This paper tackles the control of fast-scale instabilities in a buck switching power converter aiming to expand its design-space towards miniaturization. After briefly revisiting the working principle of existing chaos controllers, the paper explores an alternative approach based on amplifying the harmonic at the switching frequency. Numerical simulations show that the proposed controller can concurrently improve both fast-scale and slow-scale stability margins. Finally, the paper proposes a chaos controller combined with an output ripple reduction network and studies their interaction with the aim of achieving both low-ripple and improved stability.Preprin

Alarcón Cot, Eduardo José

El Aroudi, Abdelali

Martínez García, Herminio

Poveda López, Alberto

Rodríguez Rodríguez, Enric

IEICE Proceeding Series

English

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IEICE Proceeding Series Chaos controller for switching regulators aiming enhanced design-space towards miniaturization E. Rodriguez, A. El Aroudi, H. Martinez, A. Poveda, H. Iu, E. AlarconVol. 1 pp. 622-627 Publication Date: 2014/03/17 Online ISSN: 2188-5079©The Institute of Electronics, Information and Communication EngineersDownloaded from www.proceeding.ieice.org Chaos controller for switching regulators aiming enhanced design-spacetowards miniaturizationE. Rodriguez†, A. El Aroudi‡, H. Martinez†, A. Poveda†, H. Iu and E. Alarcon††School of Electrical, Electronic and Computer Engineering,UPC BarcelonaTech08034 Barcelona, Spain.‡Departament d’Enginyeria Electrnica, Elctrica i Automtica,Universitat Rovira i Virgili43007 Tarragona, SpainSchool of Electrical, Electronic and Computer Engineering,University of Western AustraliaAbstract—This paper tackles the control of fast-scaleinstabilities in a buck switching power converter aimingto expand its design-space towards miniaturization. Afterbriefly revisiting the working principle of existing chaoscontrollers, the paper explores an alternative approachbased on amplifying the harmonic at the switching fre-quency. Numerical simulations show that the proposedcontroller can concurrently improve both fast-scale andslow-scale stability margins. Finally, the paper proposesa chaos controller combined with an output ripple reduc-tion network and studies their interaction with the aim ofachieving both low-ripple and improved stability.1. IntroductionSwitching power converters are widely used in many ap-plications due to their high efficiency, small size and lowweight. However, the design-space when targeting minia-turization towards on-chip integration is technologicallylimited by three performance metrics: area, efficiency andoutput voltage ripple. The area occupancy is mainly limitedby the reactive components size; thus, reducing the induc-tor or capacitor parameter values results in a decrease of thearea, but also increases the natural frequency of the implicitlow-pass filter of any DC-DC converter, turning into the re-duction of the ratio between the switching frequency andthe natural frequency hence leading to high output voltageripple. This ripple is usually decreased by increasing theswitching frequency, hence, in turn, decreasing the overallconverter efficiency. Therefore, the desired low-area high-efficiency context, such as in miniaturized switching powerconverters, will upfront imply high output voltage ripple.However, the output voltage ripple amplitude confronts atwo-fold limitation: due to the strict specifications imposedby on-chip loads demanding tight regulation, but also be-cause higher output voltage ripple make the converter moreprone to exhibit the so-called fast-scale instabilities (FSI)and eventually chaotic regime [1].The control of FSI, manifested by subharmonic oscil-lation, has been traditionally tackled either by adding ex-ternal signals such as ramp compensation in current-modecontrol strategies in power electronics circuits, or addinga time-delay module in general physical nonlinear dynam-ical systems[2]. In this second case, in addition to theirimplementability constraints, in [3] it is shown that thesecontrollers are based on comb filters centered at half of theswitching frequency and that the global stability boundaryis compromised due to their additional phase lag in the sys-tem loop hence worsening the well-known slow-scale in-stability (SSI) boundary.On the other hand, the reduction of the converter outputripple has been usually tackled by relaxing circuit specifi-cations (increase reactive components values or switchingfrequency), or by using more complex topologies such asmultilevel converters [4], or by narrowband reactive filter-ing at the converter output [5].The system considered in this paper is shown in Fig. 1and the parameter