# Introduction to AC Circuits

- Oct 24, 2003

- Fundamental Definitions
- AC Circuit Analysis
- Power and Power Triangles in AC Circuits
- Power Factor Correction
- Star-Delta and Delta-Star Conversion in Three-Phase AC Circuits
- Voltage and Currents in Star- and Delta-Connected Loads
- Voltage and Current Phasors in Three-Phase Systems
- Power in Three-Phase AC Circuits
- Three-Phase Power Measurement and Data Logging
- References

## 3.8 Power in Three-Phase AC Circuits

Since the phase impedances of a balanced star- or delta-connected load contain equal currents, the phase power is one-third of the total power. As a definition, the voltage across the load impedance and the current in the impedance can be used to compute the power per phase.

Let's assume that the angle between the phase voltage and the phase current is *θ*, which is equal to the angle of the impedance. Considering the load configurations given in Fig. 3-22, the phase power and the total power can be estimated easily.

**Figure 3-22. Per-phase powers in (a) a delta-connected load and (b) a star-connected load.**

In the case of Fig. 3-22a, the total active power is equal to three times the power of one phase.

Since the line current in the balanced delta-connected loads, if this equation is substituted into equation 3.51, the total active load becomes

In Fig. 3-22b, however, the impedances contain the line currents *I*_{line} (= phase current, *I*_{phase}) and the phase voltages ). Therefore, the phase active power and the total active power are

If the relationship between the phase voltage and the line voltage () is used, the total active power becomes identical to the equation developed in equation 3.52. This means that the total power in any balanced three-phase load (Δ- or Y-connected) is given by equation 3.52.

Similarly, the total reactive and the total apparent power in the three-phase balanced ac circuits can be given by

#### Power Measurement Techniques

In the three-phase power systems, one, two, or three wattmeters can be used to measure the total power. A wattmeter may be
considered to be a voltmeter and an ammeter combined in the same box, which has a deflection proportional to *V*_{rms}*I*_{rms} cos *θ*, where *θ* is the angle between the voltage and current. Hence, a wattmeter has two voltage and two current terminals, which have +
or − polarity signs. Three power measurement methods utilizing the wattmeters are described next, and are applied to the balanced
three-phase ac load.

#### 1 Two-Wattmeter Method

This method can be used in a three-phase three-wire balanced or unbalanced load system that may be connected Δ or Y. To perform the measurement, two wattmeters are connected as shown in Fig. 3-23.

**Figure 3-23. Two-wattmeter method in star- or delta-connected load.**

In the balanced loads, the sum of the two wattmeter readings gives the total power. This can be proven in a star-connected load mathematically using the power reading of each meter as

If the difference of the readings is computed,

which is times the total three-phase reactive power. This means that the two-wattmeter method can also indicate the total reactive power in the three-phase loads and also the power factor (see Fig. 3-24).

**Figure 3-24. Three-phase voltage phasors used in the two-wattmeter method.**

#### 2 Three-Wattmeter Method

This method is used in a three-phase four-wire balanced or unbalanced load. The connections are made with one meter in each line as shown in Fig. 3-25. In this configuration, the total active power supplied to the load is equal to the sum of the three wattmeter readings.

**Figure 3-25. The wattmeter connections in the three-phase four-wire loads.**

#### 3 One-Wattmeter Method

This method is suitable only in three-phase four-wire balanced loads. The connection of the wattmeter is similar to the drawing given in Fig. 3-25. The total power is equal to three times the reading of only one wattmeter that is connected between one phase and the neutral terminal.

#### 3.8.1 Virtual Instrument Panel

The objective of this section is to understand the powers and the power measurement techniques associated with the three-phase
ac circuits. Fig. 3-26 illustrates the front panel of the VI named `Three phase power measurements.vi`.

**Figure 3-26. The front panel and brief user guide of Three phase power measurements.vi.**

#### 3.8.2 Self-Study Questions

Before studying this VI, make sure that you study and understand the power concept presented in Section 3.2.2. Open and run the custom-written VI named `Three phase power measurements.vi` in the `Chapter 3` folder, and investigate the following questions.