modeling and design rules of a two-cell buck converter under a digital pwm controller

IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 23, NO. 2, MARCH 2008 859

Modeling and Design Rules of a Two-Cell BuckConverter Under a Digital PWM Controller

Abdelali El Aroudi, Member, IEEE, Bruno Gérard Michel Robert, Angel Cid-Pastor, Member, IEEE, andLuis Martínez-Salamero, Senior Member, IEEE

Abstract—In this paper, we give a detailed analytical study ofa two-cell dc–dc buck converter. We analyse the dynamics of thesystem by using a discrete time modelling approach and consid-ering a digital controller. This controller includes a dynamic com-pensator in the form of a digital integrator for the output variableregulation. The discrete time model for the whole system is used topredict the instability of the system when some design parametersare varied. To facilitate the design, an approximated closed formdiscrete time model is derived in the form of a recurrence equationwhich accurately describes the dynamical behavior of the system.The Jury test is applied to the characteristic polynomial in orderto obtain stability boundaries in the design parameter space. Somedesign rules to obtain optimal transient behavior are also given.Numerical simulations and experimental measurements confirmthe theoretical predictions.

Index Terms—Bifurcations, discrete time model, digitalpulsewidth modulation (PWM) control, stability analysis, two-celldc–dc converters.

I. INTRODUCTION

OBTAINING accurate mathematical models for powerelectronics circuits is a traditional challenge for the

power electronics engineers. There are many efforts devotedto this research area (see [1]–[18] and references therein).However, many of the reported analysis on power electronicscircuits are based on averaging techniques [8]. Averaging isonly an approximated procedure to obtain the low frequencybehavior of the actual switching model. The averaged modelwas found to fail in predicting many of fast scale instabilitiessuch as sub-harmonic oscillations and chaotic behavior. Thisshortcoming is due to the elimination of the main nonlinearity,the switching action, of the real system by averaging the statevariables during one switching cycle. Recent findings showthat elementary switching circuits like dc–dc buck converterunder pulsewidth modulation (PWM) voltage mode control[9] or current programmed inverter [30]–[32] and dc–dc boostconverter under current programmed control [10] are proneto subharmonic oscillations that can not be understood and

Manuscript received June 27, 2007; revised September 26, 2007. This workwas supported in part by the Spanish Ministerio de Educación y Ciencia underGrant TEC-2004-05608-C02-02. Recommended for publication by AssociateEditor F. L. Luo.

A. El Aroudi, A. Cid-Pastor, and L. Martínez-Salamero are with the Depar-tament d’Enginyeria Electrònica, Elèctrica i Automàtica (DEEEA), Grupo deAutomática y Electrónica Industrial (GAEI), Universitat Rovira i Virgili, Tar-ragona, Spain (e-mail: [email protected]).

B. G. M. Robert is with the Laboratoire CReSTIC, Université de ReimsChampagne-Ardenne, Reims, France (e-mail: [email protected]).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TPEL.2007.915162

predicted by using the averaged model. This is due to the factthat when describing the system with this model, it is assumedthat the stationary operating points are equilibrium points whenactually they are limit cycles.

The more suitable tool for studying the stability of limit cy-cles in switching converters is the discrete time model. This ap-proach was applied successfully to explain the nonlinear andcomplex behavior that can occur in elementary and paralleledpower electronic circuits [11]. Nowadays, there are many worksdealing with nonlinear behavior in elementary stand alone andparalleled dc–dc converters, dc-ac inverters and ac-dc pre-regu-lators. Among others, we can quote [12]–[14] and [15] and ref-erences therein.

It has to be pointed out that discrete time modelling was an im-portant field of research in the early years of modelling of dc–dcswitchingconverters.Since thepioneeringworksofPrajoux etal.[16] in the early seventies, extended by Lee and Yu [17] and laterby Verghese et al. [18], this technique became an efficient tool todescribe the dynamic behavior of these converters. However, dueto the technological limitationsat thatmoment,nopracticaluseofsuchmodelswasdone.Now,due to thehigh performanceofmanydigital controllers, discretemodels have become increasingly im-portant. On the other hand, multi-cell converters have been de-veloped to overcome shortcomings in solid-state switching de-vice ratings so that to reduce their stress [19]. Because distributedpower sources are expected to become increasingly prevalent inthe near future, the use of such converters to control the currentand voltage output directly from renewable energy sources willprovide significant advantages because of their fast response andautonomous control. Additionally, they can also control the ac-tive and reactive power flow from an utility connected renewableenergy source. Recently, a two-cell buck converter has been alsoproposed for efficient wide-bandwidth envelope tracking in RFpower amplifiers [20]. In [19], the authors use a sliding modeapproach to design a controller for this converter. The resultingnominal operation of the overall system is a limit cycle which cannotbestudiedby the linearizedaveragedmodel [21], [22]. Inspiteof their topological simplicity, two-cell buck converters can ex-hibit complex dynamics as it is shown in this paper. In fact, it wasalready shown that a two-cell dc–dc converter under AnaloguePWM (APWM) control can present a Hopf bifurcation [23], [24],and a border collision bifurcation under a Digital PWM (DPWM)control without PI controller for the output variable [25].

