There are different electrical components like capacitors, inductors, and resistors that are used in an AC circuit. The current I and voltage V might be in-phase, lag or lead each other in these electrical components. Sometimes it is more complex and difficult to understand their relationship mathematically.

Phasor Diagram

The graphical representation that gives the relationship between two or more electrical quantities in an AC circuit, using magnitude and direction, is called a phasor diagram.

A phasor is a line with an arrowhead on one end showing the direction of the electrical quantity, and the other end of the line is pivoted at a fixed point called the origin. The length of the phasor line represents the magnitude of the electrical quantity, such as voltage and current.

A phasor is a complex number that has both magnitude and angle, the diagram which gives the relationship between the magnitude and the angle of an electrical quantity is called a phasor diagram.

Phase Difference

It is known as the difference in the phase angles of two electrical quantities. On applying the AC voltage to an inductor, the voltage reaches its maximum value at 90o before the current starts to flow at zero degrees.

But in capacitors, voltage is directly proportional to the charge between the plates of the capacitor. The current must flow to build up the voltage across the two plates of the capacitor. The current reaches its maximum value at 90o. The phase difference between voltage and current in capacitors 90o and can be represented by a phasor diagram as:

Phasor Diagram of RLC circuit

Suppose that we have an RLC circuit in which a resistor, inductor, and capacitor are connected in series with an AC voltage supply as shown:

• All the resistors, inductors, and capacitors are connected in series, so the current will be the same in all of them. So, the current phasor for all the components will be drawn along the x-axis, and we will take it as a reference to other phasors.
• In resistors both current and voltage are in the same phase. So, we draw the voltage V_R along the same axis of the current phasor.
• In inductors, the voltage leads by 90 degrees with the current. The voltage phasor for inductor V_L will be drawn perpendicular or at 90o to the current phasor.
• For capacitors, voltage lagged by 90 degrees from current. So the voltage phasor V_C for the capacitor will be drawn below the current phasor axis at 90o.

Where:

And:

Phasor Diagram for 3-Phase

Three voltages are generated by connecting three identical coils, having the same number of turns, on a rotor shaft at an angle of 120o with each other. It consists of three sinusoidal voltages 120 degrees out of phase with one another.

The phasor diagram for the three-phase voltage supply can be drawn as:

To identify each of the three phases, we use color codes red, yellow, and blue. The red one is taken as the reference phase of rotation. All three phasors rotate anti-clockwise direction with an angular speed of ω measured in radians per second. The sequence for rotation in three-phase is red to yellow and yellow to blue.

Voltage Equations for 3-Phase

Taking the red phase as a reference, the voltage equation for all three phases is as follows.

For red phase:

For yellow phase:

And for the blue phase:

Or:

Phasor Algebra

Phasor algebra is the application of mathematical operations like addition, subtraction, multiplication, and division to the phasors of various electrical quantities. With the help of phasor algebra, we can convert complex electrical circuits into simple algebraic equations and can easily solve them.

Phasor Addition

To add two or more phasors of electrical quantity, we have to split them into real and imaginary parts and add them separately. If the two phasors are in phase, then they can be added directly. For example, if V₁ = 25V and V₂ = 40V are in the same phase. We will simply add them directly and get the result V = V₁ + V₂ = 65V.

If two or more phasors are not in phase, for example, In an AC circuit two voltages across the two electrical components are as V₁ = 10V and V₂ = 20V and Voltage V₁ leads the voltage V₂ by 60o.

Horizontal and vertical components of voltage V₁ are:

So:

Similarly, the horizontal and vertical components of voltage V₂ are as:

So:

Now:

The magnitude of the resultant vector VT will be given by the resultant vector of V₁ and V₂.

Phasor Subtraction

Phasor subtraction is very similar to phasor addition:

Phasor Multiplication

Phasor multiplication can be done by using a polar form of vectors. V1 and V2 are vectors with phase angles θ₁ and θ₂ then:

And:

The phase angle of the resultant phasor will be given as:

Phasor Division

As phasor multiplication, phasor division is carried out by polar of two phasors. For illustration, if V1 and V2 are vectors with phase angles θ₁ and θ₂ then:

In polar form, we have:

The phasor resultant of two voltages will be as:

The phase angle of the phasor resultant can be found by:

Conclusion

The graphical representation of the relationship between two or more electrical quantities in an AC circuit using magnitude and direction is known as a phasor diagram. A phasor is a line with an arrowhead showing the direction and the length of the phasor is proportional to the magnitude of the electrical quantity. The other end of the phasor line is fixed to a point called the origin of the axis.