In Part I of this article (Power Electronics Technology, May


 Daniel Tobias Owen
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1 art wo A ew Way to odel urrentode ontrol Unified l uing general gain paraeter proide the olution for any peak or alleyderied  conerter. By obert heehan rincipal Application ngineer ational eiconductor anta lara alif. n art of thi article (ower lectronic echnology ay 007 the baic operation of  control wa broken down into it coponent part allowing a greater intuitie undertanding for the practical deigner. A coparion of the odulator gain wa ade to oltage operation and a iple analogy howed how the optial lopecopenation requireent could be obtained without any coplicated equation. ow unified l uing general gain paraeter are introduced along with iplified deign equation and an indepth treatent of the analyi and theory i preented. hi general ling technique explain how preiou l can copleent each other on ariou apect of the control theory. odeling ontinuouonduction ode hi article proide l and olution for fixedfrequency continuouconduction ( operation. eference [] coer the theoretical background for thi ubject proiding an exhautie analyi of the buck regulator with it aociated l and reult. o preent duplication the boot regulator of Fig. for the bai for the dicuion here. A ore rapid approach to uing thi inforation i to bypa reference [] and follow the general guideline for lope copenation decribed in the firt part of thi article. hen the iplified equation can be ued to deterine the frequency repone. A  witching regulator i a apleddata yte the bandwidth of which i liited by the witching frequency. Beyond half the witching frequency the repone of the inductor to a change in control oltage i not accurately reproduced. o quantify thi effect for linear ling the continuoutie l of reference [] uccefully placed the aplinggain ter in the cloed Fig.. hi witching l of a boot regulator topology proide an exaple for ling and iulating continuouconduction operation. ower lectronic echnology June 007
2 UO OO Fig.. For a buck regulator apling gain H ( i placed in the forward path (upper circuit and apling gain H( i placed in the cloed feedback loop path (lower circuit. feedback loop. hi allow accurate ling of the controltooutput tranfer function uing the ter H (. o accurately l the loop the unified l of reference [3] placed the aplinggain ter in the forward path. For peak or alley with a fixed lopecopenation rap thi alo accurately l the controltooutput tranfer function uing the ter F (. o deelop the theory for eulated  control reference [] ued a freh approach deriing general gain paraeter which are conitent with both l. n addition a new repreentation of the aplinggain ter for the cloed loop wa deeloped identifying liitation of the forwardpath aplinggain ter. he upper circuit in Fig. repreent the unified for of the l with being the feedforward ter. n the lower circuit i the dc audio uceptibility coefficient fro the continuoutie l. he linear l aplinggain ter a hown in Fig. are defined a: π H ( H ( ( and Q where i the witching period. he ter i new and eerged fro the deriation of the cloedloop expreion for H(. hi deriation ued lopecopenation ter other than the claic fixed rap for peak or alley. can be expreed a Q but thi ere no purpoe becaue Q would need a alue of infinity for the condition 0. o date no ethod ha been found which uccefully incorporate into the openloop expreion for H (. Ue of H ( i liited to peak or alley with a fixed lopecopenating rap for which the alue of 0. o place either aplinggain ter into the linear l for the buck boot and buckboot the following relationhip are applied: F ( F H ( and G ( G H(. he accuracy liit for the aplinggain ter i identified by coparing Q to the odulator oltage gain and the feedforward ter. Q i directly related to the lopecopenation requireent. he deriation tart with the ideal teadytate odulator gain the phyical reaon being that at the witching frequency the relatie lope are fixed with repect to the period. A change in control oltage i then related to a change in aerage inductor. Any tranfer function that i olely dependent on in the forward dcgain path will hae excellent agreeent to the witching l up to half the