v GS D 1 i S i L v D + V O + v S i D

2 Buck PWM DC DC Converer 2. Inroducion his chaper sudies he PWM buck swiching-mode converer, ofen referred o as a chopper [ 3]. Analysis is given for boh coninuous conducion mode (CCM) and disconinuous conducion mode (DCM). Curren and volage waveforms for all he componens of he converer are derived. he dc volage funcion is derived for boh he modes. Volage and curren sresses of he componens are found. he boundary beween CCM and DCM is deermined. An expression for he oupu volage ripple is derived. he power losses in all he componens and he ransisor gaedrive power are esimaed. he overall efficiency of he converer is deermined. Design examples are also given. 2.2 DC Analysis of PWM Buck Converer for CCM 2.2. Circui Descripion In general, a basic PWM converer, such as buck, boos, and buck boos converer, conains a single-pole, doublehrow swich, which conrols he energy flow from he source o he load. A circui of he PWM buck dc dc converer is depiced in Figure 2.(a). I consiss of four componens: a power MOSFE used as a conrollable swich S, a recifying diode D, an inducor, a filer capacior C. Resisor represens a dc load. Power MOSFEs are he mos commonly used conrollable swiches in dc dc converers because of heir high speeds. In 979, Inernaional Recifier paened he firs commercially viable power MOSFE, he HEXFE. Oher power swiches such as bipolar juncion ransisors (BJs), isolaed gae bipolar ransisors (IGBs), or MOS-conrolled hyrisors (MCs) may also be used. he diode D is called a freewheeling diode, a flywheel diode, oracach diode. he ransisor and he diode form a single-pole, double-hrow swich, which conrols he energy flow from he source o he load. he ask for he capacior and he inducor is energy sorage and ransfer. he swiching nework, composed of he ransisor and he diode, chops he dc inpu volage, and herefore he converer is ofen called a chopper, which producesa reduced averagevolage. he swichs is conrolled by a pulse-widh modulaor and is urned on and off a he swiching frequency f s. he duy cycle D is defined as D on on f on s on (2.) off Pulse-Widh Modulaed DC DC Power Converers, Second Ediion. Marian K. Kazimierczuk. 26 John Wiley & Sons, d. Published 26 by John Wiley & Sons, d. Companion Websie: www.wiley.com/go/kazimierczuk/modulaedpower2

Buck PWM DC DC Converer 23 S v GS D C (a) i S i v v D C (b) i v S v i D C (c) Figure 2. PWM buck converer and is ideal equivalen circuis for CCM. (a) Circui. (b) Equivalen circui when he swich is ON and he diode is OFF. (c) Equivalen circui when he swich is OFF and he diode is ON. where on is he ime inerval when he swich S is closed and off is he ime inerval when he swich S is open. Since he duy cycle D of he drive volage v GS is varied, so does he duy raio of oher waveforms. his permis he regulaion of he dc oupu volage agains changes in he dc inpu volage and he load resisance (or he load curren I O ). he circui -C- acs like a second-order low-pass filer whose corner frequency is f o (2π C). he oupu volage of he buck converer is always lower han he inpu volage. herefore, i is a sep-down converer. he buck converer bucks he volage o a lower level. Because he gae of he MOSFE is no referenced o ground, i is difficul o drive he ransisor. he converer requires a floaing gae drive. Wih he inpu curren of he converer being disconinuous, a smoohing C filer may be required a he inpu. he buck converer can operae in a CCM or in a DCM, depending on he waveform of he inducor curren. In CCM, he inducor curren flows for he enire cycle, whereas in DCM, he inducor curren flows only for a par of he cycle. In DCM, i falls o zero, remains a zero for some ime inerval, and hen sars o increase. Operaion a he boundary beween CCM and DCM is called he criical mode (CRM). e us consider he buck converer operaion in he CCM. Figures 2.(b) and (c) shows he equivalen circuis of he buck converer for CCM when he swich S is on and he diode D is off, and when he swich is off and he diode is on, respecively. he principle of he converer operaion is explained by he idealized curren and volage waveforms depiced in Figure 2.2. A ime, he swich is urned on by he driver. Consequenly, he volage across he diode is v D, causing he diode o be reverse biased. he volage across he inducor is v and herefore he inducor curren increases linearly wih a slope of ( ). For CCM, i () >. he inducor curren i flows hrough he swich, resuling in i S i when he swich is on. During his ime inerval, he energy is ransferred from he dc inpu volage source o he inducor, capacior, and he load. A ime D, he swich is urned off by he driver.

24 Pulse-Widh Modulaed DC DC Power Converers v GS D v A D A i I O Δ i D i S I SM I O I I v S D V SM D i D I DM I O I D D v D D Figure 2.2 Idealized curren and volage waveforms in he PWM buck converer for CCM.

Buck PWM DC DC Converer 25 he inducor has a nonzero curren when he swich is urned off. Because he inducor curren waveform is a coninuous funcion of ime, he inducor curren coninues o flow in he same direcion afer he swich urns off. herefore, he inducor acs as a curren source, which forces he diode o urn on. he volage across he swich is and he volage across he inducor is. Hence, he inducor curren decreases linearly wih a slope of. During his ime inerval, he inpu source is disconneced from he circui and does no deliver energy o he load and he C circui. he inducor and capacior C form an energy reservoir ha mainains he load volage and curren when he swich is off. A ime, he swich is urned on again, he inducor curren increases and hence energy increases. PWM converers are operaed a hard swiching because he swich volage waveform is recangular and he ransisor is urned on a a high volage. he power swich S and he diode D conver he dc inpu volage ino a square wave a he inpu of he -C- circui. In oher words, he dc inpu volage is chopped by he ransisor diode swiching nework. he -C- circui acs as a second-order low-pass filer and convers he square wave ino a low-ripple dc oupu volage. Since he average volage across he inducor is zero for seady sae, he average oupu volage is equal o he average volage of he square wave. he widh of he square wave is equal o he on-ime of he swich S and can be conrolled by varying he duy cycle D of he MOSFE gae-drive volage. hus, he square wave is a pulse-widh modulaed (PWM) volage waveform. he average value of he PWM volage waveform is D, which depends on he duy cycle D and is almos independen of he load for CCM operaion. heoreically, he duy cycle D may be varied from % o %. his means ha he oupu ranges from o. hus, he buck circui is a sep-down converer. In pracice, he dc inpu volage varies over a specified range while he oupu volage should be held a a fixed value. If he dc volage is increased, he duy cycle D is reduced so ha he produc D being he average value of he PWM volage remains consan. On he oher hand, if he inpu volage is reduced, he duy cycle D is increased so ha he average value of he PWM signal is consan. herefore, he amoun of energy delivered from he inpu volage source o he load can be conrolled by varying he swich on-duy cycle D. If he oupu volage and he load resisance (or he load curren I O ) are consan, he oupu power is also consan. When he inpu volage increases, he swich on-ime is reduced o ransfer he same amoun of energy. he pracical range of D is usually from 5% o 95% due o resoluion. he duy cycle D is conrolled by a conrol circui. he inducor curren conains an ac componen which is independen of he dc load curren in CCM and a dc componen which is equal o he dc load curren I O. As he dc oupu curren I O flows hrough he inducor, only one-half of he B H curve of he inducor ferrie core is exploied. herefore, he inducor should be designed such ha he core will no saurae. o avoid core sauraion, a core wih an air gap and sufficienly large volume may be required. 2.2.2 Assumpions he analysis of he buck PWM converer of Figure 2.(a) begins wih he following assumpions: () he power MOSFE and he diode are ideal swiches. (2) he ransisor oupu capaciance, he diode capaciance, and he lead inducances are zero and hereby swiching losses are negleced. (3) Passive componens are linear, ime invarian, and frequency independen. (4) he oupu impedance of he inpu volage source is zero for boh dc and ac componens. (5) he converer operaes in seady sae. (6) he swiching period f s is much shorer han he ime consans of reacive componens. (7) he dc oupu volage is consan, bu he dc inpu volage and he load resisance are variable. 2.2.3 ime Inerval: < D During he ime inerval < D, he swich S is on and he diode D is off. An ideal equivalen circui for his ime inerval is shown in Figure 2.(b). When he swich is on, he volage across he diode v D is approximaely

