48CHAPTER 2. COMPLEX NUMBERS6 V ∠ 45o-+. . . is equivalent to . . . -+6 V ∠ 225oFigure 2.35: Reversing polarity adds 180oto phase angle• REVIEW:• Polarity markings are sometimes given to AC voltages in circuit schematics in order to providea frame of reference for their phase angles.2.8Some examples with AC circuitsLet’s connect three AC voltage sources in series and use complex numbers to determine additivevoltages. All the rules and laws learned in the study of DC circuits apply to AC circuits as well(Ohm’s Law, Kirchhoﬀ’s Laws, network analysis methods), with the exception of power calculations(Joule’s Law). The only qualiﬁcation is that all variables must be expressed in complex form, takinginto account phase as well as magnitude, and all voltages and currents must be of the same frequency(in order that their phase relationships remain constant). (Figure 57,2.36)load+--+-+E1E2E322 V ∠ -64o12 V ∠ 35o15 V ∠ 0oFigure 2.36: KVL allows addition of complex voltages.The polarity marks for all three voltage sources are oriented in such a way that their stated volt-ages should add to make the total voltage across the load resistor. Notice that although magnitudeand phase angle is given for each AC voltage source, no frequency value is speciﬁed. If this is thecase, it is assumed that all frequencies are equal, thus meeting our qualiﬁcations for applying DCrules to an AC circuit (all ﬁgures given in complex form, all of the same frequency). The setup ofour equation to ﬁnd total voltage appears as such: