1 MI OpenCourseWare hp://ocw.mi.edu Power Elecronics Spring 2007 For informaion abou ciing hese maerials or our erms of Use, visi: hp://ocw.mi.edu/erms.
2 Chaper 5 Inroducion o DC/DC Converers Analysis echniques: Average KVL, KCL, P.S.S. Condiions. KCL in i2 Figure 5.1: KCL i j = 0 (5.1) 34
3 35 Average over ime: 1 ij = 0 1 i j = 0 < ij > = 0 (5.2) KCL applies o average curren as well as insananeous currens. (Derives from conservaion of charge). KVL Vn Figure 5.2: KVL V k = 0 (5.3) Average over ime: 1 Vk = 0 1 V k = 0 < V k > = 0 (5.4)
4 36 CHAPER 5. INRODUCION O DC/DC CONVERERS KVL applies o averaged variables. P.S.S. o analyze converers in Periodic Seady Sae (P.S.S.): Average KCL < i j > = 0 (5.5) Average KVL < V k > = 0 (5.6) di L from < V L > = L < > d di L in P.S.S. < > = 0 d Inducor in P.S.S. < V L > = 0 (5.7) dv L from < i C > = C < > d dv L in P.S.S. < > = 0 d Capacior in P.S.S. < i C > = 0 (5.8) If Circui is Lossless: P in = P ou (5.9) Consider he DC/DC converer from before (see Figure 5.3): q() (>0) C1 il q() = 1 I2 1 q() = 0 VL Vx C2 Vx() Pulse Widh Modulaion (PWM) d d 2 Duy Raio d d d 2 <Vx> = d Figure 5.3: DC/DC Converer
5 37 Assume L s and C s are very big, herefore: Analyze (using average relaions) in P.S.S.: < V L > = 0 v C () V C (5.10) i L () I L (5.11) < V L > = d (V 1 V 2 ) (1 d) (V 2 ) dv 1 V 2 = 0 (Since < V L >= 0, < V 2 >=< V x >= dv 1.) Consider currens: Combining: < i C2 > = 0 V 2 = dv 1 (5.12) I 1 = I 2 (5.13) < i C1 > = 0 I C1 = (I 1 I 2 d ) I 1 (1 d) = 0 I 1 = di 2 (5.14) I 1 = di 2 dv 1 = V 2 dv 1 I 1 = di 2 V 2 V 1 I 1 = I 2 V 2 (5.15)
6 38 CHAPER 5. INRODUCION O DC/DC CONVERERS herefore, power is (ideally) conserved. Noe: rick in his ype of average analysis is o be careful when one can use an average value and when one mus consider insananeous quaniies Wih he following ype of exernal nework and V 1,V 2 > 0, power flows from Swich implemenaion: buck or down converer (see Figure 5.4). C1 C2 Figure 5.4: Buck (down) Converer ype of direc converer because a DC pah exiss beween inpu and oupu in one swich sae. Suppose we change he locaion of source and load: Refine swiching funcion so q() = 1 when swich is in down posiion (see Figure 5.5). Similar analysis: By conservaion of power: < V L > = 0 (V 1 V 2 )(1 d) V 1 d = 0 1 V 2 = V 1 (5.16) 1 d I 2 = (1 d)i 1 (5.17)
7 39 q() 1 il q() = 0 I2 d d 2 Vx() - q() = 1 VL C1 Vx C2 - - d d 2 - VL - d Figure 5.5: Change he Locaion of Source and Load In his case, energy flows from 2 1 and he P.S.S. oupu volage (V 2 ) is higher han inpu volage (V 1 ). Wih he following swich implemenaion: boos or up converer. Anoher ype of direc converer (see Figure 5.6). C1 C2 Figure 5.6: Boos (up) Converer In general power flows direcion depends on: 1. Exernal nework 2. Swich implemenaion
8 40 CHAPER 5. INRODUCION O DC/DC CONVERERS 3. Conrol We may need o know all of hese o deermine behavior. he boos converer is ofen drawn wih power flowing lef o righ. However, here is nohing fundamenal abou his (see Figure 5.7). C2 C1 Figure 5.7: Boos (up) Converer Drawn Lef o Righ Boos: Swich urns on and incremenally sores energy from V 1 in L. Swich runs off and his energy and addiional energy from inpu is ransferred o oupu. herefore, L used as a emporary sorage elemen. Eiher he buck or boos can be seen as he appropriae connecion of a canonical cell (see Figure 5.8). A C B Figure 5.8: Direc Canonical Cell he direc connecion has B as he common node. he res of operaion is deermined by exernal nework, swich implemenaion and conrol. Swich implemenaion: Differen swiches can carry curren and block volage only in cerain direcions.
