BatChapter4.fm Page 128 Friday, February 21, :41 PM

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1 Bahaper4.fm Page 8 Friday, February, 003 3:4 PM INTOUTION 4. POWE SUPPY APPIATIONS 4.. inear egulaors 4.. Swiched-Mode Power Supplies 4. ONTINUOUS ONUTION MOE 4.. The Buck onverer 4.. The Boos onverer 4..3 The Buck-Boos onverer 4..4 Fourh-Order onverers 4..5 Bipolar Oupu olage onverers 4.3 ISONTINUOUS ONUTION MOE 4.3. The Buck onverer 4.3. The Boos onverer The Buck-Boos onverer 4.4 THE EFFETS OF ONETE NONIEAITIES 4.4. Inducor esisance 4.4. Transisor and iode olage rop Swich esisance 4.5 SWITH UTIIZATION FATO POBEMS In his chaper we discuss converer circuis ha are used in power elecronic sysems o change he sysem volages from one dc level o anoher dc level. Once again, swiching devices will be used o process energy from he inpu o he oupu. Since he inpu here is dc, which comes from a pos-filering sage, hese devices are normally operaed a much higher frequencies han he line frequency, reaching as high as a few hundred kiloherz. This is why such converer circuis are known as high-frequency dc-o-dc swiching converers or regulaors. The erm regulaor is used since he circui s main com-

2 Bahaper4.fm Page 9 Friday, February, 003 3:4 PM 4. Power Supply Applicaions mercial applicaion is in sysems ha require a sable and regulaed dc oupu volage. epending on wheher or no an oupu ransformer is used, high-frequency dc-o-dc swiching converers are classified as isolaed or nonisolaed. In his chaper and in he nex, he emphasis will be on he seady-sae analysis and design of several wellknown second- and fourh-order dc-o-dc converers, each having is own feaures and applicaions. We will consider hose opologies ha do no use high-frequency isolaion ransformers as par of heir power sage. Moreover, a large number of applicaions require oupu elecrical isolaion and muliple oupus ha canno be achieved using he basic opologies discussed in his chaper. The isolaed and magneically coupled opologies will be discussed in haper 5. Such opologies are he mos popular in he power supply indusry and are used in various ypes of elecronic equipmen whose design requires oupus wih elecrical isolaion and muli-oupus. A ypical block diagram of a linear regulaor power supply is shown in Fig. 4.. The fron end of he linear regulaor is a 60 Hz ransformer, T, used o provide inpu elecrical isolaion and o sep up or sep down he line volage, and his is followed by a full-wave bridge recifier o conver he ac inpu o a dc inpu by adding a large filering capacior a he inpu of he linear regulaor. This inpu o he linear regulaor,, is unregulaed dc and canno be used o drive he load direcly. Using a linear circui ha provides a sable dc oupu regulaes he dc volage a he oupu,. For many years, mos power supplies available in he marke were of he linear regulaor ype, in which a series-pass acive elemen is used o regulae he oupu volage. In general, he acive semiconducor elemen is used as a variable resisance o dissipae unwaned or excess volage. Such an arrangemen resuls in large amouns of power being dissipaed in he acive elemen, which can cause he efficiency o drop o as low as 40%. Because of his low efficiency, linear regulaors have no been used v s v in Unregulaed dc oupu egulaed dc oupu ac mains 60/50 Hz inpu isolaion ransfer v s v in ac/dc recifier bridge Filering capacior inear regulaor oad T onroller ref Typical block diagram of a linear regulaor power supply.

3 Bahaper4.fm Page 30 Friday, February, 003 3:4 PM haper 4 Nonisolaed Swich-Mode dc-dc onverers for medium- and high-power applicaions since he early 970s, when swiched-mode dc-dc converers enered he markeplace. espie he fac ha linear regulaors are simple o use and provide igh conrol, good oupu volage ripples, and a low componen coun, heir disadvanages are so numerous ha heir pracical use is limied. Because of heir high power losses, hey suffer from high hermal dissipaion, resuling in low power densiy and low efficiency. Figure 4. illusraes a simplified circui o show how he series acive elemen is conneced in a linear regulaor. represens he unregulaed inpu volage, Excess is he excess volage o be dissipaed across he linear device, and is he oupu volage. Excess, which is he difference beween and, mus be large enough o keep he ransisor in he linear acive mode, acing as a variable resisor used only o absorb he difference in he volage. This is why Excess is considered one of he key design parameers of linear regulaion. To illusrae how he linear resisor works, we presen a simplified opology showing he series elemen regulaor in Fig. 4.3, where ref is generaed by a zener diode, and and are used as a volage divider. The magniude of i B deermines how deeply he ransisor is driven in he sauraion region. The comparaor is used o compare he oupu volage wih a fixed reference volage. As he oupu volage increases, he base curren, i B, decreases and he excess volage, v E, increases, hence reducing he oupu volage. Similarly, if he oupu volage decreases, he resul is a reduced and an increased oupu. v E Excess o o _ (a) Excess (b) (a) Typical configuraion for a series acive elemen used in a linear regulaor. (b) Average volage waveforms. v E i B _ ref o o _ Simplified represenaion of a simple series elemen regulaor.

4 Bahaper4.fm Page 3 Friday, February, 003 3:4 PM 4. Power Supply Applicaions The developmen of he power semiconducor swich made i possible for power elecronics engineers o design power supplies wih much higher efficiencies compared wih linear regulaors. Since ransisors are used as swiching devices, such power supplies are known as swiched-mode power supplies or simply swiching converers. In recen years, swiching converers have become very popular due o recen advances in semiconducor echnology. Today swiching devices are available wih very high swiching speeds and very high power-handling capabiliies. I is possible o design swiched-mode power supplies wih efficiency greaer han 90% wih low cos and relaively small size and ligh weigh. Unlike linear regulaors, swiching converers use power semiconducor devices o operae in eiher he on-sae (sauraion or conducion) or he off-sae (cuoff or nonconducion). Since eiher sae will lead o low swiching volage or low swiching curren, i is possible o conver dc o dc wih higher efficiency using a swiching regulaor. Figure 4.4 shows a simplified block diagram for a swiched mode ac-o-dc power converer wih muli-oupu applicaion. ompared wih he block diagram of Fig. 4., a swiching nework and highfrequency oupu elecrical isolaion ransformer T are added. The objecive is o conrol he on ime of he power devices o regulae he dc oupu volage. The pos-filering is used o reduce he oupu volage ripple. This chaper will discuss he deailed power sage operaion of he dc-o-dc block shown in Fig. 4.4 wihou including he high-frequency isolaion ransformer. Because of he regulaion mehod used, hese converers are known as pulse-widh-modulaion (PWM) converers. v s vin Unregulaed dc inpu egulaed dc oupu ac mains v s T v in ac/dc recifier bridge Filering capacior Energy sorage elemens, Swiching nework Energy sorage elemens, ecificaion o a d ow-frequency inpu isolaion ransformer dc-dc converer Highfrequency oupu isolaion ransformer ecificaion o a d T ecificaion o a d n Block diagram of a swiched-mode power supply wih muliple oupus.

