Switchmode Boost Power Converter Using Voltage-Mode Control

Transcription

1 Switchmode Boot Power onverter Uing oltage-mode ontrol A Sunam online continuing education coure Switchmode Boot Power onverter Uing oltage-mode ontrol By Raymond. Barrett, Jr., Ph, PE EO, American Reearch and evelopment, opyright Raymond. Barrett, Jr. Page

2 Switchmode Boot Power onverter Uing oltage-mode ontrol A Sunam online continuing education coure. Switchmode Boot Power onverter Introduction and Baic Model Thi coure develop model of the Boot converter with duty cycle control. Baic operation, a practical et of example, and large/mall ignal model are dicued. onideration for feedforward control to addre line regulation and feedback control to addre load regulation of the converter are included. Figure. Ideal Boot onverter Schematic We develop a contant frequency, continuou current Boot converter deign, with the witching period defined below in figure. by T - T = T. The converter ha two conducting tate defined by period T - T = T and T T = ( )T, correponding to the two witching tate. The w node i connected uing S w to ground with the duty cycle, and a complement controlled ynchronou witch S w i applied to connect the w node to the output network during the remaining ( - ) portion of the period. Figure. Boot onverter Inductor Operating Waveform The inductor cannot upport a voltage difference acro it terminal. Intead, any hortterm voltage difference reult in a contant rate of change of current I through the inductor. With ome voltage on the capacitor, the inductor ha a voltage difference applied during the T interval and - applied during the ( - )T interval. opyright Raymond. Barrett, Jr. Page

3 Switchmode Boot Power onverter Uing oltage-mode ontrol A Sunam online continuing education coure We can equate the volt-econd product and keep a zero voltage average a: T T [.] [.] [.] [.3] Equation [.3] provide the property of the Boot converter that how that a larger output voltage can be obtained from the input voltage by controlling the duty cycle.. Switchmode Boot Power onverter Input/Output urrent Waveform A hown in figure. above, the I input current the inductor current, and i continuou and non-zero. However, a hown in figure. below, the S w current to ground during the T - T = T interval, a well a the S w current to the output voltage during the T T = ( )T interval are both dicontinuou. The I OA output current i continuou and flow through the R OA reitor a a combination of current from the S w current and the capacitor. Becaue the capacitor cannot upport a continuou current, but doe ink/ource A and tranient current, the average current to the load i identical to the average S w current. Figure. Boot onverter Switch urrent Waveform opyright Raymond. Barrett, Jr. Page 3

4 Switchmode Boot Power onverter Uing oltage-mode ontrol A Sunam online continuing education coure The average input current i identical to the I -Average current. The conducting S w current i the inductor current and ha the ame I -Average current during it ON-tate, but becaue it i non-zero only during ( )T interval, it average value i ( )I -Average over each entire period. onequently, the average I OA output current i alo equal to the ame ( )I - Average over each entire period. 3. Switchmode Boot Power onverter Input/Output Power and Efficiency We can calculate the average input power from the product of the input upply time the average I input current a follow: P I [3.] Average Similarly, we can calculate the average output power from the product of the output output time the average I OA output current a follow: P OUT OA I Average I [3.] If we inert equation [.3] for the value of in term of the input voltage into equation [3.], we find that the input and output average power level are identical: P OUT I OA I Average P [3.] The indicated % efficiency i not correct becaue we have not accounted for loe in the witching element, or the non-ideal practical component that we mut ue to implement the deign, however, very high efficiencie are achievable, often exceeding 9% efficiency in a practical deign. It i the high efficiency of the witch-mode power converter that account for the interet, depite the complexitie of the deign and control mean required to implement a practical deign. 4. Output oad urrent Range The wort-cae, highet current i determined by the mallet R OA value, and in turn, the highet I OA value. The current handling capacity of the witching device mut be ufficient to upport witching the maximum I OA value with ufficient peed to upport the witching for both T and ( )T period. opyright Raymond. Barrett, Jr. Page 4

5 Switchmode Boot Power onverter Uing oltage-mode ontrol A Sunam online continuing education coure The maximum value that the R OA reitor may attain may be contrained to determine a minimum I OA value. A minimum current value may be employed to enure continuou load current, and to enure tability requirement 5. Input/Output Ripple urrent Effect in omponent alue Selection We ee from equation [.3] that the frequency doe not enter directly into the relationhip between the input voltage and the output voltage, only the duty cycle i directly involved. In figure., we alo ee that the inductor current form a triangular waveform between the I peak current, and the I valley current. The triangular peak-to-peak current i defined to be a ripple current, and i an A waveform uperimpoed on the average or inductor current. From the fundamental differential equation decription of the behavior of an ideal inductor we have: di [5.] dt For a regime with relatively hort time, relatively large inductor value, and relatively mall voltage, we can approximate the relationhip with line egment a follow: I [5.] t And in more ueful form: I t I I [5.] From equation [5.], we ee that the volt*econd product of the applied waveform can be ued to determine the triangular ripple current between the I and I limit. To enure continuou operation, we implement the deign o that I remain non-zero. We elect an inductor value large enough to upport the volt*econd product and atify the remaining deign parameter. From the fundamental differential equation decription of an ideal capacitor we have: d I [5.3] dt There are two ditinct time interval for capacitor current, the T interval and the (-)T interval. uring the T interval, the capacitor i dicharging into the R oad load reitor opyright Raymond. Barrett, Jr. Page 5

