Designing Stable Control Loops

Transcription

1 Power Supply Design Seminar Designing Sable onrol oops Topic aegory: Feedback oop ompensaion eproduced from Texas Insrumens Power Supply Design Seminar SEM4, Topic 5 TI ieraure Number: SUP73, Texas Insrumens Incorporaed Power Seminar opics and online powerraining modules are available a: power.i.com/seminars

2 Designing Sable onrol oops By Dan Michell and Bob Mammano ABSTAT The objecive of his opic is o provide he designer wih a pracical review of loop compensaion echniques applied o swiching power supply feedback conrol. A op-down sysem approach is aken saring wih basic feedback conrol conceps and leading o sep-by-sep design procedures, iniially applied o a simple buck regulaor and hen expanded o oher opologies and conrol algorihms. Sample designs are demonsraed wih Mahcad simulaions o illusrae gain and phase margins and heir impac on performance analysis. I. INTODUTION Insuring sabiliy of a proposed power supply soluion is ofen one of he more challenging aspecs of he design process. Nohing is more disconcering han o have your lovingly crafed breadboard break ino wild oscillaions jus as i is being demonsraed o he boss or cusomer, bu insuring agains his unforunae even akes some analysis which many designers view as formidable. Pahs aken by design engineers ofen emphasize eiher cu-and-ry empirical esing in he laboraory or compuer simulaions looking for numerical soluions based on complex mahemaical models. While boh of hese approaches have a place in circui design, a basic undersanding of feedback heory will usually allow he definiion of an accepable compensaion nework wih a minimum of compuaional effor. II. STABIITY DEFINED Fig. gives a quick illusraion of a leas one definiion of sabiliy. In is simples erms, a sysem is sable if, when subjeced o a perurbaion from some source, is response o ha perurbaion evenually dies ou. Noe ha in any pracical sysem, insabiliy canno resul in a compleely unbounded response as he sysem will eiher reach a sauraion level or fail. Oscillaion in a swiching regulaor can, a mos, vary he duy cycle beween zero and % and while ha may no preven failure, i will ulimae limi he response of an unsable sysem. Perurbaion Perurbaion Sable Sysem Unsable Sysem Fig.. Definiion of sabiliy. esponse esponse Anoher way of visualizing sabiliy is shown in Fig.. While his graphically illusraes he concep of sysem sabiliy, i also poins ou ha we mus make a furher disincion beween large-signal and small-signal sabiliy. While small-signal sabiliy is an imporan and necessary crierion, a sysem could saisfy his requiremen and ye sill become unsable wih a large-signal perurbaion. I is imporan ha designers remember ha all he gain and phase calculaions we migh perform are only o insure small-signal sabiliy. These calculaions are based upon and only applicable o - linear sysems, and a swiching regulaor is by definiion a non-linear sysem. We solve his conundrum by performing our analysis using small-signal perurbaions around a large-signal operaing poin, a disincion which will be furher clarified in our design procedure discussion. Texas Insrumens SUP73

3 Uncondiionally Sable Unsable Small-Signal Sable arge-signal Unsable Fig.. arge-signal vs. small-signal sabiliy. III. FEEDBAK ONTO PINIPES The basic regulaor is shown in Fig. 3 where an unconrolled source of volage (or curren, or power) is applied o he inpu of our sysem wih he expecaion ha he volage (or curren, or power) a he oupu will be very well conrolled. The basis of our conrol is some form of reference, and any deviaion beween he oupu and he reference becomes an error. In a feedback-conrolled sysem, negaive feedback is used o reduce his error o an accepable value as close o zero as we wan o spend he effor o achieve. Typically, however, we also wan o reduce he error quickly, bu inheren wih feedback conrol is he radeoff beween sysem response and sysem sabiliy. The more responsive he feedback nework is, he greaer becomes he risk of insabiliy. Inpu Feedforward Sysem eference Fig. 3. The basic regulaor. Feedback Oupu A his poin we should also menion ha here is anoher mehod of conrol feedforward. Wih feedforward conrol, a conrol signal is developed direcly in response o an inpu variaion or perurbaion. Feedforward is less accurae han feedback since oupu sensing is no involved, however, here is no delay waiing for an oupu error signal o be developed, and feedforward conrol canno cause insabiliy. I should be clear ha feedforward conrol will ypically no be adequae as he only conrol mehod for a volage regulaor, bu i is ofen used ogeher wih feedback o improve a regulaor s response o dynamic inpu variaions. The basis for feedback conrol is illusraed wih he flow diagram of Fig. 4 where he goal is for he oupu o follow he reference predicably and for he effecs of exernal perurbaions, such as inpu volage variaions, o be reduced o olerable levels a he oupu. u 5 eference Negaive Feedback Inpus G H y Oupu Fig. 4. Flow graph of feedback conrol. y Gu GHy y( GH) Gu y G u GH Wihou feedback, he reference-o-oupu ransfer funcion y/u is equal o G, and we can express he oupu as y Gu Wih he addiion of feedback (acually he subracion of he feedback signal) y Gu yhg and he reference-o-oupu ransfer funcion becomes y G u GH If we assume ha GH, hen he overall ransfer funcion simplifies o y u H Texas Insrumens SUP73

4 No only is his resul now independen of G, i is also independen of all he parameers of he sysem which migh impac G (supply volage, emperaure, componen olerances, ec.) and is deermined insead solely by he feedback nework H (and, of course, by he reference). Noe ha he accuracy of H (usually resisor olerances) and in he summing circui (error amplifier offse volage) will sill conribue o an oupu error. In pracice, he feedback conrol sysem, as modeled in Fig. 4, is designed so ha G H and GH over as wide a frequency range as possible wihou incurring insabiliy. We can make a furher refinemen o our generalized power regulaor wih he block diagram shown in Fig. 5. Here we have separaed he power sysem ino wo blocks he power secion and he conrol circuiry. The power secion handles he load curren and is ypically large, heavy, and subjec o wide emperaure flucuaions. Is swiching funcions are by definiion, large-signal phenomenon, normally simulaed in mos sabiliy analyses as jus a wosae swich wih a duy cycle. The oupu filer is also considered as a par of he power secion bu can be considered as a linear block. The conrol circuiry will normally be made up of a gain block he error amplifier and he pulse-widh modulaor, used o define he duy cycle for he power swiches. Source Feedforward Power Sysem Power ircuiry onrol onrol ircuiry eference Fig. 5. The general power regulaor. oad Feedback IV. THE BUK ONVETE The simples form of he above general power regulaor is he buck or sepdown opology whose power sage is shown in Fig. 6. In his configuraion, a D inpu volage is swiched a some repeiive rae as i is applied o an oupu filer. The filer averages he duy cycle modulaion of he inpu volage o esablish an oupu D volage lower han he inpu value. The ransfer funcion for his sage is defined by V ON O Vi Vi d, where : T ON swich on ime T repeiive period (/fs) d duy cycle V S VO (D) VO Vi ONfS Vi d ON S PWM onrol S /f S V S Fig. 6. The buck converer. Since we assume ha he swich and he filer componens are lossless, he ideal efficiency of his conversion process is %, and regulaion of he oupu volage level is achieved by conrolling he duy cycle. The waveforms of Fig. 6 assume a coninuous conducion mode (M) meaning ha curren is always flowing hrough he inducor from he swich when i is closed, and from he diode when he swich is open. The analysis presened in his opic will emphasize M operaion because i is in his mode ha small-signal sabiliy is generally more difficul o achieve. In he disconinuous conducion mode (DM), here is a hird swich condiion in which he inducor, swich, and diode currens are all Texas Insrumens 3 SUP73