5 Bahaper4.fm Page 3 Friday, February, 003 3:4 PM haper 4 Nonisolaed Swich-Mode dc-dc onverers Seady-sae analyses of he basic direc-conneced second-order converers such as he buck (sep-down), boos (sep-up), and buck-boos (sep-up/down), and fourhorder converers such as uk and SEPI converers will be presened in his chaper. Boh he coninuous and disconinuous conducion modes of operaion, as well as some nonideal effecs will be included in he analysis. The seady-sae analysis of isolaed or magneically coupled converers (which are derived from hese basic converers) such as he flyback, forward, push-pull, half- and full-bridge, and Weinberg converers will be discussed in haper 5. Firs, we inroduce he basic concep of a swiched-mode circui and he conversion echnique in swiching converers. onsider he simples swiching volage converer, shown in Fig We assume ha he swich is ideal and i is urned on a 0 and urned off a alernaely as shown in Fig. 4.6(a), where f T is he swiching frequency. The waveforms for he oupu volage, v o, and he oupu curren, i o, are shown in Fig. 4.6(b) and (c), respecively, where is he dc inpu volage. The average oupu volage is given by T 0 o -- v T o () d 0 -- T in d T in If we le be defined as he duy raio or duy cycle, On ime Swiching period T hen he average oupu volage,, is given by 0 (4.) (4.) (4.3) If he oupu ripple facor K o, ripple is defined by he relaion K o,rms o o, ripple where, rms and are he rms and average values of v o (), respecively, deermine he oupu ripple facor for Fig ANSWE K o, ripple --- i s i o o v o A simple swiching circui.

6 Bahaper4.fm Page 33 Friday, February, 003 3:4 PM 4. oninuous onducion Mode v o Swich On Off On 0 T 0 T T (a) i s,i o (b) = / o (c) Swiching waveform for Fig I is clear from Eq. (4.3) ha he average oupu volage is less han he applied dc volage. Assuming an ideal swich, heoreically speaking, he efficiency of his converer is 00%. The drawback of his simple swiching circui is ha he oupu volage is no consan, bu a chopped dc wih high ripple volage. This resuls in high harmonics being generaed a he load. This will no be accepable for an elecronics load when he oupu mus be regulaed o a fixed dc level wih lile ripple. However, in some applicaion where a precise oupu volage is no required, such as heaing, ligh dimming, elecroplaing, and mechanical applicaions, his simple arrangemen migh be used, especially if he frequency is very high. One approach o smooh he oupu volage is o use a low-pass filer a he oupu of he circui in order o filer he high-swiching-frequency componens of he oupu volage. The filer may consis of a simple capacior and inducor. The capacior is used o hold a dc value across he oupu resisor, and he inducor is used wihin he circui o serve as a nondissipaive sorage elemen needed o sore energy from he inpu source and deliver i o he load. I can be argued ha one single-pole, single-hrow swich will no be sufficien o perform energy processing from he inpu o he oupu; raher, wo swiches or a single-pole, double-hrow swich is required. This can be easily jusified since he inducor curren canno be insananeously inerruped. When one swich is swiched, resuling in a sudden change in he inducor curren, a second swich mus be swiched so ha he coninuiy of he inducor curren is mainained. Hence, a pracical represenaion for a swiched-mode converer mus include eiher wo swiches, as shown in Fig. 4.7(a), or a single swich, as shown in Fig. 4.7(b). The inducor and capacior elemens are used as energy sorage componens o allow energy ransfer from he inpu o he oupu. The oupu capacior forms a lowpass filer o produce dc oupu volage wih lile ripple. Mos opologies, eiher isolaed or nonisolaed, will consis of one inducor and one oupu capacior hence he name second-order volage converer. Fourh-order volage converers consis of wo inducors and wo capaciors, which are considered series or combinaions of second-order converers.

7 Bahaper4.fm Page 34 Friday, February, 003 3:4 PM haper 4 Nonisolaed Swich-Mode dc-dc onverers v o _ v o _ (a) (b) Typical block diagram represenaions for swiching converers wih (a) wo independen swiches and (b) one single-pole, double-hrow swich. In second-order converers, depending on he arrangemens of he swiches and, hree possible opologies can be obained, namely buck, boos, and buck-boos, as shown in Fig Since hese swiches canno be urned on and off simulaneously, single-pole, double-hrow swiches are used, which can be eiher on or off. In he following secions hese converers will be horoughly analyzed by deriving heir seadysae characerisics. Since all swiched-mode converers have pulsaing currens and volages, he dc oupu volage can be obained by adding a low-pass filer. I will be shown ha in mos of hese converers an -ype low-pass filer secion is presen. In fourh-order circuis like he uk and SEPI, muliple low-pass filer secions are added. a b c (a) a b a b c (b) c (c) a b c (d) (a) Three possible ways o inser a low-pass filer beween a dc source and a capaciive load: (b) buck converer, (c) boos converer, and (d) buck-boos converer.