9 41 MOSFE can block posiive V and can carry posiive or negaive i (see Figure 5.9). D G D i V Body Diode S G Figure 5.9: MOSFE S BJ (or darlingon) is similar, bu negaive V blows up device (see Figure 5.10). i i Same for V V IGB Figure 5.10: BJ Combine elemens: 1. Block posiive V and carry posiive and negaive i (see Figure 5.11). i V Figure 5.11: Combine Elemens 1 2. Block posiive and negaive V and carry posiive i (see Figure 5.12).
10 42 CHAPER 5. INRODUCION O DC/DC CONVERERS i V Figure 5.12: Combine Elemens 2 3. Block posiive and negaive V and carry posiive and negaive i (see Figure 5.13). i V Figure 5.13: Combine Elemens 3 We can also consruc indirec DC/DC converers. Sore energy from inpu, ransfer energy o oupu, never a DC pah from inpu o oupu (see Figure 5.14). A B C Figure 5.14: Canonical Cell Spli capacior (see Figure 5.15):
11 43 q() = 1 q() = 0 1 q() d d 2 q() = 0 q() = 1 VL C1 C2 d Spli Capacior Figure 5.15: Indirec DC/DC Converer < V L > = 0 < V L > = V 1 d (1 d)v 2 d V 2 = V 1 (5.18) 1 d V 2 for 0 < d < 1 < < 0 (5.19) V 1 Sore energy in L(d ) from V 1. Discharge i (he oher way) in V 2. (mus have volage inversion). Buck/Boos or up/down converer (see Figure 5.16):
12 44 CHAPER 5. INRODUCION O DC/DC CONVERERS I2 Figure 5.16: Buck/Boos or up/down converer V 1 > 0 I 1 > 0 V 2 < 0 I 2 > 0 Oher indirec converers include CUK and SEPIC varians. Given conversion range < V 2 < 0, why no always use indirec vs. direc? V 1 1. Sign inversion (can fix) 2. Device and componen sresses Look a averaged circui variables (see Figure 5.17): Assume C, L are very large. I L = I 1 I 2 I L = I 1 I 2 (5.20) By averaged KCL ino doed box: Maybe couner inuiive: I 1 = average ransisor curren. I 1 I 2 = peak ransisor curren.
13 45 VC I2 iq iq IL I2 d Figure 5.17: Averaged Circui Variables By averaged KVL around loop: V C = V 1 V 2 V C = V 1 V 2 (5.21) herefore, for big L, C (see Figure 5.18): I2 Q D I2 Figure 5.18: Big L, C Indirec converer: So Q, D, L see peak curren I = I 1 I 2, Q, D, C block peak volage V = V 1 V 2. Consider direc converers (see Figure 5.19):
14 46 CHAPER 5. INRODUCION O DC/DC CONVERERS Buck Vq I2 Boos D C Q D Vd Q Vq C Figure 5.19: Direc Converers Buck: V C = V q,max = V d,max = V 1 (5.22) I L = i q,max = i d,max = I 2 (5.23) Boos: V C = V q,max = V d,max = V 2 (5.24) I L = i q,max = i d,max = I 1 (5.25) Direc converers (eiher ype): V C = V q,max = V d,max = max(v 1,V 2 ) (5.26) I L = i q,max = i d,max = max(i 1,I 2 ) (5.27) Device volage and curren sresses are higher for indirec converers han for direc converers wih same power. Inducor curren and capacior volage are also higher. Summary: For indirec converers (neglecing ripple) (see Figure 5.20):
15 47 VC I2 iq il I2 d Figure 5.20: Indirec Converers (neglecing ripple) I L = i sw,pk = i d,pk = I 1 I 2 (5.28) V C = V sw,pk = V d,pk = V 1 V 2 (5.29) For direc converers (neglecing ripple) (see Figure 5.21): Buck Vq IL Boos IL Vd Vq Figure 5.21: Direc Converers (neglecing ripple) I L = i sw,pk = i d,pk = max(i 1,I 2 ) (5.30) V C = V sw,pk = V d,pk = max(v 1,V 2 ) (5.31) Based on device sresses we would no choose an indirec converer unless we needed o, since direc converers have lower sress.
