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

342 Chapter 9 Complex numbers
have, using Equation (9.7),Ṽ = I
ωC ̸ −π/2. Alternatively,
Z = e−jπ/2
ωC = 1 (
cos π )
ωC 2 −jsinπ 2
= − j
ωC
Engineers often prefer torewrite this lastexpression as
1
jωC
The phasor diagram forthe capacitor isillustratedinFigure 9.13.
Wehaveshownhowphasorscanbemultipliedbyacomplexnumber;divisionisvery
similar.Addition ofphasors will nowbeillustrated;subtraction issimilar.Consider the
circuit shown in Figure 9.14 in which a resistor, capacitor and inductor are connected
in series and fed by an a.c. source. As this is a series circuit, the current through each
element,Ĩ,isthesamebyKirchhoff’scurrentlaw.ByKirchhoff’svoltagelawthevoltage
rise produced by the supply,Ṽ S
, must equal the sum of the voltage drops across the
elements. Therefore,
Ṽ S
=ṼR +ṼC +ṼL
Note that this is an addition of phasors so that the voltage drops across the elements
do not necessarily have the same phase. The phasor diagram for the circuit is shown in
Figure 9.15, inthiscase with |ṼL | > |ṼC |;Ĩ isthe reference phasor.
Notethatthephasoradditionoftheelementvoltagesgivestheoverallsupplyvoltage
for a particular supply current. If the magnitude of these element voltage phasors is
known then it is possible to calculate the supply voltage graphically. In practice it is
easier to convert the polar form of the phasors into Cartesian form and use algebra to
analyse the circuit.
Now,
Ṽ R
=ĨR̸
Ṽ L
=ĨωL̸
Ṽ C
=
0 =ĨR
π/2 =ĨjωL
Ĩ Ĩ
ωC ̸ −π/2 =
jωC
y
~ I x
–
p –2
V ~
v
~
I
R
~
V R
~
~
~ V S
C V C
L
~
V L
y
~
V L
~
V C
~
V S
~ ~
I V x R
Figure9.13
Phasordiagram foracapacitor.
Figure9.14
RLC circuit.
Figure9.15
Phasordiagram forthe circuit in
Figure 9.14.