Complex Vector Addition: How AC Voltages Combine

When vectors that are not aligned are added, their magnitudes do not combine in a simple arithmetic way. The resulting length depends on the angle between them.

For example, consider two vectors of unequal angles:

Vector magnitudes do not directly add when the angles differ.

AC voltages behave in the same way. If two sinusoidal sources that are 90° out of phase are connected in series, their instantaneous voltages do not simply add or cancel as DC voltages do. Instead, they are represented as complex quantities (phasors), and the combination follows trigonometric rules.

For instance, a 6‑V source at 0° added to an 8‑V source at 90° results in a total of 10 V at a phase angle of 53.13°:

The 6 V and 8 V sources add to 10 V thanks to trigonometry.

Unlike DC analysis, where voltages either fully aid or fully oppose each other, AC voltages can aid or oppose to any degree between fully aiding and fully opposing. This nuance is why a voltmeter may read 6 V and 8 V across the individual sources but only 10 V for the combined series voltage.

Without complex‑number notation, performing accurate AC circuit calculations would be cumbersome. In the next section, we’ll show how to represent these vector quantities symbolically, moving beyond simple graphical illustrations to the precise mathematics needed for rigorous analysis.

REVIEW:

• DC voltages can only directly aid or oppose each other in series.
• AC voltages can aid or oppose to any degree depending on their phase difference.

RELATED WORKSHEET:

• AC phase Worksheet