26 Pulse-Widh Modulaed DC DC Power Converers equal o, causing he diode o be reverse biased. he volage across he swich v S and he diode curren are zero. he volage across he inducor is given by v di d. (2.2) Hence, he curren hrough he inducor and he swich S is i S i v d i () d i () i () (2.3) where i () is he iniial curren in he inducor a ime. he peak inducor curren becomes i (D) ( )D i () (2.4) and he peak-o-peak ripple curren of he inducor is Δi i (D) i () ( )D ( )D f s D( D). (2.5) f s he diode volage is v D. (2.6) hus, he peak value of he diode reverse volage is V DM. (2.7) he average value of he inducor curren is equal o he dc oupu curren I O. Hence, he peak value of he swich curren is I SM I O Δi 2. (2.8) he insananeous energy sored in he magneic field in he inducor is w () 2 i2 [ 2 VI V 2 O i ()]. (2.9) he increase in he magneic energy sored in he inducor during he ime inerval o D is given by Δw (in) 2 [ i 2 (D) i2 ()]. (2.) he ime inerval o D is erminaed when he swich is urned off by he gae driver. 2.2.4 ime Inerval: D < During he ime inerval D <, he swich S is off and he diode D is on. Figure 2.(c) shows an ideal equivalen circui for his ime inerval. Since i (D) is nonzero a ha insan, he swich urns off and he fac ha he inducor curren i is a coninuous funcion of ime, he inducor acs as a curren source and urns he diode on. he swich curren i S and he diode volage v D are zero and he volage across he inducor is v di d. (2.) he curren hrough he inducor and he diode can be found as i D i D v d i (D) D ( )d i (D) D d i (D) ( D) i (D) (2.2)

Buck PWM DC DC Converer 27 where i (D) is he iniial condiion of he inducor a D. he peak-o-peak ripple curren of he inducor is Δi i (D) i () ( D) ( D). (2.3) f s Noe ha he peak-o-peak value of he inducor curren ripple Δi is independen of he load curren I O in CCM and depends only on he dc inpu volage and hereby on he duy cycle D. For a fixed oupu volage,he maximum value of he peak-o-peak inducor ripple curren occurs a he maximum inpu volage max,which corresponds o he minimum duy cycle D min.iisgivenby Δi max ( D min ). (2.4) f s he swich volage v S and he peak swich volage V SM are given by v S V SM. (2.5) he diode and swich peak currens are given by I DM I SM I O Δi 2. (2.6) his ime inerval ends a when he swich is urned on by he driver. he decrease in he magneic energy sored in he inducor during ime inerval D < is given by ΔW (ou) 2 [ i 2 (D) i2 ()]. (2.7) For seady-sae operaion, he increase in he magneic energy ΔW (in) is equal o he decrease in he magneic energy ΔW (ou). he ransien and seady-sae waveforms in converers wih commercial componens can be obained from compuer simulaions using SPICE model. 2.2.5 Device Sresses for CCM he maximum volage and curren sresses of he swich and he diode in CCM for seady-sae operaion are V SMmax V DMmax max (2.8) and I SMmax I DMmax I Omax Δi max 2 I Omax (max )D min 2f s I Omax ( D min ). (2.9) 2f s 2.2.6 DC Volage ransfer Funcion for CCM he volage and curren across a linear inducor are relaed by Faraday s law in is differenial form v di d. (2.2) For seady-sae operaion, he following boundary condiion is saisfied i () i (). (2.2) Rearranging (2.2), v d di (2.22)

28 Pulse-Widh Modulaed DC DC Power Converers and inegraing boh sides yields v d di i () i (). (2.23) he inegral form of Faraday s law for an inducor under seady-sae condiions is v d. (2.24) he average value of he volage across an inducor for seady sae is zero. hus, V (AV ) v d. (2.25) his equaion is also called a vol-second balance for an inducor, which means ha vol-second sored is equal o vol-second released. he inducor average volage for PWM converers operaing in CCM is D V (AV ) v d v d (2.26) from which D v d v d. (2.27) D his means ha he area encircled by he posiive par of he inducor volage waveform A is equal o he area encircled by he negaive par of he inducor volage waveform A,hais, V (AV ) [ D ] v d v d (2.28) D where D A v d (2.29) and Referring o Figure 2.2, which simplifies o he form D A v d. (2.3) D ( )D ( D) (2.3) D. (2.32) For a lossless converer, I I I O. Hence, from (2.32), he dc volage ransfer funcion (or he volage conversion raio) of he lossless buck converer is given by he range of M VDC is M VDC I I I O D. (2.33) M VDC. (2.34)

Buck PWM DC DC Converer 29 Noe ha he oupu volage is independen of he load resisance. I depends only on he dc inpu volage and he duy cycle D. he sensiiviy of he oupu volage wih respec o he duy cycle is S d dd. (2.35) In mos pracical siuaions, D is consan which means ha if is increased, D should be decreased by a conrol circui o keep consan, and vice versa. he dc curren ransfer funcion is given by M IDC I O I I D (2.36) and is value decreases from o as D is increased from o. From (2.8), (2.5), and (2.33), he swich and he diode uilizaion in he buck converer is characerized by he oupu-power capabiliy c p P O I O D. (2.37) V SM I SM V SM I SM V SM As D is increased from o, so does c p. 2.2.7 Boundary Beween CCM and DCM Figure 2.3 depics he inducor curren waveform a he boundary beween he CCM and he DCM, where i (). his waveform can be described by resuling in he peak inducor curren i, for < D (2.38) Δi i (D) ( )D ( )D f s ( D) f s (2.39) where M VDC D for a lossless buck converer. Hence, one obains a dc load curren a he boundary and he load resisance a he boundary I OB Δi 2 ( )D 2f s ( D) 2f s (2.4) B 2f s I OB D. (2.4) Figures 2.4 and 2.5 show he normalized load curren I OB ( 2f s ) D and he load resisance B (2f s ) ( D) a he boundary beween CCM and DCM as funcions of he duy cycle D, respecively. he plos can be obained using MAAB, described in Appendix B. i Δi max max min I OB D min D max Figure 2.3 Waveforms of he inducor curren a he boundary beween CCM and DCM a min and max.

3 Pulse-Widh Modulaed DC DC Power Converers.8 CCM I OB /( /2f s ).6.4 DCM.2.2.4.6.8 D Figure 2.4 Normalized load curren I OB ( 2f s ) a he boundary beween CCM and DCM as a funcion of he duy cycle D for buck converer. 8 B /(2 f s ) 6 4 DCM 2 CCM.2.4.6.8 D Figure 2.5 Normalized load resisance B (2f s ) a he boundary beween CCM and DCM as a funcion of he duy cycle D for buck converer.

