2.4.COMPLEX VECTOR ADDITION35on 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-shiftingthat source by 180o. (Figure 44,2.16)8 V180 deg-+8 V-+0 degThese voltage sourcesare equivalent!Figure 2.16: Example of equivalent voltage sources.2.4Complex vector additionIf vectors with uncommon angles are added, their magnitudes (lengths) add up quite diﬀerently thanthat of scalar magnitudes: (Figure 44,2.17)length = 6angle = 0 degreeslength = 8angle = 90 degreeslength = 10angle = 53.13degrees6 at 0 degrees8 at 90 degrees+10 at 53.13 degreesVector additionFigure 2.17: Vector magnitudes do not directly add for unequal angles.If two AC voltages – 90o out of phase – are added together by being connected in series, theirvoltage magnitudes do not directly add or subtract as with scalar voltages in DC. Instead, thesevoltage quantities are complex quantities, and just like the above vectors, which add up in a trigono-metric fashion, a 6 volt source at 0o added to an 8 volt source at 90o results in 10 volts at a phaseangle of 53.13o: (Figure 45,2.18)Compared to DC circuit analysis, this is very strange indeed. Note that it’s possible to obtainvoltmeter indications of 6 and 8 volts, respectively, across the two AC voltage sources, yet only read10 volts for a total voltage!There is no suitable DC analogy for what we’re seeing here with two AC voltages slightly outof phase. DC voltages can only directly aid or directly oppose, with nothing in between. WithAC, two voltages can be aiding or opposing one another to any degree between fully-aiding andfully-opposing, inclusive. Without the use of vector (complex number) notation to describe ACquantities, it would be very diﬃcult to perform mathematical calculations for AC circuit analysis.