Dynamics Buck CPL

8/13/2019 Dynamics Buck CPL

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ynamics of a Converter with a Constant Power Load2 3Vlad G rigore 2, Jari Hatonen , Jorma Kyyra 2 , Teuvo Suntio

Electronics and Telecommunications Faculty, Politehnica University of Bucharest, RomaniaInstitute of Intelligent Power Electronics,Helsinki University of Technology,Otakaari 5A, 02 150 Espoo, FinlandTel. +358 9 4514968, Fax. +358 9 460224,Email: [email protected], hatonen@ cc.hut.fi, Jorma.Kyyra@ hut.fi,

Email: [email protected] Oy, Finland

Abstract - The dynamic properties of the Buckconverter with a Constant Power L oad are studied in thispaper. The line-to-output and control-to-output transferfunctions are derived, for Voltage Mode Control andCurrent Mode Control, in Continuous Conduction Modeand Discontinuous Conduction Mode. A comparison withthe case of a resistive load is made in each case.

I. INTRODUCTIONIn most of the papers dealing with controlproperties of switch mode pow er supplies (SMPS), theload is assumed to be purely resistive. In some casesalso inductance is added. This is, of course, very oftentrue in real cases, or at least this is an accurate enoughapproximation of the load characteristics. How ever,during the recent years distributed power supply

applications have increased, [11, [2]. A distributedsystem consists of a first stage feeding a DC bus, towhich a second stage is connected.Although the first stage can be only one converter,in many cases it consists of more parallel connectedconverters. This is mainly because the redundancy ofthe system is increased when a single conversion stageis divided into smaller units, and becau se in some casesthe desired power level is achieved using more parallelconnected converters.The seco ndary side consists of converters fed fromthe DC bus. Loads having different physical locationand needing various voltage and power levels are fedby con verters connected to the same DC bus.

The distributed system is represented by thesimplified system in Fig. 1, which consists of two dcldcSMPS S M P S l and S M P S 2 , connected in series. Theauthors are aw are that this is a very simplified model ofthe distributed system, but it provides a basis foranalyzing the dynamic properties.

Fig. 1 . Simplified system.

Suppose SMPSl is a Buck converter. If the outputvoltage of SMPS1, i.e. input of SMPS2, for somereason increases, the voltage loop of the SMPS2reduces the duty cycle in order to regulate its outputvoltage to the reference value. However, this reducedduty cycle also reduces the input current of SMPS2.Therefore, an increase in the input voltage of SMPS2results into a current decrease and vice versa. Thismeans that a DCIDC converter can be modeled as aconstant power load in frequencies well below theswitching frequency. Thus, SMPS1 is a converterfeeding a Constant Power Load. It is interesting tostudy the dynam ic properties in such a situation. Wh ilethis problem can be investigated also from the point ofview of SMPS2, this paper analyzes the problem froma different perspective. The dynamic properties of theconverter feeding a DC bus (SMPSl) are investigated.It is sensible to think at such an approach, from thepoint of view of the manufacturer of rectifiers, forexample. In many cases the rectifier feeds a DC bus,and often it is the part of the distributed system onwhich the man ufacturer has control.A Buck converter feeding a Constant Power Load(CPL) is analyzed in this paper. The purpose is toderive its line-to-output and control-to-output transferfunctions, in Continuous Conduction Mode (CC M) andDiscontinuous Conduction Mode (D CM), with VoltageMode Control and with Current Mode Control. Thedynamic properties of the converter with CPL arecompared with those of the converter with resistiveload.In Section 11 of the paper various possible analysisdiscussed. After that, the model of the CPL, theaveraged model of the Buck converter and the modelof the Current Mo de Controller are presented.In Section UI ime-domain simulations using theaveraged model and the dev ice-level (switched) modelof the Buck converter with CPL are compared , in orderto validate the averaged model. The same comparisonis done also for the resistive load case. After that, the

methods are shortly reviewed and their applioability

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transfer functions of the converter with Voltage ModeControl are presented, for both CCM and DCM . Also,the Bode plots of the transfer functions, obtained bySPICE simulation using the averaged model, areincluded.In Section IV, the model of the Current ModeController is added to the model of the converterpresented in Section 11 The transfer functions arecalculated and the Bode plots, obtained by simulationusing the averaged model, are included.Finally, Section V presents some conclusions.

11. MODELINGPPROACHESA . Method ofAnalysis

The modeling approach should allow us to modelthe DC/DC converter, which is a time-variant circuit,by a time-invariant circuit. This is necessary in ord er tobe able to derive the transfer fu nctions of the converter.Also, it would be very useful if the model could beused for time-domain SPICE analysis, to determine thetransient behavior of the converter, and for small-signalanalysis to determine the Bode plots of the transferfunctions. Such a model could also be used to deriveanalytically the transfer functions.

