1. CIRCUITS and SYSTEMS – part I Prof. dr hab. Stanisław Osowski Electrical Engineering (B.Sc.) Projekt współfinansowany przez Unię Europejską w ramach Europejskiego Funduszu Społecznego. Publikacja dystrybuowana jest bezpłatnie

2. Lecture 7 Three-phase circuits

3. Definition of 3-phase circuits It is the AC power system in which three AC sources operate at the same frequency but with diffeerent phases. The individual phases of the circuit will be denoted by the letters: A, B and C.

4. Phase voltages of 3-phase symmetric generator

5. The 3-phase generator is symmetric, when all 3 phase volatges are of the same value and are shifted by 120 o to each other. The system is of positive sequence (ABC), when phase B is delayed by 120 o to phase B,while phase C is delayed by 120 o to phase (it leads phase A by 120 o ). In other case the sequence is negative (ACB). 3-phase symmetric generator

6. Complex represenatation of 3-phase generator Complex phase voltages Phasor diagram for positive sequence

7. Rotation of phases Rotation of phases in complex plane is anti-clock wise direction.

8. Between the phase (line) voltages Line-to-line voltages Phasor diagram

9. Connections of generator and load Generator Load impedances

10. Types of 3-phase networks generator and load connected in Y generator and load connected in Δ Δ generator – Y load connection Y generator – Δ load

11. Y-connected 3-phase network Point 0 – common point of generator. Point N – common point of load Symmetrical generator

12. Analysis of Y-connected network Neutral voltage U N Phase voltages of the load Load currents

13. Phasor diagram of 3-phase network There are 2 visible stars of voltages: the generator phase voltages of common point 0 and load phase voltages of common point N.

14. Balanced (symmetric) load Neutral voltage equal 0 Load currents

15. Phasor diagram at balanced load

16. Non-balanced load at zero Z N Neutral voltage U N =0 Current of neutral wire I N =I A +I B +I C Phasor diagram

17. Short circuit of the load phase Neutral voltage equal to the phase voltage of the short circuit phase (Z=0), for example Z A =0 -> U N =E A. Current of the short circuit phase I A =-(I B +I C ) Phasor diagram

18. Example Calculate the currents and voltages of the 3-phase Y-connected circuit. Assume balanced generator of the rms phase voltage equal 400V. Assume: R=40 , X C =30 , X L =60 , X 12 =10 , X 23 =20 , X 31 =20 .

19. Solution Elimination of magnetic couplings

20. Complex values of parameters Complex representation of phase voltages Complex impedances of the load

21. Currents and voltages of the load Currents Voltages

22. Phasor diagram

23. Delta connection of 3-phase network Structure of the circuit

24. Currents of Δ connected circuit Voltages of the 3-phase load Currents of the load Line currents

25. Phasor diagram of Δ connected network.

26. Example Calculate the currents and voltages of the 3-phase delta-connected circuit. Assume balanced generator of the rms phase voltage equal 200V. Assume: R=X L =X C =10 .

27. Solution The voltages of the load Load currents Line currents

28. Phasor diagram

29. Power in Δ and Y connected balanced load Structures of the circuit Power in Y-connected circuit Power in delta-connected circuit