DCDC变换器反馈控制设计

DCDC变换器反馈控制设计Typicalvariationinvg(t):100Hzripple,producedbyrectifiercircuit.Loadcurrentvariations:asignificantstep-changeinloadcurrent,suchasfrom50%to100%ofratedvalue.Circuitelementsareconstructedtosomespecifiedtolerance.EffectstopooroutputNegativefeedback:switchingregulatorsystemObjective:maintainconstantoutputvoltagev(t)=V,inspiteofdisturbancesininputandloadAtypicaloutputvoltageregulationspecification:2%forexample5V±0.1V.IntroducefeedbackcontrolNegativefeedbackconverterSmallsignalmodelofconverterOutputvoltagecanbeexpressedasLoaddisturbancemodel－－DC/DCconvertersystemdynamicmodelUsesmall-signalconvertermodelPerturbandlinearizeremainderoffeedbackloopDC/DCconvertersystemsmall-signalblockdiagram叠加原理求解－SolutionofblockdiagramLoopgainT(s)=productsofthegainsaroundthenegativefeedbackloop.ChangedintoformTerminology:open-loopvs.closed-loopOriginaltransferfunctions,beforeintroductionoffeedback(open-looptransferfunctions):Uponintroductionoffeedback,thesetransferfunctionsbecome(closed-looptransferfunctions):Theloopgain:Feedbackimprovethelineregulationopen-loopline-to-outputtransferfunction:WithadditionofnegativefeedbackFeedbackreducestheline-to-outputtransferfunctionbyafactorofIfT(s)islargeinmagnitude,thentheline-to-outputtransferfunctionbecomessmall.Closed-loopoutputimpedanceOriginal(open-loop)outputimpedance:Withadditionofnegativefeedback,theoutputimpedancebecomes:Feedbackreducestheoutputimpedancebyafactorof--

ClosedloopgainClosed-looptransferfunctionIftheloopgainislargeinmagnitude,i.e.,||T||>>1,thenT/(1+T)»T/T=1.Thetransferfunctionthenbecomeswhichisindependentofthegainsintheforwardpathoftheloop.Thisresultappliesequallywelltodcvalues:OutputisnotsensitivetoparametervariationintheforwardpathOpenloopgainexampleApproximating1/(1+T)andT/(1+T)Example:constructionofT/(1+T)Atfrequenciessufficientlylessthatthecrossoverfrequency,theloopgainT(s)haslargemagnitude.ThetransferfunctionfromthereferencetotheoutputbecomesThisisthedesiredbehavior:theoutputfollowsthereferenceaccordingtotheidealgain1/H(s).ThefeedbackloopworkswellatfrequencieswheretheloopgainT(s)haslargemagnitude.Atfrequenciesabovethecrossoverfrequency,||T||<1.ThequantityT/(1+T)thenhasmagnitudeapproximatelyequalto1,andweobtainAtfrequencieswhere||T||<1,theloophasessentiallynoeffectonthetransferfunctionfromthereferencetotheoutput.DiscussionExample:constructionof1/(1+T)Howdoesthelooprejectdisturbances?Belowthecrossoverfrequency:f<fcand||T||>1Then1/(1+T)»1/T,anddisturbancesarereducedinmagnitudeby1/||T||Abovethecrossoverfrequency:f>fcand||T||<1Then1/(1+T)»1,andThefeedbackloophasessentiallynoeffectondisturbancesinputoutputTransferfunctionTransferfunctionFactorizethedenominatorandnominatorPoles:Zeros:Foldfreq.Foldfreq.FrequencycharacteristicsBodeplots:AmplitudeplotAngleplotUnit:dBAmplitudeplotisaFoldinglinegraphMultiplyingfactorsbecomeadditionoperationinamplitudeBodeplot.Meetapole,Folddownwitharateof–20dB/decMeetazero,Foldupwitharateof+20dB/decAmplitudeBodeplot–20dB/dec–40dB/dec–20dB/decIfmeetapole,increasephasedelay90degreeoccurringbetweenfp/10and10fpIfmeetazero,leadphaseanglewith90degreeoccurringbetweenfz/10and10fzAngleBodeplot-90-1800+-ClosedlooptransferfunctionCharacteristicequationIfallrootsareinthelefthalfplane,stableIfarootintherighthalfplane,unstable.Root:StabilityofclosedloopContainsalltheinformationabouttherootsofThereforewecanknowthestabilityoftheclosedloopsystembystudying