5 zero. Each swiching period sars from he same sae (wih zero inducor curren), hus effecively reducing he sysem order by one and making small-signal sable performance much easier o achieve. Alhough beyond he scope of his opic, here may be specialized insances where he large-signal sabiliy of a DM sysem is of greaer concern han small-signal sabiliy. There are several forms of PWM conrol for he buck regulaor including, Fixed frequency (f S ) wih variable ON and variable OFF Fixed ON wih variable OFF and variable f S Fixed OFF wih variable ON and variable f S Hysereic (or bang-bang ) wih ON, OFF, and f S all variable Each of hese forms have heir own se of advanages and limiaions and all have been successfully used, bu since all swich mode regulaors generae a swiching frequency componen and is associaed harmonics as well as he inended D oupu, elecromagneic inerference and noise consideraions have made fixed frequency operaion by far he mos popular. Wih he excepion of hysereic, all oher forms of PWM conrol have essenially he same small-signal behavior. Thus, wihou much loss in generaliy, fixed f S will be he basis for our discussion of classical, small-signal sabiliy. Hysereic conrol is fundamenally differen in ha he duy facor is no conrolled, per se. Swich urn-off occurs when he oupu ripple volage reaches an upper rip poin and urn-on occurs a a lower hreshold. By definiion, his is a large-signal conroller o which small-signal sabiliy consideraions do no apply. In a smallsignal sense, i is already unsable and, in a mahemaical sense, is fas response is due more o feedforward han feedback. V. ONTOING PUSE-WIDTH MODUATION A ypical implemenaion for PWM conrol (in a form which we now call volage-mode conrol ) is illusraed in Fig. 7. From he block diagram i can be seen ha he widh of he PWM signal is deermined by he poin in ime where he sawooh, or ramp waveform (V ) crosses he volage level a he oupu of he error 5-4 amplifier (V E ). Since V raverses from zero o V P wihin a swiching period, i follows ha when V E zero, he widh of he oupu pulse will be zero, and i will increase linearly reaching % when V E V P. Therefore he duy cycle of he modulaor will be V E / V P and since, in a buck converer, he duy cycle has already been deermined o be /, he conrol gain of he modulaor is: V i V V Texas Insrumens 4 SUP73 E PWM V P V E P D eference Feedback Signal ON Error Amp T /fs Amplified Error Volage, V E Sawooh eference d ON f S V E /V P omp PWM onrol Fig. 7. Typical PWM conrol implemenaion. Noe ha if V P is made proporional o V I, a feaure which can be accomplished wih feedforward, hen he duy cycle will vary inversely proporional o he inpu volage and inpu-o-oupu volage regulaion can ideally be achieved wih no change in V E. In his analysis we have assumed complee lineariy and ha he oupu of he error amplifier, V E, is a D volage. If, in addiion o he inended D componen, V E conains excessive ripple, due o error amplifier gain a he swiching frequency, hen hose swiching frequency componens can mix wih he sawooh frequency componens causing he regulaor o exhibi large-signal swiching insabiliy, even if i has excellen small-signal sabiliy. This ype of insabiliy can cause he regulaor o produce even more ripple, usually a a subharmonic of he swiching frequency, alhough i may sill regulae a he proper oupu volage.

6 VI. HAATEISTIS OF A OADED - FITE The schemaic of Fig. 8 shows an - filer wih a load,, where he componens have been convered o impedances in he frequency domain hrough he use of aplace ransforms. The overall ransfer funcion of his nework is described by he use of Kirchhoff s law as VO s s s s (s p )(s p ) S S s Fig. 8. Frequency response of a loaded filer. By seing he ransfer funcion numeraor and denominaor each equal o zero, we can derive he roos of boh he numeraor, which are he zeros of he sysem (none in his equaion), and he denominaor which gives us he poles. This second-order expression conains wo poles, p α jω d and p α jω d where: α, ω d α, and j For lighly damped filers (ypical of swiching regulaors), we can ofen use he approximaion of: ωd ω Wih s jω, we see ha ransfer funcions are complex numbers conaining a real par and an imaginary par. The ampliude of a complex number is he square roo of he sum of he squares of he real and imaginary pars. The phase is he inverse angen (arcan) of he raio of he imaginary par o he real par. By evaluaing he ransfer funcions as a funcion of frequency, we can deermine he poin where boh he magniude and he phase make ransiions. The mos common way of doing his is o plo he gain in db ( imes he log of gain), and he phase in degrees, agains he log of frequency. These are called Bode plos and allow easy visualizaion of he characerisics ha we will use o help define sysem sabiliy. From he ransfer funcion equaions for Fig. 8, we can deermine ha: The gain and he phase for ω. The gain and he phase 9 for ω. The gain slope ω and he phase 8 for ω. For his example of a loaded - filer, Mahcad was used o draw he plos shown in Fig. 9, wih he assumpion of an arbirarily assigned se of numerical values (which we will laer use for our buck converer example): 6 µ H, 54 µ F, and. 5 Ω 3 4 from which α.85 and ω.6. Texas Insrumens 5 SUP73

7 From Fig. 9 we subsaniae ha he gain of his filer is uniy a low frequencies, experiences a resonan peak (deermined by ) a he secondorder pole frequency, and hen falls wih a slope of db/ocive ( db/decade) a higher frequencies; while he phase goes hrough a shif from zero o a 8 o lag. This higher frequency slope is someimes called a slope since, in his region, he funcion is proporional o ω, or ω. 6µ H 54µ F.5Ω 3 4 α.85 ω.6 Gain in db Phase in Degrees Frequency d I is worh reinforcing ha a sysem mus be linear before frequency-domain echniques, such as aplace ransforms and Bode plos, apply. A swiching regulaor is no even coninuous, le alone linear. Therefore, approximaions will have o be made firs, o average he swiching effecs so ha we have a coninuous sysem, and secondly, o apply a small-signal approximaion in order o assume lineariy. And, of course, all his is done under he addiional assumpions of linear passive componens and ideal swiches. VII. INEAIZING THE BUK ONVETE Wih ideal componens, a swiching regulaor is a linear circui for any given swich condiion. The concep of averaging can be applied when he swiching rae is fas wih respec o he rae of change of any oher parameers of ineres. To quanify his, we can say ha he accuracy of he approximaions is excellen up o one-enh he swiching frequency, prey good for one-hird, and one-half is he heoreical limi based on he Nyquis sampling crierion. Fig. shows he process of averaging he operaion of he circui by combining he condiion when he acive swich is closed wih ha when i is open. The relaionship beween hese wo condiions is he duy cycle of he swiching and is effec is accouned for hrough he use of a hypoheical D ransformer wih a urns raio of he duy cycle, d. Wih his model, he primary curren is now I d and he secondary volage is d. This D ransformer is an arifac from Dave Middlebrook a alech. I is no a real componen bu i is valid for your Spice library. We now have a coninuous sysem bu i is nonlinear because he ransformer urns-raio, d, is a variable - namely he conrol variable - and no a consan Frequency Fig. 9. Bode plo of a sample loaded filer. Texas Insrumens 6 SUP73

8 I i I V I i I d :d d V I Swich ondiion I [d h of he ime] I vi Vi vˆ i i I î d D dˆ Neglec "ha" producs For example, vid (Vi vˆ i )(D dˆ) vid Vi dˆ where Vˆ idˆ I I D I i V :D dˆ I D dˆ V Swich ondiion II [(-d)h of he ime] I i I d :d d I V ircui Model Using D Transformer oncep. I d and d are nonlinear erms. Fig.. Averaged buck converer. Wih an averaged, coninuous model, he nex sep is o linearize i. We do his exacly he same way we would linearize any nonlinear, coninuous sysem, namely we define he smallsignal parameers based on a large-signal operaing poin. This is shown in Fig.. Mahemaically, he linearizaion process involves separaing each variable ino is D (in capials) and signal frequency ac (wih a ha ) componens, and neglecing he producs of wo ha erms. For he example calculaion shown in Fig., he produc d is linearized abou he operaing poin, D. Fig.. inearized buck converer. The advanage of linearizaion is ha aplace ransforms (i.e. impedance conceps) apply so ha closed-form algebraic soluions can be found and ploed (e.g. Bode plos). The range over which he linear approximaions are valid depends upon he accuracy desired. In general, as long as he signals are small enough so ha he duy facor is no clamped eiher full on or full off for several swiching cycles, he linear approximaion works very well. And in any case, small-signal sabiliy as evidenced using Bode plos is sill a necessary condiion for overall sabiliy. VIII. APPYING THE INEAIZED MODE The flow diagram of he closed-loop linearized buck regulaor can be derived by applying he generalized conrol law o he linearized power circui described above, as shown in Fig.. This conrol law deermines how d, he aplace ransformed conrol variable, varies as a funcion of key circui parameers. For a second-order sysem, such as he buck regulaor wih one inpu volage, i can be expressed as: d F Î F Vˆ QVˆ i Texas Insrumens 7 SUP73

9 Tha is, he inducor curren variable, he oupu volage variable, and he inpu volage variable, can each individually conribue o he sysem conrol variable. As i perains o swiching regulaors, he expression volagemode conrol implies ha here is no curren feedback, i.e. F. urren-mode conrol means ha here is a curren loop as well as a volage loop. In eiher case, here may or may no be feedforward conrol, Q. And finally, noe ha in his case, F includes he reference and feedback summing componens. This generalized conrol law can hen be made specific o our volage-mode conrolled, buck regulaor as shown in Fig. 3, where he feedback summing poin is he differenial inpu o he Error Amplifier. v I Q d D dˆ onrol Variable Σ Generalized onrol aw : Negaive Feedback dˆ F Î F Vˆ QVˆ i Σ F I s /s V Fig.. Flow graph / schemaic of linearized buck converer wih general conrol law. ef K V E Err Amp V P F, K F, VP Q, d Modulaor D Σ / s s// Vi K V P GH s s F Fig. 3. Flow graph of linearized buck converer wih volage-mode conrol. The overall open-loop gain is equal o he produc of he individual gains of he error amplifier, he modulaor, and he oupu filer, and is shown as: Vi K V GH P s s There are several poins which could be made relaive o he above equaion: Firs, noe ha his expression is independen of he D duy facor D bu is dependen upon he D inpu volage. Hence, as would be expeced, he open-loop gain funcion is dependen upon he D operaing poin. Secondly, he second erm in he denominaor conains he load resisance,. If were o go o infiniy, his erm would go o zero indicaing an unsable sysem, however, he realiy is ha he circui would firs go o DM where is small-signal operaion becomes essenially firs order. Finally, he expression above also assumes ha he oupu volage level is equal o he reference (H ). If his is no he case, he appropriae scale facor would be added o eiher he reference or he oupu volage prior o he error amplifier inpu. Noe ha we have assumed ha GH is posiive in his expression. This is jus a simplificaion in ha we are assuming negaive feedback, so he firs 8 o is implici and insabiliy will occur a 8 o raher han he 36 o which some conrol heory exs describe. Wih Mahcad as a simulaion ool, and uilizing he parameers of our earlier example, he above general ransfer equaion can be solved as a funcion of frequency o yield he Bode plos shown in Fig. 4. This example again uses: 6 µh, 54 µf,.5 Ω and addiionally, Vi V, VO 5 V, VP V fs khz, K 5.6, and s jπ f Texas Insrumens 8 SUP73