8 Bahaper4.fm Page 35 Friday, February, 003 3:4 PM 4. oninuous onducion Mode onsider he simple PWM converer of Fig. 4.5 wih 8 and f s 50 khz. Assume an ideal swich. esign for and so ha he power delivered o he load is 5 W a an average oupu curren of.5 A. Find he oupu ripple facor. Find he converer efficiency. SOUTION For 5 W average oupu power and average oupu curren of.5 A, he average oupu volage is given by The load resisance is For 6.7 and 8, he duy cycle is The rms value for he oupu volage is The ripple facor is given by.69 K 6.7 o, ripple This circui resuls in 83% oupu volage ripple, which is by no means accepable in many dc power supply applicaions. Since he swich is ideal, he converer efficiency is 00%, which can be illusraed as follows. The average oupu power is given by and he average inpu power is given by As expeced, he average inpu and oupu powers are equal. P o I o ,.69 rms T P o ---- s v o i o d T s 0 T v o T s s 0 in T d s T in d s 0 T P in ---- s T i in d s i in d T s 0 T T s s 0 in T s 0 T s i o d v T s o d

9 Bahaper4.fm Page 36 Friday, February, 003 3:4 PM haper 4 Nonisolaed Swich-Mode dc-dc onverers epea Example 4. by assuming he swich has a.8 volage drop across i when conducing. ANSWE 0.64, 0.75, 93.6% Figure 4.9(a) and (b) shows he circui configuraion for a buck converer wih a singleand wo-swich implemenaion. Figure 4.9(c) shows he ransisor-diode implemenaion. This opology is known as a buck converer because i seps down he average oupu volage below he inpu volage. Throughou his chaper o obain he seady-sae characerisic equaions, we will assume ha power swiching devices and he converer componens are lossless. Moreover, he exac seady-sae analysis of hese converers requires solving second-order nonlinear sysems. Such analysis is complex and because of he naure of he oupu volage, i is no necessary. Since hese converers funcion is o produce dc oupu, he S i in S v o _ i v o _ (a) (b) i in i i v o _ (c) On Off 0 T T On T (d) The buck (sep-down) converer. (a) Two-swich implemenaion. (b) Single-pole, double-hrow swich implemenaion. (c) Transisor-diode implemenaion. (d) Swiching waveform for he power swich.

10 Bahaper4.fm Page 37 Friday, February, 003 3:4 PM 4. oninuous onducion Mode oupu volage v o () consiss of he desired dc and he undesired ac componens. Pracically, he oupu ripple due o swiching is very small (less han %) compared o he level of he dc oupu volage. As a resul, we will assume he oupu ripple volage is small and can be negleced when evaluaing converer volage gains, i.e., v o. In oher words, he ripple-free oupu volage assumpion is made since he oupu ime consan for he filer capacior and he oupu resisor,, is very large. Moreover, he analysis will be based on he converer operaing in he seady-sae condiion, i.e., he converer currens and volages have reached heir seady-sae values. These assumpions can be summarized and represened mahemaically as follows: Since we assume lossless componens and ideal swiching devices, he average inpu power, P in, and he average oupu power, P o, are equal: (4.4) P in P o Since we assume seady-sae operaion, he inducor curren and he capacior volage are periodic over one swiching cycle, i.e., i ( 0 ) i ( 0 T) (4.5a) v c ( 0 ) v c ( 0 T) (4.5b) where 0 is he iniial swiching ime and T is he swiching period. Since we assume ideal capaciors and inducors, he average inducor volage and he average capacior curren are zero: I T 0 c -- i T c () d 0 0 (4.6) T 0 -- v (4.7) T () d 0 0 In fac, Eq. (4.7) is a represenaion of Faraday s law, which saes ha volage ime during charging equals volage ime during discharging. This is also known as he vol-second principle. These wo relaions sugges ha he oal energy sored in he capacior or he inducor over one swiching cycle is zero. Finally, hroughou he analysis in his chaper, he ypical swiching waveform for he power devices given in Fig 4.9(d) will be used o represen he swiching acion of he power swich. For simpliciy we se he iniial swiching ime o zero, 0 0. Again, is known as he duy raio or duy cycle, defined in Eq. (4.). The power ransisor is urned on for a period of T and urned off for he remaining ime ( )T. epending on wheher he swich is urned on or off, he inducor curren will be eiher charging hrough or discharging hrough he diode, respecively. As a resul, here are wo modes of operaion. We firs consider mode, when he swich is on, shown in Fig. 4.0(a). As shown in he figure, when he swich is on, he inpu volage,, forces he diode ino he reverse bias region. To deermine he volage conversion raio, he average inpu and oupu currens, and he oupu volage, we use he inducor curren as a sae variable in he following equaion: v di d o (4.8)

11 Bahaper4.fm Page 38 Friday, February, 003 3:4 PM haper 4 Nonisolaed Swich-Mode dc-dc onverers S i i o _ (a) i i o _ (b) Equivalen circui modes for he buck converer. (a) Mode : The power swich is on. (b) Mode : The power swich is off. Equaion (4.8) can be rearranged as follows: di ( (4.9) d in ) Inegraing Eq. (4.9) from 0 o wih I ( 0) as he iniial condiion, we obain i () -- ( (4.0) in ) I ( 0) Equaion (4.0) suggess ha he inducor curren charges linearly wih a slope of ( ), where I ( 0) is he iniial inducor curren value a 0, when he swich is firs urned on. This equaion applies as long as he swich is on. However, he equivalen circui model changes when he power swich is urned off a T, resuling in he equivalen circui of mode shown in Fig. 4.0(b), during which he diode is conducing. As shown in Fig. 4.0(b), in order for he inducor curren o mainain is coninuiy, he diode is forced o conduc by becoming forward biased so ha he diode picks up he curren in he direcion shown. The diode is known as flyback or free-wheeling because of he manner in which i is forced o urn on. The resulan equaion ha describes mode operaion is (4.) Inegraing boh sides of Eq. (4.) for T wih i ( T) as an iniial condiion, we obain where I ( T) di d -- o i () ( T) I (4.) ( T) is he iniial inducor curren when he swich is firs urned off.

12 Bahaper4.fm Page 39 Friday, February, 003 3:4 PM 4. oninuous onducion Mode Equaion (4.) suggess ha he inducor curren sars discharging a T wih he slope of, as shown in Fig. 4.(a). In seady-sae operaion we have I ( 0) I ( T) (4.3) Evaluaing Eq. (4.0) a T and Eq. (4.) a T and using Eq. (4.3), we obain he following wo relaions for I ( 0) and I ( T) : I ( T) -- ( in )T I ( 0) (4.4a) I ( 0) ( ) T I (4.4b) ( T) The seady-sae curren and volage waveforms are shown in Fig. 4.. I max and I min are he inducor curren values a he insans he swich is urned off and on, respecively. Nex we use he preceding relaions o derive expressions for he volage conversion, and average inpu and oupu currens. From Eqs. (4.4) we obain (4.5) i I max = I (T) I o I min = I (0) 0 T T (a) i in I max I min I in i (b) (c) I max I o I min I o i c T (d) (e) T Seady-sae waveforms for he buck converer: (a) inducor curren, (b) inducor volage, (c) inpu curren, (d) diode curren, and (e) capacior curren.