values used are Vg = 6 V, Vre f = 3 V,R = 2.5 Ω, L = 66 nH, C = 20 nF, fs = 50 MHz, Vm =1 V, ωz1=10 Mrad/s and kp=3. These values correspondto a miniaturized converter aiming on-chip integration [4],but the selection is representative (through scaling) of anyconverter exhibiting moderately large ripple (∼50 mV).The remaining of the paper is structured as follows.First, Section II reviews briefly the time delay controllerand explores the so-called repetitive controller showingthat not only the attenuation at half of the switching fre-quency but also amplifying the harmonic at the switchingfrequency can entail the control of FSI. Then, a new con-troller is proposed, which is based on a narrow band ampli-fier (NBA) centered at the switching frequency improvingthe stability boundary in both FSI and SSI terms, apart fromsimplifying the implementability of the controller. The pa-per concludes by proposing a new topology, so-called LC- 622 -Figure 1: VMC buck converter with a compensator Gc(s)along with a FSI controller GFS (s).divider, in which the feedback ripple component is ampli-fied to obtain an enhanced stability margin while attenuat-ing the converter output voltage in order to accomplish theload ripple specifications.2. Generalized time delay feedback chaos controllers:Repetitive controllersReference [3] revisits time-delay feedback controllers(TDFC) in order to describe their operation principle andthe effect of their parameters upon stability. The workshows that the controller provides a frequency-selectivecomb-filter which attenuates the harmonics at half of theswitching frequency and its integer multiples.However, other more complex time-delay-based con-trollers have been also used for harmonic tracking appli-cations such as the so-called repetitive controller [6]. Theblock diagram of a repetitive controller is shown in Fig. 2.The s-domain transfer function of such a repetitive con-troller in terms of the parameter γ is:GFS (s) =e−sT + γe−sT − γ (1)Figure 2: Block diagram of a repetitive controller.The frequency response of a repetitive controller isshown in Fig. 3 along with that of TDFC. Note that com-paring with a simple TDFC, apart from attenuating the har-monic at half of the switching frequency, the controller is10−1 100 101 102−2−1012ω [Mrad/s]|GFS(s)|[dB]10−1 100 101 102−10−50510ω [Mrad/s]6(GFS(s))[o]Figure 3: Bode diagram representation (magnitude andphase) of the repetitive controller with γ=0.05 (solid) andTDFC (dash). Vertical dashed line indicates the switchingfrequency.amplifying the switching frequency harmonics (to improvethe tracking performance).The stability boundary as a function of the proportionalgain kp is depicted in Fig. 4, showing that the system un-der the repetitive controller is less stable in terms of SSIbut it is slightly more stable in terms of FSI. Regarding theSSI observed results, they agree with the frequency domaininterpretation, in which more phase lag is added by the con-troller. However, in terms of FSI, the improvement of thestability margin compared to TDFC can be only attributedto the amplification at the switching frequency harmonic.3. Narrow band amplifier controllerThe previous section suggests that FSI control can be ac-complished not only by attenuating the harmonic at half ofthe switching frequency but also amplifying the switchingfrequency harmonic. Thus, this section focuses on provid-ing the FSI control functionality from a feasible implemen-tation of such harmonic amplification.The transfer function of the FSI chaos controllerGFS ,NBA(s) based on a NBA is shown in Eq. 2 with ∆ξ =ξ1/ξ2 >1 and the center frequency tuned to the switchingfrequency, ωn = 2pi fs. Apart from the amplification at theswitching frequency harmonic, a key additional advantageis the fact that it notably improves the SSI boundary byadding a phase lead before the switching frequency.GFS ,NBA(s) =s2 + 2ξ1ωns + ω2ns2 + 2ξ2ωns + ω2n(2)Its effect upon FSI boundary is characterized in Fig. 6,which shows the stability boundaries surface as a func-tion of ∆ξ, ξ2 and converter parameters. These surfaces- 623 -290.02 290.04 290.06 290.08 290.1 290.122.833.2vC[V]290.02 290.04 290.06 290.08 290.1 290.120.811.21.4i L[A]290.02 290.04 290.06 290.08 290.1 290.12−0.500.511.5Time [µs]vcon[V](a)290.02 290.04 290.06 290.08 290.1 290.122.833.2vC[V]290.02 290.04 290.06 290.08 290.1 290.120.811.21.4i L[A]290.02 290.04 290.06 290.08 290.1 290.12−0.500.511.5Time [µs]vcon[V](b)0 0.05 0.1 0.1501234567kpγFSI unstable SSI unstable(c)Figure 4: Time domain wavforms with kp=5 (a) withoutFSI controller (b) with Repetitive controller with γ=0.05.