6 Switchmode Boot Power onverter Uing oltage-mode ontrol A Sunam online continuing education coure alone, and the dicharge follow the familiar exponential with a R oad time contant. uring the (-)T interval, however, there i alo added the charging current through the S w witch. The waveform during each interval can obtained by olving the differential equation explicitly, but detailed wavehape information i not neceary, only the peak-to-peak voltage ripple. We ue only the dicharge portion of the cycle, during the T interval to olve a follow: TS Road e [5.4] Uing a traight-line approximation and taking the derivative of equation for the lope of the dicharge line, we have: T S [5.5] Road Equation [5.4] offer a value for the peak-to-peak ripple voltage that can be expected to be caued by the choice of capacitor value and time interval, but it i probably more ueful expreed a the ratio: T R S oad [5.5] Additional non-ideal paraitic component are needed to decribe the power lot in the inductor and capacitor. 6. Input/Output oltage Range onideration Practical application require that we produce a controlled value for over a range of input voltage value. For intance, automotive application may require a nominal operation, but be expected to function nominally under a low battery condition below, and alo operate with tranient value in exce of 5 for a few milliecond in the cae of load-dump of highly inductive motor and olenoid device connected to that ame battery/alternator ytem. The range can be >5: for ome automotive application. Similarly, line-powered application may be expected to function correctly with common witching circuitry when powered from / main ource. The line-powered range opyright Raymond. Barrett, Jr. Page 6

7 Switchmode Boot Power onverter Uing oltage-mode ontrol A Sunam online continuing education coure may be ~85 from the low-line ource, but alo a higher than 365 under high-line ourcing. The range can be >4.5: for ome line-powered application. Although many application require a fixed output voltage, there are alo application that require a uer-programmed output voltage alo, often over a coniderable range of value. The capacitor mut withtand the highet expected output voltage under both nominal and tranient condition. The ratio of the mallet output voltage to the highet input voltage determine the mallet nominal value of duty-cycle required. ikewie, the ratio of the highet output voltage to the lowet input voltage determine the larget value of duty-cycle required. 7. Switchmode Boot Power onverter ine/oad Regulation Introduction Practical application typically require that we provide a controlled value for depite change in the input voltage. The term line regulation i ued to decribe the reulting effect of that control effort. Alo, practical application require that we provide a controlled value for depite change in the load current I OA. The term load regulation i ued to decribe the reulting effect of that control effort. Practical application ue a combined trategy for controlling the duty cycle dependent on both the and the value. That part of the control that ue the value to control the duty cycle i called a feedforward control mechanim. That part of the controller that ue the value to control the duty cycle i called a feedback control mechanim. To facilitate each form of control, a detailed mall-ignal model i developed o that the tability and performance of the control can be determined. If feed-forward control i utilized, it i deigned later and applied to the ytem to modify the model behavior after feedback i developed. However, the feedforward control leen the change in that the feedback mut deal with, making the feedback deign le demanding. It i the feedback control that require a mall-ignal model to determine gain and phae margin, a well a any compenation required to tabilize the cloed loop behavior. 8. Switchmode Boot Power onverter uty-ycle ontrol Model The Boot converter model i decribed uing two tate variable: the inductor current I and the capacitor voltage. The input voltage and the load reitance R OA are retained to expre the input and output dependencie for line and load regulation. opyright Raymond. Barrett, Jr. Page 7

8 Switchmode Boot Power onverter Uing oltage-mode ontrol A Sunam online continuing education coure Figure 8. Boot onverter Schematic uring the T Period Modeling begin with the topology defined in figure 8. during the T interval with the grounding witch S w conducting and S w OFF. We ue Kirchoff oltage aw (K) around the loop including, and, and Kirchoff urrent aw (K) at the node defined by the voltage, to write two defining equation: and [8.] I [8.] OA I Becaue, I OA, and I are not the choen tate variable, we rewrite the equation in term of the tate variable, and ue the aplace operator to obtain the equation: and I [8.] [8.3] ROA We rewrite equation [8.] and [8.3] into differential equation form, a follow: and I [8.4] [8.5] R OA opyright Raymond. Barrett, Jr. Page 8

9 Switchmode Boot Power onverter Uing oltage-mode ontrol A Sunam online continuing education coure We define a tate vector compoed of the two tate variable: I X [8.6] We then expre the two equation in matrix form uing the tate vector and build the tate matrix a the expreion of the two imultaneou equation. It i a matrix differential equation with the derivative of the tate vector X, expreed in term of the tate vector X itelf and the input voltage: X X [8.7] R OA The matrix differential equation [8.7] decribe the behavior of the Boot converter during the time T that the input upply i connected through the cloed S w witch. Figure 8. Boot onverter Schematic uring the (-)T Period We continue modeling with the topology defined in figure 8., with conduction through the ynchronou witch during the (-)T interval, again uing K and K to write two modified defining equation: and I [8.8] OA I I [8.9] A before, we rewrite the defining equation: opyright Raymond. Barrett, Jr. Page 9

10 Switchmode Boot Power onverter Uing oltage-mode ontrol A Sunam online continuing education coure and I I [8.] [8.] ROA We write equation [8.] & [8.] into explicit differential equation form, a follow: and I [8.] I [8.3] R OA Uing the tate vector a previouly defined, we expre the new matrix differential equation a follow: X X [8.4] R OA 9. Switchmode Buck Power onverter State-Space Average Model Following the practice of tate-pace averaging, we um time the component matrix in equation [8.7] plu (-) time the component matrix in equation [8.4] to provide the tatepace averaged equation: X R OA X X [9.] R OA In equation [9.] term with the factor arie from the firt interval of the witching period, and term with the (-) factor from the econd interval of the witching period. The tatevariable X i now the average for the entire witching period. We ditribute algebraically the duty-cycle dependence a follow: opyright Raymond. Barrett, Jr. Page