13 Bahaper4.fm Page 40 Friday, February, 003 3:4 PM haper 4 Nonisolaed Swich-Mode dc-dc onverers Hence, he maximum oupu volage gain is. We should poin ou ha Eq. (4.5) can be obained easily by using he vol-second principle across he inducor, which is given as follows: Mode ( inerval T) ( Inducor volage) ( ime) Mode ( inerval ( )T) (Inducor volage)(ime) 0 (4.6) where v equals ( ) and during ime inervals T and ( )T, respecively. We can make wo observaions on he buck volage gain equaion. Firs, since all he converer componens (,,, Q) are ideal, hey don dissipae any power, resuling in 00% volage efficiency. Second, he average inpu and oupu volage raio has a linear conrol characerisic curve, as shown in Fig. 4.. By varying he value of he duy cycle,, we can conrol he average oupu volage o he desired level. The inpu curren, i in, as illusraed in Fig. 4.(c), wih an average value of I in, is given by I T in -- i (4.7) T in () d 0 Since i in i in mode, we subsiue for i () from Eq. (4.0), and by evaluaing he inegral beween 0 and T, we obain I in ( (4.8) in ) T I ( 0) Using Eq. (4.4b), we obain I in -- ( I (4.9) max I min ) where I max and I min represen I ( T) and I ( 0), respecively. Similarly, by inspecion, he average oupu curren is given by v ()T v () ( )T 0 I I o I min I max o (4.0) / Ideal oupu conrol characerisic curve for he buck converer.

14 Bahaper4.fm Page 4 Friday, February, 003 3:4 PM 4. oninuous onducion Mode From Eqs. (4.4) and (4.0) we can solve for he maximum and minimum inducor currens, o obain I max -- ( )T I min -- ( )T Subsiuing hese equaions in Eqs. (4.9) and (4.0), we obain (4.) (4.) I o in I in in Hence, he curren gain is given by I ---- o --- (4.3) I in This relaion can be obained by equaing he average inpu and oupu power, o yield I in I o I in o I o From Eqs. (4.5) and (4.3), i is clear ha he curren and volage relaions for he converer are equivalen o a dc ransformer model wih a raio of, as shown in Fig The sinusoidal curve and sraigh line drawn across he ransformer windings indicae ha he ransformer is capable of ransferring ac and dc, respecively. I is clear ha for I min 0, he converer will operae in he coninuous conducion mode (ccm). To find he minimum inducor value ha is needed o keep he converer in he ccm, we se o zero and solve for : I min I min in -- ( )T cri T (4.4) I in : I o Equivalen circui represenaion for he buck converer, referred o as a dc-dc ransformer.

15 Bahaper4.fm Page 4 Friday, February, 003 3:4 PM haper 4 Nonisolaed Swich-Mode dc-dc onverers where cri is he criical inducance minimum value for a given, T, and before he converer eners he disconinuous conducion mode (dcm) of operaion. Since we have assumed ha he oupu volage has no ripple, he enire ac oupu curren from he inducor passes hrough he parallel capacior, and only dc curren is delivered o he load resisor. In pracice, he value of he oupu capacior is an imporan design parameer since i influences he overall size of he dc-odc converer and how much of he swiching frequency ripple is being removed. Having said ha, i is design pracice o choose a larger oupu capacior in order o limi he ac ripple across. Theoreically speaking, if, he capacior acs like a shor circui o he ac ripple, resuling in zero oupu volage ripple. If we assume is finie, hen here exiss a volage ripple superimposed on he average oupu volage. In order o derive an expression for he capacior ripple volage, we firs obain an expression for he capacior curren, which is given by he following relaion: As a resul, he iniial capacior curren a 0 is given by and a T, i c () i () I o I c ( 0) I ( 0) I o I max I min I c ( T) I ( T) I o I max I min The resulan capacior curren and volage are shown in Fig i c I I c(t) Q Q T T Q I c(0) v c () c (0) 0 T T T T apacior curren and volage waveforms.

16 Bahaper4.fm Page 43 Friday, February, 003 3:4 PM 4. oninuous onducion Mode The insananeous capacior curren can be expressed in erms of I from Eqs. (4.) and (4.), as shown in he following equaions: I i c () max I min I max I min I I T T 0 < T (4.5a) I i c () max I min ( )T I ( ) max I min I ( ( )T T ) I < T (4.5b) where I ( ( )T). From he capacior volage-curren relaion, i c dv ( c d), he capacior volage, v c (), can be expressed by he following inegral for 0 : v c () --- i c d c ( 0) where c ( 0) is he iniial capacior volage a 0. Subsiuing for i c () from Eq. (4.5a), we obain he following equaion: (4.6) where c ( T) is he iniial capacior volage when he swich is urned off a T. From Eq. (4.7) v c () is given by v c () I ( T) I ( T) ( )T c ( T) T < T (4.7) Since he capacior volage is in he seady sae, we have v c ( T) v c ( T) and v c ( 0) v c ( T), resuling in he following boundary condiions for he capacior volage: c ( T) c ( T) c ( 0) Since he average capacior volage is 0 v c () I I T d ( 0 ) c Evaluaing his inegral yields v c () I T I c ( 0) 0 < T Similarly, for T, he capacior volage is given by v c () --- i c d c ( T) 0 T, hen we have in general v c T o -- v T c () d v c 0 v c () d Subsiuing for and from Eqs. (4.6) and (4.7), we obain T T I ( )T c ( 0) (4.8)

17 Bahaper4.fm Page 44 Friday, February, 003 3:4 PM haper 4 Nonisolaed Swich-Mode dc-dc onverers Subsiue for I ( ( )T) in Eq. (4.8) o yield c ( 0) ( ) ( ) T (4.9) Hence, he capacior iniial values a 0 and T are equal, as expeced since he capacior curren is symmerical. Since he peak capacior volage occurs when he inducor curren is zero, we have he capacior minimum volage occurring a T, which is obained from Eq. (4.6): c, min I T I T T c ( 0) I T 8 c ( 0) and he maximum capacior volage occurring a ( )T Eq. (4.7): (4.30a) as obained from c, max I ( ) T T ( )T I ( ) T T c ( T) (4.30b) I ( ) 8 c ( T) Subsiuing for c ( 0) from Eq. (4.9) and using I ( ( )T), i can be shown ha and are expressed as follows: c, min c, max, ( ) ( ) T 4 c min (4.3a) c max Hence, he variaion in he capacior peak volage is given by and from Eqs. (4.3) we obain, ( ) T 4 c c, max c, min (4.3b) c f ( ) Someimes i is useful o express he raio of he ripple o he oupu volage, c (4.3) 0 8f This erm is known as he oupu volage ripple and represens he regulaion. As expeced, when he filering capacior and he frequency increase, he volage ripple decreases. c Anoher useful way o evaluae he expression for c wihou having o obain he exac expression for v c () is o use he oal charge, Q, deposied on he capacior curren inerval. Figure 4.4 shows he waveform for i c and v c wih areas of posiive