10 Gain in db Phase in Degrees I V K G(f )H(f ) i VP II V G(f )H(f ) i (πf ) VP S jπf Vi V VO 5V.5Ω VP V fs khz ipple < 5mVp p 6µ H 54µ F I II Frequency V K Peak i VP for K 5.6 4dB a.7khz From hese values, he Bode plos give us he gain and he phase of he open-loop ransfer funcion from which we can see ha a low frequencies, V K G(f )H(f ) i 3 db VP and a he higher frequencies, V G(f )H(f ) i ( π f ) VP a negaive slope of db/ ocave The peak gain a resonance is: V K G(f )H(f ) i 4 db a.7 khz VP A his poin we should define some erms imporan o our sabiliy analysis: A. Gain Margin The difference beween uniy gain (zero db) and he acual gain when he phase reaches 8 o. (In his case i is a posiive number.) The recommended value is -6 db o db. B. Phase Margin The difference beween 8 o and he acual phase when he gain reaches uniy gain. (In his case i is approaching zero.) The recommended value is 45 o o 6 o.. Sabiliy rieria A commonly used derivaive from he above wo definiions is ha if he slope of he gain response as i crosses he uniy-gain axis is no more han -6 db / ocave, he phase margin will be greaer han 45 o and he sysem will be sable Frequency Fig. 4. Bode plo of a W, V-o-5 V, buck converer open loop gain wih D error amp gain [K K]. Texas Insrumens 9 SUP73

11 These simulaions have made no approximaions oher han hose required for linearizing he sysem. I should be undersood, however, ha phase shif is caused no only by reacive componens bu also by ime delay, such as ransisor sorage ime or hold ime in a sampling sysem. Swich delays normally have lile effec as long as here is no explici sampleand-hold funcion, and he frequencies of ineres are well below he swiching frequency. For example, he effec of a µs delay in a khz sysem is a phase shif of less han 4 o. There will also be poenial phase lags due o he op amp and parasiic componens. Thus, alhough echnically a second-order sysem could poenially be sable since 8 o phase lag is only asympoically approached, we can expec ha his sysem, as currenly defined, will be unsable in pracice and, in any case, would suffer serious ringing under any exernal disurbance. In fac, he plos of Fig. 4 show a sysem wih essenially negaive gain margin and zero phase margin. We had assumed a value for he amplifier gain of 5.6 only because we will use his value laer, bu even if i were uniy, he gain of he modulaor alone could be enough for insabiliy since we already have 8 o phase shif jus from he oupu filer. IX. FEQUENY OMPENSATION ecognizing ha we have hus far defined wha amouns o an unsable sysem, we will now consider echniques o shape he open-loop gain funcion o provide adequae gain and phase margins. Our firs approach will be o reduce he gain a a lower frequency such ha he uniy-gain poin will be reached wih posiive phase margin. This is done by rolling off he gain of he error amplifier wih local feedback as shown in Fig. 5. Wih he addiion of ( and are wha gave us he amplifier gain of 5.6), we now have an amplifier gain of: V K OUT VIN ( s) This amouns o our original D gain wih an added pole a ω p. K(jM) (db) K(jM) (deg) UT V K V log( / ) M P / ( s) OUT IN Error Amplifier infiniy Proporional Inegral (PI) onrol K / s M P / Fig. 5. ag compensaion. V / M P V IN logm logm This single-pole compensaion is called lag compensaion and from he asympoic approximaed plos above, he phase shif changes from zero a / he corner frequency o 9 o a imes he corner frequency. igh a he corner frequency, he phase is 45 o since he denominaor is j, where he real and imaginary pars are equal. Since he ampliude of j is, he acual gain ampliude a he corner frequency is reduced by 3 db (commonly called he half-power poin). emember: The produc/division of wo complex numbers is equal o he sum/difference of heir ampliudes in db and he sum/difference of heir phase angles. Texas Insrumens SUP73

12 The ampliude of he compensaed error amplifier gain in db adds direcly o he ampliude of he oher elemens in he conrol loop, as he amplifier s phase adds o he overall phase lag. The Bode plos for lag compensaion (implemened in his form as a proporional inegraor wihou ) are shown in Fig. 6. The asympoic ransfer funcion equaions now become: A low frequencies, VI G(f )H(f ) π fvp A high frequencies, VI G(f )H(f ) 3 (π f ) VP s jπ f V The resonan peak I π fvp For a 6 db gain margin a f.7 khz, we se he peak gain equal o ½ which yields an produc equal o 99. From his we have se 67 kω and. µf. Noe ha he gainbandwidh (he uniy gain crossover frequency) is now less han 3 Hz, an order of magniude below he resonan frequency. Anoher poin of ineres is ha he gain margin of 6 db was based on he resonan peak, which in urn is dependen upon he oupu loading,, meaning ha decreasing he regulaor s load could also decrease he gain margin. In pracice, one would use he smalles load (larges ) for which he converer is sill in M. While he phase margin is now 9 o, he dominan pole has reduced he bandwidh o he poin where dynamic response will be very poor. (Noe ha in he complex plane, 9 o is he same as 7 o. The lag compensaor conribues 9 o from he D value, and above resonance, he oupu filer conribues anoher 8 o.) Gain in db Phase in Degrees I II Frequency Frequency V Peak i πfvp for 6 db gain m arg in a f.7khz, Vi 99 πfvp Sample values: 67 kω,.µ F Fig. 6. Bode plo of lag-compensaed sample buck converer open-loop gain funcion. Texas Insrumens SUP73

13 A second alernaive is lead compensaion which is described in Fig. 7. Here, insead of decreasing he gain we will increase he phase by adding a lead capacior,, o he error amplifier, inroducing a nework zero (he opposie of a pole). Of course, his also increases he gain bu if we make he break frequency of his zero, ω Z, he same as he uniy gain frequency of he uncompensaed buck regulaor, we have provided a phase margin of 45 o. The gain equaion for he error amplifier wih lead compensaion is: VOUT ( s) K VIN This amouns o he original D gain wih an added zero a ω Z UT VOUT ( s) K VIN Error Amplifier V V IN much higher han ha required for lag compensaion. I should be noed ha he error amplifier is no he only place in a swiching regulaor o experience a compensaing zero. The parasiic equivalen series resisance (ES) inheren in non-ideal capaciors seleced for use as an oupu filer will reac wih he oupu capaciance value o inroduce a circui zero o he sysem. Many designers have relied on his as a free (bu also relaively unconrolled) mehod for achieving some added posiive phase margin. (Try o ge your favorie capacior supplier o guaranee a minimum ES!) The circui of Fig. 8 combines boh a lead and a lag in an aemp o gain he bes feaures of boh namely low D error as well as higher bandwidh. In his circui, a zero capacior is placed in parallel wih he inpu resisor while a pole capacior is added in series wih he feedback resisor. The inen here is o provide a high gain a low frequencies while sill achieving accepable phase margin a crossover. The gain equaion for he error amplifier now becomes: VOUT ( s)( s) K V s IN K(jM) (db) log( / ) Valid only for K << Op Amp Gain UT Error Amplifier V V IN K(jω) (deg) 9 45 M Z / M Z / M Z logm logm Fig. 7. ead compensaion. In pracice, pure lead compensaion is physically unrealizable since he gain canno coninue o rise indefiniely due o limiaions in he open loop gain-bandwidh of he amplifier. Naurally, he gain-bandwidh of he operaional amplifier used wih lead compensaion mus be K(jM) (db) K(jω) (deg) V K V./ / / ( s)( s) OUT IN s log( / ) / Fig. 8. Achieving zero D error. logm logm Texas Insrumens SUP73