The objective of this work is to describe a DPWM controlledtwo-cell buck converter with PI compensator from the dynamicbehavior point of view. We give the different operating modesand their state space equations in a general form. Then, we de-rive a discrete time model which is able to predict accurately the

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860 IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 23, NO. 2, MARCH 2008

dynamical behavior of the system. Unusually, the controller de-sign in our paper will not be based on averaged technique. Thisaveraged model makes impossible the prediction of the fast dy-namics due to the high frequency switching. Instead, we willuse the discrete time model for the system design. The closedloop model is then used to get some useful design rules for thesystem. These rules can be used for ensuring stability of thenominal periodic regime and good performances in the transientstate.

The remainder of the paper is organized as follows. Section IIwill deal with the system description. In Section III, we obtainthe discrete time model for the two-cell dc–dc buck converter.A simplified model and its Jacobian matrix are derived in closedform. The conditions for stability are also given in the samesection. An experimental prototype is presented in Section IV. Adesign example is studied in Section V where both experimentalmeasurements and numerical simulations are shown to confirmthe theoretical study of stability in terms of some suitable designparameters. Some specifications about optimizing the transientdynamics are commented in Section VI. Our conclusions aredrawn in the last section.

II. SYSTEM DESCRIPTION

A. Power Stage Description

The studied converter is shown in Fig. 1. It is based on thewell known dc–dc buck converter. In order to reduce the stressof the semiconductor devices, voltages across the cells have tobe balanced. Therefore, the usual converter is modified by usinga controlled DC voltage source. This source is done by a simplecapacitor whose charge and discharge currents are controlled byindependent switches ( and ). Each pair formed by a switch

and a diode is activated in a complementary manner insuch a way that when is ON, is OFF and vice versa. Also,we will suppose that there exist a phase shift of between thetwo command signals in order to obtain optimum waveformsfor the inductor current [21]. Namely, the inductor current fre-quency is twice the switching frequency in this case. In orderto simplify the analysis, we will consider a passive resistive-in-ductive load. Note that if we consider a capacitor filter at theoutput, the ripple of the output capacitor voltage will be verysmall in such a way that this state variable could be consideredpractically constant. In this case, we could substitute the outputparallel RC network by an equivalent constant voltage and thevalue of the capacitor would have, practically, no effect on thefast dynamics (inductor current and flying capacitor) providedthat a small ripple at the output is guaranteed.

For each configuration, the system equations are affine andtime invariant in the form of , where isthe vector of state variables formed by the inductor currentand the capacitor voltage and are the system ma-trix and input vector, respectively, during each phase, and theoverdot stands for derivation with respect to time. There arefour different valid configurations depending on the state of theswitches and and diodes and . These are (Fig. 1):

• Configuration , (OFF, ON): During this configuration,the capacitor is discharged. The inductor is discharged ifthe averaged output voltage is greater than , while

Fig. 1. Schematic circuit diagram of a two-cell dc–dc Buck converter. (a) Cir-cuit diagram, (b) (OFF, ON) Configuration, (c) (ON,ON) Configuration, (d)(ON, OFF) Configuration, (e) (OFF, OFF) Configuration.

it is charged if it is lower. Matrices and for this con-figuration are:

(1)

• Configuration , (ON,ON): During this configuration, thecapacitor charge is maintained and the inductor is charged.Matrices and for this configuration are:

(2)

• Configuration , (ON,OFF): During this configuration,the capacitor is charged. The charge or discharge of the in-ductor depends on the same conditions as for configuration

. Matrices and for this configuration are:

(3)

• Configuration , (OFF, OFF): During this configuration,the inductor is discharged while the capacitor charge ismaintained. Matrices and for this configuration are:

(4)

B. Operating Modes

There are different operating modes for the system dependingon the duty cycles of the command signals and the phase shiftbetween them. However, it can be shown that assuming contin-uous conduction mode and phase shift , there are basicallythree operating modes for the system that can be summarized asfollows (Fig. 2).