Bodeplotisusedtoanalysisthestabilityofthesystem相位裕量增益裕量增益交越频率相位交越频率Bodegraph+-OscillationconditionandOscillationconditionToBreaktheoscillationcondition,itisrequiredwhenwhengainmargin:GM（dB）

Relativestabilityphasemargin:PM相位裕量增益裕量增益交越频率相位交越频率回路增益函数的相位为–180时的频率称为相位交越频率增益裕量是指当回路增益函数的相位为–180时，回路增益函数所能容许增加的回路增益

+-相位裕量增益裕量增益交越频率相位交越频率相位裕量:当闭环系统达到不稳定之前，其回路内所能容许增加的相位。当回路增益函数的幅值为零分贝（单位增益）时，回路增益函数的相移与–180之差。

Phasemargin+-相位裕量乃是在平面上为了使轨迹的增益交越点通过（–1，j0）点，则图必须以原点为中心顺时针所须旋转角度.增益裕量就是在平面上相位交越点对（–1,j0）点接近程度的一种量度

NaquistGraph相位裕量增益裕量增益交越频率相位交越频率-1Therelationshipbetweenphasemarginandclosed-loopdampingfactorHowmuchphasemarginisrequired?Asmallpositivephasemarginleadstoastableclosed-loopsystem,whichhascomplexpolesnearthecrossoverfrequencywithhighQ.Thetransientresponseexhibitsovershootandringing.IncreasingthephasemarginreducestheQ.Toobtainrealpoleswithnoovershootandringingrequiresalargephasemargin.ConsiderthecasewhereT(s)canbewell-approximatedinthevicinityoftheCrossoverfrequencyAsimplesecond-ordersystemClosed-loopresponseExpressedbyWherewhereissubstitutedLow-QcaseHigh-QcaseSolveforexactcrossoverfrequency,evaluatephasemargin,expressasfunctionofQQvs.phasemarginQvs.phasemarginTransientresponsevs.dampingfactorUnit-stepresponseofsecond-ordersystemT(s)/(1+T(s))whereTransientresponsevs.dampingfactorRegulatordesignTypicalspecifications:Effectofloadcurrentvariationsonoutputvoltageregulation,whichisalimitonthemaximumallowableoutputimpedanceEffectofinputvoltagevariationsontheoutputvoltageregulation,whichlimitsthemaximumallowableline-to-outputtransferfunctionTransientresponsetime,whichrequiresasufficientlyhighcrossoverfrequencyOvershootandringing.AnadequatephasemarginmustbeobtainedTheregulatordesignproblem:addcompensatornetworkGc(s)tomodifyT(s)suchthatallspecificationsaremet.+-反馈信号参考信号误差信号DC/DCpowerconvertersystemPWM调制器的传递函数为

传递函数为至输出的传递函数Buck变换器占空比至输出的传递函数Buck变换器系统原始回路增益函数

原始回路增益函数是一个二阶系统，有两个极点。

幅频图在低频段为水平线，幅值为高频段以–40dB/dec斜率下降，转折点由LC滤波器的谐振频率决定。设Buck变换器系统的参数为：输入电压输出电压，输出负载输出滤波电感值，电容值开关频率kHz，即开关周期PWM调制器中锯齿波幅值反馈分压网络传递函数可求出工作占空比：

