While the fundamental signal usd in electrical engineering is the sinusoid, it can be expressed mathematically in terms of an even more fundamental signal:
the complex exponential.
Representing sinusoids in terms of complex exponentials is not a mathematical oddity.
Fluency with complex numbers and rational functions of complex variables is a critical skill all engineers master.
Understanding information and power system designs and developing new systems all hinge on using complex numbers.
In short, they are critical to modern electrical engineering, a realization made over a century ago.
The notion of the square root of
-1
-1
originated with the quadratic formula:
the solution of certain quadratic equations mathematically exists only if the so-called imaginary quantity
-1
-1
could be defined.
Euler first used ii for the
imaginary unit but that notation did not take hold until
roughly Ampère's time.
Ampère
used the symbol ii to denote
current (intensité de current).
It wasn't until the twentieth century that the importance of complex numbers to circuit
theory became evident. By then, using
ii for current was entrenched and
electrical engineers chose ⅈ
for writing complex numbers.
An imaginary number has the form
ⅈb=-b2
b
b
2
.
A complex number, zz, consists of the ordered pair (aa,bb), aa is the real component and bb is the imaginary component (the ⅈ is suppressed because the imaginary component of the pair is always in the second position).
The imaginary number
ⅈb
b
equals (00,bb).
Note that aa and bb are real-valued numbers.
Figure 1 shows that we can locate a complex number in what we call the complex plane.
Here, aa, the real part, is the xx-coordinate and bb, the imaginary part, is the yy-coordinate.
From analytic geometry, we know that locations in the plane can be expressed as the sum of vectors, with the vectors corresponding to the
xx and
yy directions.
Consequently, a complex number
zz can be expressed as the (vector) sum
z=a+ⅈb
z
a
b
where
ⅈ indicates the
yy-coordinate.
This representation is known as the
Cartesian form of zz.
An imaginary number can't be numerically added to a real number;
rather, this notation for a complex number represents vector addition, but it provides a convenient notation when we perform arithmetic manipulations.
Some obvious terminology.
The real part of the complex number
z=a+ⅈb
z
a
b
, written
as
ℜz
z
, equals aa. We consider the
real part as a function that works by selecting that component
of a complex number not multiplied by
ⅈ.
The imaginary part of zz,
ℑz
z
, equals bb:
that part of a complex number that is multiplied by
ⅈ.
Again, both the real and imaginary parts of a complex number are real-valued.
The complex conjugate of
zz, written as
z¯
z
,
has the same real part as zz
but an imaginary part of the opposite sign.
z=ℜz+ⅈℑz
z¯=ℜz−ⅈℑz
z
z
z
z
z
z
(1)
Using Cartesian notation, the following properties easily follow.
-
If we add two complex numbers, the real part of the result equals the sum of the real parts and the imaginary part equals the sum of the imaginary parts.
This property follows from the laws of vector addition.
a1+ⅈb1+a2+ⅈb2=a1+a2+ⅈb1+b2
a1
b1
a2
b2
a1
a2
b1
b2
In this way, the real and imaginary parts remain separate.
-
The product of ⅈ and a real number is an imaginary number:
ⅈa
a
.
The product of ⅈ and an imaginary number is a real number:
ⅈⅈb=-b
b
b
because
ⅈ2=-1
2
1
.
Consequently, multiplying a complex number by ⅈ rotates the number's position by 9090 degrees.
Use the definition of addition to show that the real and
imaginary parts can be expressed as a sum/difference
of a complex number and its conjugate.
ℜz=z+z¯2
z
z
z
2
and
ℑz=z−z¯2ⅈ
z
z
z
2
.
z+z¯=a+ⅈb+a−ⅈb=2a=2ℜz
z
z
a
b
a
b
2
a
2
z
. Similarly,
z−z¯=a+ⅈb−(a−ⅈb)=2ⅈb=
2
ⅈ
ℑz
z
z
a
b
a
b
2
b
2
z
Complex numbers can also be expressed in
an alternate form, polar form, which we will find
quite useful. Polar form arises arises from the geometric interpretation of complex numbers.
