WIND ENERGY SYSTEMS - Cd3wd

Chapter 5—Electrical Network 5–5
✻
Imaginary Axis
y
✑✸
✑✑✑✑✑✑✑✑✑✑✑✑✑
✑ θ
x
Real Axis
z = x + jy
✲
.
Figure 1: Complex number on the complex plane.
and the angle is
|z| =
√
x 2 + y 2 (3)
θ =tan −1 y x
(4)
The angle is measured counterclockwise from the positive real axis, being 90 o on the
positive imaginary axis, 180 o on the negative real axis, 270 o on the negative imaginary axis,
and so on. The arctan function covers only 180 o so a sketch needs to be made of x and y in
each case and 180 o added to or subtracted from the value of θ determined in Eq. 4 as necessary
to get the correct angle.
We might also note that a complex number located on a complex plane is different from
a vector which shows direction in real space. Balloon flight in Chapter 3 was described by
a vector, with no complex numbers involved. Impedance will be described by a complex
number, with no direction in space involved. The distinction becomes important when a
given quantity has both properties. It is shown in books on electromagnetic theory that a
time varying electric field is a phasor-vector. That is, it has three vector components showing
direction in space, with each component being written as a complex number. Fortunately, we
will not need to examine any phasor-vectors in this text.
A number of hand calculators have the capability to go directly between Eqs. 1 and 2 by
pushing only one or two buttons. These calculators will normally display the full 360 o variation
in θ directly, saving the need to make a sketch. Such a calculator will be an important asset
in these two chapters. Calculations are much easier, and far fewer errors are made.
Addition and subtraction of complex numbers are performed in the rectangular form.
Wind Energy Systems by Dr. Gary L. Johnson November 21, 2001