For he wors case, I Omin I OBmax Δi max 2 Buck PWM DC DC Converer 3 max 2f s min ( D min ) 2f s min. (2.42) Hence, he minimum inducance required o mainain he CCM operaion for he duy cycle ranging from D min o D max is > min D min(max ) ( D min ) max ( D min ). (2.43) 2f s I Omin 2f s I Omin 2f s As he swiching frequency f s increases, he minimum inducance min decreases. herefore, high swiching frequencies are desirable o reduce he size of he inducor. In some applicaions, he inducance can be much higher han min in order o reduce he ripple curren hrough he inducor and he filer capacior. herefore, i is easier o reduce he oupu volage ripple, o avoid he core sauraion, and o reduce he winding and core losses. In a real converer, he efficiency η<, and herefore M VDC ηd. Since (ηd), Δi i (D) ( )D ( )D f s ( ) I OB Δi 2 ( )D V D O η 2f s 2f s I Omin I OBmax Δi max 2 ( ) V D O η f s (2.44) (2.45) ( (max )D V D O min η min (2.46) 2f s min 2f s min and ( ) ( ) > min D min (max ) V D O η min R D max η min, (2.47) 2f s I Omin 2f s I Omin 2f s where D min M VDCmin η (ηmax ). A gapped ferrie core should be used o make he inducor because he inducor curren conains a dc componen, and herefore he core may saurae. he inducance is given by μ A c N2 l g l c μ r (2.48) where N is he number of urns, A c is he core cross-secional area, l c is he magneic pah lengh (MP), and l g is he air-gap lengh. If he dc oupu curren I O and he dc inpu volage are fixed, he peak-o-peak inducor curren Δi 2I O can be made very large while mainaining he converer operaion in CCM. In his case, he ripple curren of he inducor should be limied, for example, Δi (2I O ) %. ) 2.2.8 Capaciors Capaciors are classified according o dielecric maerial used beween he conducors. he following ypes of capaciors are used in swiching-mode power supplies: we aluminum elecrolyic capaciors we analum elecrolyic capaciors

32 Pulse-Widh Modulaed DC DC Power Converers solid elecrolyic capaciors ceramic capaciors. We elecrolyic capaciors can be buil using aluminum or analum. hey are made of wo aluminum foils. A paper spacer soaked in we elecrolye separaes he wo aluminum foils. One of he aluminum foils is coaed wih an insulaing aluminum oxide layer, which forms he capacior dielecric maerial. he aluminum foil coaed in aluminum oxide is he anode of he capacior. he liquid elecrolye and he second aluminum foil ac as a cahode of he capacior. he wo aluminum foils wih aached leads are rolled ogeher wih he elecrolye soaked paper in a cylindrical aluminum case o form a we aluminum elecrolye capacior. We elecrolye analum capaciors are formed in a similar manner as we aluminum elecrolye capaciors excep ha he dielecric maerial is analum oxide. Solid elecrolyic capaciors are consruced similarly o we elecrolyic capaciors excep ha a solid dielecric maerial is used in place of a we dielecric maerial hese capaciors have moderae capaciances and a higher ripple curren raing. Elecrolyic capaciors are he mos commonly used in power elecronics because of a high raio of capaciance per uni volume and low cos. Ceramic capaciors use ceramic dielecric o separae wo conducive plaes. he ceramic dielecric maerial is composed of ianium dioxide (Class I) or barium ianae (Class II). Ceramic capaciors can be disc capaciors or mulilayer ceramic (MC) capaciors. Disc capaciors have low capaciance per uni volume. Conducive maerial is placed on he ceramic dielecric maerial forming inerlace fingers. Ceramic capaciors have lower capaciances han elecrolyic capaciors. he capaciances of ceramic capaciors are usually below μf. Ceramic capaciors have very low values of ESR. his propery reduces volage ripple and power loss. Imporan parameers of capaciors are he capaciance C, he equivalen series resisance (ESR) r C,andheseries equivalen inducance (ES) s, he self-resonan frequncy f r, and he breakdown volage V BD. he capaciance is C ε r εa (2.49) d where A is he area of each conducor, d is he hickness of he dielecric, ε r is he relaive permiiviy of he dielecric, and ε 8.85 2 F/m is he permiiviy of free space. he ESR is he sum of he resisances of leads, he resisances of he conacs, and he resisance of he plae conducors. he ES is he inducance of he leads. he self-resonan frequency is he dissipaion facor of a capacior is f r he qualiy facor of a capacior a a frequency f ω (2π)is 2π C s. (2.5) DF ωcr C. (2.5) Q C ωcr C DF. (2.52) For he buck converer, he ES is conneced in series wih he filer inducance and does no presen a problem. However, he ES can have a negaive effec in boos converer. Capaciors are raed for he breakdown volage and he maximum rms value of he ripple curren. he maximum rms ripple curren is he limi of ac curren and is dependen of he emperaure and frequency of he curren conduced by a capacior. he ripple curren flowing hrough he ESR causes power loss P C r C I 2 ac(rms),which generaes hea wihin he capacior. Elecrolyic analum capaciors have he highes values of ESR, and ceramic capaciors have he lowes ESR. he performance of elecrolyic capaciors is highly affeced by operaing condiions, such as frequency, ac curren, dc volage, and emperaure. he ESR is frequency dependen. As he frequency increases, he ESR firs

Buck PWM DC DC Converer 33 decreases, usually reaches a minimum value a he self-resonan frequency, and hen increases. For elecrolyic capaciors, he ESR decreases as he dc volage increases. I also decreases as he peak-o-peak ac ripple volage increases. he ESR is ofen measured by manufacurers a he capacior self-resonan frequency. he ESR of capaciors conrols he peak-o-peak value of he oupu ripple volage. Also, he higher he ESR of he capacior, he greaer he hea generaed due o he coninuous flow of curren hrough he ESR. his reduces he converer efficiency and life expecancy of he power supply. During aging process, he elecrolyic liquid inside he capacior gradually evaporaes, causing an increase in ESR. When a volage is applied beween he conducors and across he dielecric of a capacior, an elecric field is induced in he dielecric. he elecric energy is sored in he elecric field. he dielecric has a maximum value of he elecric field srengh E BD V BD d, resuling in a capacior breakdown volage V BD. 2.2.9 Ripple Volage in Buck Converer for CCM A model of he filer capacior consiss of capaciance C, equivalen series resisance r C, and equivalen series inducance ES he impedance of he capacior model is ( Z C r C j ω ES ) [ ( ω r ωc C jq Co ω )] r (2.53) ω r ω where he self-resonan frequency of he filer capacior is f r 2π (2.54) C ES and he qualiy facor of he capacior a is self-resonan frequency is Q Co. (2.55) ω r Cr C Figures 2.6 and 2.7 show plos of he magniude Z C and phase φ ZC of he capacior for C μf, r C 5 mω, and ES 5 nh. he filer capacior impedance is capaciive below he self-resonan frequency and inducive above he self-resonan frequency. he inpu volage of he second-order low-pass CR oupu filer is recangular wih a maximum value and a duy cycle D. his volage can be expanded ino a Fourier series v D [ 2 n sin(nπd) nπd D 2D [ sin πd πd cos nω s ] ] cos ω sin 2πD s 2πD cos 2ω sin 3πD s 3πD cos 3ω s. (2.56) he componens of his series are ransmied hrough he oupu filer o he load. I is difficul o deermine he peak-o-peak oupu volage ripple V r using he Fourier series of he oupu volage. herefore, a differen approach will be aken for deriving an expression for V r. A simpler derivaion [24] is given below. A model of he oupu par of he buck converer for frequencies lower han he capacior self-resonan frequency (i.e., f f r ) is shown in Figure 2.8. he filer capacior in his figure is modeled by is capaciance C and is equivalen series resisance (ESR) designaed by r C. Figure 2.9 depics curren and volage waveforms in he converer oupu circui. he dc componen of he inducor curren flows hrough he load resisor while he ac componen is divided beween he capacior C and he load resisor. In pracice, he filer capacior is designed so ha he impedance of he capaciive branch is much less han he load resisance. Consequenly, he load ripple curren is very small and can be negleced. hus, he curren hrough he capacior is approximaely equal o he ac componen of he inducor curren, ha is, i C i I O.

