LECTURE 9. C. Appendix

Transcription

1 LECTURE 9 A. Buck-Boos Converer Design 1. Vol-Sec Balance: f(d), seadysae ransfer funcion 2. DC Operaing Poin via Charge Balance: I(D) in seady-sae 3. Ripple Volage / C Spec 4. Ripple Curren / L Spec 5. Peak Swich Currens and Blocking Volages / Wors Case Transisor Specs B. Pracical Issues for L and C Componens 1. Inducor:L = f(i)? L= f(f sw )? a. Cos of Cores b. Inducor Core Maerials Unique o Each f sw Choice c. Core Sauraion above i(cri) B > B sa, L = f(i) B < B sa, L f(i) 2. Capacior: C(f sw, i c, v c ) a. Coss b. Dielecric Maerials (1) ε(f sw ) (2) E(breakdown) C. Appendix

2 2 A. Buck-boos Converer Design 1.Vol-Sec Balance: f(d), seady-sae ransfer funcion We can implemen he double pole double hrow swich by one acively conrolled ransisor and one passive diode conrolled by he circui currens so ha when Q 1 is on D 1 is off and when Q 1 is off D 1 is on. General form Q 1 -D 1 Swich Implemenaion Q 1 D 1 V g 1 2 i() + C R v - V g i() L + C R v - Two swich cases occur, resuling in wo separae circui opologies. Case 1: SW 1 on, SW 2 off; Transisor Q 1 is on Knowing V ou is negaive means diode D 1 is off and load is no conneced o inpu. This is a unique circui opology as given below. Only Q 1 is acive urned on/off by conrol signals. Inpu Circui Topology: V g L i Q 1 is on; V on small (2V) compared o V g. V g is across L. Oupu Circui

3 3 Topology: i C C i R R + - v o i C + i R = 0 i C = i R i C i R as hey form a loop. Laer we will see ha acually v o is negaive. Case 2: SW 2 on, SW 1 off; Transisor Q 1 is off as se by exernal conrol signal applied o Q 1. This is a second circui opology given on he nex page. Knowing i L canno change a swich since il vld =, no need o L acively conrol he diode wih any conrol signals. I is auomaically urned on by i L flow o he lef. The diode is auomaically urned off by curren flow o he righ. For DT s, SW in 1: i C i + R V g C R v o i L - For D T s, SW in 2: i L i C i R + V g L C R v - We calculae he DC Transfer Funcion f(d) via V L vs. ime and volsec balance over boh DT s and D T s. V g V o V L DT s D'T s V g DT s + V o D T s = 0 V V o D = = D' 1 D D = f(d) g This is he Buck-Boos DC ransfer funcion By symmery and power conservaion I o /I in = D /D so ha

4 4 P in = P ou neglecing losses. Example: For a buck-boos circui opology. V o = -20, V g =30. Find D and D in seady sae. V V o g D = = D D = D = V o V o V g 0. 4 and D = 0.6 Clearly D would vary wih oher PWD circui DC - DC converer opologies even for he same V g and V o. If we furher specified R L as 4Ω hen Iou = 20V/4Ω = 5A. P ou = I ou V ou = 5*20 = 100 W. for a lossless converer P in = 100W and I in = 100/30. Wha is i L? Is i I in or I ou? Acually I L will be he sum of I o and I in or 8.33 A as shown below. This rips up new sudens! 2. DC Curren Operaing Poin in a Buck-boos circui via Capacior charge balance We will show below by separae calculaion ha I L = I o + I in. I C We find ha he inducor curren is: i C + i R = 0 -V o /R V IL o DT RD' s and / or -I L - V o /R D'T s i C + i L + i R = 0 IL VgD ( D ') 2 R Example: V o = -20V, V g = 30V and R = 4 Ω.