A well known method to analyze dynamicproperties of SMPS is the state-space averaging methodproposed in [3]. The idea of this method is that theconverter is described with two separate differentialequation groups when the active switch in the converteris turned ON or OFF. These two state-space equationsare then averaged so that duty cycle d and 1 - d areused as weights respectively. Such approach leads to asmall-signal model, which cannot be used for time-domain simulations. Moreover, in this approach alsothe load is included. If this method is used in the CPLapplication, the result would be state-space equationswhere one of the state variables would be seen as aninverse value. Therefore, the model is not anymorelinear.A well suited method for our purpose is the PWMswitch averaging method [4, 51. The great m erit of thismethod is that the model is developed independent ofthe load. Averaging is applied to the PWM switch (orchangeover switch), which is the only time-variant partof the circuit. The obtained time-invariant averagedmodel can be linearized and included in the invariantpart of the converter for deriving the transfer functions.This model is also well suited for SPICE analysis. Itcan be used for time-domain analysis and, by ACsmall-signal analysis, the Bode plots of the transferfunctions can be obtained.A third analysis method could be the injected-absorbed-current method, where the model is alsodeveloped independent of the type of the load [lo], butit is not applied in this paper.

B. The Model of a Constant Power Load (CPL)A pure resistive load can be modeled with aconstant resistance if temperature effects are neglected.However, as previously explained, this approach is notvalid in many cases, for example when the load takes aconstant power. The load characteristic of a CPL is

shown in Fig. 2.

Ve,Fig. 2. Voltage and current characteristics of a constant

power load (CPL) and a resistive load.This characteristic can be easily implemented in

SPICE, as a current source controlled by the voltageacross it, described as:io = P/vo. (1)We use this model in SPICE simulations of theBuck converter with CPL.For small-signal analysis, we need to linearize thecharacteristics at point M , where the power is

P = V o l o = R e / : . If we assume that there is a smallperturbation in the output current, io = I , + then theequivalent resistance of the constant power load is alsomodified as re = Re + 5 Therefore, the resistance is:

where c2 has been neglected because it is smallcompared to other terms. The denominator is equal tothe DC value of the resistance R e , at the operatingpoint M, and in the nominator it can be assumed thatio / I , + 0 . Using the first two terms of the Taylorseries we obtain:-

3)In Fig. 3the DC m odel and the small-signal model

of the CPL are presented.

3 b)Fig. 3. Constant Power Load:a) DC model; b) small-signal model.

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The DC model is used to calculate the operatingpoint, while the small-signal model is used toanalytically determine the transfer functions of theBuck converter with CPL.In this paper, the notation R is used to define aresistive load, while Re is used to defme the equivalentresistance of the CPL.C. The Averaged Model o the Buck Converter

The schematic diagram of the Buck converter isshown in Fig. 4.

Fig. 4. The Buck converter.The PW M switch is identified in the dashed box. Itis the only time-variant part of the circuit. Its terminalsare defined as: a- active, p - passive, c - common. Theconverter can operate in Continuous Conduction Modeor Discontinuous Conduction Mode.The PWM switch is replaced with its averagedmodel. The resulting circuit is shown in Fig. 5 . Theaveraged model is described in detail in [4, 51, so hereit will be presented on ly briefly.

L I.l a = p c

o

Fig. 5. Averaged model of the Buck converter.The gain of the controlled voltage and currentsources is defined as:,u=- in DCM, (4)d+d;,u= d in CCM, 5 )

andwhere d is the averaged value of the duty-cycle overone switching period T , and d 2 is the averaged valueof the interval when the free-wheeling diode D isconducting. It is given by:

2L i ( 6 )2 =--.dT vcpNaturally, d+d2 = 1 in CCM. The model in

Fig. 5 , with the gain of the controlled sources definedby (4), can describe both CCM and DCM, if thedenominator is limited: 0 5 d + d2 5 1. This limitationcan be easily implemented in SPICE.We use this model to obtain the averaged model of

the Buck converter for SPICE simulation. It allowstime-domain analysis, and Bode plots of the transferfunctions can be generated. The validation of thismodel will be presented in Section III. The transientresponse of the device level (switched) converter andthe transient response of the averaged model will becompared. Simulations show very goodcorrespondence between the two models, both in CCMand in DCM.However, the averaged model of the PWM switchis a nonlinear model. A small-signal linearized modelmust be obtained to analytically calculate the transferfunctions of the converter. Naturally, the linearizedmodel is a function of the operating point, so twodifferent models are obtained, for CCM and D CM. T helinearization is done by considering small perturbationsaround the operating point for all the variablequantities in the circuit. The procedure is similar to theone previously described to obtain the small-signalmodel of the Constant Power Load. The equations ofthe circuit are written and the variables are replaced by

their DC values plus small perturbations(e.g. d = D + d ) . The DC terms are separated in a setof equations describing the DC operating point.Neglecting the second order terms, the remaining firstorder terms are separated into a set of equationsdescribing the small-signal model of the circuit.The small-signal model of the PW M sw itch inCCM is presented in [4] By replacing in Fig. 5 thesmall-signal model of the PWM switch in CCM, andthe small-signal model of the CPL, the small-signalmodel of the Buck converter in CCM with CPL isobtained. The model is presented in Fig. 6.