• Mode : The switching sequence is. This mode appears for duty cycles between 0.5 and 1.

• Mode : The switching sequence is. This mode appears for duty cycles less than 0.5.

EL AROUDI et al.: MODELING AND DESIGN RULES OF A TWO-CELL BUCK CONVERTER 861

Fig. 2. Steady-state operation of the system under normal periodic operation for Mode 1. Left: state variables and control signals waveforms. Right: limit cyclein the (v ; i ) state space.

• Mode : The switching sequence is . Thismode appears when both duty cycles are exactly equal to1/2.

Fig. 2 shows the waveforms of the system under steady-stateperiodic operation for Mode 1. Irrespectively of the operatingmode, the switched model of the two-cell dc–dc buck convertercan be written as follows:

(5)

It is possible to obtain an averaged model for the state variablesof the system during a switching cycle. The averaged model hasthe same expression as for the switched model but with the com-mand signal and substituted by their corresponding dutycycles and and the state variables by their correspondingaveraged values during the same cycle.

C. Digital PWM Controller

Usually, the controller of a power electronics dc–dc con-verter is designed based on a small-signal model of thesystem [5]–[7]. By linearizing the averaged model, the con-trol-to-output transfer function matrix for the open loopsystem is obtained:

(6)

where and for all variables means Laplacetransform of a small perturbation near . We can observe thatthe voltage loop is a pure integrator while the current loop isa low-pass filter. Therefore, a zero static error can be achieved

without inserting an integrator in the voltage loop. However, inorder to get a zero static error in the current loop, the controllermust incorporate an integrator in this loop. In this case, an extrastate variable , which is the integral of the current error, appearsin the system dynamics.

We stress now a first specificity of the present study relatedto the digital controller. A digital controller (DPWM) being im-plemented, the duty cycles of the command signals andare determined directly from the sampled state variablesand . Their reference values are and and beingthe digital integral of the output error. The expressions of theduty cycles in the th cycle are given by

(7a)

(7b)

where and are feedback coefficients with appropriate di-mensions. is the time constant of the digital integrator. Let usdefine a new feedback constant . is the duty cyclefor the command signal during the th switching cycle. Notethat and are only equals during the steady state but aredifferent during the transients. Based on the forward Euler ap-proximation of the digital integrator, the dynamics of is gov-erned by the following recurrence equation [26]

(8)

is the switching (sampling) period and is thecurrent error. Fig. 4 shows the block diagram of the controllerdescribed above. Note that the plant is a continuous time systemwhile its inputs (duty cycles) are discrete-time as they are de-fined cycle by cycle. This makes the whole system a hybrid dy-namical system whose behavior is difficult to analyse [27]. Adiscrete time formulation would help to overcome this problem(see Fig. 3).

III. DISCRETE TIME MODELLING APPROACH

A. Open Loop: Obtaining the Discrete Time Model of thePower Stage

An accurate study of the complex behavior of a switched dy-namical system can be carried out by constructing a nonlinearmodel which conserves the main nonlinearity of the system, i.e.,


Fig. 3. Block diagram of the DPWM used for the two-cell dc–dc buck con-verter.

Fig. 4. Command signals of S and S with a centred pattern, showing theswitching sequence for Mode M .

the switching action. In this section, we will present a system-atic way to obtain the nonlinear discrete time model for thetwo-cell dc–dc buck converter studied in this paper. Withoutloss of generality, we will suppose that the converter is workingin Mode . Then the nominal periodic behavior is character-ized by toggling among three different configurations duringfour subintervals within a switching cycle. Working in Mode

, the following switching sequence takes place for :.

A second specificity of the use of a digital controller isthe need for avoiding the disturbances of the sampling by theswitching noise of the power plant. For this reason, in a practicalsystem, the pulses of the DPWM controller are centred duringthe switching period in order to move away the switching in-stants of the sampling times. With a centred pulse, the sequencefor becomes: duringfive different subintervals (Fig. 4). Note that for an analogimplementation of the PWM controller, there is no need to usea centred pulse pattern.

Working in Mode , the two-cell converter can be describedby the following switched mode model:

(9)

During each subinterval the system equations are affine and timeinvariant. In this case, the solution during each phase intervalcan be obtained in closed form and is written as follows:

(10)

where is an appropriate initial time instant. The mapping thatrelates the state variables at the beginning of an entire cycleto is based on stroboscopic sampling at the beginning ofeach period to obtain .