14 The phase lead associaed wih canno compleely cancel ou he iniial 9 o phase lag caused by he inegraing capacior unil he frequency is reached. Therefore, in order o ensure ha he phase margin due o he zero associaed wih is no degraded, mus be a leas an order of magniude below he resonan frequency of he sysem. Tha is, he phase shif due o he zero caused by should be over before any oher phase shif in he sysem sars so ha he sysem is oherwise ransparen o he effecs of inegraing ou he D error. X. AGE-SIGNA ONSIDEATIONS We have made several references o he fac ha while small-signal sabiliy is a necessary requiremen for a sable regulaor, large-signal effecs canno be ignored. To illusrae he poenial for a large-signal problem, Fig. 9 demonsraes how he amplified oupu ripple waveform can inerfere wih he PWM operaion. For simpliciy, his illusraion assumes a consan error amplifier gain. Alhough his is no very realisic, since we have already demonsraed ha a buck converer wih consan feedback gain is likely o be small-signal unsable, i will sill serve o demonsrae he concep ha here is a limi o he amoun of gain ha he sysem can have a he swiching frequency. Beyond his limi, he sysem will exhibi large-signal insabiliy, regardless of he margins shown in he Bode plo. PWM ipple Volage V P V E m m T/ T Fig. 9. arge-signal sabiliy consideraions for volage-mode conrol. In his example, he oupu ripple volage (assumed as being largely caused by he oupu capacior ES) is amplified by he error amplifier gain o appear in invered form as V E, he volage which is compared in he PWM modulaor o he ramp waveform. A necessary condiion for largesignal sabiliy is ha he slope of V E, designaed m, mus be less ha he slope of he ramp, m. A sample problem can demonsrae he impac of his requiremen. If we assume ha he duy cycle D.5, oupu ripple.5 vols p-o-p, and V P vols, hen:.5 K. K m and m T T T Therefore, K mus be less han. In realiy, he problem can be more severe if here is an unbounded lead nework which would hen ac as a differeniaor a he swiching frequency such ha he oupu from he error amplifier migh approximae a square wave more han a riangular shape shown in his illusraion. While here is no widely acceped design crierion o define a soluion, an empirical rule of humb is o insure ha ha he oal sysem gain is db or below a he swiching frequency. In any case, his issue is highly applicaion specific and designers should make his deerminaion for each individual sysem in he course of heir evaluaion. Texas Insrumens 3 SUP73

15 XI. DESIGN POEDUE To summarize he design procedure for he lead-compensaed, volage-mode, buck regulaor, he sep-by-sep approach is demonsraed in he accompanying box. Design Procedure for ead-ompensaed Volage- Mode onrolled Buck onverer G(f )H(f ) (db) 3 V I K log V I K V P ( π f) V P V E V π π f π f S db log f fs Sep: hoose gain - bandwidh f for large - signal sabiliy ( db aenuaion a fs). Sep : hoose zero frequency f for 45 phase margin. π Sep 3 : hoose Sep 4 : hoose (πf) VP K V. for uniy gain a f f for increased low - frequency gain and "zero" dc error. Sep 5 : hoose ES o cancel lead effec of ES. 3 For EXAMPE, Using he Following Parameer Values: i. Vi V VO 5 V fs khz 6 µ H 54 µ F.5 Ω VP V 5 Sep: f khz Sep :.59 πf Sep 3 : Sep 4 : ω n ( πf ) VP V, kω > ω Sep 5 : From daa shee, ES. Ω. i 5 << inpu Z of op amp..5 kω, n K µ F ES 3 pf 8 5 pf Texas Insrumens 4 SUP73

16 In his example, we have added 3 for a highfrequency roll-off, chosen so ha he pole frequency,, cancels he zero frequency a 3. This reducion in nominal gainbandwidh is required o ensure boh small-signal (ES) sabiliy - wih 45 o phase margin and a leas db rejecion a he swiching frequency in applicaions where he ES can vary widely. Our example uses hree paralleled 8 µf solid analum capaciors where he oal capaciance is 3X and he ES is /3 of a single uni. Noe ha he value of K is much less han he maximum of calculaed for large-signal sabiliy. The calculaed Bode plos for his example are shown in Fig. from which we can see ha he gain bandwidh of his lead-compensaed circui is nearly an order of magniude greaer ha he resonan frequency of he uncompensaed sysem. This is paricularly significan in remembering ha he lag-compensaed soluion required a gain-bandwidh an order of magniude lower han resonance. The approximae equaions which are acive in defining he gain wihin he various regions of operaing frequency are given below: V I G(f )H(f ) i πf VP Vi II G(f )H(f ) VP V III G(f )H(f ) i (πf ) VP V IV G(f )H(f ) i πf V P Gain in db Phase in Degrees Frequency I II III IV Frequency The phase m argin is greaer han 45 for :.5 kω 59 kω 5 pf. µ F The gain bandwidh is greaer han khz. Fig.. Bode plo of lead-compensaed sample buck converer open-loop gain funcion wih zero D error. The final value for gain-bandwidh is.7 khz while he phase margin is nearly 55 o. The gain margin is well over - db and any higher frequency phase lag (as, for example, operaional amplifier limiaions) would cause he gain o be furher reduced. Texas Insrumens 5 SUP73

