# Scalar quantities, phasor concept

## Introduction

The available quantities for post-processing can be scalar or vector quantities.

This section deals with the scalar quantities and recalls several definitions: rotating vector, complex image, phasor…

## Rotating vector, complex image

A sinusoidal time varying scalar quantity of pulsation ω can be geometrically
represented by a vector rotating at the angular velocity **ω**.

This vector is the geometric representation of a complex number.

## Example: sinusoidal current and complex image

Let there be a sinusoidal time varying current **i(t)** of amplitude, of **ω**
pulsation (**ω = 2 π f**) and of **β** phase at the time t = 0.

The instantaneous value of this current, **i** : **i = Î.sin(ω.t + β)**

is equal to the imaginary part of the complex number **i** : **i = Im(i)**

Rotating vector with ω velocity |
Sinusoidal quantity i(t) of period T=2 π/ω |
---|---|

The complex instantaneous value of a sinusoidal current is given by the following relation:

- under a cartesian form:
**i = Î.cos(ω.t + β) + j.Î.sin(ω.t+ β)** - under an exponential form:
**i = Î.e j(ω.t+ β)**

where:

- is the modulus of the
complex value
**i** **ωt + β**is the argument (or the phase) of the complex value**i****β**is the initial phase

## Complex notation of a rotating vector

The rotating vector assigned to the sinusoidal quantity is expressed under the form
of a complex number **A**. This complex number can be written:

- under a carthesian form:
- under an exponential form:

## Phasor concept

The complex function
associated to the sinusoidal quantity **A(t)** can be decomposed into two factors as
follows:

This decomposition is presented in the table below.

The factor … | which can be written … | Corresponds to a … |
---|---|---|

rotating vector associated to the quantity
A(t) at t = 0 |
||

vector rotation by an angle
ωt |

- The factor contains information concerning the amplitude and the initial phase of the quantity
- The factor contains the information concerning the time variation of the quantity

## Phasor concept

The sinusoidal function in the time domain can be represented in the complex domain by the phasor .

## Phasor: definition

We call the phasor assigned to the quantity , i.e the phasor associated to the quantity is the rotating vector associated with this quantity at a time t=0.