We know that alternating voltages and alternating currents in a general power system are sinusoidal. But it is quite tricky to handle the instantaneous values of the alternating voltages and currents in our practical calculations. It is why a conventional method of representing sinusoidal waves comes into the picture. In this method, we represent the sinusoidal voltage and current waves with the help of vectors. Hence we call it the **vector representation of alternating quantities**.

### Concept of Vector

A vector is a physical quantity that has a specific value and a particular direction of action. We represent a vector with an arrow-headed straight line. The length of the straight line represents the absolute value of the quantity. We draw a vector with an angular inclination in respect of a reference axis. The angular inclination represents the direction of action of the vector with respect to the reference axis.

#### Vector Representation of Alternating Current and Alternating Voltage

A vector representing an alternating quantity rotates counterclockwise with the RPS equal to the frequency of the alternating quantity. The length of the vector represents the RMS value of the alternating quantity. But it is impossible to draw a rotating vector practically. Therefore we draw a static straight line at a certain angle with respect to the horizontal axis. The angle represents the phase of the alternating quantity.

## Vector Diagram of Sinusoidal Waves

We will consider an alternating current and an alternating voltage as follows. The frequency of both of the alternating quantities is the same. Therefore theoretically, the vectors representing these two quantities rotate counterclockwise with the same angular speed. There is a phase difference between these two alternating quantities. Here, the current lags the voltage by an angle α.

Therefore we first draw the voltage vector along the x-axis or the horizontal axis of the vector diagram. The physical length on a certain scale should represent the RMS value of that voltage. We have drawn the voltage vector along the x-axis because we have taken this vector as our reference. Then we will draw the current vector from the starting end of the voltage vector. Since the current lags the voltage by an angle α, we draw it with an alignment of α.

This is how we can represent alternating currents and voltages through a vector diagram. If there were more than two alternating quantities to be represented, the same technique could be applied.

Here we should remember that we cannot represent two alternating quantities of different frequencies in the same vector diagram. This is because theoretically, the vectors rotate in two different angular speeds. Therefore the angle between them cannot be a constant rather it changes continuously. In other words, the phase difference between these two quantities changes continuously. This is the reason it is next to impossible to predict the actual phase difference between these two alternating quantities.

### Addition of Two Alternating Sinusoidal Quantities

We can add two similar sinusoidal quantities. The sum of two sinusoidal waves gives another similar wave with the same frequency but with a different amplitude and phase.

The algebraic sum of the magnitude of two waves at every instant gives the magnitude of the resultant wave at that instant. The vector sum of the two original vectors gives the RMS value or the physical length of the resultant vector. The angle with the x-axis formed by the resultant vector is the phase of the resultant alternating quantity.

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