11 Switchmode Boot Power onverter Uing oltage-mode ontrol A Sunam online continuing education coure X R OA OA X X X [9.] R R OA X R OA X OA R OA X X [9.] R X X X [9.3] R OA. Boot onverter Initial Inductor hoice We chooe a a deign requirement; a Boot onverter baed on a nominal. to 4.7 range to upply 8 at with 5 milli-olt maximum ripple voltage. The converter mut upport a maximum Ampere load. We contrain the minimum load to be % of the maximum value, or milli-ampere, uing an internal load reitance. For contrating illutration, we chooe a.5 MHz witching frequency. We chooe a milli-ampere peak-to-peak current ripple in the inductor a a nominal value. From the range and the fixed 8 value, we determine that the range of duty-cycle mut be.47 to.64. At.5 MHz, the time are 9 nec to 54 nec. We ue the 4.7 value during the hortet 9 nec interval to determine the minimum inductor value that will upport that voltage with the requiite current change, a follow: I t 9 [.] opyright Raymond. Barrett, Jr. Page

12 Switchmode Boot Power onverter Uing oltage-mode ontrol A Sunam online continuing education coure H. [.] To further addre the election of the inductor, we mut conider that the Boot converter deliver a maximum of A at 8 or 8Watt and hould not diipate appreciable power in the inductor, while delivering that current. The inductor mut be capable of handling I -Average without aturation of the inductance, a well a have a low reitance. From equation [3.], we know that I oad wa derived from (-) I -Average o conequently we find the maximum average inductor current a: I OA I Average. 8A [.].64 The power lo in the reitance (R) of the inductor i: P I R.8 R [.3] We find a H Murata component (igikey # 8-34-N) that ha m R and will caue le than % lo at.8 Ampere inductor current. We conider that a acceptable. Other conideration, including price, hielding, aembly requirement, etc., can alter other component parameter, but the inductance and R requirement mut be met by whatever election i made. The Simple Boot onverter Initial apacitor hoice We determine the minimum capacitor value from equation [5.5] a follow: Min T R S oad ec 4F [.] We can meet the firt capacitor requirement with a 5 F AX multi-layer ceramic capacitor (igikey # N). Again, other device parameter mut be conidered and thee election are for illutration only.. Switchmode Boot Power onverter Small-Signal State-Space Average Model opyright Raymond. Barrett, Jr. Page

13 Switchmode Boot Power onverter Uing oltage-mode ontrol A Sunam online continuing education coure opyright Raymond. Barrett, Jr. Page 3 To model the mall-ignal behavior, we introduce a notation that repreent a operating point with capital letter, and mall ignal perturbation with the maller letter for each variable and ubtitute in the model we developed in equation [9.3], a follow: x X R x X OA v x X d [3.] We expand the term, of equation [3.], and remove any product of mall term a econd-order and mall enough to ignore, a follow: x R X R x X OA OA x X d X v [3.] From equation [3.], we ubtract the large-ignal operating-point equation given in equation [9.3], a follow: x R X R X x X x OA OA X R OA x X d X X

14 Switchmode Boot Power onverter Uing oltage-mode ontrol A Sunam online continuing education coure opyright Raymond. Barrett, Jr. Page 4 v [3.] ollecting term, we have the mall-ignal model a follow: OA v X d x R x [3.3] The tate-pace averaged mall-ignal model i truly only valid for mall ignal. ikewie, it i only valid for mall-ignal perturbation with much lower frequency than the witching frequency. Seriou aliaing effect can make the model unuable for frequencie approaching a large fraction of the Nyquit frequency (half the witching frequency). However, for analyi at lower frequencie to about % of the witching frequency, the tate-pace averaged model give good reult. 4. Small-Signal State-Space Average Model in the Frequency omain In claical aplace form, we can olve the above matrix differential equation [3.3], firt for the entire mall-ignal tate variable vector x including the inductor current and then reduced to the tranfer function for the voltage output on the capacitor alone: OA v X d x R I [4.] We have interpreted the large-ignal quantitie, and X a quai-tatic contant. Only the mall-ignal quantitie x, d, and v are treated a variable. We olve by matrix inverion a follow: OA v X d R I x [4.]

15 Switchmode Boot Power onverter Uing oltage-mode ontrol A Sunam online continuing education coure opyright Raymond. Barrett, Jr. Page 5 In equation [4.] above, the olution i compried of two factor; one factor require inverion of a x matrix and the econd factor i the um of component vector ariing from matrix algebra. We extract the matrix inverion in iolation in equation [4.] below: R I OA OA R OA R R R OA OA = R R OA OA R R R OA OA OA [4.] We define the calar polynomial equation P() originating from the determinant a follow: R P OA [4.3] A we examine the polynomial, we are reminded that during the T S interval, the inductor and capacitor are effectively diconnected. onverely, if the duty-cycle i reduced to zero,

16 Switchmode Boot Power onverter Uing oltage-mode ontrol A Sunam online continuing education coure opyright Raymond. Barrett, Jr. Page 6 the inductor and capacitor are effectively alway connected. In the event that =, then we expre P() a: R P OA [4.4] We note that equation [4.4] correctly decribe pole of econd-order reonant circuit connecting the input to the output. For the matrix inverion, we conclude with: R I OA R R P OA OA [4.5] We move the calar polynomial equation P() and (-) factor to become a multiplying factor for the mall-ignal tate vector x in the differential equation olution for the entire model and equation [4.] become equation [4.6] below a follow: OA OA v X d R R x P [4.6] In equation [4.6] above, the P() polynomial equation in the aplace operator i a calar quantity and multiplie the mall-ignal tate vector x, but there i a x matrix from the matrix inverion to be ditributed acro a x d dependency matrix and a x v dependency vector to implify the model. In equation [4.7] below, we perform the ditributive x matrix multiplication in iolation a part of the olution a follow:

17 Switchmode Boot Power onverter Uing oltage-mode ontrol A Sunam online continuing education coure d R OA R OA X ROA ROA d X [4.7] We ue the partial reult from equation [4.7] above to ubtitute in equation [4.6] to form equation [4.8] below a follow: P x d R R OA R OA OA R v OA [4.8] X We retate explicit mall-ignal equation individually, a follow: Pi di d R OA R P OA OA R OA v [4.9] R v d I d v [4.] 5. Feedforward ontrol Option for Good ine Regulation We ee that equation [4.9] indicate that the output voltage acro the capacitor ha a mallignal dependency on both the input voltage v and the duty cycle d o that we can find the condition that reduce the um to be zero, a follow: opyright Raymond. Barrett, Jr. Page 7 Pv d ff I d ff v [5.]

