THE COMPLEX OR EXPONENTIAL FORM OF A FOURIER SERIES 699

L

=

20

10

(

5

−jπn

)[

e−j

πn

(^5) −ej
πn
5
]

20
πn
[
ej
πn
(^5) −e−j
πn
5
2 j
]
i.e. cn=
20
πn
sin
nπ
5
from equation (4), page 690.
From equation (13),
c 0 =
1
L
∫ L
2
−L 2
f(x)dx=
1
10
∫ 1
− 1
20 dt

1
10
[ 20 t]^1 − 1 =
1
10
[ 20 −(−20)]= 4
c 1 =
20
π
sin
π
5
=3.74 and
c− 1 =−
20
π
sin
(
−
π
5
)
=3.74
Further values ofcnandc−n,upton = 10, are
calculated and are shown in the following table.
n cn c−n
0 4 4
1 3.74 3.74
2 3.03 3.03
3 2.02 2.02
4 0.94 0.94
5 0 0
6 −0.62 −0.62
7 −0.86 −0.86
8 −0.76 −0.76
9 −0.42 −0.42
10 0 0
A graph of|cn|plotted against the number of the
harmonic,n, is shown in Figure 74.6.
Figure 74.7 shows the corresponding plot ofcn
againstn.
Sincecnis real (i.e. nojterms) then the phase
must be either 0◦or± 180 ◦, depending on the sign
of the sine, as shown in Figure 74.8.
When cn is positive, i.e. betweenn=−4 and
n=+4, angleαn= 0 ◦.
Whencnis negative, thenαn=± 180 ◦; between
n=+6 andn=+9, αnis taken as+ 180 ◦, and
betweenn=−6 andn=−9,αnis taken as− 180 ◦.
Figures 74.6 to 74.8 together form the spectrum of
the waveform shown in Figure 74.5.
74.6 Phasors
Electrical engineers in particular often need to anal-
yse alternating current circuits, i.e. circuits con-
taining a sinusoidal input and resulting sinusoidal
currents and voltages within the circuit.
It was shown in chapter 15, page 157, that a
general sinusoidal voltage function can be repre-
sented by:
v=Vmsin(ωt+α)volts (19)
whereVmis the maximum voltage or amplitude
of the voltagev,ωis the angular velocity (= 2 πf,
wherefis the frequency), andαis the phase angle
compared withv=Vmsinωt.
Similarly, a sinusoidal expression may also be
expressed in terms of cosine as:
v=Vmcos(ωt+α)volts (20)
It is quite complicated to add, subtract, multiply
and divide quantities in the time domain form of
equations (19) and (20). As an alternative method of
analysis a waveform representation called aphasor
is used. A phasor has two distinct parts—a mag-
nitude and an angle; for example, the polar form
of a complex number, say 5∠π/6, can represent a
phasor, where 5 is the magnitude or modulus, and
π/6 radians is the angle or argument. Also, it was
shown on page 264 that 5∠π/6 may be written as
5ejπ/^6 in exponential form.
In chapter 24, equation (4), page 264, it is shown
that:
ejθ=cosθ+jsinθ (21)
which is known asEuler’s formula.
From equation (21),
ej(ωt+α)=cos(ωt+α)+jsin(ωt+α)
and Vmej(ωt+α)=Vmcos(ωt+α)
+jVmsin(ωt+α)
Thus a sinusoidal varying voltage such as in equa-
tion (19) or equation (20) can be considered to be