18 Bahaper4.fm Page 45 Friday, February, 003 3:4 PM 4. oninuous onducion Mode charge () and negaive charge (). Because of waveform symmery, T and ( )T represen he i c zero crossing imes when he capacior volage is minimum, c, min, and maximum, c, max, respecively. Hence, he capacior volage ripple is c. The oal charge sored in he capacior beween T and ( )T is obained from he following equaion: So he oal charge Q beween capacior curren i c zero crossings ( T < ( )T ) is given by Q c (4.33) However, since he oal charge is relaed o he curren according o he relaion hen we have From Eqs. (4.33) and (4.34), we obain dq dv c d d i dq d Q ( )T i d Area under he curve T T T ---T --I -- T -- --I T c I 8 Subsiuing for I ( ( )T), we obain c T f (4.34) (4.35) onsider a buck converer wih he following circui parameers: 0, 5, and I o 5 A, for f 50 khz. eermine: (a), (b) cri, (c) maximum and minimum inducor currens for 00 cri, (d) average inpu and oupu power, and (e) capacior volage ripple for 0.47 F. SOUTION 0.75 Using 3 Ω and T 0 s, he criical inducor value is given by cri T 7.5 H

19 Bahaper4.fm Page 46 Friday, February, 003 3:4 PM haper 4 Nonisolaed Swich-Mode dc-dc onverers For 00 cri 750 H0.75 mh, we have ( )T I min in I min ( 0.75) ( 0) A I max ( 0.75) ( 0) I max 5.05 A Since i is an ideal converer, he average oupu and inpu powers are given by P in P o I o ( 5) ( 5) 75 W The capacior volage ripple is given by o f ( 0.75) ( mh) ( 0.47 F) ( ) o % esign a buck converer wih he following specificaions: 0 0.5%, 0, W, f 30 khz, and 0.4. P o SOUTION In order o design his converer, we need o calculae he values for,, and. The oupu volage is given by Hence, he oupu curren is The oupu resisance is The criical inducance for ccm is given by 8 I o P o A cri T H

20 Bahaper4.fm Page 47 Friday, February, 003 3:4 PM 4. oninuous onducion Mode e us selec 600 H. Based on his value, he maximum and minimum inducor currens are given by The ripple volage is given by Solving for, I max -- ( )T ( 0.4) ( 0) A I min ( 0.4) ( 0) A o f ( f ) F edesign Example 4. o achieve an oupu ripple volage no o exceed % and an inducor curren ripple no o exceed 0% a he average load curren. ANSWE 0.5 mh, 8.33 F, 3 eermine he diode and ransisor average and rms curren values for Exercise 4.3. ANSWE.5 A, 3.75 A,.5 A, 4.33 A Show ha he expression for he peak capacior volage a ( )T is as given by Eq. (4.3b). Oher possible swich and ransisor-diode arrangemens are shown in Fig. 4.5(a) and (b), respecively. This opology is known as a boos converer since he oupu volage is higher han he inpu, as will be shown in his secion.

21 Bahaper4.fm Page 48 Friday, February, 003 3:4 PM haper 4 Nonisolaed Swich-Mode dc-dc onverers S S (a) i in i i i o i sw S (b) i c Boos converer. (a) Twoswich implemenaion. (b) Transisordiode implemenaion. Similar o he case for he buck converer, we assume all he converer componens are ideal and he ransisor swiching waveform is as shown in Fig. 4.9(d). When he swich is urned on, he equivalen circui of mode is shown in Fig. 4.6(a). This is a charging inerval, and he volage across he inducor is, and i () is given by i () -- (4.36) in I ( 0) 0 < T where I ( 0) is he iniial inducor curren value a 0.When he swich is urned off a T, he resulan equivalen mode circui is shown in Fig. 4.6(b). The inducor volage is, and i () is given by i () -- ( (4.37) in )( T) I ( T ) T < T i S (a) (b) Equivalen circui modes for he boos converer. (a) Mode : The swich is on. (b) Mode : The swich is off.

22 Bahaper4.fm Page 49 Friday, February, 003 3:4 PM 4. oninuous onducion Mode Evaluaing Eqs. (4.36) and (4.37) a T and T, respecively, and using he fac ha I ( T) I ( 0), we obain I ( T ) I ( 0) -- (4.38a) in ( T ) I ( T) I ( 0) -- ( (4.38b) in )( )T From Eqs. (4.38a) and (4.38b), he resuling volage conversion is given by o (4.39) in Hence, he volage gain is always greaer han. Also from Eqs. (4.38), he inducor ripple curren is given by I I ( T) I ( 0) I max I min -- in T Subsiuing for from Eq. (4.39), we obain I -- o ( )T Key curren and volage waveforms are given in Fig (4.40a) (4.40b) The inpu curren is he same as he inducor curren as shown in Fig. 4.7(a). Hence, he average inpu curren by inspecion is given by I I max I in min (4.4) The average oupu curren is he same as he average diode curren and is given by I I max I o min (4.4) ( ) o Since we assume an ideal converer, he average inpu and oupu powers mus be equal. Using Eqs. (4.4) and (4.4), we ge I in I o resuling in I ---- in (4.43) I o o in As wih he buck converer, he inpu-oupu curren and volage raios are equivalen o a dc ransformer wih a ransformer mode raio equal o ( ), as shown in Fig. 4.8.