34 Pulse-Widh Modulaed DC DC Power Converers 5 4 3 Z C (Ω) 2 2 2 4 6 8 f (Hz) Figure 2.6 Magniude of he capacior impedance. 9 6 3 ϕ C ( ) 3 6 9 4 5 6 7 8 f (Hz) Figure 2.7 Phase of he capacior impedance.

Buck PWM DC DC Converer 35 i i O I O i o i C C r C v C v rc v o Figure 2.8 Model of he oupu circui of he buck converer for frequencies lower han he self-resonan frequency of he filer capacior. For he inerval < D, when he swich is on and he diode is off, he capacior curren is given by i C Δi D Δi (2.57) 2 resuling in he ac componen of he volage across he ESR ( v rc r C i C r C Δi D ). (2.58) 2 i I O D i c Δ D 2 v rc D v c D v o min D max V r Figure 2.9 Waveforms illusraing he ripple volage in he PWM buck converer.

36 Pulse-Widh Modulaed DC DC Power Converers he volage across he filer capaciance v C consiss of he dc volage V C and he ac volage v c,hais,v C V C v c. Only he ac componen v c may conribue o he oupu ripple volage. he ac componen of he volage across he filer capaciance is found as v c i C C d v c () Δi ( C D ) d v 2 c () Δi ( ) 2 2C D v c (). (2.59) For seady sae, v c (D) v c (). he waveform of he volage across capaciance C is a parabolic funcion. he ac componen of he oupu volage is he sum of volage across he filer capacior ESR r C and he filer capaciance C [ v o v rc v c Δi 2 ( 2CD rc D ) r ] C v 2C 2 c (). (2.6) e us consider he minimum value of he volage v o. he derivaive of he volage v o wih respec o ime is dv ( o Δi d CD r C D ). (2.6) 2C Seing his derivaive o zero, he ime a which he minimum value of v o occurs is given by min D 2 r CC. (2.62) he minimum value of v o is equal o he minimum value of v rc if min. his occurs a a minimum capaciance, which is given by C min(on) D max. (2.63) 2f s r Cmax Consider he ime inerval D < when he swich S is off and he diode D is on. Referring o Figure 2.9, he curren hrough he capacior is i C Δi ( D) ( D) Δi (2.64) 2 resuling in he volage across he ESR [ v rc r C i C r C Δi D ( D) ] (2.65) 2 and he volage across he capacior v c C Δi 2C i C d v c (D) Δi D C D [ 2 2D (D) 2 D ( D) [ D Adding (2.65) and (2.66) yields he ac componen of he oupu volage [ v o r c Δi D ( D) ] Δi 2 2C he derivaive of v o wih respec o ime is dv o d r C Δi ( D) Δi C ] d v c (D) ( D) 2 ] v c (D). (2.66) [ 2 2D (D) 2 D ( D) ] v c (D). (2.67) [ D ( D) ]. (2.68) 2

Buck PWM DC DC Converer 37 Seing he derivaive o zero, he ime a which he maximum value of v o occurs is expressed by ( D) max r 2 C C. (2.69) he maximum value of v o is equal o he maximum value of v rc if max D. his occurs a a minimum capaciance, which is given by C min(off ) D min. (2.7) 2f s r Cmax he peak-o-peak ripple volage is independen of he volage across he filer capaciance C and is deermined only by he ripple volage across he ESR if C C min max{c min(on), C min(off ) } max{d max, D min }. (2.7) 2f s r C Hence, and C min D max 2f s r C for D min D max > (2.72) C min D min for D 2f s r min D max <. (2.73) C For he wors case, D min ord max. hus, he above condiion is saisfied a any value of D if C C min. (2.74) 2r C f s If condiion (2.7) is saisfied, he peak-o-peak ripple volage of he buck converer is V r r C Δi max r C ( D min ). (2.75) f s For seady-sae operaion, he average value of he ac componen of he capacior volage v c is zero, ha is, v c d (2.76) resuling in v c () Δi (2D ). (2.77) 2f s C Waveforms of v rc, v c, and v o are depiced in Figure 2. for hree values of he filer capaciance C. In Figure 2.(a), he peak-o-peak value of v o is higher han he peak-o-peak value of v rc because C < C min. Figures 2.(b) and (c) shows he waveforms for C C min and C > C min, respecively. For boh hese cases, he peak-o-peak volages of v o and v rc are he same. For aluminum elecrolyic capaciors, τ r C C 65 6 s. If condiion (2.7) is no saisfied, boh he volage drop across he filer capacior C and he volage drop across he ESR conribue o he ripple oupu volage. he ac componen of he volage across he filer capacior increases when he ac componen of he charge sored in capacior is posiive. he posiive charge is equal o he area under he capacior curren waveform for i C >. he capacior curren is posiive during ime inerval 2. he maximum increase of he charge sored in he filer capacior in every cycle is ΔQ Δi max 2 2 2 Δi max 8 Δi max 8f s. (2.78)

38 Pulse-Widh Modulaed DC DC Power Converers.8 V rc, V c,v o (V).4.4 v rc v o v c.8.2.4.6.8 / (a).8 V rc, V c,v o (V).4.4 v rc v o v c.8.2.4.6.8 / (b).8.4 v o V rc, V c,v o (V).4 v rc v c.8.2.4.6.8 / (c) Figure 2. Waveforms of v c,v rc,andv o a hree values of he filer capacior for CCM. (a) C < C min.(b)c C min. (c) C > C min.