The Cartesian form of a complex number can be re-written as
a+ⅈb=a2+b2aa2+b2+ⅈba2+b2
a
b
a
2
b
2
a
a
2
b
2
b
a
2
b
2
By forming a right triangle having sides
aa and
bb, we see that the real and imaginary parts correspond to the cosine and sine of the triangle's base angle.
We thus obtain the
polar form for complex numbers.
z=a+ⅈb=r∠θ
r=|z|=a2+b2
a=rcosθ
b=rsinθ
θ=arctanba
z
a
b
r
∠
θ
r
z
a
2
b
2
a
r
θ
b
r
θ
θ
b
a
The quantity rr is known as the
magnitude of the complex number
zz, and is frequently written as
|z|
z
.
The quantity θθ is the
complex number's angle.
In using the arc-tangent formula to find the angle, we must take into account the quadrant in which the complex number lies.
Convert
3−2ⅈ
3
2
to polar form.
To convert
3−2ⅈ
3
2
to polar form, we first locate the number in the complex
plane in the fourth quadrant. The distance from the origin
to the complex number is the magnitude
rr, which equals
13=32+-22
13
3
2
2
2
. The angle equals
-arctan23
2
3
or -0.588-0.588 radians
(-33.733.7 degrees).
The final answer is
13∠-33.7
13
∠
33.7
degrees.
Surprisingly, the polar form of a complex number zz can be expressed mathematically as
z=rⅇⅈθ
z
r
θ
(2)
To show this result, we use
Euler's relations that express exponentials with imaginary arguments in terms of trigonometric functions.
ⅇⅈθ=cosθ+ⅈsinθ
θ
θ
θ
(3)
cosθ=ⅇⅈθ+ⅇ-ⅈθ2
θ
θ
θ
2
(4)
sinθ=ⅇⅈθ−ⅇ-ⅈθ2ⅈ
θ
θ
θ
2
The first of these is easily derived from the Taylor's series
for the exponential.
ⅇx=1+x1!+x22!+x33!+…
x
1
x
1
x
2
2
x
3
3
…
Substituting
ⅈθ
θ
for
xx, we find that
ⅇⅈθ=1+ⅈθ1!−θ22!−ⅈθ33!+…
θ
1
θ
1
θ
2
2
θ
3
3
…
because
ⅈ2=-1
2
-1
,
ⅈ3=-ⅈ
3
, and
ⅈ4=1
4
1
. Grouping separately
the real-valued terms and the imaginary-valued ones,
ⅇⅈθ=1−θ22!+…+ⅈθ1!−θ33!+…
θ
1
θ
2
2
…
θ
1
θ
3
3
…
The real-valued terms correspond to the Taylor's series for
cosθ
θ
, the imaginary ones to
sinθ
θ
,
and Euler's first relation results. The remaining relations
are easily derived from the first.
Because of
(Reference), we see that multiplying the exponential in
Equation 3 by a real constant corresponds to setting the radius of the complex number by the constant.
Adding and subtracting complex numbers expressed in Cartesian
form is quite easy: You add (subtract) the real parts and
imaginary parts separately.
z
1
±
z
2
=
a
1
±
a
2
+ⅈ
b
1
±
b
2
±
z
1
z
2
±
a
1
a
2
±
b
1
b
2
(5)
To multiply two complex numbers in Cartesian form is not quite
as easy, but follows directly from following the usual rules of arithmetic.
z
1
z
2
=
a
1
+ⅈ
b
1
a
2
+ⅈ
b
2
=
a
1
a
2
−
b
1
b
2
+ⅈ
a
1
b
2
+
a
2
b
1
z
1
z
2
a
1
b
1
a
2
b
2
a
1
a
2
b
1
b
2
a
1
b
2
a
2
b
1
(6)
Note that we are, in a sense, multiplying two vectors to obtain another vector.