B. About a Specificity of the Association of a Centred PWMWith a Digital Controller

For the digital PWM controller, the state variables are sam-pled each switching cycle to determine the duty cycles and

with . The expressions forcan be written as follows:

(11)

where stands for the augmented vector of the sampledstate variables by taking into account the third variable (7),

and are the vectors ofthe feedback coefficients for the control signals acting onand respectively. Namely, is the proportional gain of thecurrent loop, is the gain of the voltage loop and isthe gain of the digital integrator. is the vector of the refer-ence signals given by and is thesaturation function defined by

(12)

The augmented discrete time function which relates the aug-mented state variable to defines the discrete-timemodel for the closed loop system under the DPWM control withdigital PI corrector.

The third specificity of our solution is due to the combinationof a centred PWM pattern with a digital controller. Indeed, itis worth noting that, with our centred pulse, the samples at theend of the switching periods correspond to the averaged valuesof the state variables. Therefore with a digital PI controller andcentred pulses of the driving signals, the averaged values of thestate variables are controlled to their reference values as it is thecase with an analog PI controller.

This important fact can be understood as follows. By sam-pling state variables at the beginning of an ordinary PWM pe-riod, without centred pulses, samples correspond to the minimalor the maximal values (depending on the modulation technique).If a digital PI controller is used for controlling an output vari-able, we get a zero steady state error between the sampled outputand its reference. Because those samples corresponding to anextremum of the output, and no to its average, they yield a steadystate error of the averaged output. However, with centred pulses,the samples correspond practically to the averaged values of thecontrolled output (as it can be seen on Fig. 5) and then the dig-ital PI corrector implies a practically zero state error betweenthe averaged output and its reference.

C. Closed Loop: Obtaining the Discrete Time Model With theDPWM Controller

In a practical dc–dc power electronics converter, it is alwaystrue that the switching period is smaller than the time constantsof the circuit in such a way that the matrix exponential can


Fig. 5. Effect of the centred pulse modulation on the steady state error.

be linearized. Using this approximation and neglecting secondand higher order terms in , we can obtain the followingsimple approximated model for the system:

(13)

where is the unity matrix with appropriate dimension,

(14)

It is worth to note, for the reasons explained in the previous para-graph, that the coefficients are obtained very easily. They areidentical to those used for an average state space model

(15)

Using the expression of system matrices and input vectors, a3-D discrete time model for the system is obtained. It shouldbe noted that the accuracy of this model can be improved byincluding higher order terms in the matrix exponential time se-ries. However, the obtained model will be more complex andnot suitable for controller design. Using the expression of thesystem matrices and input vectors we get

(16)

In order to further simplify the model, let us use a dimensionlessformulation. We will scale the voltages with respect to the inputvoltage , the currents with respect to the maximum current

allowable . Moreover, time is scaled with respect to theswitching period . By using the new scaled variables, the fol-lowing dimensionless model is obtained:

(17)

where is the dimensionless sampled statevariable formed by the scaled inductor current ,the scaled capacitor voltage , the integral of thescaled error and the scaled reference

. The dimensionless parameters are given byand , where . The expressions

of the duty cycles in term of the new dimensionless variablesare as follows:

(18a)

(18b)

where and are thescaled dimensionless feedback coefficients with respect to theircorresponding bases:

D. Stability Analysis and Design Rules of the Two-Cell BuckConverter

The stability of the nominal periodic orbit of the system canbe studied by analysing the stability of the fixed points ofthe three dimensional Poincaré map . These can be obtainedby enforcing the periodicity: . Once thefixed points are located, their stability analysis may be carriedout by studying the local behavior of the map near these fixedpoints. Denoting by and noting that by definition ofthe fixed point , the linearized map can be writtenas:

(19)

where is the Jacobian matrix of . A sufficient con-dition for stability is that all eigenvalues of the Jacobian matrixlie inside the unite circle.

Now we point out an important advantage of our approachjointly based on a simplified model and a digital controller. Inmany works on nonlinear phenomena in switching power elec-tronics converters, although closed form expressions of the dis-crete time model and its Jacobian matrix are obtained, they arevery complex and not suitable for system design. Moreover thefixed point is usually obtained by numerical methods. Workingwith a simplified model, this fixed point can be obtained inclosed form. By enforcing periodicity we obtain:

(20)


Fig. 6. Boundary between stable and unstable regions in the (k ; k ) and(�; k ) parameter spaces.

Fig. 7. Experimental prototype of a two-cell buck dc–dc converter. Top: Com-plete scheme. Bottom: Top view of the circuit.

which implies

(21a)

(21b)

(21c)

(21d)

(21e)

Fig. 8. Loci of the eigenvalues of the DPWM controlled two-cell buck con-verter as the parameter k is swept from 0.01 to k for different values ofL. � = T and k = 0:1.