Buck PWM DC DC Converer 39 Hence, using (2.39), he volage ripple across he capaciance C is V Cpp ΔQ C Δi max 8f s C ( D min ) 8fs 2C ( Dmin)π2 2 fo 2fs 2 (2.79) where f o (2π C) is he corner frequency of he oupu filer. he minimum filer capaciance required o reduce is peak-o-peak ripple volage below a specified level V Cpp is C min Δi max ( D min ). (2.8) 8f s V Cpp 8f 2 s V Cpp hus, C min is inversely proporional o fs 2. herefore, high swiching frequencies are desirable o reduce he size of he filer capacior. Using (2.39), he peak-o-peak volage ripple across he ESR is V rcpp r C Δi max r C ( D min ). (2.8) f s Hence, he conservaive esimaion of he oal volage ripple is V r V Cpp V rcpp ( D min ) 8fs 2C r C ( D min ). (2.82) f s 2.2. Swiching osses wih inear MOSFE Oupu Capaciance e us assume ha he MOSFE oupu capaciance C o is linear. Firs, we shall consider he ransisor urn-off ransiion. During his ime inerval, he ransisor is off, he drain-o-source volage v DS increases from nearly zero o, and he ransisor oupu capaciance is charged. Because dq C o dv DS, he charge ransferred from he inpu volage source o he ransisor oupu capaciance C o during he urn-off ransiion is Q i I d dq C o dv DS C o (2.83) yielding he energy ransferred from he inpu volage source o he converer during he urn-off ransiion as W VI p()d v I i I d i I d Q C o V 2 I. (2.84) An alernaive mehod for deriving an expression for he energy delivered from a dc source o a series R-C o circui afer urning on is as follows. he inpu curren is where τ RC o is he ime consan. Hence, i I R e τ (2.85) W VI v I i I d i I d V2 I R e τ d V2 I τ R C o V2 I. (2.86) Using dw s Qdv DS 2, he energy sored in he ransisor oupu capaciance C o a he end of he ransisor urn-off ransiion, when v DS,isgivenby W s dw s 2 Q dv DS 2 Q 2 C o V2 I. (2.87)

4 Pulse-Widh Modulaed DC DC Power Converers hus, he energy los in he parasiic resisance of he capacior charging pah is he urn-off swiching energy loss described by W urn-off W VI W s C o V 2 I 2 C ov 2 I 2 C ov 2 I (2.88) which resuls in he urn-off swiching power loss in he resisance of he charging pah P urn-off W urn-off f s W urn-off 2 f s C o V2 I. (2.89) Afer urn-off, he ransisor remains in he off-sae for some ime inerval and he charge W s is sored in he oupu capaciance C o. he efficiency of charging a linear capaciance from a dc volage source is 5%. Now consider he ransisor urn-on ransiion. When he ransisor is urned on, is oupu capaciance C o is shored ou hrough he ransisor on-resisance r DS, he charge sored in C o decreases, and he drain-o-source volage decreases from o nearly zero. As a resul, all he energy sored in he ransisor oupu capaciance is dissipaed as hea in he ransisor on-resisance r DS. herefore, he urn-on swiching energy loss is resuling in he urn-on swiching power loss in he MOSFE W urn-on W s 2 C o V2 I (2.9) P urn-on P sw(fe) W urn-on f s W urn-on 2 f s C o V2 I. (2.9) he urn-on loss is independen of he ransisor on-resisance r DS as long as he ransisor oupu capaciance is fully discharged before he urn-off ransiion begins. he oal swiching energy loss in every cycle of he swiching frequency during he process of firs charging and hen discharging of he oupu capaciance is given by W sw W urn-off W urn-on W VI C o V 2 I (2.92) and he oal swiching loss in he converer is P sw W sw f s W sw f s C o V2 I. (2.93) For a linear capaciance, one-half of he swiching power is los in he MOSFE and he oher half is los in he resisance of he charging pah of he ransisor oupu capaciance, ha is, P urn-on P urn-off P sw 2. he behavior of a diode is differen from ha of a ransisor because a diode canno discharge is parallel capaciance hrough is forward resisance. his is because a diode does no urn on unil is volage drops o he hreshold volage. However, he juncion diodes suffer from he reverse recovery a urn-off. 2.2. Swiching osses wih Nonlinear MOSFE Oupu Capaciance he MOSFE drain-o-source capaciance C ds is a nonlinear capaciance of he pn sep-juncion body-diode, which depends on he drain-o-source volage v DS. his capaciance is given by C C ds J V C B J for v v DS v DS V DS V B (2.94) B V B where C J is he zero-bias juncion capaciance and V B is he buil-in poenial barrier and i is in he range.55.9 V. From (2.94), C ds (v DS ) C ds ( ) V B C v DS V ds ( ) B. v DS (2.95)

Buck PWM DC DC Converer 4 Manufacurers of power MOSFEs usually specify he capaciances C rss C gd, C iss C gs C gd,andc oss C ds C gd a f MHz. he capaciances C rss and C oss are measured a V DS 25 V and V GS V. Hence, C ds25 C oss C rss. he oupu capaciance a v DS is C ds ( ) C J V B C ds25 25 V B V B 5C ds25 VI. (2.96) Since dq C ds dv DS, he charge ransferred from he dc inpu volage source o he drain-o-source juncion capaciance C ds during he urn-off ransiion is given by v DS v DS V Q(v DS ) C ds (v DS )dv DS C B J dv v DS V DS B V B 2C J VB (v DS V B ) 2(v DS V B )C ds (v DS ) 2C ds (v DS )v DS. (2.97) Hence, Q( ) 2( V B )C ds ( ) 2C ds ( ). (2.98) he energy ransferred from he inpu dc volage source o he converer during he urn-off ransiion is given by W VI v I i I d i V B I d Q( ) 2 ( V B )C ds ( ) 2C ds ( )V 2 I. (2.99) V B Because dw s Qdv DS 2, he energy sored in he drain-o-source capaciance C ds a v DS is v DS W s (v DS ) dw s 2 V B Hence, one obains he energy sored in C ds a v DS V B V B Qdv DS C J VB v DS V B vds V B dv DS 2 3 (v DS V B )2 C ds (v DS ) 2 3 C ds (v DS )v2 DS. (2.) W s 2 3 ( V B )2 C ds ( ) 2 3 C ds ( )V2 I. (2.) herefore, he energy los in he resisance of he charging pah of he MOSFE oupu capaciance is given by W urn-off W VI W s 2C ds ( )V 2 I 2 3 C ds ( )V2 I 4 3 C ds ( )V2 I. (2.2) Hence, he swiching power loss dissipaed in he resisance r of he pah of charging he ransisor oupu capaciance is P r P urn-off W urn-off f s W urn-off 4 3 f s C ds ( )V2 I 2 3 f s C ds25 V 3 I. (2.3) he ransisor equivalen linear oupu capaciance ha causes he same swiching power loss in he charging pah resisance r during he urn-off ransiion as he linear one is derived as producing 2 f s C eq(r) V2 I 4 3 f s C ds ( )V2 I 2 3 f s C ds25 V 3 (2.4) I C eq(r) 8 3 C ds ( ) 4C ds25 3. (2.5)