23 Bahaper4.fm Page 50 Friday, February, 003 3:4 PM haper 4 Nonisolaed Swich-Mode dc-dc onverers i I (T) I (0) 0 T T (a) v ( ) (b) I (T) i I o (c) i c I max I o I o (d) urren and volage waveforms for he boos converer. I in ( ) : I o Equivalen ransformer circui represenaion for he boos converer. Using Eqs. (4.38) and (4.4), we can solve for he maximum and minimum inducor curren values: I ( 0) I min in T (4.44a) ( ) I ( T) I max in T (4.44b) ( )

24 Bahaper4.fm Page 5 Friday, February, 003 3:4 PM 4. oninuous onducion Mode For posiive values of I max and I min, he converer will operae in he coninuous conducion mode. To solve for he minimum criical inducor value ha will keep he converer in he ccm, we se I min o zero: Imin 0 Under his boundary condiion, he criical inducor value is given by cri T ( ) (4.45) I is clear from Fig. 4.7 ha when he diode is reverse biased, he capacior curren is he same as he load curren. Since we assume he load curren is purely dc, he capacior curren is given by i c I o 0 < T i c i I o T < T The capacior curren waveform is shown in Fig. 4.7(d) and redrawn in Fig. 4.9 along wih he capacior volage waveform. Mahemaical expressions for i c can be obained direcly from his figure. The curren i c () is expressed mahemaically as i c () I ( ( )T T ) I c ( T ) T T (4.46) i c I max I o I min I o I o 0 T T v c () c (0) c (T) c = o T 0 T T apacior curren and volage waveforms for he boos converer, assuming >. I min I o

25 Bahaper4.fm Page 5 Friday, February, 003 3:4 PM haper 4 Nonisolaed Swich-Mode dc-dc onverers where I c ( T ) is he iniial i c () a T. The capacior volage for 0 < T is given by v c () --- I o d c ( 0) 0 I --- o c ( 0) where c ( 0) is he iniial capacior volage a 0. A T we have (4.47) I c ( T ) ---T o c ( 0) (4.48) Since he average capacior volage is, we can solve for c ( 0) and c ( T ) as follows: I c ( 0) T o (4.49) I c ( T) T o and he capacior volage variaion is given by I c c ( 0) c ( T) o T (4.50) For T < T he capacior volage is given by v c () T --- I ( ( )T T ) I c ( T ) d c ( T ) T I ( T) ( )T I c ( T )( )T c ( T ) The oupu ripple volage is given by Then he volage ripple is given by c I T o o T o T f (4.5) (4.5) Skech he curren waveforms for i, i in, i, i o, and i c for he boos converer wih he following parameers:.8 mh, 50, 0, 0, 47 F, and f 5 khz. Also skech he volage waveforms for v, v sw, v c, and v. SOUTION In order o skech he waveforms, we need o find, he maximum and minimum inducor currens, and he average oupu curren.

26 Bahaper4.fm Page 53 Friday, February, 003 3:4 PM Hence, he ripple is oninuous onducion Mode The duy cycle is given by o which yields 0.58 Using 0 and T s, he maximum and minimum inducor currens are given by T I max ( ) 4.94 A and I min T ( ) 3.86 A The average inpu and oupu currens are given by I I max I in min 4.4 A I o I in ( ) 6 A The capacior peak currens are given by I cmax I max I o 8.94 A I cmin I min I o 7.86 A Hence, I c ( I cmax I cmin ).074 A Noice his value mus be equal o I. The capacior volage is given by v c () =0 0 I v c () o =T T A ( 0.58) ( 66.67s) F esign a boos converer wih he following specificaions: P o 7 W, 40, 8, %, f s 35 khz. SOUTION Firs le s deermine he duy cycle, : For coninuous conducion mode, he inducance minimum value is given by cri T ( )

27 Bahaper4.fm Page 54 Friday, February, 003 3:4 PM haper 4 Nonisolaed Swich-Mode dc-dc onverers where T 8.57 s o P o ( ) cri 4.44 H We choose 00 H since should be greaer han cri for ccm operaion. The oupu ripple volage is o f ( )( 35 khz) 7.3 F eermine he average and rms curren values for he diode and ransisor in Example 4.4. ANSWE 8.4 A, A, 6 A, 9.8 A The hird possible converer ype is obained by inerchanging he diode and he inducor of he buck converer o realize he design of Fig This converer is known as a buck-boos converer since is volage gain can be less han, equal o, or S S (a) i in i S i (b) i c Buck-boos converer. (a) Swich implemenaion. (b) Transisordiode implemenaion.

28 Bahaper4.fm Page 55 Friday, February, 003 3:4 PM 4. oninuous onducion Mode greaer han. Unlike he buck and boos converers, his converer gives a negaive oupu volage when used wihou isolaion. In his analysis, we follow assumpions made in performing he seady-sae analysis for he buck and boos opologies. When he swich is on, he diode is reverse biased wih he equivalen circui of mode as shown in Fig. 4.0(a); Fig. 4.(b) gives he equivalen circui for mode when he ransisor is off. When he ransisor is urned on, he inducor curren sars charging from he source volage,, while he diode is reverse biased. The volage across he inducor is, and i is given by i () -- (4.53) in () I ( 0) where I ( 0) is he iniial inducor value ha corresponds o he minimum inducor curren value. Evaluaing Eq. (4.53) a T, when he swich is urned off, we obain he maximum inducor curren. This yields he following relaion: I max I min -- (4.54) in T Similarly, in mode he inducor curren is given by i () ( T) I (4.55) ( T ) Evaluaing Eq. (4.55) a T, and since we assume seady-sae operaion, we use I ( 0) I ( T ) o obain I ( 0) o ( T T) I ( T ) (4.56a) I max I min o ( )T (4.56b) Equaing Eqs. (4.54) and (4.56b), we obain he following volage conversion raio: o (4.57) in I is clear from his equaion ha he oupu volage can be eiher smaller or greaer han he inpu volage: > 0.5 Boos < 0.5 Buck 0.5 Uniy gain S i i (a) (b) Equivalen circuis. (a) Mode : The ransisor is conducing. (b) Mode : The ransisor is no conducing.

