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Balanced Three-Phase Circuits and Ideal Transformers ELEC 308 Elements of Electrical Engineering Dr. Ron Hayne Images Courtesy of Allan Hambley and Prentice-Hall Three-Phase Sources Most common multiple-phase AC voltage source configuration Wye connected (Y connected) ELEC 308 2 Three-Phase Sources v an t VY cost Van VY 0 o v bn t VY cost 120 o Vbn VY 120 o v cn t VY cost 120 o Vcn VY 120 o ELEC 308 3 Wye-Wye Connection ELEC 308 4 Wye-Wye Connection Lines Wires a-A, b-B, c-C Neutral Wire n-N Balanced Load Three load impedances are equal Phase Voltage or Line-to-Neutral Voltage Peak magnitude of one source, VY Line Currents Values IaA, IbB, and IcC ELEC 308 5 Wye-Wye Connection Van VY 0 I aA I L Z Z Vbn VY 120 I bB I L 120 Z Z Vcn VY 120 I cC I L 120 Z Z I nN I aA I bB I cC 0 Neutral current is ZERO in a BALANCED threephase system Can eliminate the neutral wire ELEC 308 6 Cost Advantage Three (or four) wires needed to connect a three-phase source with three loads Six wires needed to connect three singlephase sources with three loads ELEC 308 7 Three-Phase Power pt van t iaA t vbn t ibB t vcn t icC t VY I L 3 cos 2 Total power is CONSTANT with respect to time VY I L Pavg pt 3 cos 3VYrms I Lrms cos 2 VY IL where VYrms and I Lrms 2 2 ELEC 308 8 Three-Phase Reactive Power The reactive power delivered to a balanced three-phase load is given by VY I L sin 3VYrms I Lrms sin 2 VY IL where VYrms and I Lrms 2 2 Q3 ELEC 308 9 Line-to-Line Voltages Vab Van 330 o Vbc Vbn 330 o Vca Vcn 330 o ELEC 308 10 Analysis of a Wye-Wye System Balanced wye-connected 60-Hz three-phase source has lineto-neutral voltages of VY = 1000 V. Balanced wye-connected load consisting of a 0.1H inductance in series with a 50-Ω resistance. Find the line currents, the line-to-line voltages, the power, and the reactive power delivered to the load. Draw a phasor diagram showing the line-to-neutral voltages, the line-to-line voltages, and the line currents. ELEC 308 11 Analysis of a Wye-Wye System ELEC 308 12 Delta-Connected Sources Vab Vbc Vca 0 ELEC 308 13 Delta-Connected Loads Z 3ZY ELEC 308 14 Delta-Delta Connection ELEC 308 15 Power Transmission Losses Power Line Losses 2 Ploss RlineI rms Large Voltages and Small Currents Smaller Line Loss ELEC 308 16 Ideal Transformers v 2 (t ) V2 rms N2 v1 (t ) N1 N2 V1rms N1 ELEC 308 17 Ideal Transformers ELEC 308 18 Example 15.10 Determine the rms values of the currents and voltages. ELEC 308 19 Power Transmission Step-Up and Step-Down Transformers 99% Efficiency (vs. 50% with no transformers) ELEC 308 20 U.S. Power Grid ELEC 308 21 Summary Three-Phase Circuits Wye-Wye Connection Three-Phase Power Ideal Transformers U.S. Power Grid ELEC 308 22