witching frequency. Howeer any tranfer function that include in the forward dcgain path will how oe deiation at half the witching frequency. iplified ranfer Function o auption for iplification were ade during the deriation of the tranfer function. he only initial auption are the one generally accepted to be alid in a firtorder analyi. oltage ource ource and witche are ideal with no delay in the control circuit. Aplifier input are high ipedance with no ignificant loading of the preiou tage. iplification of the reult wa ade after the coplete deriation which included all ter. eference [] ha exaple for the buck regulator. o how the factored for the iplified tranfer function aue that the pole are well eparated by the loop gain. xpreion for the lowfrequency l do not how the additional phae hift due to the apling effect. he controltooutput tranfer function with the aplinggain ter accurately repreent the circuit behaior up to half the witching frequency. he linetooutput expreion for audio uceptibility are accurate at dc but dierge fro the actual repone a frequency increae. he ene gain i defined a G where G i the ene aplifier and i the ene reitor. For all tranfer function and. Z OU o include the aplinggain ter in the control to output tranfer function the ter i replaced with in the lowfrequency equation. hi repreent the cloedloop Q aplinggain ter. ncluion of thi ter in the linetooutput equation will not produce the ae accuracy of reult. For peak or alley with a fixed lopecopenating rap Q. apling Gain Q Uing a alue of Q will caue any tendency toward ubharonic ocillation to dap in one witching cycle. With repect to the cloedloop controltooutput ower lectronic echnology June
3 UO OO ode eak lope copenation Fixed lope Q ( 0. Q π ( 0 eak roportional lope For Q (inglecycle daping Q π ( ( 0. alley Fixed lope ( 0 Q π ( alley roportional lope For Q (inglecycle daping Q π ( ( ( ulatedpeak Fixed lope ( Q π ( ulatedpeak roportional lope For Q (inglecycle daping Q π ( ( able. uary of general gain paraeter. function the effectie apledgain inductor pole i gien by: f ( Q ( ( 4 Q. 4 Q hi i the frequency at which a 4degree phae hift occur becaue of the apling gain. For Q f (Q occur at 4% of the witching frequency. For Q f (Q occur at 3% of the witching frequency. For econdorder yte the condition of Q i norally aociated with bet tranient repone. he criteria for critical daping i Q 0. (δ. Uing Q ay ake an increental difference for the buck but i inconequential for the boot and buckboot with the aociated righthalfplane zero of. For the peak buck with a fixed lopecopenating rap the effectie apledgain inductor pole i only fixed in frequency with repect to change in line oltage when Q roportional lopecopenation ethod will achiee thi for other operating. o deterine the effect of reducing the lope copenation to increae the oltageloop bandwidth an eulated ower lectronic echnology June
4 UO OO Fig. 3. he lowfrequency linear l for thi buck regulator wa ade uing etrix. for a table oltage loop at the expene of underdaping the loop. With Q ubharonic ocillation i quite pronounced during tranient repone but dap at teady tate. he reader i encouraged to iulate and obere thee effect directly. A iulation exaple for the boot i proided after the linear l and tranfer function are preented. inear odel iple accurate and eaytoue linear l hae been deeloped for the buck boot and buckboot conerter topologie. ach linear l ha been erified uing reult fro it correponding witching l. n thi anner alidation for any tranfer function i poible identifying the accuracy liit of the gien linear l. General gain paraeter are lited in able. hee paraeter are independent of topology and written in ter of the terinal peak buck with proportional lopeco oltage ( and duty cycle (. penation witching circuit wa ipleented in. he coefficient for the linear l of the buck regulator A tandard type 0 Hz error aplifier wa ued for hown in Fig. 3 are: frequency copenation. With / ( µ/ µh and (0. /A the bet perforance wa achieed with OU ( OU Q for a crooer frequency of 40 khz and 4degree phae argin. By etting Q a crooer frequency of and F. wrlecentronic 0 khz wa achieed again igiwr with /4p 4degree /9/07 phae argin :3 but age OU reduced gain argin. hi appear to be the practical liit he controltooutput iplified tranfer function i: OU OU Z W! igiower ultiharger harge oputer igital aera and 4 AA or AAA  ih/i oplete with: UB lug ar lug A Wall lug Batterie apacitor aritor ranforer ower Adapter 346 onroe Ae. enilworth J el: ( Fax: ( eail: and the linetooutput iplified tranfer function i: OU O Z where OU and OU OU. he coefficient for the linear l of the  boot regulator hown in Fig. 4 are: OU OU ( OU and F. OU OU he controltooutput iplified tranfer function i: OU OU Z ower lectronic echnology June