29 Bahaper4.fm Page 56 Friday, February, 003 3:4 PM haper 4 Nonisolaed Swich-Mode dc-dc onverers i max i min i 0 T T i in i = i o v i c I o Q Q I cmin urren and volage waveforms for he buck-boos converer. Figure 4. gives ypical curren and volage waveforms for he buck-boos converer. From Fig. 4., i is clear ha he average inpu curren is given by I I max I in min (4.58a) and he average oupu curren is given by I I max I o min ( ) (4.58b) Since he conservaion of power mus hold, we use I in I o o obain I ---- in (4.59) I o o in To solve for I max and I min in erms of he converer componens, we subsiue for I o. Hence, from Eqs. (4.56) and (4.58b) we obain I max in T (4.60a) ( )

30 Bahaper4.fm Page 57 Friday, February, 003 3:4 PM 4. oninuous onducion Mode I min T (4.60b) ( ) The criical inducor value ha keeps he converer in he dcm mode can be obained by seing I min 0 in he above equaion o yield cri T ( ) (4.6) We noice ha for he same frequency of operaion and load resisance, he buck converer has he highes criical inducor limi when compared o he boos and buckboos, whereas he boos converer has he smalles cri, resuling in a wider range of inducor design. I can be shown ha he capacior curren of he buck-boos converer is he same as ha of he boos. Hence, he capacior volage is given by c () o T (4.6) o T f (a) Skech he curren and volage waveforms for a buck-boos converer wih he following parameers: 40, 60, 0.6, 0, 750 H and T s 00 s. (b) Find he rms value for i c. SOUTION e T 00 s in I max in T ( ) 40( 0.6) 40( 0.6) ( 00s) A 0( 0.6) 750H ( ) I min in T ( ) ( 0.6) ( 0.6) ( 00 s) 4.3 A 0( 0.6) 750H ( ) I I max I min in ( 0.6) 4.5 A

31 Bahaper4.fm Page 58 Friday, February, 003 3:4 PM haper 4 Nonisolaed Swich-Mode dc-dc onverers Having found P o P o I in ( 40) ( 4.5) 80 W, we can now solve for he average oupu curren: I I o W 3 A 60 Hence, we can calculae he values for I cmax and I cmin : I cmax I max I o 7.7 A I cmin I min I o.3 A T I crms -- 9d T A T T o ( T) I cmax d onsider a buck-boos converer ha supplies 75 W a I o 5 A from a 37 dc source. e T 30 s and 50 H. eermine: (a) he duy raio,, (b) I max and I min, (c) average inpu curren, (d) average diode and ransisor currens, and (e) he rms value of he capacior curren. ANSWE 0.9, 9.8 A, 4.5 A,.03 A, 5 A, A. erive he oupu volage ripple formula of Eq. (4.6) for a buck-boos converer. eermine, I max, I min, I in, I, I sw, and I crms for a buck-boos converer whose capacior curren waveform is shown in Fig. E4.9. Assume 0 H. i c (A) s 0 s Waveform for Exercise 4.9. ANSWE 0.4, 7.67 A, 5.67 A,.67 A, 4 A,.67 A, 3.3 A

32 Bahaper4.fm Page 59 Friday, February, 003 3:4 PM 4. oninuous onducion Mode Under cerain inpu and oupu volage condiions, none of he preceding hree basic converer opologies is suiable. For example, suppose he inpu volage varies beween 8 and 8 while he oupu volage is desired o be mainained fixed a. I is clear ha he volage gain varies beween /3 and 3/ while he oupu is consan a. Even hough he buck-boos converer allows he volage gain o be less or greaer han, i provides only a negaive oupu volage polariy. One opology ha is capable of providing a posiive oupu volage is known as he Single-Ended Primary Inducance onverer (SEPI), shown in Fig I is possible o cascade or cascode more han one of he basic opologies buck, boos, and buck-boos o form new opologies ha have aracive feaures ha a single opology does no. Figure 4.4(a) and (b) shows a block diagram of wo converers (a) S (b) Single-ended primary inducance converer (SEPI). (a) Singleswich, double-hrow implemenaion. (b) MOSFET-diode implemenaion. M M (a) M M (b) ombinaions of basic converer opologies. (a) ascade. (b) ascode.

33 Bahaper4.fm Page 60 Friday, February, 003 3:4 PM haper 4 Nonisolaed Swich-Mode dc-dc onverers conneced in series (cascade) and in parallel (cascode), respecively. The parallel connecion is also known as a differenial configuraion. The overall volage gains for he Fig. 4.4(a) and (b) converer arrangemens are given in Eqs. (4.63) and (4.64), respecively. M M M M M M where, in Fig. 4.4(a), M and M are given by (4.63) (4.64) and in Fig. 4.4(b) we have M M o M M Figure 4.5(a) and (b) shows he block diagram and circui implemenaion for he wo possible series cascadings of he boos converer wih a buck converer. egardless of he sequence, he volage gain for boh converers should be M M ( ). The equivalen circui represenaion for he cascade of he boos and buck converers is shown in Fig. 4.6(a). Afer careful circui manipulaion, i can be shown ha he equivalen one-swich implemenaion of Fig. 4.6(a) is as shown in Fig. 4.6(b). To illusrae how Fig. 4.6(b) is obained from Fig. 4.6(a), we assume i and i are consan curren sources as shown in Fig. 4.7(a). Figure 4.7(b) shows he equivalen circui when S is in posiions and. Nex we redraw he porion of he oupu circui as shown in Fig. 4.7(c), which is redrawn in Fig. 4.7(d). The wo modes of Fig. 4.7(d) are represened in he equivalen circui shown in Fig. 4.7(e). Similarly, he buck and boos cascade is shown in Fig. 4.8(a). The circui can be simplified hrough several sraighforward seps as shown in Fig. 4.8(b)(d). Noice he capacior in Fig. 4.8(a) is removed since regardless he sae of S and S, he capacior average curren is always zero. In Fig. 4.8(c) we combine and in series. Finally, Fig. 4.8(d) is obained using similar seps for he buck-boos cascade. Figure 4.9(a) shows a cascade of a buck and a uk converer. Show ha he volage gain is given by o (4.65) Boos Buck M M (a) Buck Boos M M (b) Block diagram represenaion for he cascade of a boos and a buck converer. (a) Boosbuck. (b) Buck-boos.

34 Bahaper4.fm Page 6 Friday, February, 003 3:4 PM 4. oninuous onducion Mode S S Boos (a) Buck i i v c (b) i v c i (c) (a) Boos-buck cascade. (b) Two-swich implemenaion. (c) One-swich equivalen circui implemenaion. SOUTION uring mode, when S and S are in posiion, and during mode, when S and S are in posiion, he inducor volages are given by T v v c T Apply he vol-second balance principle o and, o obain v v c v v v : ( ) c 0 M c : ( c ) ( ) 0 M c -----

35 Bahaper4.fm Page 6 Friday, February, 003 3:4 PM haper 4 Nonisolaed Swich-Mode dc-dc onverers a I v c b (a) I Posiion : Posiion : a a I v c I I v c I b (b) b b a I v c I I v c I a (c) b a v c b a b I I I v c I (d) a v c b I I Seps illusraing how Fig. 4.6(b) is derived. Therefore, he oal volage gain is c (e) M M Anoher sraighforward way o find he gain equaion is o realize ha he average volage beween a and a is and use his volage as an inpu o he uk converer. The swich implemenaion for Fig. 4.9(a) is shown in Fig. 4.9(b).