Buck变换器占空比至输出的传递函数原始回路增益函数幅频图的转折频率为：

幅频图低频段为幅值约为20dB水平线,高频段为斜率穿越0dB线的折线。Hz

幅频图相频图增益交越频率，

相位裕量

原始回路增益函数频率特性的相位裕量太小。虽然系统是稳定的，但存在较大的输出超越量和较长的调节时间。

通常选择相位裕量在45左右，增益裕量在10dB左右。因此需要加入补偿网路

一般原始回路增益函数不能满足系统静态和动态特性的要求，为了使系统满足静态和动态的指标，需要加入补偿网路。

虽然，补偿网路只是系统中极小的一部分，但是，对系统静态和动态特性而言却是非常重要部份，它会影响到系统的输出精度、电压调整率、频带宽度以及暂态响应。最小相位系统的幅频特性和相频特性之间存在一一对应关系，如幅频图中斜率为-20dB/dec折线对应相频图中相移为斜率为-40dB/dec折线对应相频图中相移为斜率为+20dB/dec折线对应相频图中相移为知道了幅频特性也就知道了相频特性，反之也然。最小相位系统理论水平线对应相频图中相移为为使DC/DC变换器系统满足稳定性要求，可以通过外加补偿网络使DC/DC变换器系统的回路增益函数的幅频图在增益交越频率处（增益为零dB）的斜率为-20dB/dec。相位裕量PM大于零

因为根据最小相位系统的性质，幅频图的斜率为-20dB/dec折线对应相移为-90度还须验证在相位交越频率处（相位在–180时）若相位裕量与增益裕量的值，只是稍稍大于零时，虽然对系统而言也是稳定的，不过却会具有较大的超越量和调节时间。通常选择相位裕量在45左右，增益裕量在10dB左右。增益裕量必须大于零。回路增益函数补偿网路设计

（1）超前补偿网路（2）滞后补偿网路；（3）超前–滞后补偿网路。Lag(PI)compensationImproveslow-frequencyloopgainandregulationSuitabletothe1stordersystemLagCompensationExamplecompensator:originalloopgainisselectedlessthanfotomaintainadequatephasemarginisselectedPIFirstselectcrossoverfrequencyofclosedloopLead(PD)compensatorImprovesphasemarginSuitabletothe2ndordersystemLeadcompensator:maximumphaseleadThefrequencywithmaximumphaseleador55degLeadcompensatordesignToobtainacompensatorphaseleadofatfrequencyfc,thepoleandzerofrequenciesshouldbechosenasfollows:Ifitisdesiredthatthemagnitudeofthecompensatorgainatfcbeunity,thenGc0shouldbechosenasLeadcompensationexample超前滞后补偿网络

（a）电路（b）零极点分布超前–滞后补偿网络的传递函数“超前–滞后”补偿网路输出正弦信号的相位在不同频率范围有落后又有超前于正弦输入信号的特性，它结合超前补偿与滞后补偿的特性，发挥滞后补偿特性提高静态或稳态性能，利用超前补偿特性提高相对稳定性和动态性能。，项产生超前补偿效果项产生滞后补偿效果

(c)有源超前–滞后补偿网路1RC网路所组成的超前–滞后补偿网络的增益只能衰减不能增加，增益设计不便。因此，一般采用运算放大器构成有源的超前–滞后补偿网路。-20dB/dec20dB/dec补偿网路增益

利用补偿网络幅频特性图低频的积分特性，可以使经补偿后的系统成为无差系统，使静差为零，同时减少了低频误差。利用补偿网络幅频图在至之间斜率为20dec/dB上升特性，补偿原始回路函数以斜率穿越0dB线的特性，使补偿后的回路函数以穿越0dB线，这样才能使DC/DC变换器系统具有较好的相对稳定性。因此，一般将补偿后系统的增益交越频率设定在补偿网路的与之间。有源超前—滞后补偿网络2

零点为：极点为：与有源超前–滞后补偿网路1的差异是在高频部份增加了一个极点fp3，而使其向下反折为–20dB/dec。频率fz1与fz2之间的增益可近似为

在频率与之间的增益则可近似为

一般也将补偿后系统的增益交越频率设定在补偿网路的与之间

20dB/dec-20dB/dec-20dB/dec下面以有源超前滞后补偿网络2为例介绍补偿网络