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# Power Electronics ECE 472

BSU

GPA 3.91

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This 9 page Class Notes was uploaded by Judd Okuneva on Saturday October 3, 2015. The Class Notes belongs to ECE 472 at Boise State University taught by Staff in Fall. Since its upload, it has received 18 views. For similar materials see /class/217980/ece-472-boise-state-university in Engineering Electrical & Compu at Boise State University.

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Date Created: 10/03/15

Boise State University Department of Electrical and Computer Engineering ECE 472 7 Power Electronics Fourier Series Lecture Objectives 1 To review the Fourier series expansion of periodic waveforms 2 To discuss the computation of Fourier coefficients by taking advantage of waveform sym metries even function symmetry odd function symmetry half wave symmetry and quarter wave symmetry Introduction Periodic functions occur frequently in engineering problems Some of these functions may be discontinuous such as square waveforms These functions can be represented in terms of simple sine and cosine functions leading to Fourier series named after the French physicist Joseph Fourier 1768 1830 Periodic Function A function ft is said to be periodic if it de ned for all t and there exists a positive number T called the period such that ft T ft for all t In other words such a function repeats itself every T seconds Notice that if n is an integer then ft nT ft for all if so that any integral multiple of T is also a period The smallest such number T is called the fundamental period and f 1T in hertz Hz is called the fundamental frequency Trigonom etric Fourier Series ft a0 a1 coswt a2 cos2wt ancosnwt 121 sinwt bgsin2wt bnsinnwt The class of periodic functions which can be represented by Fourier series is surprisingly large and general The following sufficient conditions cover almost any conceivable engineering application Theorem If a periodic function ft with period T is piecewise continuous and has left and right hand derivatives at every point then the above Fourier series converges to the original function at every point of continuity lt converges to the average of the left and right hand limits of the original function at any point where the original function is discontinuous Assume here that the function has a nite number of discontinuities over one period Fourier Coefficients 1 T TO mm 7 a0 1 T 1 T 2 a1 TO ftcoswt 7 TA alcos wt dt 7 3 1 T i 1 T 2 b1 TO fts1nwt 7 TA blcos wt dt 7 2 1 T 1 T T ftcosnwt ancosznwt dt 0 0 1 T 1 T T ftsinnwt T bnsinznwtdt 0 0 Summary 1 T 27r a 7 70 mm made 2 T 1 27r an 7 ftcosnwt dt 7 f0cosm9 10 T 0 7T 0 1 2 T 27r bn 7 ftsinnwt dt 7 1 09 sinm9 10 T 0 0 7T 01 919 T 2 1 T 1 T2 1 T to TO ftdt TO 2det Txgvmxi Vm 2 T 2 TZ 4V t 172 