Complex vector addition
If vectors with uncommon angles are added,
their magnitudes (lengths) add up quite differently than
that of scalar magnitudes:
If two AC voltages -- 90o out of
phase -- are added together by being connected in series,
their voltage magnitudes do not directly add or subtract as
with scalar voltages in DC. Instead, these voltage
quantities are complex quantities, and just like the above
vectors, which add up in a trigonometric fashion, a 6 volt
source at 0o added to an 8 volt source at 90o
results in 10 volts at a phase angle of 53.13o:
Compared to DC circuit analysis, this is
very strange indeed. Note that it's possible to obtain
voltmeter indications of 6 and 8 volts, respectively, across
the two AC voltage sources, yet only read 10 volts for a
total voltage!
There is no suitable DC analogy for what
we're seeing here with two AC voltages slightly out of
phase. DC voltages can only directly aid or directly oppose,
with nothing in between. With AC, two voltages can be aiding
or opposing one another to any degree between
fully-aiding and fully-opposing, inclusive. Without the use
of vector (complex number) notation to describe AC
quantities, it would be very difficult to perform
mathematical calculations for AC circuit analysis.
In the next section, we'll learn how to
represent vector quantities in symbolic rather than
graphical form. Vector and triangle diagrams suffice to
illustrate the general concept, but more precise methods of
symbolism must be used if any serious calculations are to be
performed on these quantities.
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