36 Bahaper4.fm Page 63 Friday, February, 003 3:4 PM 4. oninuous onducion Mode i i S i c S (a) i i (b) i = (c) i (d) Seps illusraing how he buck-boos converer is obained from he cascade configuraion. Show ha he cascaded buck and boos converers of Fig. 4.8(c) produce he same volage gain conversion as ha of Fig. 4.8(a), i.e., ( ), when S and S are singlepole, double-hrough swiches; assume S and S are synchronized in posiion for he T inerval and in posiion for he ( )T inerval. Figure 4.6(b) is known as a uk converer, whose volage gain is ( ), as will be discussed in deail shorly. This concep can be exended o oher cascaded opologies. The generalized fourh-order swiched-mode volage-o-volage converer is given in Fig omponens,, 3, 4, and 5 consis of a swich, a diode, wo inducors, and one capacior. By disallowing capacior loops and inducor cu-ses, he

37 Bahaper4.fm Page 64 Friday, February, 003 3:4 PM haper 4 Nonisolaed Swich-Mode dc-dc onverers a i v v c v i S S a' (a) v c i o (b) (a) ascade configuraion of buck and uk converers. (b) iode-swich implemenaion. 3 5 in 4 Generalized represenaion of fourh-order volage-o-volage converer. oal number of physically realizable opologies can be reduced. For example, all possible opologies wih componen represening a swich are shown in Fig The opology of Fig. 4.3(e) is a buck converer wih an addiional oupu filer. Figure 4.3(c) is a buck-boos converer wih an addiional oupu filer. The opologies in Fig. 4.3(b) and (f ) are physically unrealizable since he average oupu curren in each of he capaciors is zero; hence, no power is delivered o he load. On he oher hand, Fig. 4.3(d) does no have an average inpu curren. I has been shown ha oher converer opologies can be obained by combining some of he hree opologies discussed earlier. One cascade combinaion of a buck and a boos converer is known as a uk converer, given in Fig. 4.6(b) and redrawn in Fig. 4.3, named afer is invenor, Slobodan uk from he alifornia Insiue of Technology. Figure 4.3(c) shows he uk converer wih a magneically coupled inducor represenaion. The fron end of he converer is a boos, and he back end of i is a buck. Hence, we may refer o he uk converer as a boos-buck converer. Unlike he previous converers, his converer requires wo swiches and uses wo inducors, and a capacior o sore and ransfer energy from he inpu o he oupu, resuling in a higher level of

38 Bahaper4.fm Page 65 Friday, February, 003 3:4 PM 4. oninuous onducion Mode (a) (b) (c) (d) (e) (f) All possible fourh-order opologies wih a swich used as componen in Fig complexiy. ike he buck-boos, he gain of he uk converer can be less han, equal o, or greaer han, wih a posiive oupu polariy. The major advanage of his converer is ha he inpu inducor,, and he oupu inducor,, can be coupled on one magneic core srucure such ha wih he proper core gap design, he inpu and oupu swiching currens can be made zero. Oher possible combinaions of converers are shown in Fig. 4.33(a) and (b), which represen a boos cascade wih an oupu filer and a buck cascade wih an inpu filer, respecively. The analysis of he uk converer can be carried ou he same way as for he oher converer opologies. There are wo modes of operaions: mode when he swich is on, and mode when he swich is off. Mode Mode sars when he ransisor is urned on a 0 (Fig. 4.34(a)). The inducor volage is given by v di d in resuling in he following curren relaion: where I ( 0) i () is he iniial curren value in I ( 0) (4.66) (4.67)

39 Bahaper4.fm Page 66 Friday, February, 003 3:4 PM haper 4 Nonisolaed Swich-Mode dc-dc onverers i i (a) i i S (b) i i (c) uk converer wih magneically coupled inducors. (a) Two-swich implemenaion. (b) Transisor-diode implemenaion. (c) ore implemenaion. S (a) (b) (a) Boos cascade wih oupu filer. (b) Buck cascade wih inpu filer.

40 Bahaper4.fm Page 67 Friday, February, 003 3:4 PM 4. oninuous onducion Mode i i S i (a) v v v (b) (a) Mode : Transisor is urned on. (b) Mode : Transisor is urned off. Since The volage across is given by v v c i di d ( dv c d), he above equaion yields i c d v c d v c If we assume he average volage across has no ripple, hen is average value, c, is given by c (4.68) This equaion is obained by using K from he oupu hrough,,, and and seing he average inducor volages o zero. v A T, Similarly, i in I ( 0) I ( T) T in I ( 0) (4.69a) I ( T) T in I (4.69b) ( 0) Mode When he ransisor is urned off a T, urns on and he equivalen circui is given by Fig. 4.34(b). Similar analysis shows v

41 Bahaper4.fm Page 68 Friday, February, 003 3:4 PM haper 4 Nonisolaed Swich-Mode dc-dc onverers and he inducor currens are given by i () ( T) I ( T ) A T, i () ( T) I ( T ) i ( T) ( )T I ( T ) (4.70a) i ( T) ( )T I ( T ) (4.70b) Since I ( T ) I ( 0) and I ( T ) I ( 0), from Eqs. (4.69) and (4.70), we obain T o ( )T o (4.7) in The curren and volage waveforms for he uk converer are shown in Fig The average inpu curren is he same as he average inducor curren i (), given by I I max I in min (4.7) and he average oupu curren is he same as he average inducor curren i (): I I max I o min (4.73) Since he average inpu power and oupu power are equal, we can obain from he above equaions he following: I ( T) T ( ) in (4.74a) Similarly, we obain I ( 0) T ( ) in I ( T) T ( ) in (4.74b) (4.75a) of I ( 0) T (4.75b) ( ) in For coninuous inpu curren i (), we se I ( 0) 0 o obain he criical value as follows: cri ( ) T (4.76)