2 |
1 Complex Numbers for Harmonic Functions |
1.1Review of Complex Numbers
This section and the next section cover the basic formulas of complex numbers in either high schoolor college-level mathematics courses. The goal of these two sections is to refresh your knowledge of complex numbers for harmonic functions. You can skim through these two sections if you already know these formulas.
So what are complex numbers? The polynomial equation:
X2 þ 1 ¼ 0
does not have a solution that can be represented by real numbers. Instead, the solution of this polynomial equation can be represented by the complex numbers:
p
X ¼ 1
The square root of -1 is not a real number and has been defined as an imaginary number.
Hence, a combination of a real and an imaginary number is called a complex number.
For example:
Z ¼ a þ jb
where a and b are real numbers, j is the imaginary number, and the bold face character indicates that Z is a complex number.
The following are some properties of complex numbers:
• The square of the imaginary number j is 1. That is:
j2 ¼ 1
• The absolute value of a complex number is called its modulus. That is:
j j |
¼ Z ¼ |
p |
Z |
a2 þ b2 |
• The complex conjugate of Z ¼ a + jb is defined as:
Z ¼ a jb
• The addition of a complex conjugate function pair is a real number as shown:
Z þ Z ¼ a þ jb þ a jb ¼ 2a
The star symbol (*) indicates the complex conjugate of Z.
1.2 Complex Numbers in Polar Form |
3 |
1.2Complex Numbers in Polar Form
A complex number can be presented in both rectangular and polar coordinates as:
Z ¼ Ar þ jAi ¼ Ar2 þ Ai2 |
Ar |
Ai2 |
þ j |
Ai |
Ai2 |
! |
¼ Að cos ðθÞ þ j sin ðθÞÞ |
|||
Ar2 |
|
Ar2 |
|
|||||||
q |
|
þ |
|
|
|
|
þ |
|
|
|
|
q |
|
q |
|
|
|||||
where:
|
¼ q |
|
|
|
1 |
Ai |
|
|
|
|
|
|
|
|
||
A |
A2 |
þ |
A2 |
; θ |
¼ |
tan |
|
; A |
r ¼ |
A cos θ |
; A |
i ¼ |
A sin |
θ |
Þ |
|
|
r |
i |
|
|
Ar |
|
ð Þ |
|
|
ð |
||||||
Note that the complex number in polar coordinates (shown above) is formulated using trigonometric functions such as cosine and sine functions. The complex number in polar coordinates can be further expressed in a complex exponential format using Euler’s formula. Euler’s formula shows the relationship between the trigonometric functions and complex exponential functions:
e jθ ¼ cos ðθÞ j sin ðθÞ
Based on Euler’s formula, a complex number can be presented as a complex exponential function in polar form as:
Z ¼ Að cos ðθÞ þ j sin ðθÞÞ ¼ Aejθ
4 |
1 Complex Numbers for Harmonic Functions |
Imag
Real
-
Based on the equation above, a real or imaginary number can be expressed in the complex exponential format as:
1 ¼ cos ð0Þ þ j sin ð0Þ ¼ e jð0Þ
π |
π |
π |
j ¼ cos 2 |
þ j sin 2 ¼ e |
jð2Þ |
1 ¼ cos ðπÞ þ j sin ðπÞ ¼ e jðπÞ |
||
π |
π |
π |
j ¼ cos 2 þ j sin 2 |
¼ e jð 2Þ |
|
1.3Four Equivalent Forms to Represent Harmonic Waves
A summary of the four equivalent forms for simple harmonic motion is shown in this section. The derivations of these four equivalent forms are shown in the following sections.
Four equivalent forms can be used to describe simple harmonic motion. The following is a summary of four equivalent forms used to describe simple harmonic motions:
(Form 1) |
Ac cos (ωt) As sin (ωt) |
|
|
|
||||||||
(Form 2) |
A cos (ωt + ϕ) |
|
|
|
|
|
|
|||||
(Form 3) |
21 |
Ae jðωtþϕÞ þ Ae jðωtþϕÞ |
jAs |
e jωt |
|
|||||||
(Form 4) |
21 |
|
Ac |
þ |
jAs |
ejωt |
þ ð |
Ac |
|
& |
||
|
|
½ð |
|
Þ |
|
|
Þ |
|
||||
Real trigonometric function Real trigonometric function Complex conjugate function pair
Complex conjugate function pair
where A, Ac, and As are real numbers and have the relationships shown in the following table. In the figure below, A is a positive number; Ac and As are either positive or negative numbers. Therefore, the phase angle ϕ is between π and π, and
1.3 Four Equivalent Forms to Represent Harmonic Waves |
5 |
|||
ϕ ¼ tan |
1 |
As |
can be computed by the function atan2(As, Ac) in most programming |
|
|
Ac |
|||
codes. |
|
|
|
|
Note that Form 1 and Form 2 are formulated using real numbers (R). Form 3 and Form 4 are formulated using complex numbers (C). In addition, Form 1 and Form 4 are formulated using implicit phase (IP) of ϕ. Form 2 and Form 3 are formulated using explicit phase (EP) of ϕ. According to these properties, the four forms above can be rearranged into the following table:
|
|
|
|
Real |
|
|
|
|
|
Complex |
|||||
|
|
Trigonometric Function |
|
|
|
|
|
Conjugate Function Pair |
|||||||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
cos |
sin |
1 |
|
+ |
|
|
|
||||||
|
|
|
|
|
|
|
|
|
|
||||||
|
|
2 |
|
|
|||||||||||
|
|
|
|
Form 1: |
|
|
|
|
|
||||||
|
|
|
|
|
|
|
|
|
Form 4: |
|
|||||
|
|
|
|
|
|
|
|
|
|
|
|
|
|||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
1 |
|
|
|
|
||||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
||
|
|
|
Form 2: |
2 |
|
|
|
|
|||||||
|
|
|
|
|
|
|
|
Form 3: |
|
||||||
|
|
|
|
|
|
|
|
|
|
|
|
|
|||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
R: Real Trigonometric Function |
IP: Implicit Phase of |
||||||||||||
|
|
C: Complex Conjugate Function |
EP: Explicit Phase of |
||||||||||||
where: |
|
|
|
|
|
|
|
|
|
|
|
|
|||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
||
|
|
Geometric Relationship |
|
|
|
|
|
Imag |
|||||||
|
|
|
|
|
|
|
|
|
|
|
|
||||
|
Geom 1 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Geom 2 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Geom 3 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Geom 4 |
|
= sin |
|
|
|
|
|
|
|
Real |
||||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
||
Note that Form 3 can also be expressed as:
(Form 3) 12 Ae jðωtþϕÞ þ Ae jðωtþϕÞ ¼ 12 Aejωtejϕ þ Ae jωte jϕ (Form 3.5)
Also, Form 3 and Form 4 are often expressed with a Re[] function which takes the real part of the complex number as Form 3’ and Form 4’.
(Form 3) |
21 |
|
Ae jðωtþϕÞ |
þ Ae jðωtþϕÞ |
¼ Re |
Ae jðωtþϕÞ |
jAs e |
|
|
(Form 3’) |
|||||||||
(Form 4) |
2 |
Ac |
jAs |
|
e |
jωt |
Ac |
jAs |
|
e |
|
|
= Re |
Ac |
jωt |
|
(Form 4’) |
||
|
1 |
½ð þ |
|
Þ |
|
þ ð |
|
Þ |
|
jωt |
& |
|
½ð þ |
Þ |
& |
|
|||
|
|
|
|
|
|
|
|
|
|
|
|||||||||