Download Complex Number Arithmetic (Review)

Complex Number Arithmetic (Review)
By the mid-16th century, Italian mathematicians were experimenting with solutions to such equations
(x2 + 1 = 0) that involved the square roots of negative numbers. Swiss mathematician Leonhard Euler
introduced the symbol 'i' for √-1 in 1777. The result of the operation √-1 cannot be expressed in terms
of real (i.e., familiar) numbers, and so represents a different type of number called an imaginary number.
In ECE practice it is common to use the letter 'j' to represent √-1 since the mathematical convention of
using the letter 'i' for 'imaginary' conflicts with the established use of 'i' to represent electrical current
(from the French 'intensité').
The composite number Z = A + jB, where A and B are real numbers, is called a complex number.
Some important relationships and definitions:
a) If two complex numbers are equal then their real and imaginary parts are separately equal.
Thus given A+jB = C+jD, form (A-C)2 = [j(B-D)2] = -(B-D)2. The conclusion stated follows
on noting that the square of a real number not equal zero is positive.
b) Addition and subtraction of complex numbers are performed subject to the rules of ordinary
arithmetic, with the real and imaginary parts operated on separately.
Example:
(3 + j7) + (-5 + j9) = (3 - 5) + j (7 + 9) = -2 + j 16.
Multiplication and division also follow the rules of ordinary arithmetic with some
straightforward adaptations. It is helpful to keep in mind that j2 = (√-1)(√-1) = -1. A
multiplication example is:
Example:
(3 + j7) (-5 + j9) = (3)(-5) +(3)(j9) + (j7)(-5) + (j7)(j9)
= -15 + j27 -j35 -63 = -78 - j8
The complex conjugate of a complex number Z, indicated commonly as Z*, is formed by
changing the sign of the imaginary part. Thus if Z=A+jB then Z* = A-jB. Note that the
relationship is reciprocal, i.e., Z* is the conjugate of Z, and Z =Z** is the conjugate of Z*.
Example:
If Z = 2 - j7; Z* = 2 + j7
If Z = 3 + j13; Z* = 3 - j13
A very useful relationship in performing division is (verify by performing the multiplication):
Given Z = A + jB, then ZZ* = (A+ jB)(A - jB) = A2 + B2 = | A + jB|2
Example:
c) Euler's Theorem asserts ejθ = cosθ + j sinθ. (An interesting special case is ejπ = -1(two
transcendental numbers and the imaginary operator combine to result in a negative real
number)!)
Circuits Complex Arithmetic Notes
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M H Miller
d) Just as real numbers can be identified with points along a 'number line' complex numbers can
be identified with points in a plane. A complex number Z written in the form A + jB is said
to be written in 'rectangular' form. The real part is interpreted as the abscissa and the
imaginary part is interpreted as the ordinate. (A complex number is sometimes written as an
ordered pair (real, imaginary) as Cartesian coordinates.) An alternative form is the 'polar'
form, which is related to the rectangular form algebraically and geometrically as follows:
Example:
e) The 'parallelogram' relationship for combining complex numbers is illustrated below. Be
careful not to refer to the complex numbers as 'vectors'. They do not necessarily represent a
physical quantity with magnitude and direction. They are properly called 'phasors'.
f) From Euler’s theorem the following often useful identities can be determined:
ejθ = cosθ + j sinθ.
sinθ. = (ejθ − e-jθ)/2j
cosθ. = (ejθ + e-jθ)/2
sinhθ. = (eθ − e-θ)/2
coshθ. = (eθ + e-θ)/2
Circuits Complex Arithmetic Notes
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M H Miller
Several well-known trigonometric formulas are relatively easily derived using these
relationships. For example:
ej2θ =( ejθ)2 = cos2θ + j sin 2θ = ( cosθ + j sinθ.)2 = cos2θ – sin2θ + 2j sinθ. cosθ
and equating real and imaginary parts
cos2θ = cos2θ – sin2θ
sin2θ. = 2jsinθ. cosθ
Note that j = ejπ/2 . Hence if Z = A ejθ is a complex number in polar
form then jZ = ej(θ+π/2). Hence multiplication by the complex
operator j is equivalent to rotation of the polar
representation counterclockwise by 90º.
Incidentally multiplication by –1 is equivalent to multiplication by j x j,
corresponding to a counterclockwise rotation through 90º twice, i.e., by
180º.
g)
Another interesting use of complex arithmetic is illustrated by the following
calculation of the cube roots of 1.
You should be prepared to perform the operations listed below readily:
Add, subtract, multiply, divide complex numbers in rectangular form.
Multiply, divide complex numbers in polar form.
Convert rectangular form to polar form and polar form to rectangular
form.
Circuits Complex Arithmetic Notes
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M H Miller