18 Switchmode Boot Power onverter Uing oltage-mode ontrol A Sunam online continuing education coure opyright Raymond. Barrett, Jr. Page 8 We olve equation [5.], uing d ff to repreent the feed-forward component of the dutycycle variation a follow: ff ff v d I d [5.] ff v I d [5.] ff v I d [5.3] From equation [5.3], we found a ingle-pole relationhip that i required to eliminate the mall-ignal upply perturbation effect on the output voltage. We defer any implementation dicuion but note that we would require a duty-cycle control ignal with right-half-plane characteritic defined in equation [5.3] to provide optimum feed-forward line regulation. 6. Feedback ontrol Option for Good oad Regulation From equation [4.] we iolate the mall-ignal dependency of the v capacitor voltage on the duty-cycle. We retain only the dependency of the capacitor voltage v on the feedback duty cycle d a follow: d I d v P [6.] I d v P [6.] P I d v [6.]

19 Switchmode Boot Power onverter Uing oltage-mode ontrol A Sunam online continuing education coure opyright Raymond. Barrett, Jr. Page 9 R I d v Boot OA [6.3] The objective of feedback control i to maintain the value of, o we can treat that quantity a quai-tatic. Becaue the output capacitor i not involved in any current, we can expre the oad current through R oad a the average of the duty-cycle modulated inductor current or (-)I and from Ohm aw: oad R I [6.4] oad R I [6.5] R R Boot OA oad [6.6] R R Boot OA oad [6.7] Equation [6.7] expree the mall-ignal dependency of the output voltage on the dutycycle. We defer dicuion of how the duty-cycle variation i developed from the feedback. We contruct a typical block diagram for the development of open and cloed-loop behavior a follow:

20 Switchmode Boot Power onverter Uing oltage-mode ontrol A Sunam online continuing education coure Figure 6. Boot onverter Small-Signal oltage-mode ontroller Block iagram The PWM ontroller convert the error voltage into the clocking ignal with the duty-cycle d for control of the Boot witching. In that repect, i a mall-ignal quantity itelf. The ref ignal i typically a value developed from a Bandgap or ome form of oltage reference but not necearily at the ame level a the deired output voltage. The Attenuator reduce the voltage value o that it can be compared to the ref value and thu produce the error ignal. We model both the PWM ontroller T PWM and the Attenuator T ATTEN a wide-bandwidth tranfer function to produce an open-loop tranfer function: In tandard form: T T PWM T T T T Boot T ATTEN PWM [6.8] PWM T ATTEN opyright Raymond. Barrett, Jr. Page ATTEN A Z Boot Z [6.9] Thu far, our open-loop frequency dependencie lie in the operating point variable and, and the econd-order combination of the component, but the damping factor alo depend on the equivalent value of R OA on damping the tranfer function. From the tandard form of the denominator, we have: ABoot [6.]

21 Switchmode Boot Power onverter Uing oltage-mode ontrol A Sunam online continuing education coure ROA Z [6.] R OA From equation [6.9] we know that the zero lie in the right-half plane and cannot be cancelled by what would be an untable pole in the PWM controller. From equation [6.], we know that the location of the zero depend on the R oad reitance a well a the dutycycle. [6.] R OA z [6.3] [6.4] z R R OA OA Boot P-Z Summary Rload = 8 Rload = 8 A Boot f z f z z f z k rad/.7khz.7m rad/ 3kHz.65.7M rad/.3mhz k rad/ 3.kHz 5k rad/ 8.kHz.883.5M rad/ 8kHz.883 Table 6. Boot Small-Signal Tranfer Function Pole and Zero Behavior 7. Boot Small-Signal Tranfer Function Bode Plot ariation The econd-order Pole locu depend on the component choice made earlier to addre the ripple concern, but alo depend on the operating point duty-cycle from the turn-on intant up to the nominal wort-cae operating point. In addition, both the reonant peaking a well a a right-half plane zero depend alo on the equivalent R oad load reitor. We do not explicitly control the election of the duty-cycle, but rather accept it a impoed by the required operating point and a a reult of the control function. ikewie, we do not explicitly control the election of the R oad load reitor, rather it i the reult of the load requirement placed on the converter. Both are exogeneou or external quantitie that we opyright Raymond. Barrett, Jr. Page

22 Switchmode Boot Power onverter Uing oltage-mode ontrol A Sunam online continuing education coure mut provide a compenation cheme for Boot converter pole-zero variation in the deign of the controller. We how Bode plot of our mall-ignal model a we vary the exogenou duty-cycle and R oad load reitor variable over their expected range of operation. Figure 7. below how the expected variation in gain (increae with larger ), a well a the decreae in reonant frequency (decreae with larger ), but alo how the effect of the varying right-half plane z zero adding 9 o to the aymptotic phae at frequencie above the reonance. Figure 7. Boot Small-Signal Tranfer Function Bode Plot: < <.7, Amp oad Becaue figure 7. above how imulation with the greatet load current correponding to R oad = 8, it alo preent the greatet damping and the reonant peaking i relatively mall. It i alo difficult to dicern the magnitude effect of the right-half plane z zero. We would expect to encounter uch a ituation whenever there are wide variation in the value, but R oad i contant. opyright Raymond. Barrett, Jr. Page

