1. Phasor Method Aug 24, 2011USC

2. Outline Review of analysis of DC (Direct Current) circuits Analysis of AC (Alternating Current) circuits – Introduction – Challenge of analysis of AC circuits Phasor method – Idea and concept – Advantage Conclusions Next… 2

3. Review of Analysis of DC circuits DC circuits 3 LL C R + - LL C R + - Inductor: Capacitor: Resistor: Short Open Pure Resistive t u i 0 +

4. Review of Analysis of DC circuits Complete solution for DC circuits 4 E – + G R3R3 R4R4 R2R2 R1R1 Unknown variable: 6 Voltages (b) 6 Currents (b) 12 (2b) Constraint Equations: Elements: 6 (b) Network: KCL: 4-1=3 (n-1) KVL: 6-3=3 b-(n-1) 6 (b) 12 (2b)=12 (2b) As number of braches grows: Too many variables! Too many equations! As number of braches grows: Too many variables! Too many equations!

5. Review of Analysis of DC circuits Summary of DC circuits analysis methods – Circuit simplification Equivalent transformation of resistors Equivalent transformation of sources – General analytical methods Node-voltage method (suitable for fewer nodes) Mesh-current method (suitable of fewer meshs) – Theorem Superposition (linear circuits) Thevenin and Norton equivalent 5 The purpose of circuit analysis method: To reduce the number of variables and equations The purpose of circuit analysis method: To reduce the number of variables and equations

6. Introduction of AC circuits Why AC? – Generation, transmission, distribution and consumption of electric energy are all in steady state sinusoidal. 6 t u i 0 +  AC (Alternating current) Sinusoidal steady state analysis – Any signal can be thought of as superposition of sinusoidal signals.

7. Introduction of AC circuits Challenge 7 Inductor: Capacitor: Resistor: with analysis of AC circuit LL C R + - + + - - The +,-,*,/ operation with trigonometric function is not easy!

8. Review of Analysis of DC circuits Summary of DC circuits analysis methods – Circuit simplification Equivalent transformation of resistors Equivalent transformation of sources – General analytical methods Node-voltage method (suitable for fewer nodes) Mesh-current method (suitable of fewer meshs) – Theorem Superposition (linear circuits) Thevenin and Norton equivalent 8

9. Introduction of AC circuits 9

10. Phasor Method 10 Hint:

11. Phasor Method 11 Charles Proteus Steinmetz German-American mathematician and engineer (1865 – 1923) In 1893, he introduced the phasor method to calculation of AC circuits GE required him to submit a itemized invoice. They soon received it. It included two items: 1.Marking chalk "X" on side of generator: $1. 2.Knowing where to mark chalk "X": $999.

12. Phasor Method 12 Trigonometric functionPhasor Domain transform Inverse transform

13. Phasor Method 13 Complex operation: Sum/Subtraction: Multiplication/Division:

14. Phasor Method 14 Sinusoidal expression Trigonometric calculation Phasor （ Complex ） Result (Phasor) Complex Operation transform Inverse transform Result (sinusoidal) Time DomainPhasor Domain

15. Phasor Method 15 Trigonometric calculation Complex Operation equivalent

16. Phasor Method 16 Trigonometric calculation equivalent Complex Operation

17. Phasor Method 17 Example:

18. Conclusions The trigonometric function involved in the sinusoidal steady-state circuits is not convenient to calculation. By projecting trigonometric function to phasor domain, the calculation can be dramatically simplified. 18

19. Quiz 1- problem1 19 Convert the following instantaneous currents to phasors, using cos(wt) as the reference. Give your answer in polar form. (1). 2).

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21. Review of Analysis of DC circuits Summary of DC circuits analysis methods – Circuit simplification Equivalent transformation of resistors Equivalent transformation of sources – General analytical methods Node-voltage method (suitable for fewer nodes) Mesh-current method (suitable of fewer meshs) – Theorem Superposition (linear circuits) Thevenin and Norton equivalent 21

22. Review of Analysis of DC circuits Summary of DC circuits analysis methods – Circuit simplification Equivalent transformation of resistors Equivalent transformation of sources – General analytical methods Node-voltage method (suitable for fewer nodes) Mesh-current method (suitable of fewer meshs) – Theorem Superposition (linear circuits) Thevenin and Norton equivalent 22

23. 23 For the circuit shown below, compute the voltage across the load terminals. I=125 0° A 240 0 ° V LOAD

24. Power Aug 24, 2011USC

25. Review of Phasor 25 Questions: 1. What is the main idea of Phasor method?

26. Review of Phasor 26 LL C R + - + + - - + -

27. Power 27 Instantaneous Power Average Power Real Power Active Power Reactive Power Complex Power Apparent Power

28. Power 28

29. Power: Pure Resistive 29

30. Power: Pure Inductive 30

31. Power: Pure Capacitive 31

32. Average Power 32

33. Example 2.1 33

34. Complex Power 34

35. Power Triangle 35

36. Power Triangle 36