17 We can use Mahcad o plo oher characerisics of our demonsraion example as shown in Fig.. These plos show oupu impedance (a measure of load regulaion) and audiosuscepibiliy (line regulaion) as a funcion of frequency. The benefis of feedback o dynamic performance are graphically illusraed by he comparaive curves in hese plos. / I O / Oupu Impedance Wihou Feedback Frequency Audiosuscepibiliy Wihou Feedback Wih Feedback Frequency Wih Feedback Fig.. Oupu impedance and audiosuscepibiliy for he lead-compensaed sample buck converer. XII. PEAK UENT-MODE ONTO The volage-mode algorihm which we have been using as a model for he preceding analysis achieves is PWM conrol by comparing he error amplifier command signal wih an arificiallygeneraed sawooh, or ramp waveform. Wih peak curren-mode (/M) conrol, his comparison ramp is derived from he oupu inducor curren waveform and hus forms an inner curren feedback loop. While here is sill an ouer volage loop, is funcion is o program he oupu inducor curren raher han he duy cycle direcly. Wih he open loop characerisics of a programmable curren source, curren-mode conrol effecively hides he inducor wihin he inner loop, changing he resonan wo-pole oupu filer o a single lower frequency dominan pole, plus a higher frequency pole a or beyond he gain-bandwidh of he sysem. In so doing, he ask of compensaion is significanly eased. A simplified block diagram of his opology is shown in Fig. and is operaion is described as follows: A fixed-frequency oscillaor iniially ses he lach which urns on he power swich, causing inducor curren o rise according o: di (V V ) i o. d This curren is sensed and convered o a volage ramp which is compared o he error signal from he volage error amplifier. When he curren ramp crosses he error signal, he comparaor reses he lach urning off he power swich, allowing he inducor curren di V o decay according o o. While boh d DM and M operaion are possible, his example assumes M. The power swich is held off by he lach unil he clock iniiaes he nex cycle. Texas Insrumens 6 SUP73

18 I S S Drive :N S I I S S V v E m -m PWM lock ach omp S I S /N V E K ef V E Ideal analog waveform PWM lock conrols urn-on Peak curren conrols urnoff During he on ime, I S I d d/f S /f S V E S I S /N S I S /N K Fig.. Peak curren-mode conrol implemenaion for buck converer. f F S ( D) fsnk F ( D)S D Q Vi urren sensing can be accomplished wih eiher a sensing resisor or a curren sense ransformer (as used in his example) bu, in eiher case, a proporionae volage waveform is derived for he comparaor. One problem wih curren-mode conrol is ha his waveform ofen also conains leading-edge noise spikes caused by parasiic capaciance and diode recovery. These spikes need o be conrolled by eiher blanking or filering or else he comparaor may rese he lach righ a he beginning of he power pulse. Noe ha his example acually measures swich curren raher han inducor curren since he only informaion needed for conrol is he peak value of he inducor curren and his is usually more convenienly done in series wih he power swich. Wha happens o he inducor curren afer he swich opens is, in principle, unimporan. However, in M applicaions where he duy cycle may exend beyond 5%, a largesignal, sub-harmonic insabiliy can occur which is described in Fig. 3. Acual analog waveform for m < m (D >.5) Any perurbaion δ will resul in swiching insabilily (swiching frequency subharmonics) Need o add exernal ramp m 3 for sabilizaion m 3 V E m3 -m m m 3 m I -m Effecive analog waveform wih sabilizing ramp m 3 Fig. 3. Peak curren-mode swiching insabiliy. The waveforms in he upper porion of his illusraion show he swich curren in a M applicaion wih a rising inducor curren slope equal o m and a falling slope (when he swich is off) of m. Under sable operaing condiions, he end of m has o mach he beginning of m. Wih less han 5% duy cycle, -m < m and any small perurbaion (δ) which migh occur during he swich on-ime will be reduced by he ime he nex period sars and evenually die ou wih successive swiching cycles. Bu wih more han 5% duy cycle, -m > m and δ will be Texas Insrumens 7 SUP73

19 larger a he sar of he nex period, resuling in regeneraive oscillaion. The cure for his is he addiion of an addiional ramp (m 3 ) on op of he curren waveform so ha he conroller sees an up-slope of ( m m3 ) and a downslope of ( m m 3 ). (m m ) If m 3 is chosen so ha m 3 >, hen m 3 m < m m3 and he sysem will be sable. The minimum value of m 3 for he buck converer (for he impracical condiion of jus borderline sabiliy) is hen (m - m ) V (D ) m i S 3 N To deermine a more pracical value for m 3, we firs need o deermine he overall closed loop gain of he sysem because his added ramp also has he effec of reducing he gain of boh he volage and curren loops by a facor of: m γ m m3 (m m ) Wih he consrain ha m 3 >, hen i is also implied ha γ < ( D). (This D expression is rue for any of he hree basic swiching regulaor opologies.) As shown in Fig. 4, he generalized conrol law firs presened back in Fig. can be used o develop individual expressions for he smallsignal gains for boh he curren and volage loops. This sems from he fac ha, in accordance wih basic flow graph heory, he oal loop gain G H of a sysem wih wo ouching loops can be expressed as he sum of he individual open-loop gains, namely G H GH. Noe ha he s erm in he numeraor of G H(s ) provides he opporuniy for lead compensaion, which is why he inner curren loop in curren-mode conrol can be hough o compensae he ouer volage loop. Any ransfer funcion can be expressed T as [ GH] where T is he ransfer funcion wihou feedback and GH is he feedback relaed characerisic equaion of he sysem. Gain in db Gain in db urren oop: γfs s D GH s s s jπf Frequency Volage oop: γfs NK D S GH s s Frequency Fig. 4. Individual loop gains for linearized buck converer wih peak curren-mode conrol. Texas Insrumens 8 SUP73

