The complex or exponential form of a Fourier series 653

The complex coefficient is given by equation (12):

cn=

1

L

∫ L

2

−L 2

f(t)e−j

2 πLnt

dt

=

1

10

∫ 1

− 1

20e−j

2 π 10 nt

dt=

20

10

[

e−j

πnt

5

−jπn

5

] 1

− 1

=

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 644.
From equation (13),
c 0 =
1
L
∫ L
2
−L 2
f(x)dx=
1
10
∫ 1
− 1
20dt

1
10
[20t]^1 − 1 =
1
10
[20−(− 20 )]= 4
c 1 =
20
π
sin
π
5
=3.74and
c− 1 =−
20
π
sin
(
−
π
5
)
=3.74
Furthervaluesofcnandc−n,upton=10,arecalculated
andareshowninthefollowingtable.
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 71.6.
Figure 71.7 shows the corresponding plot of cn
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 71.8.
Whencnis positive, i.e. betweenn=−4andn=+ 4 ,
angleαn= 0 ◦.
Whencnis negative, thenαn=± 180 ◦; betweenn=+ 6
andn=+9,αnis taken as+ 180 ◦, and betweenn=− 6
andn=−9,αnis taken as− 180 ◦.
Figures 71.6 to 71.8 together form the spectrum of the
waveform shown in Figure 71.5.

71.6 Phasors

Electrical engineers in particular often need to anal-

yse alternating current circuits, i.e. circuits containing

a sinusoidal input and resulting sinusoidal currents and

voltages within the circuit.

It was shown in chapter 14, page 143, that a general

sinusoidal voltage function can be represented 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 with

v=Vmsinωt.

Similarly,asinusoidal expressionmayalsobeexpressed

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

waveformrepresentationcalledaphasorisused. Apha-

sor has two distinct parts—a magnitude 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, andπ/6 radians is the angle or argument.

Also, it was shown on page 228 that 5∠π/6maybe

written as 5ejπ/^6 in exponential form.

In chapter 21, equation (4), page 228, it is shown that:

ejθ=cosθ+jsinθ (21)

which is known asEuler’s formula.