Three-Phase Circuit Analysis: Voltages, Currents, and Phasors

1. Lesson 35Per Phase Analysis

2. Learning Objectives • Derive the relationship between line to line voltages and line to neutral voltages in a balanced Y-Y three phase circuit. • For a balanced Y-Y three phase circuit convert the line to line voltage phasor to the line to neutral voltage phasor and vice versa. • Plot the phasors of line to line voltages and line to neutral voltages for a Y-Y three phase circuit. • Derive the relationship between line current and phase load current in a balanced Y- Δ three phase circuit. • For a balanced Y- Δ three phase circuit convert the line current phasor to phase load current phasor and vice versa. • Plot the phasors of line current and phase load current in a balanced Y- Δ three phase circuit.

3. Review Y-Y system Y Load Generator Lines

4. Determining a line voltage for a three-phase generator. Line and phase voltages of the Y-connected three-phase generator. PHASE/LINE VOLTAGES

5. Line/Phase Voltages for a Wye Circuit Using the following relationship we can also convert back… Therefore, given any voltage at a point in a balanced, 3-phase Y system, you can determine the remaining 5 voltages by inspection.

6. Law of Sines • The ratio of the length of a side to the sine of its corresponding opposite angle is constant • Applying the law of sines to the vector EAB: EAN EBN

7. Nominal voltages • If phase voltage Van= 120 V then line voltages Vab= 208 V. • This is referred to as 120/208-V system. • Two other common nominal voltages are 220/380-V and 347/600-V.

8. Currents for a Wye Circuit • For Y loads, line current and phase current are the same. Line current Phase current

9. Wye-Wye Single Phase Equivalent Original circuit Single-phase equivalent

10. Example Problem 1 EAB = 2080 V . Find the phase voltages and line currents.

11.  load definitions Review Y-Y For  loads,phase voltage and line voltage are the same thing. Line currents are the currents in the line conductors. Phase currents are the currents through phases .

12. Line/Phase Currents for a Delta Circuit Iab Ica Iab Ibc Line current Phase current

13. Line/Phase Currents for a Delta Circuit • Relationship between line and phase currents • For a balanced  system, the magnitude of line current is 1.732 times the magnitude of the phase current and line current lags phase current by 30. • Therefore, given any current at a point in a balanced, 3-phase  system, you can determine the remaining 5 currents by inspection.

14. Line/Phase Currents for a Delta Circuit Phasor diagram

15. Equivalent Ckt to solve Wye-Delta Problem • This is just an equivalent circuit of the load side! • What is missing from the equivalent circuit? • Ia , Eab , Ean • You can only use the equivalent circuit to solve for load side voltages and currents: • Vab, Iab • Notice that this equivalent circuit uses LINE VOLTAGE • #1 mistake with DELTA circuits is using the wrong kind of voltage!

16. Example Problem 2 Ia = 41.66.9 A . Determine the phase currents in the load. Determine the supply line voltage EAB and the supply phase voltage EAN.

17. Line Impedances in 3 Phase AC Impedances in transmission lines complicate three phase analysis Voltage drop in the transmission line must be accounted for: EAN will not be the same as Van VAa

18. Wye-Wye Single Phase Equivalent Original circuit Single-phase equivalent • Notice that the neutral line impedance drop is ignored on the single phase equivalent. • No voltage drop occurs across this neutral line since in reality, no current flows through it.

19. Wye-Wye Single Phase Equivalent Original circuit + VAa - Single-phase equivalent • Use KVL on the single phase equivalent to find unknowns: EAN –VAa –Van = 0

20. Example Problem 3 EAB = 2080 V. Determine: a. Line Currents b. Load phase voltages

21. Delta To Wye (∆→Y) Conversion A Delta circuit with line impedances cannot be solved without first performing a ∆→Y conversion. The Delta load can be converted to an equivalent Wye load

22. Equivalent Delta Ckt with LINE IMPEDANCES

23. Example Problem 4 EAB = 2080 V. Determine: a. Find load phase voltage Vab b. Find the load phase current Iab .