From (21-b), we derive that at the fixed point .This means that, at the steady state, duty cycles must be equalsto get stationary behavior. From (21-c), we obtain that the sta-tionary dimensionless sampled inductor current is

. From (21-d) and (21-e), we obtain that the steadystate dimensionless voltage is , andfinally, the stationary extra variable due to the digital integratoris .

The approximated Jacobian matrix can be easily ob-tained as follows:

(22)

By performing the partial derivatives and evaluating the resultat the fixed point, this matrix becomes

(23)

From (23), the characteristic polynomial equation can be writtenin the following form:

(24)

where is a second order polynomial given by

(25)

and is the first eigenvalue of the Jacobian matrix which isgiven by

(26)


Fig. 9. Inductor current bifurcation diagrams of the DPWM controlled two-cell buck converter taking parameter k as a bifurcation parameter. (a) Approximateddiscrete time model, (b) Direct circuit simulation from PSIM. k �

= k .

The coefficients of the second order polynomial are asfollows:

(27)

The first condition for stability is that the modulus of the realeigenvalue must be less than one

(28)

Equation (28) implies an upper limit for as well aspositivness of this coefficient. This upper limit depends solelyon the reference current, capacitance and switching period

. We can observe that the smaller the reference current andthe switching period are and the bigger the capacitance is, thebigger this upper bound is.

Note that we can ameliorate the transient dynamics of theclosed loop system by imposing that qual to zero. The valueof giving rise to a zero eigenvalues is

(29)

On the other hand, the Jury criterion [26] applied to the secondorder polynomial implies the following conditions forstability:

(30a)

(30b)

(30c)

From (30a) we infer . Equation (30b) impliesan upper limit for

(31)

This critical value depends on inductor , load resistance, switching period , time constant of the digital integrator

and input voltage . Note that the load resistance and the inputvoltage may undergo variations in a certain range. In this casethe upper value of both of them should be considered in orderto obtain a robust stability condition.

Equation (30-c) implies both an upper and a lower value of

(32)

However the lower value imposed by this condition can beomitted as it is negative.

Also, the critical value is greater than and then(32) could be entirely omitted. Note, however, that

(33)

In this case, condition (32) will be imposed and it will corre-spond to a purely proportional controller in the current loop.Note also, as a usual result, that the digital controller eliminatesthe steady state error at the expense of narrowing the stabilityrange of . As the critical value of does not depend onand vice versa, the stable zone is expected to be a rectangle inthe parameter space . The stable region becomes widerfor smaller switching periods , input voltages and load re-sistances and bigger inductances . These properties give riseto the stability regions represented in Fig. 6 which shows the sta-bility boundary in the parameter spaces and . Thedynamics of the system outside this region can present subhar-monics and chaotic oscillations as it will be shown later. Beforepresenting the numerical simulations, let us present a laboratoryprototype used to confirm experimentally the analytical predic-tions and the simulation results.

IV. EXPERIMENTAL PROTOTYPE OF A TWO

CELL DC–DC BUCK CONVERTER

A prototype of a two-cell dc–dc buck converter is built inorder to confirm our analytical results and numerical simula-


Fig. 10. State variables and command signals of the DPWM controlled two-cell buck converter before (k = 1:2) and after (k = 1:7) losing stability. Top:Direct circuit simulation from PSIM. Bottom: Experimental measurements. k �= k .

tions. The nominal values of parameters are as follows: inputvoltage V, capacitance F, inductance

H, load resistance , reference currentA (nominal output power is about 11.25 W), switching periodis s and time constant of the PI controller . Thecapacitor voltage and the inductor current are sensed using thedifferential amplifiers INA111. A shunt resistormeasures the inductor current. The control signals are obtainedby using the AD718 operational amplifiers. The feedback gains

and are considered as design parameters that should be ad-justed to obtain a stable behavior. They are tuned by means ofpotentiometers. The voltage reference is obtained from the inputvoltage by means of a voltage divider implemented by two re-sistors. The commercially available S&H IC LF398 is used inour experiment to sample the control signals and process theresults in order to decide the duty cycles of the command sig-nals by means of two comparators. The comparators inputs arethe sampled control voltages and two externally generated tri-angular signals. These signals have the same amplitude and arephase-shifted during the switching period. The phase shift of

is done by simply inverting one triangular signal withrespect to the other and adjusting its off-set. The comparatorsoutputs provide the digital signals which are buffered by twonon-inverting gates from IC CD4050. Then they are appliedto the MOSFET gates through dedicated MOS drivers IR2125.The buffers are useful to be compatible with the standard logicinputs of the drivers. The IR2125 provides a high pulse currentdesigned for a minimum driver cross-conduction in order to re-duce power losses. Moreover, the floating bootstrap technologyof its outputs is well adapted to our converter. Switches and