42 Pulse-Widh Modulaed DC DC Power Converers During he urn-on ransiion, all he energy sored in he ransisor oupu capaciance is los in he MOSFE on-resisance r DS W urn-on W s 2 3 C ds ( )( V B )2 2 3 C ds ( )V2 I. (2.6) hus, he MOSFE urn-on swiching loss is P sw(fe) P urn-on W urn-on f s W urn-on 2 3 f s C ds ( )V2 I 3 f s C ds25 V 3 I. (2.7) he ransisor equivalen linear oupu capaciance ha causes he same swiching power loss in he MOSFE on-resisance during he urn-on ransiion as he linear one can be obained as 2 f s C eq(fe) V2 I 2 3 f s C ds ( )V2 I 3 f s C ds25 V 3 (2.8) I resuling in C eq(fe) 4 3 C ds ( ) 2C ds25 3. (2.9) he oal swiching energy loss in each cycle of he swiching frequency is W sw W urn-off W urn-on W VI 2C ds ( )V 2 I (2.) and he oal swiching loss in he converer is P sw W sw f s W sw 2f s C ds ( )V2 I f s C ds25 V 3 I. (2.) he ransisor equivalen linear oupu capaciance C eq(sw) ha produces he same amoun of he swiching loss as he nonlinear one a a given can be derived as yielding f s C eq(sw) V 2 I 2f s C ds ( )V 2 I f s C ds25 V 3 I (2.2) C eq(sw) 2C ds ( ) C ds25 VI. (2.3) he urn-off swiching power loss is wice as high as he urn-on swiching power loss for he MOSFE wih a nonlinear oupu capaciance. he raio of hese losses is P urn-off 2. (2.4) P urn-on Example 2. A power MOSFE IRF5 wih V B.77458 V, C rss 25 pf, and C oss pf is operaed in he buck PWM converer a V and f s khz. Find: C ds25, C J, C ds ( ), Q( ), W sw, P sw, C eq(sw), W urn-on, P sw(fe), C eq(fe), W urn-off, P urn-off,andc eq(r). Soluion: he ransisor drain-o-source capaciance a V DS 25 V is C ds25 C oss C rss 25 75 pf. (2.5) he zero-bias drain-o-source capaciance is C J C ds25 25 V B 75 25 432.75 pf. (2.6).77458

Buck PWM DC DC Converer 43 he drain-o-source capaciance a V is C ds ( ) C J V B 432.75 2.77458 he charge ransferred from he dc inpu source o C ds during he urn-off ransiion is he swiching energy is he swiching loss is 37.93 pf. (2.7) Q( ) 2( V B )C ds ( ) 2 (.77458) 37.93 2 7.6447 nc. (2.8) W sw W VI 2V 2 I C ds ( ) 2 2 37.93 2 758.6nJ. (2.9) P sw 2f s V 2 I C ds ( ) 2 3 2 37.93 2 75.86 mw. (2.2) he equivalen linear swiching capaciance is C eq(sw) 2C ds ( ) 2 37.93 2 75.86 pf. (2.2) he energy los during he urn-on ransiion is equal o he energy sored in C ds a he end of he urn-off ransiion when v DS. his energy is W urn-on W s 2 3 V2 I C ds ( ) 2 3 2 37.93 2 252.87 nj. (2.22) he swiching power loss in he MOSFE is P sw(fe) 2 3 f s V2 I C ds ( ) 2 3 3 2 37.93 2 25.287 mw. (2.23) he equivalen linear urn-on capaciance is C eq(fe) 4 3 C ds ( ) 4 3 37.93 2 5.57 pf. (2.24) he energy los in he resisance of he charging pah of C ds during he urn-off ransiion is he urn-off swiching loss is W urn-off 4 3 V2 I C ds ( ) 4 3 2 37.93 2 55.73 nj. (2.25) P urn-off 4 3 f sv 2 I C ds( ) 4 3 3 2 37.93 2 5.573 mw. (2.26) he urn-off equivalen linear capaciance is C eq(r) 8 3 C ds ( ) 8 3 37.93 2.pF. (2.27) 2.2.2 Power osses and Efficiency of Buck Converer for CCM An equivalen circui of he buck converer wih parasiic resisances is shown in Figure 2.. In his figure, r DS is he MOSFE on-resisance, R F is he diode forward resisance, V F is he diode hreshold volage, r is he ESR of he inducor, andr C is he ESR of he filer capacior C. he slope of he I D V DS curves in he ohmic region is equal o he inverse of he MOSFE on-resisance r DS. he MOSFE on-resisance r DS increases wih emperaure because he mobiliy of elecrons μ n K 2.5 decreases wih emperaure in he range from o 4 C, where K is a consan. ypically, r DS doubles as he emperaure rises by C.

44 Pulse-Widh Modulaed DC DC Power Converers i S r DS i r I O i C i D C R F V F r C Figure 2. Equivalen circui of he buck converer wih parasiic resisances and he diode offse volage. he large-signal model of a diode consiss of a baery V F in series wih a forward resisance R F. he volage across he conducing diode is V D V F R F I D. If a line is drawn along he linear high-curren porion of he I D V D curve (or log(i D ) V D ) exending o he V D -axis, he inercep on he V D -axis is V F and he slope is R F. he hreshold volage V F is ypically.7 V for silicon (Si) pn juncion diodes, and V F 2.8 V for silicon carbide (SiC) pn juncion diodes. he hreshold volage V F.3.4 V for silicon Schoky diodes and V F 2Vfor silicon carbide Schoky diodes. he hreshold volage V F of silicon diodes decreases wih emperaure a he rae of 2 mv/ C. he series resisance R F of pn juncion diodes decreases wih emperaure, while resisance R F of Schoky diodes increases wih emperaure. he conducion losses will be evaluaed assuming ha he inducor curren i is ripple free and is equal o he dc oupu curren I O. Hence, he swich curren can be approximaed by i S { IO, for < D, for D < (2.28) which resuls in is rms value and he MOSFE conducion loss I Srms i 2 S d D I 2 O d I O D (2.29) P rds r DS I 2 Srms Dr DS I2 O Dr DS P R O. (2.3) he ransisor conducion loss P rds is proporional o he duy cycle D a a fixed load curren I O.AD, he swich is off for he enire cycle and herefore he conducion loss is zero. A D, he swich is on for he enire cycle, resuling in a maximum conducion loss. Assuming ha D max min as for he lossless converer, he maximum MOSFE conducion power is P rdsmax D max r DS I 2 Omax D max r DS P R Omax r DS P min min R Omax. (2.3) min Assuming ha he ransisor oupu capaciance C o is linear, he swiching loss is expressed by P sw f s C o V 2 I f s C o V2 O M 2 VDC f s C o P M 2 O. (2.32) VDC he maximum swiching loss is P sw(max) f s C o V 2 Imax f s C o V2 O M 2 VDCmin f s C o min V2 Imax P V 2 O. (2.33) O

Buck PWM DC DC Converer 45 Excluding he MOSFE gae-drive power, he oal power dissipaion in he MOSFE is P FE P rds P ( sw 2 Dr DS I2 O 2 f s C DrDS o V2 I f s C o R ) P 2M 2 O. (2.34) VDC Similarly, he diode curren can be approximaed by {, for < D i D (2.35) I O, for D < yielding is rms value I Drms i 2 D d I 2 O d I O D (2.36) D and he power loss in R F P RF R F I 2 Drms ( D)R F I2 O ( D)R F P R O. (2.37) he average value of he diode curren is I D i D d which gives he power loss associaed wih he volage V F D I O d ( D)I O (2.38) P VF V F I D ( D)V F I O ( D)V F P. (2.39) O hus, he overall diode conducion loss is ( P D P VF P RF ( D)V F I O ( D)R F I 2 O ( D) VF R ) F P R O. (2.4) he diode conducion loss P D decreases, when he duy cycle D increases a a fixed load curren I O.AD, he diode is on for he enire cycle, resuling in a maximum conducion loss. A D, he diode is off for he enire cycle and herefore he conducion loss is zero. he maximum diode conducion loss is ( VF P Dmax ( D min ) R ) ( F P R Omax V )( O VF R F min max min ) P Omax. (2.4) ypically, he power loss in he inducor core can be ignored and only he copper loss in he inducor winding should be considered. he inducor curren can be approximaed by leading o is rms value i I O (2.42) I rms I O (2.43) and he inducor conducion loss P r r I 2 rms r I2 O r P R O. (2.44) he maximum power loss in he inducor is P rmax r I 2 Omax r P R Omax. (2.45) min

