2. Chapter 5 Steady-State Sinusoidal Analysis 2008.1 0 Electrical Engineering and Electronics II Scott

3. 1. Identify the frequency, angular frequency, peak value, rms value, and phase of a sinusoidal signal. 2. Solve steady-state ac circuits using phasors and complex impedances. Main Contents 3. Compute power for steady-state sinusoidal ac circuits.

4. 4. Find Thévenin equivalent circuits for steady-state ac circuits. 5. Determine load impedances for maximum power transfer. 6. Solve balanced three-phase circuits. Main Contents

5. The importance of steady-state sinusoidal analysis Electric power transmission and distribution by sinusoidal currents and voltages Sinusoidal signals in radio communication All periodic signals are composed of sinusoidal components according to Fourier analysis

6. 5.1 Sinusoidal Currents and Voltages

7. Parameters of Sinusoidal Currents and Voltages V m is the peak value, unit is volt ω is the angular frequency, unit is radians per second f is the frequency ， unit is Hertz (Hz) or inverse second. θ is the phase angle, unit is radian or degree.

8. Frequency Angular frequency To uniformity, This textbook expresses sinusoidal functions by using cosine function rather than the sine function. 。 Parameters of Sinusoidal Currents and Voltages

9. Root-Mean-Square Values or Effective Values Average Power

10. RMS Value of a Sinusoid The rms value for a sinusoid is the peak value divided by the square root of two. However, this is not true for other periodic waveforms such as square waves or triangular waves.

11. A voltage given by is applied to a 50Ω resistance. Find the rms value of the voltage and the average power delivered to the resistance.

12. Complex Numbers 3 forms of complex numbers Arithmetic operations of complex numbers

13. Rectangular form a b ImIm Re 0 |A| Imaginary unit Real part Imaginary part Magnitude Angle Conjugate

14. a b ImIm Re 0 |A| Conversion between Rectangular and polar Polar form

15. Conversion

16. Euler’s Identity a b ImIm Re 0 |A| Exponential form

17. Rectangular form Polar form Exponential form Three forms

18. Arithmetic Operations Given Identity Adding and subtracting

19. Arithmetic Operations Given Product

20. Arithmetic Operations Given Dividing

21. Phasor Definition Sinusoidal steady-state analysis is greatly facilitated if currents and voltages are represented as vectors or Phasors. rms Phasor or effective phasor. In this book, if phasors are not labeled as rms, then they are peak phasors. 5.2 PHASORS

22. 正弦量可以表示为在复 平面 complex plane 上 按逆时针方向旋转的相 量的实部 real part 。 Angular velocity Phasors as Rotating Vector

23. Sinusoids can be visualized as the real-axis projection of vectors rotating in the complex plane. The phasor for a sinusoid is a ‘snapshot’ of the corresponding rotating vector at t = 0. θ

24. Phase Relationships Alternatively, we could say that V 2 lags V 1 by θ. (Usually, we take θ≤180 o as the smaller angle between the two phasors.) Then when standing at a fixed point, if V 1 arrives first followed by V 2 after a rotation of θ, we say that V 1 leads V 2 by θ. To determine phase relationships from a phasor diagram, consider the phasors to rotate counterclockwise.

25. Phase Relationships

27. Adding Sinusoids Using Phasors Step 1: Determine the phasor for each term Step 2: Add the phasors using complex arithmetic. Step 3: Convert the sum to polar form. Step 4: Write the result as a time function.

28. Example: Solution: Adding Sinusoids Using Phasors

30. Exercise Answers Adding Sinusoids Using Phasors

31. 5.3 COMPLEX IMPEDANCES By using phasors to represent sinusoidal voltages and currents, we can solve sinusoidal steady-state circuit problems with relative ease. Sinusoidal steady-state circuit analysis is virtually the same as the analysis of resistive circuits except for using complex arithmetic.

