IE1206 Embedded Electronics

Transcription

1 E06 Embedded Elecronics Le Le3 Le4 Le Ex Ex P-block Documenaion, Seriecom Pulse sensors,, R, P, serial and parallel K LAB Pulse sensors, Menu program Sar of programing ask Kirchhoffs laws Node analysis Two-erminals RR AD Le5 Ex3 K LAB Two-erminals, AD, omparaor/schmi Le6 Le8 Ex6 Le3 Ex4 Ex5 Le0 Le7 Le9 Le Le Ex7 Display Wrien exam K3 LAB3 Transiens PWM Phasor PWM P AP/ND-sensor K4 LAB4 LP-filer Trafo Sep-up, R-oscillaor L-osc, D-moor, P PWM Display of programing ask Trafo, Eherne conac

2 Easy o generae a sinusoidal volage Our enire power grid works wih sinusoidal volage. When he loop roaes wih consan speed in a magneic field a sine wave is generaed. So much easier, i can no be

3 The sine wave wha do you remember? y T period Y Y ˆ op, RMS ampliude ime y ( Yˆ sin( ω ω πf f Y T Yˆ

4 f a sine curve does no begin wih 0 he funcion expression has a phase angle ϕ. (. Phase ϕ y( Yˆ sin( ω + ϕ Specify his funcion mahemaically: y u( 6sin( π ϕ 3 u( 0 3 6sin( ϕ ϕ arcsin 0,5 rad ( 30 6 u( 6sin( ,5

5 Apples and pears? n circui analyses i is common (eg. in exbooks o expresses he angle of he sine funcion mixed in radians ω [rad] and in degrees ϕ [ ]. This is obviously improper, bu pracical (!. The user mus "conver" phase angle o radians o calculae he sine funcion value for any given ime. (You have now been warned u( 6sin( ?? onversion: x[ ] x[rad] 57,3 x[rad] x[ ]0,07

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7 Mean and effecive value All pure A volages, has he mean value 0. More ineresing is he effecive value roo mean square, rms. med T 0 T u( d 0 0 T u( T d

8 (. Example. RMS. The rms value is wha is normally used for an alernaing volage.,63 V effecive value gives he same power in a resisor as a,63 V pure D volage would do. RMS, effecive value T u( d ( 3 T + ( ,63 V

9 Sine wave effecive value Ex..3 sin has he mean value ½ Therefore: ˆ sin ( x RMS, effecive value sin ( x dx Effecive value is ofen called RMS ( Roo Mean Square.

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11 Addiion of sinusoidal quaniies y ˆ A sin( ω + ϕ y ˆ A sin( ω + ϕ y + y?

12 Addiion of sinusoidal quaniies When we shall apply he circui laws on A circuis, we mus add he sines. The sum of wo sinusoidal quaniies of he same frequency is always a new sine of his frequency, bu wih a new ampliude and a new phase angle. ( Ooops! The resul of he raher laborious calculaions are shown below cos( Â cos( Â sin( Â sin( Â arcan sin cos( Â Â Â Â ( ( ( sin( Â ( sin( Â ( ϕ ϕ ϕ ϕ ω ϕ ϕ ϕ ω ϕ ω y y y y y

13 Sine wave as a poiner A sinusoidal volage or curren, y( Yˆ sin( ω can be represened by a poiner ha roaes (counerclockwise wih he angular velociy ω [rad/sec]. Wikipedia Phasors

14 Simpler wih vecors f you ignore he "revoluion" and adds he poiners wih vecor addiion, as hey sand a he ime 0, i hen becomes a whole lo easier! Wikipedia Phasors hp://en.wikipedia.org/wiki/phasors

15 Poiner wih complex numbers A A volage 0 V ha has he phase 30 is usually wrien: maginary axis 0 30 ( Phasor Once he vecor addiions require more han he mos common geomerical formulas, i is insead preferable o represen poiners wih complex numbers. z a + jb Real axis e j30 0cos jsin 30 n elecriciy one uses j as imaginary uni, as i is already in use for curren.

16 Phasor Sinusoidal alernaing quaniies can be represened as poiners, phasors. amoun phase A poiner (phasor can eiher be viewed as a vecor expressed in polar coordinaes, or as a complex number. is imporan o be able o describe alernaing curren phenomena wihou necessarily having o require ha he audience has a knowledge of complex numbers - hence he vecor mehod. The complex numbers and -mehod are powerful ools ha faciliae he processing of A problems. They can be generalized o he Fourier ransform and Laplace ransform, so he elecro engineer s use of complex numbers is exensive.

