Introduction
If I needed to describe the distance between
two cities, I could provide an answer consisting of a single
number in miles, kilometers, or some other unit of linear
measurement. However, if I were to describe how to travel
from one city to another, I would have to provide more
information than just the distance between those two cities;
I would also have to provide information about the
direction to travel, as well.
The kind of information that expresses a
single dimension, such as linear distance, is called a
scalar quantity in mathematics. Scalar numbers are the
kind of numbers you've used in most all of your mathematical
applications so far. The voltage produced by a battery, for
example, is a scalar quantity. So is the resistance of a
piece of wire (ohms), or the current through it (amps).
However, when we begin to analyze
alternating current circuits, we find that quantities of
voltage, current, and even resistance (called impedance
in AC) are not the familiar one-dimensional quantities we're
used to measuring in DC circuits. Rather, these quantities,
because they're dynamic (alternating in direction and
amplitude), possess other dimensions that must be taken into
account. Frequency and phase shift are two of these
dimensions that come into play. Even with relatively simple
AC circuits, where we're only dealing with a single
frequency, we still have the dimension of phase shift to
contend with in addition to the amplitude.
In order to successfully analyze AC
circuits, we need to work with mathematical objects and
techniques capable of representing these multi-dimensional
quantities. Here is where we need to abandon scalar numbers
for something better suited: complex numbers. Just
like the example of giving directions from one city to
another, AC quantities in a single-frequency circuit have
both amplitude (analogy: distance) and phase shift (analogy:
direction). A complex number is a single mathematical
quantity able to express these two dimensions of amplitude
and phase shift at once.
Complex numbers are easier to grasp when
they're represented graphically. If I draw a line with a
certain length (magnitude) and angle (direction), I have a
graphic representation of a complex number which is commonly
known in physics as a vector:
Like distances and directions on a map,
there must be some common frame of reference for angle
figures to have any meaning. In this case, directly right is
considered to be 0o, and angles are counted in a
positive direction going counter-clockwise:
The idea of representing a number in
graphical form is nothing new. We all learned this in grade
school with the "number line:"
We even learned how addition and subtraction
works by seeing how lengths (magnitudes) stacked up to give
a final answer:
Later, we learned that there were ways to
designate the values between the whole numbers marked
on the line. These were fractional or decimal quantities:
Later yet we learned that the number line
could extend to the left of zero as well:
These fields of numbers (whole, integer,
rational, irrational, real, etc.) learned in grade school
share a common trait: they're all one-dimensional.
The straightness of the number line illustrates this
graphically. You can move up or down the number line, but
all "motion" along that line is restricted to a single axis
(horizontal). One-dimensional, scalar numbers are perfectly
adequate for counting beads, representing weight, or
measuring DC battery voltage, but they fall short of being
able to represent something more complex like the distance
and direction between two cities, or the amplitude
and phase of an AC waveform. To represent these kinds of
quantities, we need multidimensional representations. In
other words, we need a number line that can point in
different directions, and that's exactly what a vector is.
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REVIEW:
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A scalar number is the type of
mathematical object that people are used to using in
everyday life: a one-dimensional quantity like
temperature, length, weight, etc.
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A complex number is a mathematical
quantity representing two dimensions of magnitude and
direction.
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A vector is a graphical
representation of a complex number. It looks like an
arrow, with a starting point, a tip, a definite length,
and a definite direction. Sometimes the word phasor
is used in electrical applications where the angle of the
vector represents phase shift between waveforms.
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