Practical_Antenna_Handbook_0071639586

724 A p p e n d i x A : U s e f u l M a t h
A.6 Imaginary Numbers and the Complex Plane
Suppose, in the process of solving a math problem, you had to find the square root of
–1. What would you do? (Dropping math and enrolling in another field of study is not
an option!)
For most of us (and for purposes of this book), the only useful response is to say, “I
live in the real world, and I have no idea what the square root of –1 is or even what it
means, but I’m going to give it a shorthand notation so that I can easily work with it
whenever it comes up. In fact, I’m going to call it j.”
So j it is. That is,
j = − 1
(A.6.1)
In some older texts, i is used instead. (That sentence is one of the few known instances
of when it is grammatically correct to say “I is . . .”)
For clarity, let’s reverse the process and obtain the square of j:
2
j × j = j = − 1 × − 1 = − 1
(A.6.2)
To simplify the design and analysis of electronic circuits, scientists and engineers
invoke a graphical mathematical tool called the complex plane. Because it is a plane,
every point on it can be defined in relation to its two orthogonal axes. In particular, we
call the x axis the axis of real numbers and the y axis the axis of imaginary numbers. In
common usage, we speak of the real axis and the imaginary axis.
Thus, a number such as 3 or –7 or 6.14 or –5/8 lies at an appropriate spot on the x or
real axis. A number such as +j or –2pj or jl/2 lies at a similarly appropriate point on the
y or imaginary axis. But what about numbers that lie somewhere in the rest of the space,
or plane, between the real and imaginary axes? Just like ordinary graphing techniques,
we can represent them as having both an x value and a y value by writing them as (3,2)
or (–p,6) or any other appropriate combination, where it’s understood that the second
number is the imaginary component of the complex number being described.
Another way we can format the presentation of complex numbers is as A + jB. Thus,
(3,2) on the complex plane can be written as 3 + 2j, and (–p,6) can be written as –p + 6j.
In general, this is the notation used throughout this book.
When we add or subtract complex numbers, we add the real parts together and
separately we add the imaginary parts together.
Example A.6.1 What is the sum of (3 + 2j) and (–p + 6j)?
Solution
( 3 + 2 j) + ( −π + 6 j) = (3 − π ) + j( 2 + 6 ) = (3 − π ) + 8j
(A.6.3)
When discussing antennas and other electronic circuits, we can’t get very far without
dealing with inductances and capacitances, introduced in Chap. 3. There we ob-