17 peak/effecive value - phasor maginary axis z a + The phasor lenghs corresponds o sine peak values, bu since he effecive value only is he peak value scaled by / so i does no maer if you coun wih peak values or effecive values - as long as you are consisen! jb Real axis

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19 The inducor and capacior couneracs changes The inducor and capacior couneracs changes, such as when connecing or disconnecing a source o a circui. Wha if he source hen is sinusoidal A which is hen changing coninuously??

20 Alernaing curren hrough resisor A sinusoidal curreni R ( hrough a resisor R provides a proporional sinusoidal volage drop u R ( according o Ohm's law. The curren and volage are in phase. No energy is sored in he resisor. Phasors R and R become parallel o each oher. i R ( R sin( ω ur ( ir ( R ur ( R R sin( ω R ˆ R R Vecor phasor ˆ R R R omplex phasor The phasor may be a peak poiner or effecive value poiner as long as you do no mix differen ypes.

21 Alernaing curren hrough inducor L L L L L L L u i L u i + ω π ω ω ω ω ω sin( ˆ cos( ˆ ( d ( d ( sin( ˆ ( L L L L L A sinusoidal curren i L ( hrough an inducor provides, due o self-inducion, a voage drop u L ( which is 90 before he curren. Energy sored in he magneic field is used o provide his volage. When using complex poiners one muliplies ωl wih j, his roaes he volage poiner +90 (in complex plane. The mehod auomaically keeps rack of he phase angles! L L L L j j X L ω Vecor phasor omplex phasor The phasor L will be ωl L and i is 90 before L. The eniy ωl is he amoun of he inducor s A resisance, reacance X L [Ω].

22 Alernaing curren hrough capacior sin( ˆ cos( ˆ ( sin( ˆ ( d ( ( ( d d d ( d π ω ω ω ω ω u i i u i q u Q A sinusoidal curren i ( hroug a capacior will charge i wih he volage drop u ( ha lags 90 behind he curren. Energy is sorered in he elecric field. Vecor phasor Phasor is /(ω and i lags 90 afer. The eniy /(ω is he amoun of he capacior s A resisance, reacance X [Ω]. ω

23 omplex phasor and he sign of reacance f you use complex phasor you ge he -90 phase by dividing (/ω wih j. The mehod wih complex poiner auomaically keeps rack of he phase angles if we consider he capacior reacance X as negaive, and hence he inducor reacance X L as posiive. - j X omplex phasor ω ω

24 L+ in series 5jΩ 4jΩ + jω L 4jΩ 5jΩ jω

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26 Reacance frequency dependency X L [Ω] X L [Ω] f [Hz] f [Hz] X L ω L X ω ω π f

27 log LOG LOG plo ( X L scale [ Ω] log( scale [ Ω] X log( f scale [Hz] log( f scale [Hz] Ofen elecronics engineers use log-log scale. The inducor and capacior reacances will hen boh be "linear" relaionship in such chars.

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29 R L n general, our circuis are a mixure of differen R L and. The phase beween and is hen no ±90 bu can have any inermediae value. Posiive phase means ha he inducances dominaes over capaciances, we have inducive characer ND. Negaive phase means ha he capaciance dominaes over he inducances, we have capaciive characer AP. The raio beween he volage and curren, he A resisance, is called impedace Z [Ω]. We hen have OHM s A law: AP Z

30 Phasor diagrams n order o calculae he A resisance, he impedance, Z, of a composie circui one mus add currens and volages phasors o obain he oal curren and he oal volage. Z The phasor diagram is our "blind sick" in o he A World!

31 Ex. Phasor diagram (.5 Elemenary diagrams for R L and A a cerain frequency f he capacior has he reacance X and he resisor R has he same amoun (absolue value, R [Ω]. se he elemenary diagrams for R and as building blocks o draw he whole circui phasor diagram (for his acual frequency f.