23 Switchmode Boot Power onverter Uing oltage-mode ontrol A Sunam online continuing education coure Figure 7. Boot Small-Signal Tranfer Function Bode Plot: < <.7,. Amp oad Simulation in figure 7. are imilar to thoe of figure 7., but under a minimum load condition correponding to the R oad = 8 value. Figure 7. how the expected variation in gain, a well a the decreae in reonant frequency. The peaking and rapid phae change at reonance are more pronounced and preent control challenge. It i le difficult to dicern the magnitude and phae effect of the right-half plane z zero. Again, we would expect to encounter uch a ituation whenever there are wide variation in the value, but R oad i light, but contant. Figure 7. Boot Small-Signal Bode Plot: =>,. to Amp oad Figure 7. imulation above are performed with contant = and reult in no variation in gain or the reonant frequency, but how the R oad variation caue variation in reonant peaking and phae lope around the reonant frequency, and the frequency variation of the magnitude and phae of the right-half plane z zero above the reonance alo change. We expect to encounter thi ituation with = whenever nearly equal the target value and both are contant, but R oad i changing. opyright Raymond. Barrett, Jr. Page 3

24 Switchmode Boot Power onverter Uing oltage-mode ontrol A Sunam online continuing education coure Figure 7.3 Boot Small-Signal Bode Plot: =>.7,. Amp oad Figure 7.3 imulation above are imilar to thoe of figure 7. but are performed with contant = 7 correponding to the greatet expected difference between and the target value. Again, we encounter no variation in gain or the reonant frequency, but how the R oad variation caue variation in reonant peaking and phae lope around the reonant frequency, a well a frequency variation of the magnitude and phae of the right-half plane z zero above the reonance. Figure 7.4 Boot Small-Signal Bode Plot: =,.7; oad =., Amp Figure 7.4 how the extreme value for the expected variation in gain, reonant frequencie, peaking, the range of phae lope through the reonant frequencie, and the frequency variation of the magnitude and phae aociated with the right-half plane zero above the reonance. The PWM controller mut function over thi entire range of behavior. 8. icrete-time effect of a Pule-Width Modulator (PWM) opyright Raymond. Barrett, Jr. Page 4

25 Switchmode Boot Power onverter Uing oltage-mode ontrol A Sunam online continuing education coure A Pule Width Modulator (PWM) block add a dicrete-time ampling effect with an equivalent T ZOH () Zero-Order-Hold (ZOH) tranfer function within the loop. State-pace averaging prevent u from having an average value for a cycle of the PWM until the cycle i complete. To model the PWM, we utilize the well-known ZOH behavior inide the loop, with an average half-cycle delay at the ampling rate, applied to each ample. Figure 8. Zero-Order Hold Magnitude and Phae A conequence introduced by the ZOH i the magnitude notch introduced by the ZOH at the Nyquit frequency. No magnitude information i available at the Nyquit rate. The magnitude envelope i the hape of a coine with the argument equal to the ratio: ZOH f co [8.] f The ZOH delay behavior alo add additional phae delay in the phae repone for the loop caued by the full-cycle ampling delay. The average delay i one-half-cycle (8 o ) at the ampling frequency implie half that value (9 o ) at the Nyquit frequency. In figure 8., we how the linear phae to the right on a linear frequency cale. The d/d i identically the contant delay. For convenience, the phae i alo hown on a logarithmic frequency cale in the center of the illutration o that phae and magnitude (from the left illutration), can more eaily be aociated. 9. Pole-Zero ompenation for the.5 MHz Boot onverter oop We have een that the particular example i compoed of an behavior with modification for a Nyquit notch and delay, both related to the ampling inherent in the witching/averaging nature of the converion. Thee effect notwithtanding, the network i till dominated by duty-cycle and R oad variation of the Boot mall-ignal behavior. opyright Raymond. Barrett, Jr. Page 5

26 Switchmode Boot Power onverter Uing oltage-mode ontrol A Sunam online continuing education coure opyright Raymond. Barrett, Jr. Page 6 A T T T Z Boot ATTEN PWM [9.] We introduce a Pole-Zero (PZ) compenation cheme a well a the ZOH model, within the PWM a follow: Int P P Z Z Z ZOH PZ ZOH PWM T T T T [9.] The purpoe of the PZ compenator i to introduce an integrator at Int to increae the loop gain at low frequencie to reduce error, and effectively introduce phae hift to reduce the total phae introduced by the Boot behavior at lower frequencie a follow A T T T Z Int Pole Pole Pole z omp Zero z omp Boot ZOH ATTEN [9.] We deign the PZ compenator with a complex-conjugate omp-z zero pair with location % below the lowet expected reonance frequency. We place the PZ double Pole real pole obtained by making Pole =, at a higher frequency and depend on the remaining Int integrator magnitude lope to bring the gain to unity. The relatively mall gain of the Boot function prompt the addition of the ubtantial gain of the integrator to reap the reward of the feedback control for error reduction. But, imply adding the integrator gain alo increae the unity-gain frequency of the open loop and, with that increae in unity-gain frequency, we could alo loe phae margin. Pole Int Zero Z omp Z omp T PZ j [9.3]

27 Switchmode Boot Power onverter Uing oltage-mode ontrol A Sunam online continuing education coure PZ Zero 9 [9.4] ompz o ompz j tan tan Pole We chooe to place the two pole frequencie far above the zeroe. The intention i to ue the phae lead of the zero pair to compenate for the two-pole lag of the varying reonance and defer replacing the Pole two pole lag until above the unity-gain frequency. We employ a tate-variable filter hown in chematic form in figure 9. below to realize the appropriate PZ compenation. The active filter realization i ueful for obtaining the complex-conjugate omp-z zero location and relatively large phae lope that are available a a conequence. The Pole real pole-pair location i eaily realized by the topology and relatively inexpenive amplifier can be employed. Figure 9. State-ariable Filter Schematic for PZ ompenation The State-ariable filter hown in figure 9. above can eaily be tuned uing the following deign equation: R.ec [9.5] Int f Int Int Hz [9.6] Int 8 Int R ec [9.7] Pole Pole Pole opyright Raymond. Barrett, Jr. Page 7

