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.