Illustrating AC Voltage Addition Using Complex Numbers

In this tutorial we explore how three alternating‑current (AC) voltage sources combine when connected in series, using complex‑number arithmetic to determine the net voltage. The approach demonstrates that all familiar DC circuit laws—Ohm’s law, Kirchhoff’s voltage and current laws, and network analysis techniques—carry over to AC analysis provided every quantity is expressed in phasor (complex) form and all sources operate at the same frequency.

We begin by arranging the three sources as shown in the diagram below. The polarity marks are chosen so that the phasor directions add directly, enabling us to write the total voltage as a simple vector sum:

KVL allows the addition of complex voltages.

Although the problem statement does not specify an explicit frequency, the implicit assumption is that all three sources share the same frequency. This satisfies the requirement that their phase relationships remain constant, allowing the use of DC‑style rules.

The algebraic expression for the total voltage is:

V_total = V₁ + V₂ + V₃

Graphically, the phasor addition is illustrated in the figure below.

Graphic addition of vector voltages.

The resulting vector originates at the start of the 22‑V source and terminates at the tip of the 15‑V source, as shown in the next figure.

Resultant is equivalent to the vector sum of the three original voltages.

To compute the magnitude and phase of the resultant without resorting to a diagram, we convert each polar phasor to rectangular form, add them, and then convert back. The calculation yields:

V_total = 36.8052 V ∠ –20.5018°

In practical terms, a voltmeter would display the 36.8052 V magnitude only, whereas an oscilloscope could be used to determine the –20.5018° phase shift relative to the 15‑V reference source. The same principle applies to AC ammeters, which report the magnitude of current but not its phase angle.

Rectangular representation is convenient for arithmetic but has no direct empirical analogue in measurement; therefore, we routinely convert the final result back to polar form for comparison with real‑world instruments.

To validate the analytic result, we run a simple SPICE simulation. A 10 kΩ resistor is included merely to avoid an open‑circuit error. The chosen frequency of 60 Hz is arbitrary because resistors exhibit frequency‑independent behavior, though other elements such as capacitors and inductors would introduce frequency dependence.

Spice circuit schematic.

v1 1 0 ac 15 0 sin
v2 2 1 ac 12 35 sin
v3 3 2 ac 22 -64 sin
r1 3 0 10k
.ac link 1 60 60            ; I'm using a frequency of 60 Hz
.print ac v(3,0) vp(3,0)     ; default value
.end
freq           v(3)           vp(3)
6.000E+01      3.681E+01      -2.050E+01

The simulation confirms the analytic result: a total voltage of 36.81 V ∠ –20.5°. This seemingly counter‑intuitive outcome—obtaining just over 36 V from 15 V, 12 V, and 22 V sources in series—illustrates how phase relationships can cause constructive or destructive interference in AC circuits.

Next, we examine the effect of reversing the polarity of the 12‑V source. Reversing the leads changes the vector direction but not the nominal phase angle; mathematically this can be represented either by adding 180° to the angle (12 V ∠ 215°) or by negating the magnitude (–12 V ∠ 35°). Both approaches yield the same rectangular result:

In rectangular form, the new sum is:

V_total = 30.4964 V ∠ –60.9368°

We verify this with SPICE:

ac voltage addition
v1 1 0 ac 15 0 sin
v2 1 2 ac 12 35 sin            ; reversal of node numbers 2 and 1
v3 3 2 ac 22 -64 sin
r1 3 0 10k .ac lin 1 60 60
.print ac v(3,0) vp(3,0)
.end
freq           v(3)           vp(3)
6.000E+01      3.050E+01      -6.094E+01

Review

• All DC circuit laws apply to AC circuits—except power calculations—provided all variables are in complex form and share a common frequency.
• Reversing an AC source’s polarity can be expressed either by adding 180° to its angle or by negating its magnitude.
• Measured values in AC circuits correspond to the polar magnitudes of calculated phasors; rectangular values are useful only for intermediate algebraic steps.

Related Worksheets

• AC Network Analysis Worksheet
• Kirchhoff’s Laws Worksheet
• Ohm’s Law Worksheet