5 5 IL = ( 20) = ( 4)( 0. 6) A which is I in + I ou Plos of he DC volage ransfer funcion M(D) and I(D) shown below are no dependen on our choice of he swiching frequency explicily. Again a jier in he swiching frequency can cause big changes in V ou. Oupu is invered! V Vg = M( D) As D 1 M(D) - (Won occur in pracice) I Vg R = I( D) As D 1 I(D) + Won occur in pracice N.B. for HW #2, how valid is he claim ha M(D) and I(D) are no f(f sw )? Be specific and quaniaive.

6 6 Wha abou M(D) or f(d) being sensiive o he level of he load curren? For HW #2, also explain how simple power conservaion can ell us ha I(in) = M(D) I(ou) as we oulined. 3. Choice of C value via ripple volage spec across C Choice of C value via ripple volage spec across C is vc icd =. C Knowing charge balance occurs in seady sae. Again V o is invered wih respec o V g. V ou DT s D'T s -(Vo/R)(1/C) (IL + V/R)(1/C) V DT 2 vc = o RC V DT C o s 2 vr C required o employ in order o have specified V ou ripple s Example: V o = -20, v = 1/2% Here we have a igher v spec of 0.1V. C required for various f sw f: 40KHz 400KHz 4MHz C: 250µF 25µf 2.5µF Which C is smaller and cheaper? Wha are he pracical f limis for capaciors? Do capaciors have any losses? Finally, all capaciors have parasiic inducance associaed wih hem due o wire leads on he capaciors. This inroduces resonan frequencies. A ypical case migh be he 25 µf capacior wih L(lead

7 7 parasiic) = 20 nh (usually 5nH/cm of lead wire) of f R = 225 khz (w R = 1 LC = 2πf R). Wha occurs if f R is close o he swiching frequency f s? 4. L value requiremen via ripple curren specificaion for quasi-saic condiions. The ripple curren hrough L is il balance occurs in seady sae. VL d =, and knowing vol-sec L I dc Vg /L V g /L DT s D'T s 2 i L VgDTs 2 il = L VgDTs L 2 i L L required for given i L ripple is a funcion of f sw Example: I DC = 8.33A, I = 10% = 0.83 A, D =0.4 L required a f sw f: 40 khz 400 khz 4 MHz L: 179 µh 17.9 µh 1.79 µh Which L is smaller and cheaper? Wha limis he f sw for inducors? Do inducors have losses? 5. Peak currens and volages versus ransisor specs The peak on curren / peak off volage specificaion mus be me by he swiches. i L values effec maximum values of I peak in he swiches employed. i > i(criical) kills a solid sae swich in nanoseconds. When swiches urn-off peak sand-off volages can also damage swiches. v(ripple) ses V(peak) values.

8 8 Transisors are raed by boh I DC (max) - Depends on hea sink and power in TR. I peak (max) - If his is exceeded, TR is dead. No second chances As well as by maximum rms values. Diodes are he same as regards i > i(criical). Consider he i L waveform given below vs. ime. i L I dc i L I peak I DC is NOT I rms DT s D'T s For complex waveforms i L is measured from he I DC baseline and is so defined hroughou. Some ypical waveforms and rms values: i L I dc D I peak Irms = Ipeak D i L I dc i D I peak i Irms = Idc D I Forunaely, in Appendix I of Erickson s ex (pgs ) here are summarized many common waveforms and associaed RMS values. Hence, he definiions of peak currens, effecive DC currens, and rms are all unique. Likewise manufacurers spec shees for devices will give all hree curren spec s. dc I peak = I DC (during DT s ) + i(during DT s ) 10-60% of I DC Device loss: P av = I rms V on,rms per cycle

9 9 Noe: I DC above is no I rms P av = f sw P rms,cycle Now we are using D o vary V o via duy cycle conrol of applied volage V g. Laer, in Chaper 11, we will inroduce curren conrol of PWM dc-dc converers. One nice feaure of curren conrol is ha we can limi i peak by i conrol i max. Tha is, if i conrol is exceeded he ransisor is urned off and peak curren damage can never occur. i c =i max of TR specs DT s D'T s Example: DC operaing poin P ou of buck-boos for V o = 20V and R = 4 Ω is 100W. I dc 8.33 A Consider hese ac condiions during DT s : 10% ripple 50% ripple i = A i = 4.17 A I pk = 9.17 A I pk = 12.5 A Wha abou he cos of ransisors and diodes o handle he peak currens? vs. The cos of addiional value inducors o reduce I pk? In his buck-boos circui is I pk he same for he diode and he ransisor?

