Phasor Method

1. USC Phasor Method Aug 24, 2011

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…

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

4. R1 R2 G R3 R4 – + E Review of Analysis of DC circuits • Complete solution for DC circuits Unknown variable: 6 Voltages (b) 12 (2b) 6 Currents (b) Constraint Equations: • As number of braches grows: • Too many variables! • Too many equations! Elements: 6 (b) KCL: 4-1=3 (n-1) 6 (b) Network: KVL: 6-3=3 b-(n-1) 12 (2b)=12 (2b)

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 • The purpose of circuit analysis method: • To reduce the number of variables and equations

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

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

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

9. Introduction of AC circuits

10. Phasor Method Hint:

11. Phasor Method Charles Proteus Steinmetz • 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: Marking chalk "X" on side of generator: $1. Knowing where to mark chalk "X": $999. German-American mathematician and engineer (1865 – 1923)

12. Phasor Method Trigonometric function Phasor Domain transform Inverse transform

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

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

15. Phasor Method equivalent Complex Operation Trigonometric calculation

16. Phasor Method equivalent Trigonometric calculation Complex Operation

17. Phasor Method 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.

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

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

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

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

24. USC Power Aug 24, 2011

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

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

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

28. Power

29. Power: Pure Resistive

30. Power: Pure Inductive

31. Power: Pure Capacitive

32. Average Power

33. Example 2.1

34. Complex Power

35. Power Triangle

36. Power Triangle