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.
- REVIEW:
- 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.
- A complex number is a
mathematical quantity representing two dimensions of magnitude and
direction.
- 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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