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.
- REVIEW:
- DC voltages can only either directly
aid or directly oppose each other when connected in series. AC
voltages may aid or oppose to any degree depending on the phase
shift between them.
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