1. TUTORIAL ON HARMONICS Theory and Computation Techniques C.J. Hatziadoniu: hatz@siu.edu

2. AC Drive Harmonics Harmonic Sources: Power converter switching action Motor own generated harmonics (spatial distribution of windings, stator saturation) Transformer/inductor iron core saturation Harmonics flowing between generator and motor sides

3. Potential Problems due to Harmonics Power losses and heating: reduced efficiency, equipment de-rating Over-voltage and voltage spiking, due to resonance: insulation stressing, limiting the forward and reverse blocking voltage of power semiconductor devices, heating, de-rating EMI: noise, control inaccuracy or instability Torque pulsation: mechanical fatigue, start-up limitation

4. Power Loss and Heating Losses into the resistive and magnetic components Resistive losses: skin effect Magnetic losses: Eddy currents and hysterisis losses increase with frequency

5. Over-Voltage, Over-Current (due to resonance) Capacitor loss due to harmonics (insulation loss)

6. Interference with Control, EMI

7. What Are Harmonics? Technical Description A high frequency sinusoidal current or voltage produced by certain non-linear and switching processes in the system during normal periodic operation (steady state); The harmonic frequency is an integer multiple of the system operating frequency (fundamental). The non-sinusoidal part in a periodic voltage or current is the harmonic ripple or harmonic distortion—comprised of harmonic frequencies. Mathematical Definition Sine and cosine functions of time with frequencies that are integer multiples of a fundamental frequency Harmonic sine and cosine functions sum up to a periodic (non- sinusoidal) function Terms of the Fourier series expansion of a periodic function;

8. Harmonic Analysis What is it? Principles, properties and methods for expressing periodic functions as sum of (harmonic) sine and cosine terms: Fourier Series Fourier Transform Discrete Fourier Transform Where is it used? Obtain the response of a system to arbitrary periodic inputs; quantify/assess harmonic effects at each frequency Framework for describing the quality of the system input and output signals (spectrum)

9. Superposition A LTI system responds linearly to its inputs  i1   o1,  i2   o2 a  i1 +b  i2  a  o1 +b  o2 For sinusoidal inputs:

10. Application preview: DC Drive Find the armature current i o (t) below

11. DC Voltage Approximation

12. Source Superposition

13. Output Response

14. Procedure to obtain response Step 1: Obtain the harmonic composition of the input (Fourier Analysis) Step 2: Obtain the system output at each input frequency (equivalent circuit, T.F. frequency response) Step 3: Sum the outputs from Step 2.

15. Fundamental Theory Outline Harmonic Fundamental Theory—Part a: Periodic Signals—sinusoidal function approximation Fourier Series—definition, computation Forms of the Fourier Series Signal Spectrum Applications of the FS in LTI Wave Form Quality of Periodic Signals

16. Measures Describing the Magnitude of a Signal Amplitude and Peak Value Average Value or dc Offset Root Mean Square Value (RMS) or Power

17. Amplitude and Peak Value Peak of a Symmetric Oscillation

18. Non-Symmetric Signals Peak-to-peak variation

19. Average Value Signal=(constant part) + (oscillating terms)

20. Examples

21. AC Signals Zero Average Value

22. DC and Unidirectional Signals

23. Root Mean Square Value (RMS) For periodic signals, time window equals one period

24. Remarks on RMS RMS is a measure of the overall magnitude of the signal (also referred to as norm or power of the signal). The rms of current and voltage is directly related to power. Electric equipment rating and size is given in voltage and current rms values.