46 Pulse-Widh Modulaed DC DC Power Converers Using (2.3), (2.57), and (2.64), he rms curren hrough he filer capacior is found o be I Crms i 2 C d Δi ( D) 2 2fs (2.46) and he power loss in he filer capacior P rc r C I 2 Crms r C Δi2 2 he maximum power loss in he capacior is P rcmax r C Δi2 max 2 he overall power loss is given by r C V2 ( D)2 O 2f 2 r C V2 O ( D min )2 2f 2 s 2 r C ( D)2 P s 2 2fs 2 O. (2.47) 2 r C ( max ) 2 2f 2 s 2 P Omax. (2.48) P S P rds P sw P D P r P rc Dr DS I 2 O f s C o V2 I ( D)(V F I O R F I 2 O ) r I2 O r C Δi2 2 [ DrDS f s C o R ( VF ( D) R ) F r r C R ] ( D)2 P M 2 V VDC O 2fs 2 O. (2.49) 2 hus, he converer efficiency is η P O P I P O P O P S Dr DS( D)R F r P S P O ( D)V F D Dr DS( D)R F r ( D)V F f sc o M 2 VDC r C ( D) 2 2f 2 s 2 D 2 2fs 22. (2.5) f sc o r C ( D) 2 For D, he swich is off and he diode is on, yielding he converer efficiency η R Fr For D, he swich is on and he diode is off, resuling in he converer efficiency V F. (2.5) η r DSr. (2.52) If he inducor peak-o-peak curren ripple Δi ( D) (f s ) D( D) (f s ) is aken ino accoun, he rms value of he swich curren is given by D I Srms 3 (I2 Smin I Smin I Smax I2 Smax ) I O D ( ) 2 Δi (2.53) 2 I O where I Smin I O Δi 2andI Smax I O Δi 2. Similarly, he rms value of he diode curren is I Drms D 3 ( I 2 Dmin I Dmin I Dmax ) I2 Dmax IO D 2 ( ) 2 Δi (2.54) I O

Buck PWM DC DC Converer 47 where I Dmin I O Δi 2andI Dmax I O Δi 2. he rms value of he inducor curren is ( I rms I 2 3 min I min I max ) I2 max IO ( ) 2 Δi. (2.55) 2 I O For example, for Δi I O., I rms.7i O,andforΔi I O.5, I rms.48i O. Assuming ha he resisances r, r DS,andR F are consan and frequency independen, he conducion power loss in he MOSFE is given by [ P rds r DS I 2 Srms r DS DI2 O ( ) ] 2 Δi r DS D [ ( ) ] 2 Δi P 2 I O 2 I O. (2.56) O he conducion power loss in he diode forward resisance is [ P RF R F I 2 Drms R F ( D)I2 O ( ) ] 2 Δi R [ F ( D) ( ) ] 2 Δi P 2 I O 2 I O. (2.57) O Assuming ha he inducor resisance r is independen of frequency, he power loss in he inducor winding is given by [ P r r I 2 rms r I2 O ( ) ] 2 Δi r [ ( ) ] 2 Δi P 2 I O 2 I O. (2.58) O he overall power loss is { [ DrDS ( D)R P S F r Hence, he converer efficiency is η 2 ( ) ] 2 Δi I O f s C o M 2 VDC ( ) ] 2 Dr DS( D)R F r [ Δi ( D)V F f sc o 2 I O M 2 VDC ( ) ] 2 Dr DS( D)R F r [ Δi ( D)V F f sc o 2 D D ( D)V F r C ( D)2 2f 2 r C ( D) 2 2f 2 s 2 s 2 r C ( D) 2 D 2 2fs 22 } P O. (2.59). (2.6) For example, for Δi I O., [ P r r I 2 rms r I2 O ( ) 2 ] ( r 2 I 2 O ).8333r 2 I 2 O. (2.6) For Δi I O.2, [ P r r I 2 rms r I2 O ( ) 2 ] ( r 2 5 I 2 O ).333r 3 I 2 O. (2.62) In he buck converer, par of he dc inpu power is ransferred direcly o he oupu and is convered o ac power, which is hen convered back o dc power. I can be shown ha he amoun of power which is convered o ac power is P AC ( D)P O (2.63) and he amoun of he dc power ha direcly flows o he oupu is P DC DP O. (2.64)

48 Pulse-Widh Modulaed DC DC Power Converers 2.2.3 DC Volage ransfer Funcion of ossy Converer for CCM he dc componen of he inpu curren is I I i S d I O d DI O (2.65) leading o he dc curren ransfer funcion of he buck converer M IDC I O I I D. (2.66) his equaion holds rue for boh lossless and lossy converers. he converer efficiency can be expressed as η P O P I I O I I D M VDC M IDC M VDC D from which he volage ransfer funcion of he lossy buck converer is M VDC η M IDC ηd For D, M VDC η<. From (2.68), he on-duy cycle is Dr DS( D)R F r Dr DS( D)R F r D ( D)V F D D M VDC η D ( D)V F f sc o M 2 VDC r C ( D) 2 2f 2 s 2 (2.67) ( ) 2 2fs 22. (2.68) f sc o C ( D) 2 η. (2.69) he duy cycle D a a given dc volage ransfer funcion is higher for he lossy converer han ha of a lossless converer. his is because he swich S mus be closed for a longer period of ime for he lossy converer o ransfer enough energy o supply boh he required oupu energy and he converer losses. Subsiuion of (2.69) ino (2.5) gives he converer efficiency where and ( VF N η M VDC r C 6f r DS R F s 22 ( M2 VDC r C 3f 2 s 2 R F r V F f s C o M 2 VDC ( D η 2 R F r η N η D η (2.7) ) {[ ( VF M VDC r C 2f 2 s 2 V F f s C o M 2 VDC r C 6f r DS R )] 2 F s 22 ) } 2 r C 2f 2 s 2 (2.7) ). (2.72) 2.2.4 MOSFE Gae-Drive Power When he ransisor is driven by a square-wave volage source, he MOSFE gae-drive power is associaed wih charging he ransisor inpu capaciance, when he gae-o-source volage increases, and discharging his

Buck PWM DC DC Converer 49 capaciance when he gae-o-source volage decreases. Unforunaely, he inpu capaciance of power MOSFEs is highly nonlinear and herefore i is difficul o deermine he gae-drive power, using he ransisor inpu capaciance. In daa shees, a oal gae charge Q g sored in he gae-o-source capaciance and he gae-o-drain capaciance is given a a specified gae-o-source volage V GS (usually, V GS V) and a specified drain-o-source volage V DS (usually, V DS.8 of he maximum raing). Using a square-wave volage source o drive he MOSFE gae, he energy ransferred from he gae-drive source o he ransisor is W G Q g V GSpp. (2.73) his energy is los during one cycle of he swiching frequency f s for charging and discharging he MOSFE inpu capaciance. hus, he MOSFE gae-drive power is P G W G f s W G f s Q g V GSpp. (2.74) he gae-drive power P G is proporional o he swiching frequency f s. he power gain is defined by k p P O. (2.75) P G he power-added efficiency (PAE) incorporaes he gae-drive power P G by subracing i from he oupu power P O and is defined by η PAE P O P G. (2.76) P I If he power gain k p is high, η PAE η. If he power gain k p <, η PAE <. he oal efficiency is defined by he average efficiency is defined by η P O P I P G. (2.77) η AVG P OAVG. (2.78) P IAVG In order o deermine his efficiency, he probabiliy-densiy funcions of he average inpu and oupu powers are required. 2.2.5 Gae Driver Boh he gae and he source of he MOSFE in he buck converer are conneced o wo ho poins. herefore, i is difficul o drive he ransisor. he driver is usually an inegraed circui, which requires a power supply and one end erminal of he power supply should be conneced o ground. One opion is o connec he driver beween he gae and ground. In his case, KV is yielding When he MOSFE is on, v DS, resuling in v G v GS v DS (2.79) v GS v G v DS. (2.8) v GS v G. (2.8)

