Environmental Engineering Reference
In-Depth Information
Imaginary
numbers
jV
Real numbers
-
V
V
-
jV
Figure A.14
Complex number plane
Imaginary
V
jb
V
q
Real
d
Figure A.15
Rectangular - polar conversion
It is also possible to write
V
=
V
(cos
θ
+ j sin
θ
). A famous expression known as Euler's
theorem states that cos
θ
+ j sin
θ
= e
j
θ
; therefore an alternative expression for
V
in polar form
is
V
e
jθ
(A.17)
V
=
Circuit calculations involving the addition or subtraction of complex numbers are carried
out conveniently using rectangular coordinates. Polar coordinates are more convenient when
multiplications and divisions are required. Since complex numbers are legitimate mathemati-
cal entities, just like scalar numbers they can be added, subtracted, multiplied, divided,
squared, inverted and so on, just like any other kind of number.
Addition and subtraction with complex numbers in rectangular form is easy. For addition,
the real components are simply added to determine the real component of the sum and the
imaginary components are added to determine the imaginary component of the sum. For