are two HEXFET from IR. In order to reduce conductinglosses, we chose a low voltage MOSFET with higher current ca-pability than necessary and then a very low static on-resistance(55 V–81 A–12 m ). With a total gate charge less than 130 nC,the IRFP054N is effectively driven by the IR2115. Diodesand are implemented in two BYV32-200. These ultrafastrectifiers (200 V–32 A) have a maximum recovery time of 35ns. This is in good accordance with the rise time, turn-on delaytime and fall time of the MOSFET. The complete scheme and apicture of the implemented circuit are depicted in Fig. 7.


Fig. 11. Stability regions in the (k ; k ) and (k ; �) parameter spaces obtained from direct numerical circuit simulation using PSIM package.

V. RESULTS

Our concern in this section is the confirmation of our analyt-ical results by means of numerical simulations and experimentalmeasurements. We will try to make clear the effects of the vari-ation of some parameters on the stability of the system. Mainly,the inductor current loop gain , the voltage loop gain andthe time constant of the integrator . The illustration of the re-sults is supported by time domain waveforms obtained fromcomputer simulations and experimental measurements. In orderto check the analytical expressions for the results obtained in theprevious section, let us consider the same values of parametersas in the experimental prototype.

First, the eigenvalues of the approximated Jacobian matrixare obtained. A set of eigenvalues loci is plotted for three dif-ferent values of as the parameter is increased from 0.01 to

and the results are shown in Fig. 8. For each case we ob-serve that at one of the eigenvalues of the Jacobianmatrix crosses the unit circle at the point , indicating aperiod doubling or flip bifurcation. Another important conclu-sion from Fig. 8 is that there are two values of that give riseto equal eigenvalues as parameter is increased from 0.01 upto the critical value. For both values of it is expected that thetransient response of the system would be critically damped butit will be faster in the case that the eigenvalue is close to theorigin. These remarks will be addressed later in Section VI.

The bifurcation diagram corresponding to the variation of pa-rameter is shown in Fig. 9 using the approximated model[Fig. 9(a)]2 and from PSIM simulations [Fig. 9(b)]. The pa-rameter is taken equal to . PSIM simulations are alsocarried out in order to confirm the theoretical results. In orderto perform the bifurcation diagrams with PSIM, we use thePARAM SWEEP block for time domain simulation. The dataare stored in a file and then are loaded by using Matlab. Thesamples at each switching period are represented in terms of thebifurcation parameter. It can be shown that practically there isno significant qualitative discrepancy between the results.

Time domain waveforms are also used to check the results.Fig. 10 shows the state variables and the command signals ofthe DPWM controlled two-cell dc–dc buck converter before andafter losing stability. It can be observed that the results obtained

Fig. 12. Sampled and hold waveforms of the inductor when a pulse with anamplitude 0.1 A is applied.

from simulations match very well with those measured in thelaboratory.

Finally, in order to show the influence of other feedback co-efficients rather than the current gain, we will consider the vari-ation of the voltage gain and the time constant of the digitalintegrator . Fig. 11 shows a 3D representation of the sampledinductor current in and parameter spaces. Forthis figure the parameter sweep is used for one parameter (or in this case) and the process is repeated for another familyparameter ( in this case). The data are saved in different filesand a Matlab program is used to load all files and represent thesampled data at the switching period in a 3D mesh graph. Stablezones are characterized by the same value of sampled state vari-ables for all switching cycles. In Fig. 11, stable regions are in-dicated by clear flat areas whose colour is determined by thereference value of the displayed state variable (inductor currentin this case).

The appearance of multiple colours in the same area meansdifferent values of the sampled state variable in this region andthen unstable behavior for the system. Note from Fig. 11 that inthe stable zone the sampled inductor current is well regulated to


Fig. 13. Response of the system to a rectangular pulse in the current reference when k is selected near the optimum value and k = k . Left: numericalsimulations from PSM. Right: Experimental measurements. The reference is at �5 div for all channels.

Fig. 14. Effect of the output capacitor on the response of the system to a rectangular pulse in the current reference when k is selected near the optimum value.Left: C = 10 �F. Right: C = 20 �F.

its reference value ( A) while it undergoes fluctuationsin the unstable area of the design parameter space. Note also thatin the parameter space the stable zone is a rectangle asit is expected from the analysis.