20 For our wo-loop sysem, he feedback relaed characerisic equaion becomes G H GH. However, as long as G H is never (in his case he curren loop phase shif is never greaer han 9 o ), we can divide boh sides of he characerisic equaion by G H o form a new singleloop characerisic equaion G H where he new singleloop open-loop gain funcion is: { GH} G H GH [ GH ] This single-loop represenaion is illusraed wih he flow graph shown in Fig. 5 and resuls in a new open-loop volage gain equaion: fsγ N K D S GH f γ s S s D ef v E K Err Amp f S γ N S ( D) V I (D/)(-γ / ( D)) Σ (s f S γ/ ( D))(s/) Fig. 5. Single-loop flow graph of linearized buck converer wih peak curren-mode conrol. for he original circui wih he curren-loop closed. In oher words, insead of a resonan circui wih a double-pole a, we now have a new power circui wih a dominan low frequency pole a, and anoher f higher frequency pole a s γ. ( D) If we make he assumpion ha GH (which is he case wih f f sγ ) he single-loop gain can now be π( D) reduced o simply: N K G H S GH G H s and he sysem is essenially firs order. An addiional feaure achieved wih he combinaion of he volage and curren loops ino he single-loop flow graph of Fig. 5, we can see ha he gain block associaed wih he inpu volage goes o zero for γ D, corresponding o, heoreically, zero audiosuscepibiliy. Thus, γ D implies an opimum ramp for m 3 ( /)( S /N ), which is independen of D and is greaer han he minimum requiremen previously discussed. If we assume ha we have applied his opimum amoun of slope compensaion such ha γ -D, hen he generalized equaion given in he Fig. revers o he same simplified form ha we have calculaed earlier for all f << f S / π. Texas Insrumens 9 SUP73

21 Design Procedure for Peak urren-mode onrolled Buck onverer V E 3 V G(f )H(f) (db) log N K S π For Opimum amp π N K f S f S π f f S log f db Sep: hoose S/N so ha he effecive peak - o - peak curren analog (including ramp) SD(Vi VO / ) is well above he noise level, e.g..5 vols. NfS Sep : hoose second pole fs/ π o correspond o uniy gain crossover frequency for 45 phase margin. Aenuaion a fs will auomaically be db. f f Sep 3 : hoose K S S for uniy gain a f f S. N π Sep 4 : hoose for increased low - frequency gain and " zero"dc error. Sep 5 : hoose 3 ES o cancel lead effec of ES. For EXAMPE, Using he Following Parameer Values: Vi V 6 µ H VO 5 V 54 µ F fs khz D 5/.5 Ω V m O S 3 N S fs Sep:.3 S Ω, N Turns N D(Vi VO / ) Sep : f fs / π 3.8 khz (asympoic approximaion esimae) fs S Sep 3 : K.8 << inpu Z of op amp. kω, N 9 Sep 4 :.5 7 pf ES Sep 5 : From daa shee, ES. Ω, 3. 7 kω pf Texas Insrumens SUP73

22 XIII. DESIGN POEDUE FO PEAK UENT- MODE BUK EGUATO To illusrae he design procedure for a buck regulaor using a single-loop peak curren-mode conrol algorihm, a ypical example using a curren sense ransformer and slope compensaion is presened in he accompanying sidebar box and he resuls are shown in he Bode Plos of Fig. 6. This example assumes ransformer curren sensing and slope compensaion, he laer because, even hough he nominal inpu-o-oupu volage raio predics a duy cycle of less ha 5%, operaing exremes and circui olerances could poenially push he duy cycle higher and, of course, we have also seen he benefi of improved audio suscepabiliy. The design crierion here is o achieve a leas 45 o of phase margin, which corresponds o seing he higher frequency pole a he uniy gain crossover frequency, wih a gain of - db a he swiching frequency. This corresponds o an asympoically prediced gain-bandwidh of f S /, or approximaely f S /π, as compared o he gainbandwidh of approximaely f S / for our lead compensaed volage-mode conrol example. Noe ha in push-pull applicaions, where he possibiliy of signal reinforcemen a f S / exiss, addiional aenuaion a he swiching frequency may be necessary o preven subharmonic oscillaions. More sophisicaed analyses, simulaions and/or empirical sudies could help o adequaely address hese large-signal issues for nonlinear sysems. Gain in db Phase in Degrees Frequency The gain bandwidh is greaer han 5kHz Frequency The Phase Margin is 45 o for: S Ω 7 kω N urns 7 pf kω 3 pf Fig. 6. Bode plo of peak curren-mode conrolled sample buck converer open-loop gain funcion wih opimum ramp. The compensaion-relaed benefis of currenmode conrol are seen in he plos of Fig. 6, which shows wha is essenially a firs-order sysem wihin he gain-bandwidh. The acual gain-bandwidh of 5 khz (f S /4) is less han he asympoic predicion (f S /π) because he curves are acually rounded a he break poins. Texas Insrumens SUP73

23 XIV. THE BOOST ONVETE The basic boos converer opology is shown in Fig. 7 operaing in he coninuous conducion mode, (M). The operaion of his circui is a wo-sep process where, wih he swich on, energy is added o he inducor as curren di V increases according o i. When he swich d is opened, he curren is shuned o he diode and he inducor discharges wih a decreasing curren di (V V ) according o o i. For seady-sae d operaion, he average curren in he inducor mus equal he D curren in he load and he average dc volage across he inducor mus equal zero. As disinguished from he buck regulaor where he filer inducor and capacior coninuously work ogeher as a eam, in a boos (and flyback) opology hey essenially alernae in a bucke-brigade mode. While energy is being sored in he inducor, he capacior is working alone o deliver energy o he load, and hen wih he swich open, he inducor replenishes he delivered energy by recharging he capacior o prepare i for he nex cycle. Since he average (D) volage across he inducor mus be zero, we can hen wrie (for M) vid (Vo vi)( d), v which gives us o d ( ) V - ON v PWM onrol /f S S S Fig. 7. The boos converer. Equal Areas An imporan characerisic of boh he boos and he flyback opologies in M is he presence of a righ-half-plane zero, a characerisic which gives a gain increase bu wih a phase lag. This HP zero is caused by he delay beween conrolling and delivering energy o he load. For example, a sudden increase in load curren causes a droop in he oupu volage, upon which he conroller calls for an increase in he swich duy cycle o sore more energy in he inducor. Increasing he swich on-ime, however, causes a decrease in he off-ime, meaning ha less energy is going ino he capacior and he oupu volage hus falls furher. Evenually a new balance is reached o regulae a he new load bu his HP delay is almos impossible o compensae and usually requires a relaively low frequency sysem gain rolloff for simple volagemode conrol. Texas Insrumens SUP73

