A.2. ARITHMETIC REPRESENTATION OF
COMPLEX NUMBERS
A complex number C is represented in a number of
different ways in the literature, such as
Equation A1'
Equation A1''
Equation A1'''
Equations
(A1'') and (A1''')
remind us that the complex number C can also be considered the tip of a
phasor on the complex plane, with magnitude M, in the direction of ø degrees
relative to the positive real axis as shown in Figure A2.
(We'll avoid calling phasor M a
vector because the term vector means different things in different
contexts. In linear algebra, vector is the term used to signify a
onedimensional matrix. On the other hand, in mechanical
engineering and field theory, vectors are used to signify
magnitudes and directions, but there are vector operations (scalar or dot
product, and vector or
crossproduct) that don't apply to
our definition of a phasor.) The relationships between the
variables in this figure follow the standard trigonometry of right
triangles. Keep in mind that C is
a complex number, and the variables R, I,
M, and ø are all real numbers. The magnitude of C, sometimes called the modulus of C, is
Equation A2
and, by definition, the phase angle, or argument, of C is the arctangent of I/R,
or
Equation A3
The variable ø in Eq. (A3) is a general angle term. It can
have dimensions of degrees or radians. Of course, we can convert
back and forth between degrees and radians using p radians = 180°. So, if ø_{r} is in radians and
ø_{d} is in degrees,
then we can convert ø_{r} to degrees by the
expression
Equation A4
Likewise, we can convert ø_{d} to radians by the
expression
Equation A5
The exponential form of a complex number has an
interesting characteristic that we need to keep in mind. Whereas
only a single expression in rectangular form can describe a single
complex number, an infinite number of exponential expressions can
describe a single complex number; that is, while, in the
exponential form, a complex number C can be represented by C = Me^{j}^{ø}, it can
also be represented by
Equation A6
where n =
±1, ±2, ±3, . . . and ø is in radians. When
ø is in degrees, Eq.
(A6) is in the form
Equation A7
Equations
(A6) and (A7) are
almost selfexplanatory. They
indicate that the point on the complex plane represented by the tip
of the phasor C remains unchanged
if we rotate the phasor some integral multiple of 2p radians or an integral multiple of
360°. So, for example, if C =
Me^{j}^{(20}^{°}^{)},
then
Equation A8
The variable ø, the angle of the phasor in
Figure A2,
need not be constant. We'll often encounter expressions containing
a complex sinusoid that takes the form
Equation A9
Equation
(A9) represents a phasor of magnitude M whose angle in Figure A2 is
increasing linearly with time at a rate of w radians each second. If w = 2p, the phasor
described by Eq. (A9) is
rotating counterclockwise at a rate of 2p radians per second—one revolution per second—and
that's why w is called the radian
frequency. In terms of frequency, Eq. (A9)'s phasor is rotating
counterclockwise at w = 2pf radians
per second, where f is the cyclic
frequency in cycles per second (Hz). If the cyclic frequency is
f = 10 Hz, the phasor is rotating
20p radians per second. Likewise, the
expression
Equation A9'
represents a phasor of magnitude M that rotates in a clockwise direction
about the origin of the complex plane at a negative radian
frequency of –w radians per
second.
