- 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
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
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.
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.
Star-Connected Balanced Load
Delta-Connected Balanced Load
Phase current: I1p = I1L, I2p = I2L, I3p = I3L
Line current: IL = I1L = I2L = I3L
Line current: IL = I1L = I2L = I3L and Ip = I12p = I23p = I31p
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.