28 Switchmode Boot Power onverter Uing oltage-mode ontrol A Sunam online continuing education coure f khz [9.8] Pole 8 Pole RUnit RQ Pole [9.9] R Zero f Zero RUnit f [9.] Pole f RZero f Pole f Pole.. khz [9.] R Zero 5 Unit R QZero Zero f f Zero Pole R Unit [9.] f Zero RUnit..3 [9.3] Zero f Pole RZero Figure 9. State-ariable Filter PZ ompenation Bode Plot. Open oop Behavior for the.5 MHz Boot onverter We how et of three Bode plot for the Boot converter: firt, in iolation, econd, for the Boot converter with the PZ compenation included, and third, with the ZOH effect included. The et of three plot are preented tarting with the tartup condition at = and R oad = 8 light load, followed by R oad = 8 maximum load, then under the nominal operating condition with =.7 and R oad = 8 light load, followed by R oad = 8 maximum load. opyright Raymond. Barrett, Jr. Page 8

29 Switchmode Boot Power onverter Uing oltage-mode ontrol A Sunam online continuing education coure Figure.a Boot onverter Bode Plot: =, 8 oad Figure.b Boot with PZ ompenation Bode Plot: =, 8 oad Figure.c Boot with PZ & ZOH ompenation Bode Plot: =, 8 oad Figure.c how that the open loop gain croe unity everal time during thee condition, but the phae margin i greater than 5 o in all cae. We have included the cae with = to enure tability at tartup, but the condition are expected to be tranitory. opyright Raymond. Barrett, Jr. Page 9

30 Switchmode Boot Power onverter Uing oltage-mode ontrol A Sunam online continuing education coure Figure.a Boot onverter Bode Plot: =, 8 oad Figure.b Boot with PZ ompenation Bode Plot: =, 8 oad Figure.c Boot with PZ & ZOH ompenation Bode Plot: =, 8 oad Figure.c above how that the open loop gain again croe unity everal time during thee condition, but the phae margin i greater than 5 o in all cae. We have included the cae with = to enure tability at tartup under maximum load, but thee condition are expected to be tranitory, too. Figure.a Boot onverter Bode Plot: =.7, 8 oad opyright Raymond. Barrett, Jr. Page 3

31 Switchmode Boot Power onverter Uing oltage-mode ontrol A Sunam online continuing education coure Figure.b Boot with PZ ompenation Bode Plot: =.7, 8 oad Figure.c Boot with PZ & ZOH ompenation Bode Plot: =.7, 8 oad Figure.c above how that the open loop gain again croe unity at ~khz under thee condition, but the phae margin i greater than 5 o up to that frequency. Figure.3a Boot onverter Bode Plot: =.7, 8 oad Figure.3b Boot with PZ ompenation Bode Plot: =.7, 8 oad opyright Raymond. Barrett, Jr. Page 3

32 Switchmode Boot Power onverter Uing oltage-mode ontrol A Sunam online continuing education coure Figure.3c Boot with PZ & ZOH ompenation Bode Plot: =.7, 8 oad Figure.3c above how that the open loop gain again croe unity at ~khz under thee condition, but the phae margin i greater than 5 o up to that frequency.. Pule-Width Modulator It i common practice to ue a triangular waveform and a comparator to provide the Pule- Width Modulation (PWM) function hown in figure 6.. ariou triangular wave hape have been ued from awtooth to ymmetrical triangle, but all tranlate a voltage at the comparator input into a duty-cycle. We develop the following waveform: Figure. Pule Ocillator at.5mhz opyright Raymond. Barrett, Jr. Page 3

33 Switchmode Boot Power onverter Uing oltage-mode ontrol A Sunam online continuing education coure Figure. The Pule Ocillator with Two elayed Replica We produce a pule ocillator timing reference hown in figure. a a hort duration pule train with the period equal to the T S ampling period. In thi example, the.5mhz pule ocillator ha a T S period of 4 nanoecond and a duration of ~ nanoecond. The two delayed verion hown in figure., one deignated with a T min delay, and the econd with a longer T max delay relative to the T pule pule ocillator reference waveform improve timing accuracy for control purpoe. Implementation that control timing of thee delay i poible with le than % uncertainty of each delay. We define the delay between the pule ocillator and the pule deignated a the horter delay a T on-min for reaon that will alo become apparent. We employ a Flip/Flop a a Pule-Width Modulator (PWM). We initiate all PWM period with the T pule waveform. In each PWM period the initiation by the T pule waveform i called T Start. The period of the PWM i controlled o that it terminate on one of three event, the waveform deignated a the T min waveform, the next T max waveform, or a pule that occur between the two. The T on-min duration i ~5 nanoecond for a minimum duty-cycle of 5/4 = ~.5% but the T off-min = T S - T max duration i ~ nanoecond for a maximum duty-cycle of (4-)/4 = ~75%. To initialize the PWM Flip/Flop, we employ a reet pule hown in figure 6. that i derived from tart-up logic deignating that a logic high mean the implementation i not ready and converely the logic low enable the PWM Flip/Flop. Figure. The Superviory Reet Signal to Permit Timing to ommence The pule ocillator periodically reet a ramp generator and produce typical awtooth waveform a follow: opyright Raymond. Barrett, Jr. Page 33

34 Switchmode Boot Power onverter Uing oltage-mode ontrol A Sunam online continuing education coure Figure.3 Sawtooth Ocillator at.5mhz We obtain the ramp from a current ource charging a capacitor and periodically dicharge the capacitor to zero volt under control of the PWM. Figure.4 Simple Sawtooth Ocillator Schematic The lope i given by: dramp I ramp Slope [.] dt ramp For the I ramp current of milli-ampere and a ramp value of pf, the Slope i: Slope d dt ramp I ramp ramp 3 ec [.] In 4nec T S, we attain a 4 ignal magnitude and the pule ocillator reet the capacitor voltage to zero, tarting a new cycle at.5 MHz. Uing the ame charging/dicharging circuitry, we can cale the behavior of the awtooth waveform by chooing different capacitor value. Similarly, we can alo cale the ramp rate Slope by controlling the charging current. Increaing the current increae the ramp rate, while decreaing the current decreae the Slope. opyright Raymond. Barrett, Jr. Page 34

