Small-signal model of BumbleBee output voltage controller for DC/DC Converter. Stability analysis of BumbleBee method.

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1 Jacek KACZMAREK Politechnika Kozalińka, Wydział Elektroniki i Informatyki Small-ignal model of BumbleBee output voltage controller for DC/DC Converter. Stability analyi of BumbleBee method. Strezczenie. Artykuł przedtawia model małoygnałowy i analizę tabilności nowej cyfrowej metody terowania przetwornicami DC/DC BumbleBee (Trzmiel). W metodzie tej napięcie wyjściowe przetwornicy tabilizowane jet pośrednio, poprzez utrzymywanie na tałym poziomie ilości energii elektrycznej zgromadzonej w elementach L i C przetwornicy. Przewidywane dobre właściwości metody zotały potwierdzone wynikami badań ymulacji komputerowych oraz tetami przeprowadzonymi na zbudowanej przetwornicy Buck terowanej przez proceor TMS30F8335, przeznaczony do zatoowań w energoelektronice. (Małoygnałowy model regulatora napięcia przetwornicy DC/DC projekt BumbleBee. Analiza tabilności metody). Abtract. During two previou MIXDES conference author preented new method of controlling DC/DC converter, which ue law of conervation of energy. Developed theory wa confirmed by both imulation reult and meaurement conducted on the contructed prototype. Now in the following paper, the mall-ignal model of the propoed controller i preented along with proof of tability for preented method of tabilization of converter output voltage. In the paper the reult of meaurement that allow determination of the accuracy of the model are alo preented. Słowa kluczowe: przetwornica DC/DC, cyfrowe terowanie przetwornicą DC/DC, model małoygnałowy, tabilność metody. Keyword: circuit tability analyi, converter work model BumbleBee controller, digital power control, DC/DC. Introduction The idea behind BumbleBee method aume, that the output voltage of the DC/DC converter can be tabilized indirectly, by tabilizing amount of energy accumulated in the DC/DC converter. Earlier paper preented by author []-[8] contained reult of imulation of uch circuit and the prototype characteritic meaurement [9-11]. Meaurement reult were agreeable with imulation and both confirmed author aumption. Up till now the analytical proof of tability of propoed olution wa not preented. In the paper author preent mall-ignal controller model, which allow invetigation of tability of propoed control method. Thank to BumbleBee controller model, analytical decription of tate variable of convertercontroller ytem in both Laplace and time domain wa derived. For the mathematical decription of ideal Buck converter it popular mall-ignal model wa ued, derived by averaging and linearization of differential equation that decribed operation of ideal converter during ON and OFF phae. Small-ignal model of Buck converter Ideal Buck converter i preented in the figure below: Figure 1 Ideal Buck converter And it model i hown in the next figure: Thi mall-ignal model wa choen becaue tranmittance function of ideal Buck converter are characterized by Vc voltage phae hift of 180 degree, which in turn caue tightening of the tability condition. In the preented mall-ignal model of Buck converter the input ignal are input voltage and duty ratio of PWM ignal. The output ignal are voltage drop on the capacitor and inductor current value. Symbol ^ over the variable name mean that it i a mall-ignal variable. Tranmittance function which define dependencie between input and output of the converter circuit are given below: (1) () (3) (4) vc ( hvc_d( VIN 1 L 1 L C il ( VIN 1 C hil_d( L 1 L C vc ( hvc_( D ( 1 il ( hil_( ( D L 1 L C 1C L 1 L C Converter and it model operating point i defined by contant component of each ignal during teady tate: VIN contant component of the ource voltage, converter load reitance, D duty ratio of PWM ignal which control K1 witch. Decription of dependencie between input and output ignal in mall-ignal model of buck converter i hown in the following equation: Figure Small ignal model of Buck converter (5) vc ( hvc_( ( hvc_d il( hil_( ( hil_d 3 PRZEGLĄD ELEKTTECHNICZNY (Electrical Review), ISSN , R. 86 NR 11a/010

