# Introduction to AC Circuits

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## 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.

Equation 3.50 Equation 3.51 Since the line current in the balanced delta-connected loads, if this equation is substituted into equation 3.51, the total active load becomes

Equation 3.52 In Fig. 3-22b, however, the impedances contain the line currents Iline (= phase current, Iphase) and the phase voltages ). Therefore, the phase active power and the total active power are

Equation 3.53 Equation 3.54 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

Equation 3.55 Equation 3.56 #### 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 VrmsIrms 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

Equation 3.57 If the difference of the readings is computed,

Equation 3.58 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.

Equation 3.59  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.

 1: A balanced three-phase, three-wire star-connected load is connected to a three-phase supply. The line voltage is 400 V. The load comprises an impedance of 100 + j100 Ω per phase. Set these parameters and select the suitable circuit to determine the total active, reactive, and apparent power by using the VI provided. 2: Assume that the load in question 1 is a four-wire circuit. Use three power measurement methods and confirm your findings manually. 3: Three 10 μF capacitors are connected in star (Y) across a 2300 V (rms, line voltage), 60 Hz line. Calculate the line current, the active power, the reactive power, and the apparent power by using the VI. 4: A three-phase heater dissipates 15 kW when connected to a 208 V, three-phase line. Determine the value of each resistor if they are connected as star. 5: An industrial plant draws 600 kVA from a 2.4 kV line at a power factor of 0.8 lagging. What is the equivalent line-to-neutral impedance of the plant? 6: An electric motor having a power factor 0.82 draws a current of 25 A from a 600 V three-phase ac supply. Find the active power supplied to the motor. 7: Each phase of a delta-connected load comprises a resistor of 50 Ω and a capacitor of 50 μF in series. The three-phase load is connected to a 440 V (rms, line voltage) and 50 Hz three-phase star (Y)-connected supply. Calculate the phase and line currents, the power factor, the total active power, and total apparent power. Observe the phasor diagrams using the VI given in the previous section, 3phase phasors.vi. Hint: First, transform the delta-connected load to a star-connected equivalent (see Section 2.5). Then perform the power calculations. A7: Answers: 5.46 A (rms), 9.46 A (rms), 0.62, 4480 W, 7240 VA