32. Sinusoidal response of resistance Relationship between voltage and current U u i I 0 tt then ：

33. 2.Phasor relation and diagram Substituting for current and voltage phasors Phasor diagram Sinusoidal response of resistance

34. Plot the phasor diagram phasors expression in Ohm’s Law Sinusoidal response of resistance

35. No doubt, p≥0,Resistance always absorbs energy. Power Instantaneous power Sinusoidal response of resistance

36. 0 tt v i p V rms I rms 2V rms I rms Power Sinusoidal response of resistance

37. Average Power ——Average value of instantaneous power in a period Average Power represents the consumed power, is also named as Real power. Power Sinusoidal response of resistance

38. u 、 i have the same frequency ， u leads i by 90 o u π 2π2π 0 ωtωt Sinusoidal response of Inductance Relationship between voltage and current

39. Definition Reactance 电抗－感抗 Effective Value Sinusoidal response of Inductance

40. i v L jωLjωL Phasor diagram ： Phasor relations Sinusoidal response of Inductance Complex impedance

41. Power 0 tt u i p V rms I rms Sinusoidal response of Inductance

42. 0 tt i UI 0 tt p u i u i u i u i u ——magnitude of magnetic field increases ， inductance absorbs energy ——magnitude of magnetic field decreases ， inductance supplys energy Power Sinusoidal response of Inductance

43. Inductance is an energy-storage element rather than an energy-consuming element Average Power Power Sinusoidal response of Inductance

44. The power flows back and forth to inductances and capacitances is called reactive power Q, it is the peak instantaneous power 。 Unit: Var ( 乏 ) Reactive Power Reactive power is important because it causes power dissipation in the lines and transformers of a power distribution system. Specific Charge is executed by electric-power companies for reactive power. Sinusoidal response of Inductance

45. i v C Current leads voltage by 90° i  22 0 Sinusoidal response of capacitance Relationship between voltage and current

46. i v C Definition Reactance 电抗－容抗 Relationship between voltage and current Sinusoidal response of capacitance

47. i v C -jωC Phasor diagram Sinusoidal response of capacitance

48. 0 tt u i p U rms I rms PowerPower Instantaneous power Sinusoidal response of capacitance

49. 0 tt UI 0 tt p u i i u i u i u i u ——capacitance charges ， it absorbs energy ——capacitance discharges ， it supplys energy PowerPower Sinusoidal response of capacitance

50. Average Power PowerPower Capacitance is an energy-storage element rather than an energy-consuming element. Sinusoidal response of capacitance

51. Assuming the same current acts on the inductance and capacitance respectively, the initial phase is 0 ， then Reactive Power PowerPower Sinusoidal response of capacitance

52. It is given that the frequency of sinusoidal source is 50Hz, the rms or effective value is 10V, capacitor is 25μF, determine the rms value of current. If the frequency is 5000Hz, then what is the rms value of current now ？ i u C when f =50Hz When f =5000Hz The higher frequency is under fixed rms value of voltage, the bigger the rms value of current flowing through capacitor. Solution

53. Summary ele me nt circuit Relationship between u and i Ohm’s Law Complex impedance Phasors diagram R L C R i v i v C i v L

54. Inductance

55. Capacitance

56. In phase Resistance

60. True or False 1. ？ ？ 3.Complex j45  ？ Peak Value ？ ？ minus 2. 4.

61. Complex Impedances Complex Impedance Two- terminal circuit of zero sources + - Magnitude Angle

62. Complex Impedances of R L C

63. The Laws of Phasor Form (KCL) ： (KVL) ：

64. Voltage-Divider Z eq =Z 1 +Z 2 Equivalent Impedance Z1Z1Z1Z1 Z2Z2Z2Z2 + – + + – – Z + – Complex Impedances in Series

65. Z + – Z1Z1Z1Z1 Z2Z2Z2Z2 + – Current-Divider Equivalent Impedance Complex Impedances in Parallel