25. Examples of Signal RMS

26. Effect of DC Offset New RMS= SQRT [ (RMS of Unshifted) 2 +(DC offset) 2 ]

27. Examples of Signals with equal RMS

28. RMS and Amplitude Amplitude: Local effects in time; Device insulation, voltage withstand break down, hot spots RMS: Sustained effects in time; Heat dissipation, power output

29. Harmonic Analysis: Problem Statement Approximate the square pulse function by a sinusoidal function in the interval [–T/2, T/2]

30. General Problem Find a cosine function of period T that best fits a given function f(t) in the interval [0,T] Assumptions: f(t) is periodic of period T

31. Approximation Error Error: Method: Find value of A that gives the Least Mean Square Error Objective: Minimize the error e(t)

32. Define the average square error as : E is a quadratic function of A. The optimum choice of A is the one minimizing E. Procedure

33. Optimum Value of A Set dE/dA equal to zero Find dE/dA:

34. EXAMPLE: SQUARE PULSE

35. A geometrical interpretation Norm of a function, error, etc is defined as:

36. Shifted Pulse

37. Average Square Error : Approximation with many harmonic terms

38. Harmonic Basis The terms From an orthogonal basis Orthogonality property:

39. Optimum coefficients The property of orthogonality eliminates the cross harmonic product terms from the Sq. error For each n, set

40. Optimum coefficients Obtain the optimum expansion coefficients:

41. Example—Square Wave Pulse

46. nAB 1 4/  0 200 3 -4/3  0 400 5 4/5  0 600 n ±4/n  0

47. Waveform Recovery n=1 n=1-3n=1-5n=1-7n=1-9

48. Example: Sawtooth

49. Odd Symmetry

50. nAnAn BnBn 000 10 2/  20 -1/  30 2/3  40 -1/2  50 2/5  60 -1/3  70 2/7  n0

51. Periodic Approximation

52. Approximation of the Rectified sine A periodic signal= (constant part)+ (oscillating part)

53. Average Value

54. Harmonic Terms

55. Summary

56. Numerical Problem: DC Drive

57. Input Harmonic Approximation Average or dc component Harmonic Expansion Truncated Approximation (n=2, 4, and 6)

58. Equivalent Circuit

59. Superimpose Sources: DC Source

60. Superposition: n=2, f=120 Hz

61. Superposition: n=4, f=240 Hz

62. Superposition: n=6, f=360 Hz

63. Summary Freq., Hz V o ampl, VI o ampl, A Z a magn,  Power loss, W 0 (dc) 216.166.114,369.2 120 144.136.93.9680.8 240 28.83.787.617.14 360 12.31.0811.35.583 RMS24071.1 Total Power Loss 5,057.7 Output Power (66.1A)(150V) 9,915

64. Output Time and Frequency Response

65. Generalization: Fourier Series The Fourier theorem states that a bounded periodic function f(t) with limited finite number of discontinuities can be described by an infinite series of sine and cosine terms of frequency that is the integer multiple of the fundamental frequency of f(t): Where is the zero frequency or average value of f(t).

66. Waveform Symmetry Half Wave Symmetry Quarter Wave Symmetry Odd Even

67. Half Wave Symmetry Half-wave symmetry is independent of the function shift w.r.t the time axis Even harmonics have zero coefficient

68. Square Wave

69. Triangular

70. Saw Tooth—Counter Example

71. Quarter Wave Symmetry Half wave and odd symmetry Half wave and even symmetry

72. Half-wave: odd and even

73. Quarter Wave Symmetry Simplification

74. Forms of the Fourier Transform Trigonometric Combined Trigonometric Exponential

75. Trigonometric form Combined Trigonometric

76. Exponential

77. Relations between the different forms of the FS

78. Summary of FS Formulas

79. Time Shift

80. Example: SQP -90° Shift originalshifted n2|C n | nn  n  1 4/  0  200- 3 4/3  400- 5 4/5  0  600- 7 5/7  n 4/n  (n-1)  /2-  /2

81. Example: SQP -60° Shift originalshifted n2|C n | nn  n  1 4/  0  200- 3 4/3  400- 5 4/5  0  600- 7 5/7  n 4/n  (n-1)  /2(n-3)  /6