10 10 i L () 8.33A 12.5A peak for i = 0.5I dc 9.17A peak for i = 0.1I dc i D () 8.33A DT s T s Inducor curren DT s Diode curren T s B. Pracical Issues for Inducive and Capaciive Componens We alked briefly abou skin effec in wires a high frequencies in lecure 3. Now we briefly alk abou capaciors and inducors a high frequencies. I is worhwhile o know early ha he circui elemens are no wha we firs imagine bu are raher very complex in heir behavior due o parasiics and non-linear effecs. 1. Inducors (coss, sauraion, maerials) Copper wire is wound around a magneic core L N = µ 2 N urns = µa magneic relucance of flux L pah in H -1 N A = µ 2 I appears L f(i L ) For a fixed L we can rade he amoun of copper wire (N 2 ) for he amoun of iron core (A) o achieve a desired value of L. We can also rade copper wire vs. core maerial choice depending on he size, weigh and cos requiremens. Core permeabiliy iself varies wih

11 11 frequency and he erm Ni=H. Where N is he number of wire urns on he core and I is he curren in he wire. a. Big L coss maerial and money: (1) N 2 - number of urns of wire: coss in copper. (2) A/ - Area of magneic maerial/lengh coss in core size. Noe you can rade core for copper o he exen we don saurae he core. (3) Higher µ maerial a given frequency coss. No maerials have high µ above 1 MHz. b. Various core maerials for f sw : (1) 60 Hz - 20 khz Iron cores are O.K., µ = 1000 (2) khz powdered iron, meal-glass, µ = 100 (3) khz use ferrie cores, µ = Losses in Cores Eddy currens ~f 2 Hyserisis ~f These losses limi upper f sw o MHz for presen cores. Perhaps wih ime low loss cores which operae a 10 Mhz can be found. f c. Sauraion of flux Acually he inducance L(i L ) a high currens and for i > i(criical) L will suddenly decrease precipiously. This may cause higher currens and hese kill solid sae devices as well ha are in series wih he inducance.

12 12 B B sa slope µ o slope µ r µ o H We wan o operae a H < H(criical) or B below B(sauraion). µ = µ r µ 0 only if B < B sa wih B sa unis Wb/m 2 Tesla; Core Maerial Maximum B sa f sw (max) due o losses Iron ~1-2 Tesla khz Powdered iron ~½ o 1 Tesla 40 khz Meal-glass ~½ Tesla 100 khz Ferrie ~¼ - ½ Tesla MHz There is an apparen B max *f max produc ha no core maerials will exceed oday. See chapers in Erickson. Finally, in any analysis of magneic maerials ry o include parasiic inducor effecs as well due o flux leakage from he core. Tha is flux will leak ou from a ransformer core, for example, and cause parasiic inducor ha is locaed before he ideal ransformer. This causes los on unexpeced volages in ransformer circuis due o LEAKAGE INDUCTANCE. 2. Capacior is εa d = f( f sw ) a. Coss Dielecric maerial choice for ε(f sw ) o achieve high C values. Low f caps ε High f caps ε C = f(f sw )

13 13 V c ε d V c /D E c mus no exceed breakdown of maerial Vacuum caps are bes bu since ε = ε 0 hey are large and cosly. b. Capacior Dielecric Maerials ε(f) mached o f sw Maerial choice for ε is compaible wih E(breakdown) Loss vs. f The op circui in he figure below shows he circui model for a capacior including: R w (wire losses due o skin effecs a f sw ) >> R wire L w (wire inducance) which is ypically 500nH/m or 5nH/cm. Beware L w of 5nH/cm wih a di/d = 50A/200ns hrough a capacior wih lead lenghs of only 8cm we can drop 100 V across L w even before we place any volage across C. Moreover, we could have a series resonan circui a w 1 = if R leak is large. L C w

14 14 By simplifying he model as shown, we can derive he equivalen series resisance (ESR) used by C manufacurers. ESR = R w w RleakC 2 1 wr an δ= w C (ESR) If R w is small hen: an δ= wc 2 w RleakC 2 = 1 wc leak C which measures capacior loss R leak In erms of known measuremens usually an δis specified for a capacior so: (ESR) = an δ/wc The ESR of a capacior will decrease as w increases for a fixed an δ.

