Higher Engineering Mathematics

THE COMPLEX OR EXPONENTIAL FORM OF A FOURIER SERIES 701

L

either the real or the imaginary part ofVmej(ωt+α),
depending on whether the cosine or sine function is
being considered.

Vmej(ωt+α)may be rewritten asVmejωtejαsince
am+n=am×anfrom the laws of indices, pageX.
The ejωt term can be considered to arise from
the fact that a radius is rotated with an angu-
lar velocityω, andα is the angle at which the
radius starts to rotate at timet=0 (see Chapter 15,
page 157).
Thus,Vmejωtejαdefines aphasor. In a particular
circuit 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 690

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. rotating anticlockwise) and a
positive value ofα, and the other having a negative
value of angular velocity (i.e. rotating clockwise)
and a negative value ofα, as shown in Figure 74.9.

The two phasors are

1

2

Vm∠αand

1

2

Vm∠−α.

Imaginary axis

(^0) Real axis
Vm
Vm
α
α
ω
ω
(^12)
21
Figure 74.9
From equation (11), page 691, the Fourier repre-
sentation 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.
It can thus be considered that these terms represent
phasors, those with positives powers being phasors
rotating with a positive angular velocity (i.e. anti-
clockwise), 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-rotating component, 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=8 cos 2t (b)v=8 cos (2t−1.5)
(a) From equation (3), page 690,
cosθ=
1
2
(ejθ+e−jθ)
Hence,
v=8 cos 2t = 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 2 rad/s, and another phasor
of length 4 and rotating clockwise (i.e. in the
negative direction) with an angular velocity of
2 rad/s. Both phasors have zero phase angle.
Figure 74.10 shows the two phasors.