 25 V and V2 = 30 V respectively, where the voltage phasor V1 is rotating ahead (leading) of the voltage V2 by some angle, say φ = 650.

Since voltage phasor V1 leads the other voltage phasor V2, consider V2 phasor as a reference. Hence, draw the phasor V2(OA) on the reference axis as shown in the figure below. From the endpoint(A) of the first phasor, draw the other phasor V1(AB) with an angle 650 with respect to the reference axis.

out of phase - phasor diagram

Now join the origin(O) and endpoint of the final phasor(B). OB is the resultant phasor, whose magnitude is given by the length of OB(VT = 55 V) and phase angle is given by \angle AOB = \phi_r.

Analytical method

Using this method, the phasors are represented in rectangular form. In rectangular form, a phasor is represented by a generalized equation Z = x ± jy, where ‘x’ is the real part and ‘y’ is the imaginary part. Using this generalized equation, a sinusoidal voltage is expressed as,

    \[v = V_m cos \phi + j V_m sin \phi \]

Here, the real part is x = V_mcos\phi and the imaginary part is y = V_msin\phi

Let A and B are two phasors, which are represented in rectangular form as,

    \[A = p + jq, B = r + js\]

For phasor addition, add the real and imaginary part of the two phasors individually as, ,

    \[A + B = (p + r) + j(q + s) \]

Similarly, for phasor subtraction, the resultant phasor is obtained as,

    \[A-B = (p - r) + j(q - s) \]

Learn about the rectangular form of representation of phasor.

Phasor addition using rectangular form

Consider two voltages V1 = 25 V and V2 = 30 V respectively, where the voltage V1 leads the voltage V2 by an angle φ = 650.

Using the generalized expression for rectangular form, the voltage phasor V1 and V2 can be written as

    \[V_1 = 25 cos \phi + j 25 sin \phi \]

    \[30 cos \phi + j 30 sin \phi \]

Since V1 is leading, the voltage phasor V2 is along the reference axis(which is the horizontal zero axis). It has a horizontal component but no vertical component. Thus the rectangular form for voltage V2 is given by,

    \[V_2 =  30 cos 0^0 + j30 sin 0^0 = 30 + j0 \]

The voltage V1 = 20 V leads voltage V2 by 65o, hence it has both horizontal and vertical components. The rectangular form for voltage V1 is given by,

    \[V_1 =  25 cos 65^0 + j25 sin 65^0 = 10.56 + j22.65 \]

The resultant voltage phasor is obtained by adding the above two phasors as,

    \[V_T = V_1 + V_2 = (30 + j0) + (10.56 + j22.65)\]

    \[V_T = (30 + 10.56) + j(0 + 22.65) \]

    \[ V_T = 40.56 + j22.65 V \]

The resultant voltage phasor has both real part x = 40.56 and imaginary part y = 22.65, which are horizontal and vertical components respectively.

Using Pythagorus theorem, the magnitude of the total voltage can be obtained as,

    \[V_T = \sqrt{(x)^2 +(y)^2} \]

    \[V_T = \sqrt{40.56^2 + 22.65^2} = 46.46 V\]

The resultant phase angle is obtained from the formula,

    \[\phi_r = tan^{-1}\left(\frac{y}{x}\right) = tan^{-1}\left(\frac{22.65}{40.56}\right) = 29.18^0 \]

Learn about the Polar form of representation of phasor.

FAQs

What is phasor?

Phasor is the representation of a sinusoidal waveform at any instant of time. Phasor is a rotating vector that has both magnitude and direction. It is represented by means of a straight line with an arrow at one end, where the straight line represents the magnitude and the arrow represents the direction.

What is Phasor Diagram?

Phasor Diagram is a graphical representation of the relation between two or more alternating quantities in terms of magnitude and direction. It depicts the phase relationship between the sinusoidal waveforms having the same frequency.

How to draw phasor diagram?

Choose a phasor as a reference and draw it along the horizontal axis. At that instant, draw the other phasors one after another from the origin point. The phasor drawn behind the reference phasor is said to have a lagging phase difference. Similarly, the phasor that is drawn ahead of the reference phasor is said to have the leading phase difference.

Categories

0 Comments

Submit a Comment

Your email address will not be published. Required fields are marked *

Shares
Share This