082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

340 Chapter 9 Complex numbers
9.8 PHASORS
Electrical engineers are often interested in analysing circuits in which there is an a.c.
power supply. Almost invariably the supply waveform is sinusoidal and the resulting
currentsandvoltageswithinthecircuitarealsosinusoidal.Forexample,atypicalvoltage
isof the form
v(t) =Vcos(ωt +φ) =Vcos(2πft +φ) (9.6)
whereV is the maximum or peak value, ω is the angular frequency, f is the frequency,
and φ is the phase relative to some reference waveform. This is known as the time
domain representation. Each of the voltages and currents in the circuit has the same
frequency as the supply but differs inmagnitude and phase.
In order to analyse such circuits it is necessary to add, subtract, multiply and divide
these waveforms. If the time domain representation is used then the mathematics
becomes extremely tedious. An alternative approach is to introduce a waveform representation
known as a phasor. A phasor is an entity consisting of two distinct parts: a
magnitude and an angle. It is possible to represent a phasor by a complex number in
polarform.Thefixedmagnitudeofthiscomplex number corresponds tothemagnitude
of the phasor and hence the amplitude of the waveform. The argument of this complex
number, φ, corresponds to the angle of the phasor and hence the phase angle of the
waveform. Figure 9.10 shows a phasor for the sinusoidal waveform of Equation (9.6).
The time dependency of the waveform is catered for by rotating the phasor anticlockwise
at an angular frequency, ω. The projection of the phasor onto the real axis
gives the instantaneous value of the waveform. However, the main interest of an engineer
is in the phase relationships between the various sinusoids. Therefore the phasors
are‘frozen’atacertainpointintime.Thismaybechosensothatt = 0oritmaybechosensothataconvenientphasor,knownasthereferencephasor,alignswiththepositive
realaxis.Thisapproachisvalidbecausethephaseandmagnituderelationshipsbetween
the various phasors remain the same at all points in time once a circuit has recovered
from any initial transients caused by switching.
Sometextbooksrefertophasorsasvectors.Thiscanleadtoconfusionasitispossible
todividephasorswhereasdivisionofvectorsbyothervectorsisnotdefined.Inpractice
this is not a problem as phasors, although thought of as vectors, are manipulated as
complex numbers, whichcan bedivided. We willavoidthese conceptual difficulties by
introducing a different notation. We will denote a phasor byṼ, which corresponds to
V ̸ φ incomplexnumbernotation(seeFigure9.10).Thus,forexample,acurrenti(t) =
Icos(ωt+φ)wouldbewrittenĨinphasornotationandI̸ φincomplexnumbernotation.
Many engineers use the root mean square (r.m.s.) value of a sinusoid as the magnitude
of a phasor. The justification for this is that it represents the value of a d.c. signal
that would dissipate the same amount of power in a resistor as the sinusoid. For
y
v
V
~
V
f
x
Figure9.10
Illustration ofthe phasorṼ =V̸ φ where ω =angular
frequency with which the phasor rotates.