# Introduction to AC Circuits

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## 3.6 Voltage and Currents in Star- and Delta-Connected Loads

A three-phase ac system consists of three voltage sources that supply power to loads connected to the supply lines, which can be connected to either delta (Δ) or star (Y) configurations as stated previously.

In three-phase systems, the voltages differ in phase 120°, and their frequency and amplitudes are equal. If the three-phase loads are balanced (each having equal impedances), the analysis of such a circuit can be simplified on a per-phase basis. This follows from the relationship that the per-phase real power and reactive power are one-third of the total real power and reactive power, respectively.

It is very convenient to carry out the calculations in a per-phase star-connected line to neutral basis. If Δ-Y, Y-Δ, or Δ-Δ connections are present, the parameters on Δ side(s) are transformed to Y-connection, and computations are carried out.

Two three-phase load connections that are commonly used in the ac circuits were given in Fig. 3-15. In this section, the voltage and the current functions are examined while the three-phase loads are connected to the star-connected three-phase supplies, shown in Fig. 3-18.

Figure 3-18. Two common balanced-load connections in three-phase ac circuits.

In Fig. 3-18, ν1s, ν2s, ν3s, ν1p, ν2p, ν3p, ν12p, ν23p, and ν31p are the phase voltage functions, and ν12, ν23, and ν31 are the line-to-line voltages (or simply line voltages). Similarly, i1p, i2p, i3p, i12p, i23p, and i31p are the phase currents, and i1L, i2L, and i3L are the line currents.

The phase voltages of a three-phase supply can be given as

Equation 3.42

Equation 3.43

Equation 3.44

In the case of sinusoidal steady-state operation, similar expressions can be written for the current waveforms with identical phase difference θ, which depend on the phase angle of the balanced load inductances.

Equation 3.45

Equation 3.46

Equation 3.47

A three-phase load is balanced when the line voltages are equal in magnitude and mutually displaced in phase by 2π/3 in radians and the line currents are equal. In a balanced three-phase system, there is a very simple relationship between the line and phase quantities, which can be obtained from the phasor quantities or the time-varying expressions of the voltages and the currents.

The voltage and current relationships in three-phase ac circuits can be simplified by using the rms values (I and V) of the quantities. Refer to Fig. 3-18, and study Table 3-1.

#### Table 3-1. Voltage and current relationships in three-phase circuits.

Phase current: I1p = I1L, I2p = I2L, I3p = I3L

Line current: IL = I1L = I2L = I3L

Phase current:

Line current: IL = I1L = I2L = I3L and Ip = I12p = I23p = I31p

Phase voltage:

Line voltage: VL = V12 = V23 = V31

Phase voltage: V12 = V12p, V23 = V23p, V31 = V31p

Line voltage: VL = V12 = V23 = V31 and Vp = V1p = V2p = V3p

The voltages across the impedances and the currents in the impedances are 120° out of phase.

#### 3.6.1 Virtual Instrument Panel

Fig. 3-19 shows the front panel of the VI named Voltage and currents in delta/star loads.vi. The VI provides a visual aid to understanding the definitions of phase and line voltages and phase and line currents in the delta- and the star-connected ac systems that contain the loads as well as the ac supplies. In addition, the instantaneous voltage and currents are displayed in the front panel of the VI.

Figure 3-19. The front panel and brief user guide of Voltage and currents in delta/star loads.vi.

#### 3.6.2 Self-Study Questions

Open and run the custom-written VI named Voltage and currents in delta/star loads.vi in the Chapter 3 folder, and investigate the following questions.

 1: Show that the line voltage Vline in the three-phase system is times the phase voltage Vphase, and verify the result by using the VI for a given phase voltage. 2: Study the concept in question 1 this time for the line currents and the phase currents in the case of a delta-connected three-phase load. 3: In question 2, find out the angles in degrees between the phase and the line quantities on the supply side and the load side. 4: Use the single-phase equivalent circuit in each load configuration and calculate the phase currents for given values of the voltage and the load impedance. 5: Three incandescent lamps rated 60 W, 120 V (rms) are connected in the delta form. What line voltage is needed so that the lamps burn normally (at rated conditions)? What are the line and phase currents in the circuit? Hint: First calculate and set the resistance of the lamps using the controls provided. 6: Three load resistors are connected in the delta form. If the line voltage is 415 V (rms) and the line current is 100 A (rms), calculate the current in each resistor, the voltage across the resistors, and the resistance of each resistor. Verify the results analytically. 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-connected supply. Calculate the phase and line currents. A7: Answer: 5.46 A (rms), 9.46 A (rms)

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