(c) Stability boundary as a function of γ and kp with arepetitive controller (solid) and TDFC (dashed line). FSIstability boundary without controller (γ=0), kp=4.3.have been obtained from the discrete-time model [7, 8] ofthe system, the well-known complex mathematical model10−1 100 101 102−202468ω [Mrad/s]|GFS(s)|[dB]10−1 100 101 102−20−1001020ω [Mrad/s]6(GFS(s))[o]Figure 5: Bode diagram representation (magnitude andphase) of an NBA tuned to the switching frequency (ver-tical dashed-line) with ∆ξ=2 and ξ2=0.1 (solid) and ξ2=0.5(dash).commonly used to predict subharmonic oscillation bound-ary with high accuracy. Note that depending upon the valueof the two parameters, the obtained results are much morebetter than the one obtained in delay-based controllers (seeFig. 4).As it was mentioned at the beginning of this work, theindirect aim of a chaos controller, apart from obviously im-proving the stability margin, is that such improvement al-lows to expand the design-space towards miniaturization(for instance reducing the inductance value, which is re-lated to the system area), without losing stability.Simulations in Fig. 6 show the stability boundary asa function of ∆ξ and ξ2 and the inductance (related toarea/volume occupancy) and switching frequency (relatedto efficiency).Both obtained surfaces are similar and the stabilityboundary is clearly improved, allowing to reduce both pa-rameter values without losing stability, when ξ2 <1 and∆ξ > 1 (a narrow band amplification) with regards to thecase of not using such controller (∆ξ = 1). Note that the pa-rameter ξ2, which has a direct effect upon the width of theamplification band, has an important effect upon the sta-bility boundary so that as it increases, the FSI boundary isworsened. This suggests that the benefits of the NBA con-troller on stability are limited by such bandwidth, namelywhen both harmonics at the switching frequency and itshalf are amplified the FSI boundary is not modified. Onthe other hand, it is possible to observe that the higher theamplification is, the more stable the system is in terms ofFSI.- 624 -(a)(b)(c)Figure 6: FSI boundary surfaces, obtained from thediscrete-time model, of a VMC buck converter with anNBA as a function of parameters ∆ξ, ξ2 and (a) the propor-tional gain kp, (b) the inductance L and (b) the switchingfrequency fs.(a)101 102 103−60−50−40−30−20−1001020ω [krad/s]6(G(s))(b)101 102 103−200−180−160−140−120−100−80−60−40−20020ω [krad/s]6(G(s))(c)Figure 7: (a) Frequency response of the buck switchingconverter under voltage-mode control with a PWM andan additional NBA controller and output notch filter. (b)Magnitude and (c) phase of the control-to-output (solid)and control-to-feedback (dashed) transfers function withthe output notch filter. Vertical dashed-line indicates theswitching frequency.- 625 -4. Combining chaos controller with output ripple re-ductionThe improvement in stability terms when a chaos con-troller is used implies a clear step forward towards circuitminiaturization without losing stability. However, this willimply an increase in the converter output ripple, which isnot desirable in order to keep the DC-DC nature of thesystem. Therefore, the advantage of the NBA chaos con-troller could be invalidated because of the expansion of thedesign-space, namely the area or switching frequency re-duction could be limited by load specifications in terms ofripple.The natural way for tackling this excess of ripple wouldbe to add a ripple attenuation module to the converter out-put, but this is not trivial since this should be based on mod-ifying the switching frequency harmonic, which is the sameworking principle of the NBA chaos controller, and henceit could cancel the advantage of such approach.This section proposes a joint approach to obtain high sta-bility margin in term of FSI, that allows reducing the con-verter reactive components, and at the same time reducingthe output voltage ripple to accomplish low-ripple perfor-mance as required by DC-DC converters.The frequency domain representation of the buck switch-ing converter under voltage-mode control with a PWM andan additional NBA controller and output notch filter aredepicted in Fig. 7. Both the control-to-feedback and thecontrol-to-output transfer functions show that the output ofthe LC network does not affect the feedback transfer func-tion but adds a notch in the output transfer function hencereducing the amount of ripple delivered to the load. Addi-tionally the NBA will be added as a chaos controller in thefeedback loop.In Fig. 8 results from using only the NBA (obtainedin the last section) and this approach are compared. Theresults are obtained from the discrete-time model devel-oped. Note that a similar stability boundary is obtained asa function of the duty cycle, but the output voltage ripple islargely reduced by canceling the harmonic at the switchingfrequency.The