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

9.8 Phasors 341
example, in the case of a current signal, Irms 2 R is the average power dissipated by the
sinusoidIcos(ωt +φ)inaresistorR.Forthecaseofasinusoidalsignalther.m.s.value
ofthesignalis1/ √ 2timesthepeakvalueofthesignal(seeSection15.3,Example15.4,
foraproof of this).We will not adopt thisapproach butitisacommon one.
We start by examining the phasor representation of individual circuit elements. In
order todo thiswe need a phasor form of Ohm’s law. This is
Ṽ
=ĨZ (9.7)
where Ṽ is the voltage phasor, Ĩ is the current phasor and Z is the impedance of an
element or group of elements and may be a complex quantity. Note that phasors and
complex numbers are mixed together in the same equation. This is a common practice
because phasors areusually manipulated as complex numbers.
9.8.1 Resistor
Experimentally it can be shown that if an a.c. voltage is applied to a resistor then the
currentisinphasewiththevoltage.Theratioofthemagnitudeofthetwowaveformsis
equal to the resistance,R. So, givenĨ = I̸ 0,Z=R̸ 0, using Equation (9.7) we have
Ṽ=IR̸ 0.This isillustratedinFigure 9.11.
9.8.2 Inductor
Foraninductorweknowfromexperimentthatthevoltageleadsthecurrentbyaphaseof
π/2, and so the phase angle of the impedance is π/2. We also know that the magnitude
oftheimpedanceisgivenby ωL.So,givenĨ =I̸ 0,Z=ωL̸ π/2,usingEquation(9.7)
we haveṼ =IωL̸ π/2. An alternative way of representingZ for an inductor is to use
the Cartesian form, that is
(
Z=ωLe jπ/2 = ωL cos π )
2 +jsinπ 2
= jωL
Thisisusefulwhenphasorsneedtobeaddedandsubtracted.Thephasordiagramforan
inductor isillustratedinFigure 9.12.
9.8.3 Capacitor
For a capacitor it is known that the voltage lags the current by a phase of π/2 and the
magnitude of the impedance is given by 1
ωC . So givenĨ = I̸ 0,Z= 1
ωC ̸ −π/2, we
y
y
p –2
~
I
V ~
x
~
I x
V ~
Figure9.11
Phasordiagram foraresistor.
Figure9.12
Phasordiagram foran inductor.