16 48 CHAPER 5. INRODUCION O DC/DC CONVERERS 5.1 Ripple Componens and Filer Sizing Now, selecing filer componen sizes does depend on ripple, which we have previously negleced. Les see how o approximaely calculae ripple componens. o eliminae 2 nd order effecs on capacior volage ripple: 1. Assume inducor is (Δi pp 0). 2. Assume all ripple curren goes ino capacior. Similarly, o eliminae 2 nd order effecs in inducor curren ripple: 1. Assume capaciors are (ΔV C,pp 0). 2. Assume all ripple volage is across he inducor. We can verify assumpions aferwards. Example: Boos Converer Ripple (see Figure 5.22) id id Figure 5.22: Boos Converer Ripple Find capacior (oupu) volage ripple (see Figure 5.23): Assume L, herefore, i 1 () I 1. So a ripple model for he oupu volage is (see Figure 5.24):
17 5.1. RIPPLE COMPONENS AND FILER SIZING 49 id Acual Waveform i D Including Ripple Id=<id>=(1-D) D Figure 5.23: Capacior Volage Ripple ~ id D D ~ id C R ~ V (1D) ΔVCpp 2 ΔVCpp 2 ~ VC D o V 2. Figure 5.24: Ripple Model wih Capacior If we assume all ripple curren ino capacior 1 2πf sw C R or Ṽ2 is small recpec Le us calculae he ripple: herefore, o limi ripple: i = C dv C d D 1 D ΔV C,pp = I 1 d 0 C (1 D)D = I 1 (5.32) C
18 50 CHAPER 5. INRODUCION O DC/DC CONVERERS (1 D)D C I 1 (5.33) ΔV C,pp Now le us find he capacior volage ripple (see Figure 5.25): Zi Source Impedance Vx C1 Vx Acual Vx Including Ripple D <Vx>=(1D) Figure 5.25: Ripple Replace V x wih equivalen source and eliminae DC quaniies (see Figure 5.26). ~ Vx D D ~ ~ i1 Vx (1D) ~ i1 Δ ipp 2 D Δ ipp 2 Figure 5.26: Ripple Model wih Inducor Neglecing he drop on any source impedance ( Z i 2πf sw L). 1 D Δi L,pp = (1 D)V 2 d L 0 D(1 D) = V 2 (5.34) L
19 5.1. RIPPLE COMPONENS AND FILER SIZING 51 herefore, we need: L D(1 D) (5.35) Δi pp Energy sorage is one meric for sizing L s and C s. Physical size may acually be deermined by one or more of: energy sorage, losses, packing consrains, maerial properies. o deermine peak energy sorage requiremens we mus consider he ripple in he waveforms. Define ripple raios (see Figure 5.27): ΔV C,pp R C = (5.36) 2V C Δi L,pp R L = (5.37) 2I L his is essenially % ripple: peak ripple magniude normalized o DC value. Xpk X Δ Xpp 2 Δ Xpp 2 Figure 5.27: Ripple Raios Specificaion of allowed ripple and converer operaing parameers deermines capacior and inducor size requiremens. herefore: V C,pk = V C (1 R C ) (5.38) i L,pk = I L (1 R L ) (5.39)
20 52 CHAPER 5. INRODUCION O DC/DC CONVERERS So from our previous resuls (boos converer): C L (1 D)D 2R C V C (5.40) D(1 D) 2R L I 1 (5.41) he ripple raios also deermine passive componen energy sorage requiremens and semiconducor device sresses. So les calculae he required energy sorage for he capacior: 1 CV 2 E C = 2 C,pk 1 (1 D)D V 2 = 2 2RC V C (1 R C ) 2 C = DI 2 V 2 (1 R C ) 2 4f sw R C DP o (1 R C ) 2 = (5.42) 4f sw R C So required capacior energy sorage increases wih: 1. Conversion raio 2. Power level and decreases wih swiching frequency. Similar resul for inducor energy sorage: (1 D)P o (1 R L ) 2 E L = (5.43) 4f sw I can be shown ha direc converers always require lower energy sorage han indirec converers. R L
21 5.2. DISCONINUOUS CONDUCION MODE 53 able 5.1: Effec of Allowed Ripple on Swiches Converer ype Value L, C Finie L, C Direc Indirec i sw,pk, i d,pk V sw,pk, V d,pk i sw,pk, i d,pk V sw,pk, V d,pk max( I 1, I 2 ) max( V 1, V 2 ) I 1, I 2 V 1, V 2 max( I 1, I 2 )(1 R L ) max( V 1, V 2 )(1 R C ) ( I 1, I 2 )(1 R L ) ( V 1, V 2 )(1 R C ) Consider effec of allowed ripple on swiches (see able 5.1): Define a meric for swich sizing (qualiaive only): For a boos converer:. Swich Sress P arameer(ssp ) = V sw,pk i sw,pk (5.44) SSP = max(v 1,V 2 )(1 R C )max(i 1,I 2 )(1 R L ) herefore, SSP ges worse for: = V 2 (1 R C )I 1 (1 R L ) P o = (1 R C )(1 R L ) 1 D V 2 = P o (1 R C )(1 R L ) (5.45) V 1 Large power Large conversion raio Large ripple 5.2 Disconinuous Conducion Mode Consider he waveform of he boos converer (see Figure 5.28):