main drawback of this proposed configuration is itsadditional required circuitry: the NBA plus the reactivecomponents of the output notch. In addition, the notch in-ductor should drive all the output current hence requiringlarge area and the circuit also requires accurately control-ling the center frequency and the gain of the amplifier andthe output notch since a mismatch between both center fre-quencies (such as those due to the effect of temperature orparasitic elements) could limit the benefits of the approachin both output voltage ripple and fast-scale stability terms.5. ConclusionsThis work has addressed controllers for fast-scale insta-bility (FSI) in a buck switching power converter in pur-0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.91020304050607080DL[nH]stableFSI unstable(a)0 0.2 0.4 0.6 0.8 10102030405060Vˆo[mV]D(b)Figure 8: (a) FSI boundary curves, obtained from thediscrete-time model, of a VMC buck converter with a NBAcontroller and output notch filter (solid) and a conventionalbuck (dash) as a function of the inductance L and the dutycycle D and (b) the output voltage ripple Vˆo in both casesas a function of the duty cycle D. L=66 nH, Ln=2 nHCn=5 nF, ξ2=0.1 and ∆ξ=2.suit of expanding the design-space when targeting minia-turization. The control of such instabilities would allowthe reduction of reactive component values or the switch-ing frequency without losing stability. With this purpose,this paper has proposed a FSI controller based on a narrowband amplifier (NBA) centered at the switching frequencyharmonic which does not compromise the global stabilityboundary. A final extension has been considered to com-bine the NBA controller with an output ripple reductionnetwork in the power topology with the aim of concurrentlyachieving a low-ripple at the output of the converter and animproved global stability of the closed loop system.- 626 -AcknowledgmentsThis work is funded by FPU-grant of ministry of Scienceand Innovation of Spain under grants DPI2010-16481 andTEC2010-15765.References[1] Rodriguez, A. E. Aroudi, F. Guinjoan, and E. Alarcon,“A ripple-based design-oriented approach for predict-ing fast-scale instability in DC-DC switching powersupplies,” IEEE Transactions on Circuits and SystensI: Regular Papers, vol. 59, pp. 215–227, 2012.[2] K. Pyragas, “Continuous control of chaos by self-controlling feedback,” Physics Letters A, vol. 170, pp.421–428, 1992.[3] E. Rodriguez, E. Alarcn, H. Iu, and A. El Aroudi, “Afrequency domain approach for controlling chaos inswitching converters,” in IEEE International Sympo-sium on Circuits and Systems (ISCAS’10), 2010, pp.2928–2931.[4] G. Villar and E. Alarcon, “Monolithic integration ofa 3-level DCM-operated low-floating-capacitor buckconverter for DC-DC step-down donversion in stan-dard CMOS,” in IEEE Power Electronics SpecialistsConference (PESC’08), 2008, pp. 4229–4235.[5] E. Alarcon, G. Villar, S. Ferrandez, F. Guinjoan, andA. Poveda, “Ripple-reduction tuned filtering switchingpower converter topology,” in IEEE Power ElectronicsSpecialists Conference (PESC’04), 2004.[6] G. Escobar, P. Martinez, J. Leyva-Ramos, and P. Mat-tavelli, “A negative feedback repetitive control schemefor harmonic compensation,” IEEE Transactions on In-dustrial Electronics, vol. 53, no. 4, pp. 1383 –1386,2006.[7] M. di Bernardo and F. Vasca, “Discrete-time maps forthe analysis of bifurcations and chaos in DC/DC con-verters,” IEEE Transactions on Circuits and Systems I:Fundamental Theory and Applications, vol. 47, no. 2,pp. 130–143, 2000.[8] A. El Aroudi, M. Debbat, L. Giral, G. Olivar, L. Be-nadero, and E. Toribio, “Bifurcations in DC-DCswithching converters: reivew of methods and applica-tions,” International Journal of Bifurcation and Chaos,vol. 15, no. 5, pp. 1549–1578, 2005.- 627 -

Chaos controller for switching regulators aiming enhanced design-space towards miniaturization

This paper tackles the control of fast-scale instabilities in a buck switching power converter aiming to expand its design-space towards miniaturization. After briefly revisiting the working principle of existing chaos controllers, the paper explores an alternative approach based on amplifying the harmonic at the switching frequency. Numerical simulations show that the proposed controller can concurrently improve both fast-scale and slow-scale stability margins. Finally, the paper proposes a chaos controller combined with an output ripple reduction network and studies their interaction with the aim of achieving both low-ripple and improved stability

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Chaos controller for switching regulators aiming enhanced design-space towards miniaturization

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