Complex arithmetic provides a unique way of defining vector multiplication.
What is the product of a complex number and its conjugate?
zz¯=a+ⅈba−ⅈb=a2+b2
z
z
a
b
a
b
a
2
b
2
.
Thus,
zz¯=r2=|z|2
z
z
r2
z
2
.
Division requires mathematical manipulation. We convert the
division problem into a multiplication problem by multiplying
both the numerator and denominator by the conjugate of the
denominator.
z1z2=a1+ⅈb1a2+ⅈb2=a1+ⅈb1a2+ⅈb2a2−ⅈb2a2−ⅈb2=
a
1
+ⅈ
b
1
a
2
−ⅈ
b
2
a
2
2+
b
2
2=
a
1
a
2
+
b
1
b
2
+ⅈ
a
2
b
1
−
a
1
b
2
a
2
2+
b
2
2
z1
z2
a1
b1
a2
b2
a1
b1
a2
b2
a2
b2
a2
b2
a
1
b
1
a
2
b
2
a
2
2
b
2
2
a
1
a
2
b
1
b
2
a
2
b
1
a
1
b
2
a
2
2
b
2
2
(7)
Because the final result is so complicated, it's best to
remember
how to perform
division—multiplying numerator and denominator by the
complex conjugate of the denominator—than trying to
remember the final result.
The properties of the exponential make calculating the product and ratio of two complex numbers much simpler when the numbers are expressed in polar form.
z
1
z
2
=
r
1
ⅇⅈ
θ
1
r
2
ⅇⅈ
θ
2
=
r
1
r
2
ⅇⅈ
θ
1
+
θ
2
z
1
z
2
r
1
θ
1
r
2
θ
2
r
1
r
2
θ
1
θ
2
(8)
z
1
z
2
=
r
1
ⅇⅈ
θ
1
r
2
ⅇⅈ
θ
2
=
r
1
r
2
ⅇⅈ
θ
1
−
θ
2
z
1
z
2
r
1
θ
1
r
2
θ
2
r
1
r
2
θ
1
θ
2
To multiply, the radius equals the product of the radii and
the angle the sum of the angles. To divide, the radius equals
the ratio of the radii and the angle the difference of the
angles. When the original complex numbers are in Cartesian
form, it's usually worth translating into polar form, then
performing the multiplication or division (especially in the
case of the latter). Addition and subtraction of polar forms
amounts to converting to Cartesian form, performing the
arithmetic operation, and converting back to polar form.
When we solve circuit problems, the
crucial quantity, known as a transfer function, will always be
expressed as the ratio of polynomials in the variable
s=ⅈ2πf
s
2
f
. What we'll need to understand the circuit's effect
is the transfer function in polar form. For instance, suppose
the transfer function equals
s+2s2+s+1
s
2
s
2
s
1
(9)
s=ⅈ2πf
s
2
f
(10)
Performing the required division is most easily accomplished
by first expressing the numerator and denominator each in
polar form, then calculating the ratio. Thus,
s+2s2+s+1=ⅈ2πf+2-4π2f2+ⅈ2πf+1
s
2
s
2
s
1
2
f
2
-4
2
f
2
2
f
1
(11)
s+2s2+s+1
=4+4π2f2ⅇⅈarctanπf1−4π2f22+4π2f2ⅇⅈarctan2πf1−4π2f2
s
2
s
2
s
1
4
4
2
f
2
f
1
4
2
f
2
2
4
2
f
2
2
f
1
4
2
f
2
(12)
s+2s2+s+1
=4+4π2f21−4π2f2+16π4f4ⅇⅈarctanπf−arctan2πf1−4π2f2
s
2
s
2
s
1
4
4
2
f
2
1
4
2
f
2
16
4
f
4
f
2
f
1
4
2
f
2
(13)
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