82. SPECTRUM: SQ. Pulse (amplitude=1)

83. SPECTRUM: Sawtooth (amplitude=1)

84. SPECTRUM: Triangular wave (amplitude=1)

85. SPECTRUM: Rectified SINE (peak=1)

86. Using FS to Find the Steady State Response of an LTI System Input periodic, fundamental freq.=f 1 =60 Hz Voltage Division

87. Square Pulse Excitation Harm. Order Circ. TF |H(n)|, <H(n) Inp. |Uin(n)|, <Uin(n) Out. |Uout(n)|, <Uout(n) 010 0 0 0 0 10.4686-1.083 1.2732 0 0.5967 -1.083 20.2564-1.3115 0 0 0 0 30.1741-1.3958 0.4244 -3.1416 0.0739 1.7458 40.1315-1.4389 0 0 0 0 50.1055-1.4651 0.2546 0 0.0269 -1.4651 60.0881-1.4826 0 0 0 0 70.0756-1.4952 0.1819 -3.1416 0.0137 1.6464 80.0662-1.5046 0 0 0 0 90.0588-1.5119 0.1415 0 0.0083 -1.5119 100.053-1.5178 0 0 0 0 110.0482-1.5226 0.1157 -3.1416 0.0056 1.619 120.0442-1.5266 0 0 0 0 130.0408-1.53 0.0979 0 0.004 -1.53 140.0379-1.5329 0 0 0 0 150.0353-1.5354 0.0849 -3.1416 0.003 1.6061

88. SQUARE PULSE Excitation

89. Rectified SINE Wave Harm. Order Circ. TF |H(n)|, <H(n) Inp. |Uin(n)|, <Uin(n) Out. |Uout(n)|, <Uout(n) 010 0.63660 0 10.4686-1.083 0000 20.2564-1.3115 0.08493.14160.02181.8301 30.1741-1.3958 0000 40.1315-1.4389 0.02023.14160.00271.7027 50.1055-1.4651 0000 60.0881-1.4826 0.00893.14160.00081.659 70.0756-1.4952 0000 80.0662-1.5046 0.0053.14160.00031.637 90.0588-1.5119 0000 100.053-1.5178 0.00323.14160.00021.6238 110.0482-1.5226 0000 120.0442-1.5266 0.00223.14160.00011.615 130.0408-1.53 0000 140.0379-1.5329 0.00163.14160.00011.6087 150.0353-1.5354 0000

90. Rect. SINE wave

91. Total RMS of A Signal Rewrite the FS as: Nth harmonic rms (except for n=0) Total rms of the wave form:

92. Total RMS and the FS Terms For ac wave forms (A 0 =0) it is convenient to write: Using the orthogonality between the terms:

93. Waveform Quality-AC Signals Total Harmonic Distortion Index

94. Waveform Quality-DC SIgnals (A 0 ≠0) Ripple Factor

95. Example: W.F.Q. of the circuit driven by a Sq.P. Harm. Order Inp. Rms: |Uin(n)|/√2 Out. Rms: |Uout(n)|/√2 0 00 1 0.9002880.421931 2 00 3 0.3000960.052255 4 00 5 0.1800290.019021 6 00 7 0.1286230.009687 8 00 9 0.1000560.005869 10 00 11 0.0818120.00396 12 00 13 0.0692260.002828 14 00 15 0.0600330.002121 RMS 1.00 (excact) 0.4258 %THD 48.43 (exact) 13.6 √(0.4258 2 – 0.4219 2 )=0.0575 {

96. Example: WFQ of the circuit driven by a rect. sine Harm. Order Inp. Rms: |Uin(n)|/√2 Out. Rms: |Uout(n)|/√2 0 0.6366=U in (0)0.6366=U out (0) 1 00 2 0.0600330.015415 3 00 4 0.0142840.001909 5 00 6 0.0062930.000566 7 00 8 0.0035360.000212 9 00 10 0.0022630.000141 11 00 12 0.0015567.07E-05 13 00 14 0.0011317.07E-05 15 00 RMS 0.707 (exact) 0.6368 %RF 48.35 (exact) 2.5