2 Principle of Bumblebee controller operation In the propoed method tability of output voltage i enforced indirectly, by tabilizing the amount of energy accumulated in L and C element of the converter [-6]. The method wa teted by computer imulation along with meaurement of the prototype [7-11]. In the baic verion of the dicued method, DSP proceor pecialized for energo-electronic application TMS30F8335 control the operation of Buck converter baed on the following equation [9-11]: (6) Vin ( nt ) Vin( nt ) Vc( nt ) t L Vin( nt ) Il( nt ) t on on T... Introduction of mall-ignal variable for converter load reitance wa abandoned, becaue aumed mall-ignal model of Buck converter uppoe that thi value doen t change. The chematic diagram of the block model of BumbleBee controller i hown in the Figure 3. Small-ignal model, Buck converter Bumblebee controller Small-ignal model of the ytem: Buck converter controlled by BumbleBee controller i hown in the picture below: where: Vin(nT) ource voltage value, Vc(nT) - voltage drop on the C capacitor, Il(nT) - current flowing through inductor, - required value of output voltage Vc, - load reitance, L inductance of converter inductor, t on ON phae duration in ingle work cycle, n number of the converter work cycle, T converter cycle duration. Converter work cycle i atime range < n T, (n+1) T> Right ide of the equation (6) define how much energy hould converter take from the voltage ource during ON phae in a ingle work cycle. In the preented verion of thi method, it i a contant value, equal to the amount of energy taken by converter load during teady tate in one converter work cycle. Left ide of the equation decribe the amount of energy that the converter may acquire during ON phae time ton taking into account voltage value on the input and output of the converter and the value of inductor current [],[5],[6]. Above equation i olved at the beginning of each work cycle. The variable that i etimated i ton time which determine duty ratio of the PWM ignal which control the converter operation [4],[7], [10], [11]. Small-ignal model of the Bumblebee controller By introducing linearization and mall-ignal variable into equation (6) for Vin, Il, ton and Vc ignal, and after taking into account dependency: t on d T, and that in teady tate VC = a mall-ignal decription of controller in time domain wa obtained: (7) d D K il K vc K v in cont il vc Figure 4 Small-ignal model of the Buck converter BumbleBee controller ytem The difference in interpretation between inductor current for mall-ignal model of converter and BumbleBee controller wa overlooked on purpoe. Converter model calculate the average value of inductor current, and the BumbleBee controller aume that it the minimal value of inductor current in one converter work cycle. Thi difference can be compenated by introducing an amplification block with contant gain between the output of converter inductor current and input in BumbleBee controller. The gain value of uch block i aumed in uch a way, that during teady tate for aumed operating point the average value would be converted to a minimal value. The circuit with uch correction wa analyzed mathematically and the reult confirmed that the influence of thi implification wa negligible. Thi implification i introduced in order to obtain clearer image of preented material. Mathematical decription in Laplace domain i decribed by the following equation ytem: (9) vc ( hvc_( ( hvc_d( il ( hil_( ( hil_d( d D K il K vc K cont il vc Figure 3 Block chematic of mall-ignal BumbleBee controller Value of coefficient (7) becaue of the editorial iue are hown at the end of article in Table 1. In Laplace domain controller i decribed by following dependency: (8) Dcont d( K il il( Kvc vc ( K ( It i compoed of dependencie which decribe mallignal Buck converter model (5) and equation decribing mall-ignal BumbleBee controller model in Laplace domain (8). Equation ytem (9) allow computation of tate ^ variable of the converter vc, il and value of duty ratio for PWM ignal d which control the converter. By ubtitution of dependencie decribing tranmittance function of converter and BumbleBee controller model, thee ignal can be rewritten in form of complex rational function. ^ PRZEGLĄD ELEKTTECHNICZNY (Electrical Review), ISSN , R. 86 NR 11a/010 33

3 (10) (11) (1) L0 VC vc( M0 M1 M L0vc M0 M1 M M3 ( 3... L0il L1il il(... 