TUTORIAL ON HARMONICS Theory and Computation Techniques C

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

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

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

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

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

Interference with Control, EMI

What Are Harmonics? l. Technical Description A high frequency sinusoidal current or voltage produced by certain nonlinear and switching processes in the system during normal periodic operation (steady state); l l 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. l. Mathematical Definition l Sine and cosine functions of time with frequencies that are integer multiples of a fundamental frequency l Harmonic sine and cosine functions sum up to a periodic (nonsinusoidal) function l Terms of the Fourier series expansion of a periodic function;

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

Superposition l A LTI system responds linearly to its inputs l ui 1 uo 1, ui 2 uo 2 l l aui 1+bui 2 auo 1+buo 2 For sinusoidal inputs:

Application preview: DC Drive Find the armature current io(t) below

DC Voltage Approximation

Source Superposition

Output Response

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.

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

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

Amplitude and Peak Value l Peak of a Symmetric Oscillation

Non-Symmetric Signals l Peak-to-peak variation

Average Value l Signal=(constant part) + (oscillating terms)

Examples

AC Signals l Zero Average Value

DC and Unidirectional Signals

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

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

Examples of Signal RMS

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

Examples of Signals with equal RMS

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

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

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

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

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

Optimum Value of A Find d. E/d. A: Set d. E/d. A equal to zero

EXAMPLE: SQUARE PULSE

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

Shifted Pulse

Approximation with many harmonic terms Average Square Error :

Harmonic Basis The terms From an orthogonal basis Orthogonality property:

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

Optimum coefficients l Obtain the optimum expansion coefficients:

Example—Square Wave Pulse

n A B 1 4/p 0 2 0 0 3 -4/3 p 0 4 0 0 5 4/5 p 0 6 0 0 n ± 4/np 0

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

Example: Sawtooth

Odd Symmetry

n An Bn 0 0 0 1 0 2/p 2 0 -1/p 3 0 2/3 p 4 0 -1/2 p 5 0 2/5 p 6 0 -1/3 p 7 0 2/7 p n 0

Periodic Approximation

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

Average Value

Harmonic Terms

Summary

Numerical Problem: DC Drive

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

Equivalent Circuit

Superimpose Sources: DC Source

Superposition: n=2, f=120 Hz

Superposition: n=4, f=240 Hz

Superposition: n=6, f=360 Hz

Summary Freq. , Hz Vo ampl, V Io ampl, A Za magn, W Power loss, W 0 (dc) 216. 1 66. 1 1 120 144. 1 36. 9 3. 9 680. 8 240 28. 8 3. 78 7. 61 7. 14 360 12. 3 1. 08 11. 35 . 583 71. 1 Total Power Loss RMS 240 Output Power (66. 1 A)(150 V) 4, 369. 2 5, 057. 7 9, 915

Output Time and Frequency Response

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).

Waveform Symmetry l Half Wave Symmetry l Quarter Wave Symmetry Odd l Even l

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

Square Wave

Triangular

Saw Tooth—Counter Example

Quarter Wave Symmetry l Half wave and odd symmetry l Half wave and even symmetry

Half-wave: odd and even

Quarter Wave Symmetry Simplification

Forms of the Fourier Transform l Trigonometric l Combined Trigonometric l Exponential

• Trigonometric form • Combined Trigonometric

• Exponential

Relations between the different forms of the FS

Summary of FS Formulas

Time Shift

Example: SQP -90° Shift original shifted n 2|Cn| qn -np/2 1 4/p 0 -p/2 2 0 0 - 3 4/3 p p -p/2 4 0 0 - 5 4/5 p 0 -p/2 6 0 0 - 7 5/7 p p -p/2 n 4/np (n-1)p/2 -p/2

Example: SQP -60° Shift original shifted n 2|Cn| qn -np/3 1 4/p 0 -p/3 2 0 0 - 3 4/3 p p 0 4 0 0 - 5 4/5 p 0 p/3 6 0 0 - 7 5/7 p p 2 p/3 n 4/np (n-1)p/2 (n-3)p/6

SPECTRUM: SQ. Pulse (amplitude=1)

SPECTRUM: Sawtooth (amplitude=1)

SPECTRUM: Triangular wave (amplitude=1)