VI. DYNAMIC OPTIMIZATION

The main concern in the Section III was the stability of thefixed point and therefore of the underlying periodic orbit. Inthe stability region, the system can have different dynamicperformances such as settling time and resonant behavior.Fig. 12 shows the simulated sampled and hold inductor currentby Matlab for different values of when a 0.1 A step is appliedfrom the steady state reference A. We can observethat the transient time increase when is small as well aswhen it tends to the critical value . There is an optimumvalue of for which the settling time is minimum. Note that

for low values of the transient behavior does not presentoscillation but it is very slow (See the upper graph of Fig. 12).For a certain value of , the system exhibits optimum transientas it is illustrated in the middle plot of Fig. 12. For values closeto but inferior than , the system present oscillations as itis shown by the lower plot of the same figure. In this section,we will try to derive some expressions that optimise the settlingtime in terms of circuit parameters. The linearization of thesystem gives

(33)

This implies that [28], [29]

(34)


where is a positive polynomial in and is the spectralradius of . Namely, the spectral radius of isdefined as . The faster responsecorresponds to the minimum spectral radius of the Jacobian ma-trix. In our system and for , one of the eigenvaluesis null and the spectral radius will be minimum if the remainingtwo eigenvalues are real and equal. That is, the second orderpolynomial has a zero discriminant. Two different valuesof lead to a zero discriminant. These are given by

(35)

Both and give rise to equal eigenvalues buthas an optimal settling time because the modulus of the

eigenvalues is near to the origin.Numerical simulations from PSIM and experimental mea-

surements shown in Fig. 13 confirm that a good response is ob-tained when .

Finally in order to check the effect of the output capacitor onthe results obtained in this work, we obtain for the same condi-tions, the response of the system for two values of capacitors.In Fig. 14, we can observe that for a value of F, thesystem response does not change except from a filtering effect atthe output voltage. However, as the output capacitor is increasedbeyond F, the system could present overdamped re-sponse.

VII. CONCLUSION

In this paper, we have given a description of the dynamicalbehavior of a two-cell dc–dc buck converter. Different oper-ating modes are possible depending on the duty cycle of thedriving signals and phase-shift between them. We have pre-sented a systematic way to obtain the discrete time model of thesystem. This model can be used to study accurately the stabilityof the periodic behavior of the system. Based on this study, theboundary between the stable and the unstable region can be ob-tained. The simplified model has been derived based on somerealistic assumptions. The stability region has been determinedby analysing the characteristic polynomial of the Jacobian ma-trix of the discrete time model. Different design parameters havebeen considered and the stability region has been located in thecorresponding parameter space. Some design rules can be for-mulated from the derived stability conditions. The results on thestability analysis by using the Jacobian matrix have been con-firmed by time domain simulations and bifurcation diagramsobtained from different approaches. Namely, they have been ob-tained from an approximated model, from numerical simulationusing PSIM and experimental measurements. A DPWM con-troller has been considered and the mechanism for losing sta-bility has been shown to be a period doubling. A good agree-ment between numerical simulation and theoretical predictionhas been obtained. It was observed that regardless of the op-erating mode, the same approximated model is obtained andtherefore the critical values of the design parameters that de-fine the boundary of stability are independent of the operating

mode. An experimental prototype was built to verify the analyt-ical and simulation results. The implementation uses a commer-cially available S&H IC and has very advantageous character-istics from a practical point of view. The digital controller wasbuilt without the use of a microprocessor that could decreasethe system reliability and increase the cost of the controller. Theapproach here reported can be extended to other multi-cell con-verters. A general framework extending discrete-time modellingapproach for multi-cell dc–dc converters is in preparation and itwill be reported in a future work.

ACKNOWLEDGMENT

The authors wish to thank J. María Bosque for implementingthe experimental prototype of a two-cell dc–dc buck converterand the anonymous reviewers for their valuable comments andsuggestions to improve an original version of this work.

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Abdelali El Aroudi (M’00) was born in Tangier,Morrocco, in 1973. He received the M.S. degree inphysical science from Faculté des sciences, Univer-sité Abdelmalek Essadi, Tetouan, Morocco, in 1995,and the Ph.D. degree from Universitat Politècnica deCatalunya, Barcelona, Spain, in 2000.