24 XV. DEVEOPING THE SMA-SIGNA BOOST MODE As we did wih he buck opology, a smallsignal model for analysis needs o be developed from he equivalen circui by a process of averaging and linearizing. Fig. 8 shows he developmen of he average model by he process of firs deriving an equivalen circui for each sae. Swich condiion I corresponds o S on and S off while swich condiion II has S off and S on. Then using he duy cycle relaionships, one can deermine he average volage across he swich branches of he loops involving inducors, and he average currens hrough he swich branches ino he nodes involving capaciors. These relaionships are hen conneced using a D ransformer wih he appropriae urns raio. I i I i I I V V Swich condiion I [(d)h of he ime] Fig. 9 hen shows he linearizaion process which also follows as i did for he buck bu, unlike he buck, he open loop gain funcion is dependen upon he D duy facor. I i I i I ( - d) I VO d (- d): I ( - d) ( - D) I d ( - D): I ( - D) Fig. 9. inearized boos converer. The closed loop flow graph for he resulan linearized boos converer, using volage-mode conrol, is presened in Fig. 3. From his flow graph, we can derive he overall closed loop gain equaion as: ( D) s V K GH i VP( D) s ( D) s V V Swich condiion II [(-d)h of he ime] I i I I ( - d) ef v E (( - D) / - s) K V P ( - D) Err Amp Modulaor & HP Zero ( - D)/ Σ s s/ ( - D) / ( - d) V Fig. 3. Flow graph of linearized boos converer wih volage-mode conrol. (- d): ircui model using D ransformer concep Fig. 8. Averaged boos converer. Texas Insrumens 3 SUP73

25 Noe ha due o he duy facor dependency, he erm ( D) appears in he erms relaing o boh he resonan frequency and he HP zero. The duy facor dependen pole is easy o raionalize as he and are only conneced during he ( d) porion of he period. There is also no quesion abou he circui s uncompensaed insabiliy since he HP zero conribues an added 9 o of phase shif beyond resonance, for a oal of 7 o. As we have already deermined, his phase lag is relaed o he exra ime delay (as compared o he buck) in geing energy from he source ou o he load, however, i is also a funcion of he load. The ligher he load (larger ), he furher he HP pole moves ou in frequency and he less i will impac circui behavior. In sabilizing he volage-mode conrolled boos converer, generally only lag compensaion is applicable because a se of fixed-frequency lead neworks would work for only a very narrow range of inpu volage and oupu loading. Thus, he ypical volage-mode boos converer exhibis very low gain-bandwidh and a poor dynamic response. In paricular, any sudden line or load perurbaion will resul in a damped oscillaion a he effecive resonan frequency of he converer. The lag nework mus be designed for he lowes resonan frequency, corresponding o he maximum D, and a he lighes load for which he circui is sill in M operaion. These argumens apply o virually all nonbuck opologies. The similariies are eviden in comparing he boos flow diagram above wih he flyback shown in Fig. 3. The gain equaion for an equivalen flyback opology is: ( D) s NDV K GH i N D VP( D) s ( D) s N D( - D)/N ef v E ND (( - D) /N D - s) K Σ V P ( - D) s s/ ( - D) /N Err Amp Modulaor & HP Zero Fig. 3. Flow graph of flyback converer wih volage-mode conrol. In his equaion, is he ransformer inducance refleced o he primary, and N is he secondary-o-primary urns raio. Wih sabiliy issues and closed-loop dynamics virually he same as he boos converer, he design procedures are effecively he same and, by exension, can also be applied o he uk and SEPI opologies. As we shall see, he use of curren-mode conrol will significanly improve performance, allowing hese opologies o operae wih a much higher gain-bandwidh. XVI. UENT-MODE ONTO FO THE BOOST ONVETE The boos converer wih curren/mode conrol is shown schemaically in Fig. 3 and 33 Fig. 3 wih peak /M, and Fig. 33 wih average /M implemenaion. As we have already menioned, hese circuis are shown assuming he use of curren ransformers for curren sensing. If a resisor was used for sensing, S sill applies bu N in he equaions becomes uniy. Peak /M conrol allows sensing swich curren in place of inducor curren and he swich off-ime allows ime for he curren sense ransformer o rese. Wih average /M, we use he enire inducor waveform, which conains a D componen ha would saurae a single curren ransformer. Therefore, wo are shown in he schemaic, alernaing in measuring swich and diode curren. These wo signals are hen summed o reconsruc he oal equivalen inducor curren. A curren sense resisor in he inpu or reurn line may be a more pracical soluion in all excep higher power applicaions bu, again, his jus means ha N and we have kep he equaions consisen. Texas Insrumens 4 SUP73

26 lock PWM V E K I PWM Drive ach :N I S S omp d/f S /f S S I S /N S I S /N I S S S I S /N S v E V K lock conrols urn-on Peak curren conrols urn-off I S I during on ime Fig. 3. Peak curren-mode conrol for boos converer. :N :N S I S Drive S V K Gnd PWM d v E /V P d I omp Sawooh V P v E K S S I /N v E I S I I S I S V K Fig. 33. Average curren-mode conrol for boos converer. While here are disinc differences beween peak and average curren-mode conrol, hey are primarily large-signal characerisics, such as: Peak /M ofen requires slope compensaion where average /M does no. Average /M requires an addiional error amplifier. Peak /M offers some inpu volage feed forward (bu no generally as effecive as in he buck). There is a peak-o-average conrol error wih peak /M. Average /M ofen allows painless crossing of he M / DM mode boundary. Peak /M is highly subjec o noise riggering. Naurally, hese differences cause some applicaions o be inherenly beer suied o one or he oher and, in pracice, differen applicaions may impose significanly differen design crieria (e.g., a slow volage loop in a PF applicaion vs. a high-speed volage regulaor). However, while hese disincions beween peak and average conrol can influence individual large-signal design crieria, heir small-signal performance and closed-loop requiremens can be virually idenical for he same applicaion. Fig. 34 illusraes he large-signal differences beween peak and average /M conrol for he boos converer. Wih average /M, he inducor curren ripple creaes an opposie-phased signal a he oupu of he added curren error amplifier (K), amplified by is closed-loop gain. Again, large-signal crierion demands ha m < m bu dynamic curren loop response is improved as m approaches m. Wih average /M conrol, slope compensaion is unnecessary so he slope equaions are: VP m VPf S T K (V V ) K V D m s o i s o N N VPf SN K < SVO D Wih peak /M conrol, we need o consider slope compensaion in he same manner as wih he buck converer. Here he exernal ramp crierion is: VO S(D ) m3 > which corresponds o γ < (-D)/D where gamma is he gain reducion facor inroduced earlier. Alhough adding an exernal compensaing ramp improves line regulaion for he boos opology, i is no possible o null ou inpu volage sensiiviy compleely as i was wih he buck regulaor. Therefore, here is no opimum ramp slope for he boos converer bu seing γ ( D) as we did for he buck is sill a reasonable decision since (-D) < (-D)/D. Texas Insrumens 5 SUP73