5 Pulse-Widh Modulaed DC DC Power Converers If he gae-o-source volage v GS in he on sae is 5 V, he on-gae volage is v G(ON) v GS(ON) 5V o V. (2.82) For example, if 5V,v G(ON) 5 5 V o v G(ON) 5 5 V. However, if V, v G(ON) 5 5 V o V. When he MOSFE is off, he diode is on, v D, and v G(OFF) v GS(OFF). If v G(OFF), v GS(OFF) v G(OFF) V. his high volage will break he SiO 2 dielecric in he gae. 2.2.6 Design of Buck Converer for CCM Design a PWM buck converer operaing in CCM o mee he following specificaions: 28 ± 4V, 2 V, I Omin A,I Omax A, f s khz, and V r %. Soluion: he minimum, nominal, and maximum values of he inpu volage min 24 V, nom 28 V, and max 32 V. he maximum and minimum values of he dc oupu power are and he minimum and maximum values of he load resisance are P Omax I Omax 2 2 W (2.83) P Omin I Omin 2 2 W. (2.84) min 2.2 Ω (2.85) I Omax and max 2 2 Ω. (2.86) I Omin he minimum, nominal, and maximum values of he dc volage ransfer funcion are M VDCmin M V DCnom 2.375 (2.87) max 32 2.43 (2.88) nom 28 and M VDCmax 2.5. (2.89) min 24 Assume he converer efficiency η 85%. he minimum, nominal, and maximum values of he duy cycle are D min M VDCmin η.375.85.44 (2.9) and D nom M V DCnom η D max M VDCmax η.43.56 (2.9).85.5.588. (2.92).85

Buck PWM DC DC Converer 5 Assuming he swiching frequency f s khz, he minimum inducance ha is required o mainain he converer in CCM is ( ) ( ) R D max η min 2.44.85 min 44.8 μh. (2.93) 2f s 2 5 e us use a sandard value of he inducane 5 μh/r.5 Ω. he maximum inducor ripple curren is he ripple volage is Δi max ( D min ) f s 2 (.44).346 A. (2.94) 5 5 6 V r 2 2 mv. (2.95) If he filer capaciance is large enough, V r r Cmax Δi max and he maximum ESR of he filer capacior is r Cmax V r 2 3 89.5 mω. (2.96) Δi max.346 e r C 5 mω. he minimum value of he filer capaciance a which he ripple volage is deermined by he ripple volage across he ESR is C min max { Dmax, D } min 2f s r C 2f s r C D max 2f s r C.588 58.8 μf. (2.97) 2 5 5 3 Pick C μf/25 V/5 mω. he corner frequency of he oupu low-pass filer is f o 2π C 2π 2.25 khz. (2.98) 5 6 6 hus, f s f o 2.25 44.4. he bandwidh of he converer is approximaely equal o he corner frequency. he volage and curren sresses of power MOSFE and diode are V SMmax V DMmax max 32 V (2.99) and I SMmax I DMmax I Omax Δi max.427.735 A. (2.2) 2 2 An Inernaional Recifier IRF5 power MOSFE is seleced, which has V DSS V, I SM 4 A, r DS 55 mω, C o pf, and Q g 63 nc. Also, an MBR6 Schoky barrier diode is chosen, which has I DM 2 A, V DM 6 V, V F.4 V, and R F 25 mω. he power losses and he efficiency will be calculaed a he minimum load resisance min.2 Ω and he maximum dc inpu volage max 32 V, which correspond o he minimum duy cycle D min.44. he conducion power loss in he MOSFE is P rds D min r DS I 2 Omax.44.55 2 2.426 W (2.2) and he swiching loss is P sw f s C o V 2 Imax 5 2 32 2. W. (2.22) Hence, he oal power loss in he MOSFE is P FE P rds P sw 2 2.426.5 2.43 W. (2.23)

52 Pulse-Widh Modulaed DC DC Power Converers However, he maximum conducion power loss in he MOSFE occurs a he minimum dc inpu volage min 24 V, min.2 Ω, andd max.588. hus, P rdsmax D max r DS I 2 Omax.588.55 2 3.234 W. he diode loss due o V F is he diode loss due o R F is and he oal diode conducion loss is P VF ( D min )V F I Omax (.44).4 2.236 W (2.24) P RF ( D min )R F I 2 Omax (.44).25 2.398 W (2.25) he power loss in he inducor dc ESR r 5 mω is he power loss in he capacior ESR is he oal power loss is P D P VF P RF 2.236.398 3.634 W. (2.26) P r r I 2 Omax.5 2 5W. (2.27) P rc r C (Δi max )2 2.5.4272 2.8 W. (2.28) P S P rds P sw P D P r P rc 2.426. 3.634 5.8.78 W (2.29) and he efficiency of he converer a full load is P η O 2 9.55%. (2.2) P O P S 2.78 If he assumed efficiency is much differen han he calculaed one in (2.2), a nex ieraion sep is needed wih a new assumed converer efficiency. Noe ha he maximum conducion power loss in he MOSFE occurs a min 24 V and min.2 Ω and is given by P rds D max r DS I 2 Omax.588.55 2 3.234 W. (2.2) Assuming ha he peak-o-peak gae-o-source volage is V GSpp 6 V, he MOSFE gae-drive power is P G f s Q g V GSpp 5 63 9 6.8mW. (2.22) he efficiency η of he designed buck converer was compued from (2.7) hrough (2.72) over he enire range of he specified operaing condiions. Nex, he duy cycle D was compued from (2.69), using he calculaed efficiency η. he plos of η and D as funcions of, I O,and are shown in Figures 2.2 hrough 2.7 for r DS 55 mω, R F 25 mω, V F.4 V,r 5 mω, r C 5 mω, 4 μh, C o pf, and f s khz. he converer efficiency η decreases as he load curren I O increases (or he load resisance decreases). he minimum efficiency η min occurs a he maximum load curren I Omax and he maximum dc inpu volage max.he duy cycle D decreases when increases, and D increases when I O increases (or decreases). 2.3 DC Analysis of PWM Buck Converer for DCM Equivalen circuis for he PWM buck converer operaing in he DCM are depiced in Figure 2.8. Idealized curren and volage waveforms are shown in Figure 2.9. A ime when he swich is urned on, he inducor curren is zero. For he ime inerval < D, he swich is on and he diode is off as depiced in Figure 2.8(b). he volage across he diode is. he volage across he inducor is, which causes he inducor curren o increase