SPECTRUM: Rectified SINE (peak=1)

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

Square Pulse Excitation Harm. Order Inp. |Uin(n)|, <Uin(n) Circ. TF |H(n)|, <H(n) Out. |Uout(n)|, <Uout(n) 0 1 0 0 0 1 0. 4686 -1. 083 1. 2732 0 0. 5967 -1. 083 2 0. 2564 -1. 3115 0 0 3 0. 1741 -1. 3958 0. 4244 -3. 1416 0. 0739 1. 7458 4 0. 1315 -1. 4389 0 0 5 0. 1055 -1. 4651 0. 2546 0 0. 0269 -1. 4651 6 0. 0881 -1. 4826 0 0 7 0. 0756 -1. 4952 0. 1819 -3. 1416 0. 0137 1. 6464 8 0. 0662 -1. 5046 0 0 9 0. 0588 -1. 5119 0. 1415 0 0. 0083 -1. 5119 10 0. 053 -1. 5178 0 0 11 0. 0482 -1. 5226 0. 1157 -3. 1416 0. 0056 1. 619 12 0. 0442 -1. 5266 0 0 13 0. 0408 -1. 53 0. 0979 0 0. 004 -1. 53 14 0. 0379 -1. 5329 0 0 15 0. 0353 -1. 5354 0. 0849 -3. 1416 0. 003 1. 6061

SQUARE PULSE Excitation

Rectified SINE Wave Harm. Order Inp. |Uin(n)|, <Uin(n) Circ. TF |H(n)|, <H(n) Out. |Uout(n)|, <Uout(n) 0 1 0 0. 6366 0 1 0. 4686 -1. 083 0 0 2 0. 2564 -1. 3115 0. 0849 3. 1416 0. 0218 1. 8301 3 0. 1741 -1. 3958 0 0 4 0. 1315 -1. 4389 0. 0202 3. 1416 0. 0027 1. 7027 5 0. 1055 -1. 4651 0 0 6 0. 0881 -1. 4826 0. 0089 3. 1416 0. 0008 1. 659 7 0. 0756 -1. 4952 0 0 8 0. 0662 -1. 5046 0. 005 3. 1416 0. 0003 1. 637 9 0. 0588 -1. 5119 0 0 10 0. 053 -1. 5178 0. 0032 3. 1416 0. 0002 1. 6238 11 0. 0482 -1. 5226 0 0 12 0. 0442 -1. 5266 0. 0022 3. 1416 0. 0001 1. 615 13 0. 0408 -1. 53 0 0 14 0. 0379 -1. 5329 0. 0016 3. 1416 0. 0001 1. 6087 15 0. 0353 -1. 5354 0 0

Rect. SINE wave

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

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

Waveform Quality-AC Signals Total Harmonic Distortion Index

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

Example: W. F. Q. of the circuit driven by a Sq. P. Inp. Rms: |Uin(n)|/√ 2 Harm. Order Out. Rms: |Uout(n)|/√ 2 0 0 0 1 0. 900288 0. 421931 2 0 0 3 0. 300096 0. 052255 4 0 0 5 0. 180029 0. 019021 6 2 – 0. 42192)=0. 0575{ 0 √(0. 4258 0 7 0. 128623 0. 009687 8 0 0 9 0. 100056 0. 005869 10 0 0 11 0. 081812 0. 00396 12 0 0 13 0. 069226 0. 002828 14 0 0 15 0. 060033 0. 002121 RMS 1. 00 (excact) 0. 4258 %THD 48. 43 (exact) 13. 6

Example: WFQ of the circuit driven by a rect. sine Inp. Rms: |Uin(n)|/√ 2 Harm. Order Out. Rms: |Uout(n)|/√ 2 0 0. 6366=Uin(0) 0. 6366=Uout(0) 1 0 0 2 0. 060033 0. 015415 3 0 0 4 0. 014284 0. 001909 5 0 0 6 0. 006293 0. 000566 7 0 0 8 0. 003536 0. 000212 9 0 0 10 0. 002263 0. 000141 11 0 0 12 0. 001556 7. 07 E-05 13 0 0 14 0. 001131 7. 07 E-05 15 0 0 RMS 0. 707 (exact) 0. 6368 %RF 48. 35 (exact) 2. 5