3 M0 M1 M M3 L0il L1il ( M0 M1 M L0d L1d Ld... 3 M0 M1 M M3 L0d L1d ( M0 M1 M Value of coefficient in equation (10), (11), (1) are hown in Table. Analyi of tability of mall-ignal model: Buck converter Bumblebee controller Function (10), (11), (1) decide about the tability of the ytem with cloed feedback loop decribed by equation ytem (9).Thoe function (10), (11), (1) can be rewritten in following form: (13) x x( Hxx_cont( Hxx_( ( where: xx i a variable name: vc, il, d Hxx_cont( contant independent from the input ignal, reponible for the phae trajectory of the ytem. Hxx_( tranmittance function decribing influence of the input ignal v c( on the invetigated ignal. By analyzing the olution of equation ytem (9) one may come to the following concluion: Denominator of Hxx_cont( function are identical, The value of M0 coefficient i alway zero, Denominator of Hxx_( function are identical, Between denominator of Hxx_cont i Hxx_( function the following dependency can be derived: (14) denomhxx_cont denom Hxx_( where: denomhxx_cont denominator of Hxx_cont function, denomhxx_( denominator of Hxx_( function, The poition of tranmittance function Hxx_( pole i decribed by dependency: (15) M0 M1 M 0 That why three pole decide about the tability of the ytem. Firt pole in function Hxx_cont S0=0, and pole S1 and S which occur imultaneouly in tranmittance function Hxx_( i Hxx_cont. S1 and S pole are quare root of equation (15). Occurrence of S0 pole for Hxx_cont function in the middle of coordinate ytem doen t caue intability of the ytem, becaue Hxx_cont function aren t tranmittance function. They are not dependent on any tate variable which decribe invetigated ytem. They decribe (in the time domain) waveform of xx ignal from the moment of turn-on until the moment when teady tate occur, when the operating point doen t change. Invere tranformation of complex function k/ in time domain equal k and i not dependent on time. S0=0 pole decide about DC component value of following ignal: v c, î l, d during teady-tate. The imaginary part of S0 pole i alway equal to zero and doen t caue appearing of component variable in form of in(ω t+φ) function in time domain. Therefore about the tability of whole ytem decide the poition of S1 and S pole. Sytem will be table if the real part of S1 and S pole will be le than zero in Laplace domain. Thi i decribed by below dependency: (16) Re( S 1) 0and Re( S) 0 Analyi of condition (16) do not allow for the tatement that the ytem will be alway table, independently of the value of element that the converter conit of, and the value of coefficient which decide about it operating point (for example:, VIN,, T etc.). It i neceary to include value of thoe element to check the poition of S1 and S pole. Alo uage of Hurwitz criteria doen t give ynonymou anwer, that the ytem will alway be table, independently of converter parameter and foreeen area of change of operating point. That why author teted if the condition (16) i fulfilled for Buck converter of all important converter manufacturer (Linear Technologie, Maxim, Texa Intrument etc.). Thi condition wa alway fulfilled in whole operating area foreeen by the manufacturer for the particular type of converter. In propoed BumbleBee method, a in uage of PI and PID controller it i eential to invetigate if change in operating point do not caue the intability. Verification of accuracy of mall-ignal model: Buck converter Bumblebee controller Accuracy of the model wa verified on bae of comparion between imulation reult of Buck converter BumbleBee controller ytem with preented mall-ignal model of ame ytem. During the imulation of Buck converter Bumblebee controller ytem ideal converter wa modeled by differential equation during ON and OFF phae, and BumbleBee controller calculated duration of ON phae ton baed on equation (6). For imulation a Matlab-Simulink package wa ued. Author preent influence of load reitance on accuracy of the model, becaue all the other factor: change of ource voltage value, reference voltage, value of C and L element and operating frequency of the converter doen t change the accuracy of the model. The tet ignal wa a teep function of ource voltage value change decribed by following dependency: (17) Vin( t) 4 1 [ 1 Heaviide( 0,01 t)... Heaviide( 0,005 t)] where: Heaviide(k +t) function Heaviide. Tet were conducted for Buck converter with following parameter: referential voltage =1V, L=100μH, C=470μF, and operating frequency 100kHz. For the nr 1 tet, =0,5Ω, for the econd one it raie four time up to Ω. Number 1 tet for load reitance =0,5Ω i hown in figure 6. Number tet for load reitance =Ω i hown in figure PRZEGLĄD ELEKTTECHNICZNY (Electrical Review), ISSN , R. 86 NR 11a/010

