Higher Engineering Mathematics, Sixth Edition

The complex or exponential form of a Fourier series 655

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 either
the real or the imaginary part ofVmej(ωt+α), depend-
ing on whether the cosine or sine function is being
considered.
Vmej(ωt+α)may be rewritten as Vmejωtejα since
am+n=am×anfrom the laws of indices, page 1.
The ejωtterm can be considered to arise from the fact
that a radius is rotated with an angular velocityω,and
αis the angle at which the radius starts to rotate at time
t=0 (see Chapter 14, page 143).
Thus,Vmejωtejαdefines aphasor. In a particular cir-
cuit the angular velocityωis the same for all the
elements thus the phasor can be adequately described
byVm∠α, as suggested above.

Alternatively, if

v=Vmcos(ωt+α)volts

and cosθ=

1

2

(

ejθ+e−jθ

)

from equation (3), page 644

then v=Vm

[

1

2

(

ej(ωt+α)+e−j(ωt+α)

)]

i.e. v=

1

2

Vmejωtejα+

1

2

Vme−jωte−jα

Thus,vis the sum of two phasors, each with half the
amplitude, with one having a positive value of angular
velocity(i.e.rotatinganticlockwise)andapositivevalue
ofα, and the other having a negative value of angular
velocity (i.e. rotatingclockwise) and a negative value of
α, as shown in Figure 71.9.

The two phasors are

1

2

Vm∠αand

1

2

Vm∠−α.

From equation (11), page 645, the Fourier representa-
tion of a waveform in complex form is:

cnej

2 πnt

L =cnejωnt for positive values ofn

(

sinceω=

2 π

L

)

and cne−jωnt for negative values ofn.

Real axis

Imaginary axis









0

12 Vm

1

(^2) Vm
Figure 71.9
It can thus be considered that these terms represent pha-
sors, those with positives powers being phasors rotating
with a positive angular velocity (i.e. anticlockwise), and
those with negative powers being phasors rotating with
a negative angular velocity (i.e. clockwise).
In the above equations,
n=0 represents a non-rotatingcomponent, since e^0 =1,
n=1 represents a rotating component with angular
velocity of 1ω,
n=2 represents a rotating component with angular
velocity of 2ω, and so on.
Thus we have a set of phasors, the algebraic sum of
which at some instant of time gives the magnitude of
the waveform at that time.
Problem 7. Determine the pair of phasors that
can be used to represent the following voltages:
(a)v=8cos2t (b)v=8cos(2t−1.5)
(a) From equation (3), page 644,
cosθ=
1
2
(ejθ+e−jθ)
Hence,
v=8cos2t= 8
[
1
2
(
ej^2 t+e−j^2 t
)]
=4ej^2 t+4e−j^2 t
This represents a phasor of length 4 rotating anti-
clockwise (i.e. in the positive direction) with an
angular velocity of 2rad/s, and another phasor
of length 4 and rotating clockwise (i.e. in the