35 Switchmode Boot Power onverter Uing oltage-mode ontrol A Sunam online continuing education coure Figure.3 Pule Width Modulation (PWM) icuion Waveform We have taken libertie with the magnitude in figure.3 to enable clarity in the dicuion that follow. We have retained the.5m Hz T S timing of a 4nec cycle, but caled the awtooth peak magnitude to to enable dicuion of imilar triangle in the following dicuion. For the example illutrated in figure.3, the lope i: dramp Slope 3 [.4] dt 4nec ec And, indirectly, the Peak peak voltage in thi cae i: dramp Peak TS 3.4 ec dt ec [.5] At the precribed 3/ec lope, we require nec to reach the 3.3 witching point of the comparator et by the 3.3 feedback voltage. It take the full 4nec to reach the Peak peak value. The comparator output waveform duty cycle i: T ON nec [. 6] T 4nec S The comparator waveform i the deired control waveform required for Boot converter witching duty cycle control. The behavior i a conequence of the traight-line relationhip of the right triangle from the origin of the awtooth to the peak, or termination voltage value. The ratio of the to Peak voltage i proportional to the comparator T ON witching time to T S pule ocillator period. We control the comparator duty cycle by etablihing the feedback voltage a a proportion of the peak voltage. onverely, the Slope ratio of voltage provide the mall ignal gain of the modulator. Peak opyright Raymond. Barrett, Jr. Page 35

36 Switchmode Boot Power onverter Uing oltage-mode ontrol A Sunam online continuing education coure We have variation of the PWM cheme that we will employ for the duty-cycle control of the Boot converter. Firt, we cale the awtooth lope changing it peak magnitude, and conequently cale the requiite feedback voltage to a fraction of the voltage hown: [. 7] Slope T Peak We introduce the ame notation for mall-ignal quantitie that we have ued previouly to form: d K v [. 8] PWM S K PWM [. 9] Slope TS We ubtract the large-ignal operating point relationhip given in equation [.8] from the compoite of large and mall-ignal component given in equation [.9], a follow: d d K PWM v K PWM K PWM v [. ] opyright Raymond. Barrett, Jr. Page 36

37 Switchmode Boot Power onverter Uing oltage-mode ontrol A Sunam online continuing education coure. The Boot onverter loed oop Behavior We cloe the Boot converter loop and add a oft-tart ramp to the Ref ignal o that it take ~4ec to reach the deired 8 regulation point. The oft-tart feature i included for two primary reaon: firt, it control the input current reulting from initially charging the output capacitor to the operating voltage, and econd, it allow the feedback loop to avoid aturation and recovery during the tart. Ref tart from, and after ~4ec reache the deired 8 regulation point. A imple R filter enure mooth tranition at the tart and finih of the tranition. Figure. Boot onverter oltage-mode Tracking In figure., we ee the output in blue. The R filter applied to the Ref ignal effectively prevent udden tracking change. Figure. Boot onverter oltage-mode oad urrent We initiated oft-tart ramp to the Ref ignal and include an R OA = 56 value o that the Boot converter tart under load and i delivering.5 Ampere following tartup. At the 5 econd mark, we apply the full R OA = 8 o that the Boot converter i required to deliver. Ampere for econd before returning to half load. We make thi tep load opyright Raymond. Barrett, Jr. Page 37

38 Switchmode Boot Power onverter Uing oltage-mode ontrol A Sunam online continuing education coure change o that we can invetigate the tranient behavior of the loop a a load i applied and removed. Figure. Boot onverter oltage-mode Inductor urrent uring the oft-tart ramp the inductor current follow the derivative of the output voltage and charge the output capacitor, a well a delivering the.5 Ampere to the R OA = 56. The oft-tart ramp to the Ref ignal terminate at ~4 econd and the inductor current decreae upplying the remaining charge to bring the capacitor to a full charge. Thereafter, the inductor current (with it uperimpoed ripple) i providing the average load current. With a load tep at 5 econd, and it removal at 7 econd, the inductor current repond to the controller ignal. Figure.3 Buck onverter oad Effect on the Output oltage The change of R OA from 56 to R OA = 8 and back to R OA = 56 caue a rapid change in the voltage. The change from the.5 Ampere load to a. Ampere load at 5 ec i initially upplied from the charge tored in the output capacitor. A oon a the capacitor voltage decreae though, an error voltage develop and the error amplifier caue the feedback to correct the output error back to the value of 8. opyright Raymond. Barrett, Jr. Page 38

39 Switchmode Boot Power onverter Uing oltage-mode ontrol A Sunam online continuing education coure Similarly, the inductor current change from a. Ampere load back to a.5 Ampere load at 7 ec mut initially be aborbed into the capacitor, charging it to a higher voltage. A oon a the capacitor voltage increae, the error voltage developed at the error amplifier caue the feedback to correct the voltage back to the target value of 8. Figure.4 Boot onverter oad Error oltage The error amplifier caue the feedback to correct the voltage error to zero. Error i correlated with change in the tate variable, Becaue the voltage error i the ource of feedback, the voltage error i aociated with change in the econd tate variable, the I inductor current. We ee that there i a tracking error difference during the firt ~4 econd that i required to bring the Boot converter to it required average I inductor current. at = 8 with the R OA = 56 value. At 5 econd and again at 7 econd there are the change in R OA and that caue new error voltage to be developed and change in average I inductor current. Figure.5 Boot onverter Switching oltage Waveform The Boot converter cloed-loop repond to the error ignal by temporarily increaing or decreaing the average voltage difference acro the inductor and conequently it average current. At the increae in load current and conequent decreae in output voltage, the opyright Raymond. Barrett, Jr. Page 39