15 15 Example #1: A 100 µf elecrolyic C has 5 cm long leads and inernal L of 15 nh. We are given an δ= 0.2, consan for all f < 100 khz. Find: w(resonance) of C L oal = * 5 nh = 40 nh w R = 1 40*100 = 80 khz Choose f sw well below w R, say 20 khz and find ESR here. ESR(20 khz) = an δ/wc = 8 mω Example #2: A 2 µf C has an L(oal) = 25 nh and an δ= 0.01 is consan from 50 Hz o 200 khz. Find he resonan frequency. w R = 1 25nH *2µ F = 0.7 MHz Calculae ESR a 120 Hz and 120 khz ESR(120 Hz) = an δ/wc = 6 Ω ESR(120 khz) = an δ/wc = 6 mω Again for a fixed an δesr decreases as f increases. Exra Credi: For Homework #2 please review he properies of pracical dielecric capaciors in he range of 0.1 o 1 MHz. Talk abou an δand realisic R for real capaciors. C. Appendix 1 RMS Values of Commonly-Observed Converer Waveforms The waveforms encounered in power elecronics converers can be quie complex, conaining modulaion a he swiching frequency and ofen also a he ac line frequency. During converer design, i is ofen necessary o compue he rms values of such waveforms. In his appendix, several useful formulas and ables are developed which allow hese rms values o be quickly deermined.

16 16 RMS values of he doubly-modulaed waveforms encounered in PWM recifier circuis are discussed in secion A 1.1 Some common waveforms DC, Fig A 1.1: rms = I i() I 0 DC plus linear ripple, Fig A 1.2: i rms = I I i() ι I 0 Ts

17 17 Square wave, Fig. A 1.3: rms = I pk i() Ipk -Ipk Cener-apped bridge winding waveforms, Fig. A1.10: 1 rms = I pk 1+ D 2 i() Ipk 1/2 Ipk 1/2 Ipk 0 DTs Ts (1+D)Ts 2Ts 0

18 18 General sepped waveform, Fig. A1.11: rms = D2 I1 2 + D2 I i() I2 I1 D1Ts D2Ts 0 Ts A 1.2 General piecewise waveform For a periodic waveform composed of n piecewise segmens as in Fig. A 1.12, he rms value is rms = n D k u k k= 1 i() Triangular segmen Consan segmen Trapezoidal segmen ec. 0 D1Ts D2Ts D3Ts Ts Where D K is he duy cycle of segmen k, and u k is he conribuion of segmen k. The u k s depend on he shape of he segmens several common segmen shapes are lised below:

19 19 1. Consan segmen, Fig A 1.13: uk = I1 2 i() I Triangular segmen, Fig. A 1.14: uk = 1 I 3 12 i() I Trapezoidal segmen: D3 = ( 01. µ s)( 10µ s) = u3 = ( I1 2 + I1I2 + I2 2 ) / 3 = 148A 2 4. Consan segmen D4 = ( 5µ s)( 10µ s) = 0. 5 u4 = I2 2 = ( 2) 2 = 4A 2

20 20 5. Triangular segmen D5 = ( 0. 2µ s)( 10µ s) = u5 = I2 2 / 3 = ( 2) 2 / 3 = 13. A 2 6. Zero segmen u 6 = 0 The rms value is 6 rms = Dkuk = 376. A k= 1 Even hough is duraion is very shor, he curren spike has a significan impac on he rms value of he curren wihou he curren spike, he rms curren is approximaely 2.0 A.