32 Try i your self

33 Example. Phasor diagram. reference phase ( horizonal 3 R R R + 4 R 5 R R 6 +

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35 mpedance Z The circui A resisance, impedance Z, one ge as he raio beween he lengh of and phasors. The impedance phase ϕ is he angle beween and phasors. The curren is before he volage in phase, so he circui has a capaciive characer, AP. ( Somehing else had hardly been o wai since here are no coils in he circui

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37 omplex phasors, -mehod + 90 arg(j j arg( j j 90 arg(j j j 0 arg( L L L L R R X L L X R R ω ω ϕ ω ω ϕ ω ϕ omplex phasors. OHM s law for R L and. omplex phasors. OHM s law for Z. ] m[ ] m[ arcan ] Re[ ] m[ arcan arg( ] Re[ ] Re[ arg( arg( arg arg( Z R X Z Z Z Z Z Z Z ϕ n fac, here will be four useful relaionships! Re, m, Abs, Arg

38 Ex. omplex phasors. 0 V 30 µf R 0 Ω f 50 Hz

39 Ex. omplex phasors. 0 V 30 µf R 0 Ω f 50 Hz jπ f jπ j

40 Ex. omplex phasors. 0 V 30 µf R 0 Ω f 50 Hz Z R// jπ f jπ j R 0( 0 j (0 + 0 j 5 5 j R j (0 + 0 j

41 Ex. omplex phasors. 0 V 30 µf R 0 Ω f 50 Hz Z R// jπ f jπ j R 0( 0 j (0 + 0 j 5 5 j R j (0 + 0 j

42 Ex. omplex phasors. 0 V 30 µf R 0 Ω f 50 Hz Z R// jπ f jπ j R 0( 0 j (0 + 0 j 5 5 j R j (0 + 0 j

43 Ex. omplex phasors. 0 V 30 µf R 0 Ω f 50 Hz Z R// jπ f jπ j R 0( 0 j (0 + 0 j 5 5 j R j (0 + 0 j Z + Z R // 0-0 j+ (5-5 j 4-3 j ( + 3 j ( + 3 j 0,4 +, j 0, 4 +, j 0,4 +,,6

44 Ex. omplex phasors. 0 V 30 µf R 0 Ω f 50 Hz Z R// jπ f jπ j R 0( 0 j (0 + 0 j 5 5 j R j (0 + 0 j

45 Ex. omplex phasors. 0 V 30 µf R 0 Ω f 50 Hz Z R// jπ f jπ j R 0( 0 j (0 + 0 j 5 5 j R j (0 + 0 j 4 j (0,4 +, j (-0 j 4 j + ( 4,65

46 Ex. omplex phasors. 0 V 30 µf R 0 Ω f 50 Hz Z R// jπ f jπ j R 0( 0 j (0 + 0 j 5 5 j R j (0 + 0 j

47 Ex. omplex phasors. Volage divider: 0 V 30 µf R 0 Ω f 50 Hz Z R// jπ f jπ j R 0( 0 j (0 + 0 j 5 5 j R j (0 + 0 j Z R // + Z R // j -0 j+ (5-5 j j 0-3 j (+ 3 j (+ 3 j j j ,94

48 Ex. omplex phasors. 0 V 30 µf R 0 Ω f 50 Hz Z R// jπ f jπ j R 0( 0 j (0 + 0 j 5 5 j R j (0 + 0 j

49 Ex. omplex phasors. 0 V 30 µf R 0 Ω f 50 Hz Z R// jπ f jπ j R 0( 0 j (0 + 0 j 5 5 j R j (0 + 0 j j -0 j 0,4 + 0,8 j 0,4 + 0,8 j 0,4 + 0,8 0,89

50 Ex. omplex phasors. R 0 V 30 µf R 0 Ω f 50 Hz Z R// jπ f jπ j R 0( 0 j (0 + 0 j 5 5 j R j (0 + 0 j

51 Ex. omplex phasors. R 0 V 30 µf R 0 Ω f 50 Hz Z R// jπ f jπ j R 0( 0 j (0 + 0 j 5 5 j R j (0 + 0 j R R j 0 0,8 + 0,4 j R 0,8 + 0,4 j 0,8 + 0,4 0,89

52 You ge he phasor char by ploing he poins in he complex plane!

53 Roae diagram When we draw he phasor diagram i was naural o have as reference phase (horizonal, wih he mehod was he naural choice of reference phase (real. Because i is easy o roae he char, so, in pracice, we have he freedom of choosing any eniy as he reference. arg( arg(8 + 4j arcan (cos( 6,7 + jsin( 6, ,7 Muliply he all complex numbers by his facor and he roaion will ake effec!

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55 Summary Sinusoidal alernaing quaniies can be represened as poiners, phasors, amoun phase. A poiner (phasor can eiher be seen as a vecor expressed in polar coordinaes, or as a complex number. alculaions are usually bes done wih he complex mehod, while phasor diagrams are used o visualize and explain alernaing curren phenomena.

56 Noaion x Xˆ X X X nsan value Top value omplex phasor Absolue value, he amoun

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