# Introduction to AC Circuits

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## 3.7 Voltage and Current Phasors in Three-Phase Systems

As can be seen in equations 3.42, 3.43, and 3.44, the voltage in Phase 1 reaches a maximum first, followed by Phase 2 and then Phase 3 for sequence 1, 2, 3. This sequence should be evident from the phasor diagram of the three-phase source where the phasors should pass a fixed point in the order 1-2-3, 1-2-3, . . . .

In this section, the variation of the phasors in a three-phase ac circuit will be examined. The phasors are obtained by selecting one phase voltage as a reference with a phase angle of zero and determining the phase angles of the other two phases. Since the amplitudes and the frequencies of the voltage sources are equal, the phasors have equal lengths.

The phase and the line voltage phasors in the three-phase system can also be represented in polar form. Note that similar phasor representation can be given for the current waveforms if the phase angle between the voltage and the current is known.

Equation 3.48

Equation 3.49

where VPh is the phase voltage.

Fig. 3-20 shows the sequence and position of the voltage phasors in a three-phase star-connected ac circuit. As shown in the figure, there are two different ways to illustrate the phase and line voltages in the phasor form. The dotted lines in the figure demonstrate the line voltage phasors starting from origin, which is also used in this study.

Figure 3-20. The three-phase phasor representation.

#### 3.7.1 Virtual Instrument Panel

The objective of this section is to understand the phasors and the phase sequences in the three-phase balanced ac circuits.

In the VI named 3phase phasors.vi, two types of phasors are studied: for phase voltages and for line voltages.

As demonstrated in the front panel in Fig. 3-21, since one phase is always the reference, changing the phase angle affects all phasors equally and they rotate in the same direction as expected. On the front panel of the VI, the phase and the line voltages are also displayed in the time domain.

Figure 3-21. The front panel and brief user guide of 3phase phasors.vi.

#### 3.7.2 Self-Study Questions

Open and run the custom-written VI named 3phase phasors.vi in the Chapter 3 folder, and investigate the following questions.

 1: Use the knobs provided on the front panel to vary the voltage amplitude, frequency, and phase angle. Observe the changes in the voltage and current phasors and the waveforms. 2: Vary the phase angle clockwise and counterclockwise and observe the direction of the rotation of the phasors. Did the phase angles change, and if so, why? 3: Read the magnitude of each phasor from the phasor graph and compare with the values displayed in the time domain. Note: To make the comparison easy, you may display either phase or line quantities simultaneously, but not both.