5 UO OO and the linetooutput ipliﬁed tranfer function i: OU OU Z where OU OU Fig. 4. he lowfrequency linear l for thi boot regulator wa ade uing etrix. OU Z Fig.. he lowfrequency linear l for thi buckboot regulator wa ade uing etrix. OU the optial Q at one input oltage. he controltooutput gain plot in Fig. 6 how only a light deiation between the two l at half the witching frequency where fw 00 khz. For the iulation lope copenation wa et for Q he choice of iulation progra i iportant ince not all progra calculate paraeter with the ae degree of accuracy. For witchingl iulation i able to produce Bode plot directly fro the witching l. hi progra wa ued to produce the witchingl iulation reult. he lowfrequency l wa ade with etrix which i the generalpurpoe iulator for the etrix/ progra. hi iulator only handle aplace equation for in nuerical for where the nuerator order ut be equal to or le than the denoinator order. pice i uch better uited for linear l with aplace function in paraeter for. t i ore accurate than the etrix/ progra but cannot produce Bode plot directly fro the witching l. pice or a progra with iilar capability ay be ued to obtain the iulation reult for the linear l. OU OU OU OU OU and. OU OOU Boot egulator iulation xaple For the peak boot conerter exaple coparion of reult fro the witching circuit of Fig. were ade to the linear l of Fig. 4 uing the aplinggain ter H(. o ue the forwardpath aplinggain ter lope copenation wa ipleented with a ﬁxed rap. he reult will be lightly different if a proportional rap i ued a thi odiﬁe the odulator gain ter and feedforward ter. For an actual bootconerter ipleentation with a ﬁxed rap it i only poible to get ower lectronic echnology June 007 and the linetooutput ipliﬁed tranfer function i: he controltooutput ipliﬁed tranfer function i: OU OU Z where he coefﬁcient for the linear l of the  buckboot regulator hown in Fig. are: OU OU ( OU OU and F. OU OU and. out OU OU 30
6 UO OO Fig. 6. hi coparion of controltooutput tranfer function for a peak boot conerter uing ﬁxedlope copenation reeal the witching and linear l behae iilarly with a light dicrepancy at 00 khz which i half the witching frequency. Uniﬁed odulator odeling opology n art of thi article the criteria for  control wa conidered. hi led to the linear l with the gain ter being eaily identiﬁed. he iportance of the concept of a the odulator oltage gain cannot be oertated. ot linear l for  control hae allowed the ath to deﬁne the l. n reference an intuitie undertanding of the odulator wa ued to drie the ath. By algebraic anipulation both the aeraged i Buck i i Boot i ig Buckboot i i ig ( ( OFF ( OFF ( (OF OFF F / O OFF ( OFF ( / (OF OFF F O ote: OFF ; O able. orrection and clariﬁcation for reference [3]. l and continuoutie l were redeﬁned to ﬁt the for of the uniﬁed l. obining the uniﬁedl gain block with the threeterinal W witch reulted in the linear l ued here. A new cloedloop aplinggain ter ha been deﬁned that accoodate any ﬁxedfrequency peak or alleyderied operating. iitation of the forwardaplinggain ter ha been identiﬁed proiding direction for further deelopent in linear ling. ech eference. heehan obert ulated urrent ode ontrol for Buck egulator Uing aple and Hold echnique ower lectronic echnology xhibition and onference 0 October 006. An updated erion of thi paper which include coplete appendix aterial i aailable fro ational eiconductor orp.. idley.b. A ew ontinuouie odel for urrent ode ontrol ranaction on ower lectronic ol. 6 ue pp an F.. and iddlebrook.. A Uniﬁed odel for urrentrograed onerter ranaction on ower lectronic ol. 0 ue 4 pp ower lectronic echnology June
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