vector addition

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 quantitie: (Figure

Vector magnitudes add like scalars for a common angle.
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: (Figure )

“In phase” AC voltages add like DC battery voltages.
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 (180^o out of phase) are added together, their magnitudes (lengths) subtract just like positive and negative scalar quantities subtract when added: (Figure

Directly opposing vector magnitudes subtract.
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: (Figure )

Opposing AC voltages subtract like opposing battery voltages.
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 0^o and the other a phase angle of 180^o. 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: (Figure )

Opposing voltages in spite of equal phase angles.
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 (0^o), 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. (Figure )

Just as there are two ways to express the phase of the sources, there are two ways to express the resultant their sum.
The resultant voltage can be expressed in two different ways: 2 volts at 180^o with the (-) symbol on the left and the (+) symbol on the right, or 2 volts at 0^o 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 180^o. (Figure )

Example of equivalent voltage sources.

Complex vector addition

If vectors with uncommon angles are added, their magnitudes (lengths) add up quite differently than that of scalar magnitudes: (Figure )

Vector magnitudes do not directly add for unequal angles.
If two AC voltages — 90^o 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 0^o added to an 8 volt source at 90^o result in 10 volts at a phase angle of 53.13^o: (Figure )

The 6V and 8V sources add to 10V with the help of trigonometry.
Compared to DC circuit analysis, this is very strange indeed. Note that its 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.