Currently he is a lecturer in the Departamentd’Enginyeria Electrónica, Elèctrica i Automàtica(DEEEA) of Escola Tècnica Superior d’ Enginyeria(ETSE), Universitat Rovira i Virgili (URV), Tar-ragona, Spain. His research interests are in the field

of structure and control of power conditioning systems for autonomous systems,power factor correction, stability problems, nonlinear phenomena, chaoticdynamics, bifurcatons and control. He is a Reviewer for the InternationalJournal of Control, IET Electric Power Applications, International Journalof Systems Science, and Circuits, Systems and Signal Processing. He haspublished about 80 papers in scientific journals and conference proceedings.He is a member of the GAEI research group (Rovira i Virgili University) onIndustrial Electronics and Automatic Control whose main research fields arepower conditions for vehicles, satelites and renewable energy.

Dr. El Aroudi is a reviewer for IEEE TRANSACTIONS ON CIRCUITS AND

SYSTEMS PART. I AND II, IEEE TRANSACTIONS ON POWER ELECTRONICS, andIEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS

Bruno Robert received the M.Sc. degree in elec-trical engineering and power electronics from theEcole Normale Superieure of Cachan and Universityof Paris VI, Paris, France, in 1988, and Ph.D.degree in computer science, automation and signalprocessing from the University of Reims, France, in1993.

Since then, he is an assistant professor at the Uni-versity of Reims. He is 15 years experienced in thefield of power electronic and electric drives. Since1997, he investigates nonlinear dynamics of electric

drives and power stages by applying chaos theory. His main research areas aretransient modes of quasi-resonant converters, bifurcation analysis and chaoscontrol in current inverters and time series analysis and control of step mo-tors. These studies include modelling, simulation and control aspects and exper-imental investigations. He is co-author of more than 40 journal and conferencepapers. He has been a Reviewer for International Journal of Bifurcation andChaos, International Journal of Control, Chaos Solitons and Fractals and is apermanent Reviewer of European Power Electronics and Drives Journal. Heis a member of the CReSTIC Laboratory, University of Reims-Champagne-Ar-denne.

Dr. Robert is a Reviewer for the IEEE TRANSACTIONS ON CIRCUITS AND

SYSTEMS PART. I AND II.

Angel Cid-Pastor (M’06) received the B.S. degreein electrónica industrial and the B.S. degree inautomática y electrónica industrial from the Uni-versitat Rovira i irgili, Tarragona, Spain, in 1999and 2002, respectively, the M.S. degree in designof microelectronics and microsystems circuits fromInstitut National des Sciences Appliquées, Toulouse,France, in 2003, and dual Ph.D. degrees from theUniversitat Politècnica de Catalunya, Barcelona,Spain, in 2005, and from the Institut Nationaldes Sciences Appliquées, LAAS-CNRS Toulouse,

France, in 2006.He is currently an Associate Professor at the Departament d’Enginyeria Elec-

trónica, Elèctrica i Automàtica, Escola Tècnica Superior d’Enginyeria, Univer-sitat Rovira i Virgili, Tarragona, Spain. His research interests are in the field ofpower electronics and renewable energy systems.

Luis Martínez-Salamero (SM’06) received theM.S. degree in telecomunicación and the Ph.D. de-gree from the Universidad Politécnica de Cataluña,Barcelona, Spain, in 1978 and 1984, respectively.

From 1978 to 1992, he taught circuit theory,analog electronics and power processing at EscuelaTécnica Superior de Ingenieros de Telecomuni-cación de Barcelona. From 1992 to 1993, he wasVisiting Professor at the Center for Solid State PowerConditioning and Control, Deparment of ElectricalEngineering, Duke University, Durham, NC. He is

currently a Full Professor with the Departamento de Ingeniería Electrónica,Eléctrica y Automática, Escuela Técnica Superior de Ingeniería, UniversidadRovira i Virgili, Tarragona, Spain. From 2003 to 2004, he was a visitingscholar at the Laboratoire d’Architecture et d’Analyse des Systèmes (L.A.A.S),Research National Center (CNRS), Toulouse, France. He has published a greatnumber of papers in scientific journals and conference proceedings and holdsa U.S. patent on the electric energy distribution in vehicles by means of abidirectional dc-to-dc switching converter. He is the Director of the GAEI, aresearch group on Industrial Electronics and Automatic Control whose mainresearch fields are power conditioning for vehicles, satellites and renewableenergy. His research interests are in the field of structure and control of powerconditioning systems for autonomous systems.

Dr Martínez-Salamero is currently the president of the Spanish Chapterof the IEEE Power Electronics Society. He was Guest Editor of the IEEETRANSACTIONS ON CIRCUITS AND SYSTEMS for the special issue on Simulation,Theory and Design of Switched-Analog Networks. He has been distinguishedlecturer of the IEEE Circuits and Systems Society from 2001 to 2002.

modeling and design rules of a two-cell buck converter under a digital pwm controller

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