Representation of AC Current and Voltage by Rotating Vectors

We have seen many electrical circuits which have resistors connected across an AC source, inductors across an AC source, capacitors across an AC source and also the combination of any two or all three of these components connected across an AC source. In case of a resistor, the current across a resistor is in phase with the voltage source. But in case of an inductor or a capacitor, the current either lags or leads the voltage source by any certain value. Now here is where we use the concept of phasors to relate the current and voltage.

Phasors

A phasor is a vector that is used to represent a sinusoidal signal. When we have more than one sinusoidal signal with the same frequency, different phase and different amplitude then we can use this phasor diagram to represent the phase difference between these sinusoidal signals.
The magnitude of the phasor represents the peak value of the voltage and the current.
The term “phasor” can also be applied to impedance and related complex quantities that are not time dependent.
The international standard is that phasors always rotate in the counterclockwise direction

We know that any sinusoidal signal in a general form can be represented as:

$$ A(t)=A_{m}\sin \left ( \omega t+\phi \right )$$

Where A_m=Peak Amplitude, ω = Angular Frequency and ϕ= Phase Shift. Now to represent the above sinusoidal signal using phasor. Now, this phasor is nothing but a vector which rotates around its origin at a constant speed of rad/s in an anticlockwise direction.

phasors Source: Wikipedia

So now as you can see in the above diagram as the vector rotates in an anticlockwise direction at a speed of rad/s and if we take the projection of this vector in Y-axis then we get the instantaneous value of this sinusoidal signal. If we dot all the values which are possible for this phasor then we can reproduce the sinusoidal signal.

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Rules for Drawing a Phasor Diagram

Rule 1: The length of the phasor is directly proportional to the amplitude of the wave depicted.
Rule 2: In circuits which have combinations of L, C & R in Series it is customary to draw the phasor representing Current horizontally and call this the Reference phasor. This is because the current in a series circuit is common to all the components.
Rule 3: In parallel circuits, where L, C and R are connected in parallel the phasor representing the Supply Voltage is always drawn in the Reference direction. This is because in a parallel circuit it is the supply voltage that is common to all components.
Rule 4: The direction of rotation of all phasors is considered to be Anticlockwise.
Rule 5: In any one diagram, the same type of value(RMS, peak etc)is used for all phasors, not a mixture of values.

Phasor Representation

There are three ways of phasor representation in mathematical form:

Polar Form: Suppose we have a phasor which has an amplitude of V_m and makes an angle with the horizontal axis. So in the polar form, we can represent it as V_m.
Rectangular Form :In this form we can represent any phasor as complex number like A+iB

$$r=\sqrt{\left ( A^{2}+B^{2} \right )}$$

$$ \phi =\arctan \left ( \frac{B}{A} \right )$$

Exponential Form : Here we represent the phasor in the form of as $$ V_{m}e^{j\phi }$$

Solved Questions for You

Question: The direction in which the phasor diagram rotates.
Solution: Anticlockwise.

Question: Mention the basic representation of phasor diagram
Solution:

Polar Form
Rectangular Form
Exponential Form

Question: Which property of a sine wave does the length of a phasor represent?
Solution: Amplitude

Question: In a phasor diagram, the frequency of each of the waves shown by the phasors is:

The same for all waves, and shown by the point of origin of the phasors.
Different for each wave, and proportional to the different lengths of the phasors.
Different for each wave, and shown by the change in angle of the phasors.
Not shown.

Solution: (d) Not shown.

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