40 Switchmode Boot Power onverter Uing oltage-mode ontrol A Sunam online continuing education coure feedback loop repond by momentarily cauing a longer PWM waveform to change to a higher inductor voltage to increae the inductor current. ikewie, at the decreae in load current and conequent increae in output voltage, the feedback loop repond by momentarily forcing a lower average inductor voltage to increae the inductor current. Under udden load increae, a -feedback ignal to the PWM increae o that the PWM the duty cycle become longer for everal cycle, hence the inductor voltage i at it maximum value equal to the difference between the upply voltage and the voltage for a longer time. Figure.6 Boot onverter PWM Switching oltage Waveform at oad ecreae Under udden load decreae, a -feedback ignal to the PWM decreae o that the PWM duty cycle become horter for everal cycle, hence the inductor voltage i at it minimum value equal to the - voltage for a longer time. uring the udden application and the udden removal of the tep load, the error may be ufficient to caue T Max or T Min duty cycle in repone and the maximum rate of change of inductor current occur. Becaue the duty cycle value are limited at the maximum or minimum for ome number of cycle, the loop ha no feedback control and i operating open-loop for a hort time. uring the open-loop interval, the integrator till attempt to exert control over the PWM, but ucceed only in accumulation a ignal that mut be unwound before the duty-cycle control i again valid. More complex controller can be contructed to limit the integrator from accumulating uch value and haten the recovery time. 3. Boot onverter with Feed-Forward and Feedback ontrol Mechanim We introduced the requirement for optimal feedforward control in equation [5.3], but found that a right-half plane pole wa required. We ignore the frequency dependence and introduce a ub-optimal feedforward control intead. In figure 3., we illutrate the combination in the Block iagram Schematic, a follow: opyright Raymond. Barrett, Jr. Page 4

41 Switchmode Boot Power onverter Uing oltage-mode ontrol A Sunam online continuing education coure Figure 3. Boot onverter Small-Signal Feed-Forward/Feedback ontroller v d ff [3.] In place of equation [5.3], we introduce equation [3.], a relationhip to reduce the mallignal v upply perturbation effect on the output voltage. We how that the PWM can be modified to produce the correction directly. To develop the modification we reviit the PWM chematic and waveform. Figure 3. Modified Simple Sawtooth Ocillator Schematic The lope i modified to be given by: dramp I ramp GSope Slope [3.] dt ramp ramp opyright Raymond. Barrett, Jr. Page 4

42 Switchmode Boot Power onverter Uing oltage-mode ontrol A Sunam online continuing education coure Figure 3. Pule Width Modulation (PWM) icuion Waveform The large-ignal duty-cycle ignal produced by the PWM i: Ramp [3. ] G T Slope We include mall-ignal uperpoition a follow: Slope S S Ramp v d [3. ] G T v Ramp v d G SlopeTS v v [3. 3] Performing long diviion in the firt term and eliminating higher-order term, we have: Ramp v d v G SlopeTS [3. 4] We ubtract the large-ignal contribution a follow: Ramp v Ramp d v G SlopeTS G SlopeTS [3. 5] Ramp v d v d d ff G SlopeTS [3. 6] opyright Raymond. Barrett, Jr. Page 4

43 Switchmode Boot Power onverter Uing oltage-mode ontrol A Sunam online continuing education coure The reulting PWM relationhip provide a capability to add mall-ignal feedforward control uing the v term to the feedback control uing the v Ref term. Ramp d vre f [3. 7] G T Slope S d ff Ramp v [3. 8] G T Slope S Figure 3.3 Maximum and Minimum Feedforward Slope ependence on In figure 3., we ee that the addition of the lope dependence effectively modifie the dutycycle with a contant -feedback voltage value. Although we cannot infer that the magnitude i optimum, it doe alter the PWM duty-cycle in the correct direction to provide ome feedforward control. 4. Boot onverter Feed-Forward Improvement In figure 4., we how the waveform aociated with the introduction of a khz inuoidal perturbation added to the previouly quiet upply a follow: Figure 4. Boot onverter with khz A Perturbation opyright Raymond. Barrett, Jr. Page 43

44 Switchmode Boot Power onverter Uing oltage-mode ontrol A Sunam online continuing education coure In figure 4., we have added the khz Sinewave to the nominal input without the benefit of feedforward and obtained the repone hown below: Figure 4., I, and I oad Repone to with, khz Sinuoid Perturbation A hown in figure 4., the Boot converter running cloed-loop convert the input to the output with the ratio determined by the feedback duty-cycle value. The input perturbation i 4 peak-to-peak, and the repone at the output i ~ peak-to-peak. The Boot converter generate the nominal 8 output from a nominal input, o we hould expect an open-loop ytem to generate a diturbance with the ratio 8/ * 4 peakto-peak or > 8 peak-to-peak diturbance diturbance about 4X to ~ peak-to-peak. The cloed-loop feedback i reponible for reducing the diturbance about 4X to ~ peak-topeak. The reult i conitent with a loop-gain magnitude at khz of ~4, and i about all that can be expected of the compenated feedback loop we have employed. Figure 4., I, and I oad Feedforward Repone to with, khz Sinuoid Perturbation We add the modification to the PWM Slope to make the Slope current equal G Slope *, and obtained a awtooth with modified Slope and conequent feedforward a hown in figure 4. above with a modet reduction in the diturbance on the output from about 4X at ~ peak-to-peak about 4X to ~. peak-to-peak. The feedforward control i perhap uboptimal, but nonethele effective. opyright Raymond. Barrett, Jr. Page 44