4 Table 1. Coefficient in equation (7),(8) D cont K il K vc K D VIN D ( ) T ( T VIN T ) D D D ( ) Table Coefficient in equation (10), (11), (1) L0 vc [D ( ) ] M0 0 M1 M M3 L0vc M0 M1 M ( D) ( T) C ( T) C C ( T) 1 D ( D) ( T) C ( T) C ( T) L0 il D ( ) L1 il C (D ( ) L0il L1il L0 d L1 d L d L0d L1d 1 D 1C D (D ( ) ) (D ( ) ) C (D ( ) ) 1[ D ( D) D ( T) )] 1[ C D D ( T) )] Vertical intermittent line mark the moment when the teep change of ource voltage occur. Plot 1 how output voltage of the converter Vc(t). Plot how inductor current il(t). Plot 3 how duty ratio of PWM ignal which control operation of converter d(t). Red line mark imulation reult of Buck converter BumbleBee controller ytem. Blue line mark imulation reult of mall-ignal model of Buck converter and BumbleBee controller. Figure 5 Source voltage Vin(t) Concluion Up till now tability of the method and predicted propertie of tabilized output voltage were confirmed by reult of computer imulation and meaurement reult of converter prototype which controller ha been deigned according to author algorithm [9-11]. Through the ue of mall-ignal model of BumbleBee controller developed by author, tability of propoed olution ha been proved, through analyze of pole location. Analytical computation of tranmittance function pole poition interchangeably ettle, that the method i table for the Buck converter that are ued up to date (value of element and range of operation point change. Linearized model of thi controller, depite it implicity reflect very preciely the waveform of output voltage Vc(t). Waveform with much higher dynamic of change i.e. inductor current il(t) and ignal controlling the converter d(t) are modeled with leer accuracy. It becaue of the bigger change that occur in thee ignal during unteady tate, where the influence of linearization of equation (6) become viible. PRZEGLĄD ELEKTTECHNICZNY (Electrical Review), ISSN , R. 86 NR 11a/010 35

5 Accuracy of the model, epecially for inductor current i decreaing with the value of converter load reitance. It i becaue of the bigger difference in inductor current between teady and unteady tate, but doen t influence the modeling of output voltage Vc(t) in a clearly viible way. The imple contitution of BumbleBee controller allow it future ue in controlling the converter. It will allow more than tenfold increae in the operating frequency for a DSP controlled converter while maintaining very good propertie of output voltage. At thi moment author conider taking into account the change of the converter load reitance in controller model. Figure 6 Number 1 tet for load reitance =0,5Ω REFERENCES [1] Ramanarayanan V., Reource Centre of the Power Electronic Group, SMPS. Coure Note by V.Ramanarayanan. Switched Mode Power Converion Coure Note (E6 1) file E6105.pdf [] Kaczmarek J., Mazurek A., New concept of dc/dc converter digital control baed on law of conervation of energy project Bumblebee. Proceeding of the 14th International Conference Mixed Deign of Integrated Circuit and Sytem, 1 (007), [3] Kaczmarek J., Mazurek A., Comparion of claic dc/dc converter with converter equipped with analog-digital regulator baed on law of conervation of energy (Bumblebee type). Proceeding of the 14th International Conference Mixed Deign of Integrated Circuit and Sytem, 1 (007), [4] Kaczmarek J., Mazurek A., Compenation of calculation duration On converter output voltage in digitally controled converter baed on law of conervation of energy- project Bumblebee. Proceeding of the 14th International Conference Mixed Deign of Integrated Circuit and Sytem,1 (007), [5] Kaczmarek J., Mazurek A., Nowa idea terowania przetwornicami DC/DC. VI Krajowa Konferencja Elektroniki, (007), [6] Kaczmarek J., Mazurek A., Nowa koncepcja cyfrowego terowania przetwornicami DC/DC.. VI Krajowa Konferencja Elektroniki, (007), [7] Kaczmarek J., Mazurek A., Wpływ czau obliczeń algorytmu w nowej metodzie terowania napięciem wyjściowym przetwornicy DC/DC. VI Krajowa Konferencja Elektroniki, (007), [8] Kaczmarek J., Mazurek A., Porównanie nowej metody terowania przetwornicami DC/DC z metodą terowania wykorzytującą kontrolę prądu cewki. VI Krajowa Konferencja Elektroniki, (007), [9] Kaczmarek J., Mazurek A., Badanie prototypu cyfrowo terowanej przetwornicy DC/DC z wykorzytaniem algorytmu Bumblebee, VII Krajowa Konferencja Elektroniki, (008), [10] Kaczmarek J., Mazurek A., Badanie prototypu cyfrowo terowanej przetwornicy DC/DC z wykorzytaniem algorytmu Bumblebee, Elektronika - kontrukcje, technologie, zatoowania,9 (008), [11] Kaczmarek J., Mazurek A., Meaurement reult of BUCK converter prototype digitally controlled by algorithm uing law of conervation of energy Project Bumblebee, Proceeding of the 15th International Conference Mixed Deign of Integrated Circuit and Sytem, 1 (008), Autor: mgr inż. Jacek Kaczmarek, Politechnika Kozalińka, Wydział Elektroniki i Informatyki, ul. Śniadeckich, Kozalin, Kaczmarek.j@gmail.com; Figure 7 Number tet for load reitance =Ω 36 PRZEGLĄD ELEKTTECHNICZNY (Electrical Review), ISSN , R. 86 NR 11a/010