Simple vector addition
Remember that vectors are mathematical
objects just like numbers on a number line: they can be added,
subtracted, multiplied, and divided. Addition is perhaps the easiest
vector operation to visualize, so we'll begin with that. If vectors with
common angles are added, their magnitudes (lengths) add up just like
regular scalar quantities:
Similarly, if AC voltage sources with the
same phase angle are connected together in series, their voltages add
just as you might expect with DC batteries:
Please note the (+) and (-) polarity
marks next to the leads of the two AC sources. Even though we know AC
doesn't have "polarity" in the same sense that DC does, these marks are
essential to knowing how to reference the given phase angles of the
voltages. This will become more apparent in the next example.
If vectors directly opposing each other
(180o out of phase) are added together, their magnitudes
(lengths) subtract just like positive and negative scalar quantities
subtract when added:
Similarly, if opposing AC voltage sources
are connected in series, their voltages subtract as you might expect
with DC batteries connected in an opposing fashion:
Determining whether or not these voltage
sources are opposing each other requires an examination of their
polarity markings and their phase angles. Notice how the polarity
markings in the above diagram seem to indicate additive voltages (from
left to right, we see - and + on the 6 volt source, - and + on the 8
volt source). Even though these polarity markings would normally
indicate an additive effect in a DC circuit (the two voltages
working together to produce a greater total voltage), in this AC circuit
they're actually pushing in opposite directions because one of those
voltages has a phase angle of 0o and the other a phase angle
of 180o. The result, of course, is a total voltage of 2
volts.
We could have just as well shown the
opposing voltages subtracting in series like this:
Note how the polarities appear to be
opposed to each other now, due to the reversal of wire connections on
the 8 volt source. Since both sources are described as having equal
phase angles (0o), they truly are opposed to one another, and
the overall effect is the same as the former scenario with "additive"
polarities and differing phase angles: a total voltage of only 2 volts.
The resultant voltage can be expressed in
two different ways: 2 volts at 180o with the (-) symbol on
the left and the (+) symbol on the right, or 2 volts at 0o
with the (+) symbol on the left and the (-) symbol on the right. A
reversal of wires from an AC voltage source is the same as
phase-shifting that source by 180o.
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