an TA ftCOsnwt dt TA vacosnwt dt Tm SIIO 4Vms1nnw 7 0 for n 17 27 37W 7quot 71w mu 2 T 2 T2 4Vm 608nm TZ bn fO smmat dt 7 TO QVmsmnwt dt 7 T 77h 4Vm 7man 1 i2vm1licosn7r1 T 71 71w mr forn 173757 VLW 0 form 274767 4V 1 1 1 U05 Vm7msinwhkgsin3wtgsin5wtsin7wt 7T Even Function Symmetry it ft for all t 2 T2 17r a0 TA mm 7 f0d0 7r 4 TZ 2 7r an 7 ftcosnwtdt 7 f0cosm9d0 for n 1273 T 0 7T 0 bn 0 1 05 t Odd Function Symmetry it 7 ft for all 25 a0 0 an 4 TZ 2 7r bn 7 ftsinnwtdt 7 f0sinm9d0 for n 1273 T 0 7T 0 t blsinwtbgsin2wt bnsinnwt Half Wave Symmetry ft T2 7 a0 an 7 7 ft for all t 0 for n 1375 2 f0cosm9 10 W 0 3 f0sinm9 d0 7r 0 T2 3 ftcosnwt dt T 0 4 TZ T ftsinnwt dt 1375 0 form alcoswtachSBwta50085wtblsinwtbgsin3wtb5sin5wt Quarter Wave Symmetry Even or Odd Function Symmetry plus Half Wave Symmetry a i 0 4 8n 4 7r2 f ftsinmutdt 7 f0sinm9 d0 135 0 7r 0 form 7 b1 sinwt bgsin3wt b5sin5wt blsin 7T w t Zgtbgsin3w 257z 4 13581nSwltt7I 4 57139 blsinlt 77r wt 5 bgsin lt3wt731 2 b5sinlt5w 77 2 1 cos wt 7 3 cos 3wt 5 cos 5wt Vm sinwhk siant sinSwtnl 7139 Change of y axis 2571 gt t OWhent T I 4 Vmsinw 257z 17181113411 257 7139 4 3 Vm 1 sin any 7 sin lt3wt 7 77quot 4V 1 1 Vm 1 7m cos wt 7 3 cos 3wt 3 cos 5wt 7 7r Vt ZVm O O l T 2T 1 2 V t Vm O i T 2T t 2 Wm 1 T 1 772 1 T TO ftdt TO 2det TXQme7 Vm vt 7 Vm b1 sinwt 1 3 sin 3wt 1 5 sin 5wt T4 T4 T4 E vtsinnwt dt vt7Vmsinnwt dt Vmsinnwt dt T 0 T 0 T 0 svm 70087141121774 8Vmicosn7r2 4Vm for n 17 37 T mu 0 T mu 717139 5 17181115 25 T J 5 4 gt sinlt5wt7577rgt EE225 MATLAB SCRIPT ON FOURIER SERIES SPRING 2007 Half Wave Recti ed Waveform ZMATLAB script for synthesing a halfwave rectified waveform Zfrom its Fourier coefficients Vm 1 T 1 w 2piT a0 Vmpi for m 1401 tm m 1001 vm a0 Vm2sinwtm for n 2250 an 2Vmpin1n1 Vm vm ancosnwtm end end plottvgrid Full Wave Recti ed Waveform ZMATLAB script for synthesing a fullwave rectifed waveform Zfrom its Fourier coefficients Vm 1 T 1 w 2piT a0 2Vmpi for m 1401 tm m 1001 vm a0 for n 2250 an 4Vmpin1n1 Vm vm ancosnwtm end end plottvgrid Useful Trigonometric Results 81112 0 1 C082 0 1 sin 20 28in0cos0 c0820 c082078in20 12c0820 1728in20 6082 0 1 cos 20 2 sing 0 17 cos 20 2 cosa 1 cos a cos b 7 sin a sin b cosa 7 b cosacosb 8ina8inb 8ina b 8inacosb cosasinb 8ina 7 b 8inacosb 7 8ina8inb 1 cosacosb cosa 7 b cosa 13 1 8ina8inb cosa 7 b 7 cosa 13 1 8inacosb sina b 8ina 7 19 1 cosasinb sina b 7 8ina 7 19 1 T 7 cosnwtdt 0forn 1273 T 0 1 T 7 sinnwtdt 0forn 1273 T 0 1 T 2 1 T 1 1 TA cos mat dt 7 T 0 1 cos2nwt dt 7 5 1 T 1 T 1 1 TO 8in2 mat dt T 0 5H 7 cos2nwt dt 5 1 T 1 T 1 7 cos mat cos mwt dt 7 7cosn 7 mt cosn mt T 0 T 0 2 1 T 1 1 T 7 sin mat 8in mwt dt T 0 1 T 7 8in mat cos mwt dt T 0 To for n for n cosn 7 mt 7 cosn mt dt 1 T 1 1 7 781nn mt 81nn 7 mt dt T 0 2 1273 1273 0form 71 71 0form 71 71 Oformy n

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