22 54 CHAPER 5. INRODUCION O DC/DC CONVERERS q() Swiching Funcion for Diode qd() VL D L q() C R D il D Figure 5.28: Boos Converer Waveforms V 1 D Δi L,pp = (5.46) L V 2 = R(1 D) (5.47) Δi L,pp 2 R L = I 1 = V 1 D(1 D)R 2V 2 L = D(1 D) 2 R 2L (5.48) R L as R,L (5.49) (see Figure 5.29 for an illusraion) Evenually peak ripple becomes greaer han DC curren: boh swich and diode off for par of cycle. his is known as Disconinuous Condiion Mode (DCM). I
23 5.2. DISCONINUOUS CONDUCION MODE 55 il As R Increases il As L Decreases D D Figure 5.29: Changing R and L happens when R L > 1. R L = R L > R D(1 D) 2 R 2L 1 2L D(1 D) 2 (5.50) A ligh load (big R and low power) we ge DCM. Ligher load can be reached in CCM for larger L. DCM occurs for: L D(1 D)2 R 2 (5.51) he minimum inducance for CCM operaion is someimes called he criical inducance. D(1 D) 2 R L CRI,BOOS = (5.52) 2 For some cases (e.g. we need o operae down o almos no load), his may be unreasonably large. Because of he new swich sae, operaing condiions are differen (see Figure 5.30).
24 56 CHAPER 5. INRODUCION O DC/DC CONVERERS DCM q(), q D () VL D (DD2) - D (DD2) il Mus Be Zero in Remaining ime D (DD2) Figure 5.30: Differen Operaing Condiions Volage conversion raio: < V L > = 0 in P.S.S. V 1 D (V 1 V 2 )D 2 = 0 where D 2 < 1 D. V 1 (D D 2 ) = V 2 D 2 How does his compare o CCM? In CCM: V 2 1 = 1 D V 1 V 2 D D 2 = V 1 D 2 D = 1 (5.53) D 2
25 5.2. DISCONINUOUS CONDUCION MODE 57 1 D D = 1 D D = 1 (5.54) 1 D Since V 2 = 1 D and D 2 < 1 D, V 2 is bigger in DCM. V 1 D 2 V 1 Eliminaing D 2 from equaions, can be shown for boos: V D 2 R = 1 (5.55) V L herefore, conversion raio depends on R, f sw, L,... unlike CCM. his makes conrol ricky, as all of our characerisics change for par of he load range. How do we model DCM operaion? Consider diode curren (see Figure 5.31). IL id id() ipk id I2 D (DD2) Figure 5.31: DCM Operaion Model V 1 D i pk = L D 2 = Δ
26 58 CHAPER 5. INRODUCION O DC/DC CONVERERS Δi = L V D L = L V 2 V 1 V 1 D D 2 = (5.56) V 2 V 1 < i ou > = < i d > 1 1 = (D 2 )(i pk ) 2 1 V 1 V 1 D 1 = ( D )( ) 2 V 2 V 1 L V 1 2 D 2 = 2(V 2 V 1 ) Model as conrolled curren source as a funcion of D. So DCM someimes occurs under ligh load, as dicaed by sizing of L. (5.57) Someimes we can no pracically make L big enough. Mus handle conrol (changes from CCM o DCM). Also, we ge parasiic ringing in boh swiches (see Figure 5.32). Vx Vx Ideal L Rings wih Parasiic C s D D2 Figure 5.32: Parasiic Ringing Someimes people design o always be in DCM. Inducor size becomes very small and we can ge fas di (see Figure 5.33). d
27 5.2. DISCONINUOUS CONDUCION MODE 59 CCM Desired i DCM di/d limied so canno respond fas. Figure 5.33: Design in DCM In his case we ge: 1. Very fas di capabiliy. d 2. Simple conrol model i ou = f(d). 3. Small inducor size (E L R L = 1) Bu we mus live wih: 1. Parasiic ringing 2. High peak and RMS currens 3. Need addiional filers DCM is someimes used when very fas response speed is needed (e.g. for volage regulaor modules in microprocessors), especially if means are available o cancel ripple (e.g. inerleaving of muliple converers). In many oher circumsances DCM is avoided, hough one may have o operae in DCM under ligh-load condiions o keep componen sizes accepable.