-

I I

Fig. 6. Small-signal model of the Buck converter in CCM,with CPL.The small-signal model of the PWM switch inDCM is presented in [SI. By replacing it in the Buckconverter, the small-signal model of the Buckconverter in DCM with CPL, presented in Fig. 7, isobtained.p*+:;1 -Re CO1 - gogfv c k d

PFig. 7 . Small-signal model of the Buck converter in DCMwith CPL.

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The parameters of the PWM switch model in DCM,for this particular situation, are:1 D 2 I 2 0go = - ( I - D ) , g f =-Re RE?gi =--R, I - D

1- DDki = 2 I , k , =2-1 7)

D. The Model of the Current Mode ControllerTo analyze the Buck converter with Current ModeControl, also the model of the Current Mode controllermust be obtained. Such model calculates the value ofthe duty cycle d , based on the current reference and onthe average values of inductor current and on the inputand output voltages. Like in the case of the PWMswitch, it is desirable to have a model, which describesboth CCM and DCM.The principle of Current Mode Control in CCM isshown in Fig. 8 .

I

Fig. 8 . Current mode control in CCM.At the beginning of each period, the switch S isturned ON. The switch is turned OFF when the currentthrough the inductor reaches a control signal. The

control signal is obtained from a reference current i,A compensating ramp ( - m a ) is added to i, , in orderto prevent instability, when D>0.5 [SI. It can be seenthat the value of the inductor current at the end of theswitching period is not necessarily equal to the currentat the beginning of the switching period. Thus, themodel developed based on this waveform takes intoaccount the transients [6]. The averag e inductor currentsatisfies the relation:

8)d 2 ( v i o ) T I - )2v,T+ 1i , - m a d T = i + ?)ILL LLThe principle of the Current Mode C ontrol in DCMis presented in Fig. 9.The average inductor current inDCM satisfies the relation:i, -m,dT)(d + d2) =

d 2 ( v i- o ) T ( I - d - 2 ) 2 ~ , T (9)= i + +2L 2LIt can be seen from (8) and (9) that the CurrentMode controller can be described by (9), f ( d + d 2 )

t>dT d2TFig. 9. Current mode control in DCM.

is limited to un ity. Equation (9) can be implem ented inSPICE, and the SPICE model of the Current ModeController is thus obtained. It has as an input the inputand ou tput voltages, the reference current, the averageinductor current and d 2 , and as an output the dutycycle d.The sm all-signal models for each type ofconduction are obtained by linearizing (8) and (9).While the small-signal model of the Current Modecontroller in CCM can be derived ind ependently of theconverter, the model in DCM is dependent on theconverters operating point. It is determined inconjunction with the equations describing the m odel ofthe converter in DCM.

111. VO LTA GE ODECONTROLWe present in this section the open-loop line-to-output and control-to-output transfer functions, whenthe controlled variable is the duty cycle d . We refer tothis situation as to Voltage Mode Control.In the simulations made to compare the CPL case tothe resistive load case, the operating point was set so

that R, = R .A . Validation of the model

The transient response of the device level(switched) converter was compared with the transientresponse of the averaged model shown in Fig. 5 . Theconverter has the following parameters: 5 = 150 V,V, = 54 V, D = 0.36, L = 2 0 0 p H , C = O O p F . Thecomparison was done for both a CPL with P = 300W

R e = 9.7252 at V, = 54V ), and for a resistive loadR = Re = 9.7252 After reach ing the operating point,

the input voltage V = 150 V was stepped up to 160V.The transient responses in the CPL case are shown

inF ig . 10.The upper trace shows the output voltage of theswitched and averaged model. The middle trace show sthe inductor current in the switched model (shiftedupwards for clarity), and the lower trace shows theinductor current in the averaged models. The transientresponses match very well. It can be seen that themodel describes both CCM and DCM and that thesystem is unstable.

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Constant power load P=300W...............................................................................

, w

4w

w -

ICCM DCM

5111) ...............................................................................

0 U ( 3 a ) o U ( 3 b )

?I . 0- I ( L I , )I ( L 1 t . )Y y

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. ., ,S 1 o m 2 w 3.ms 4.911

,>:

B. Transfer functions in CCMThe small-signal model in Fig. 6is used to calculateThe line-to-output transfer functions is:the transfer functions.

where 4 = V:/P = D2Y2/P. he transfer functionI O ) has two po les in the right half plane, meaning thatthe system is unstable. The real parts is positive,1/(2ReC) and the poles are complex valued ifRe > , /L/(4C) . The resonance frequency isf o = 1/2nz/LC = 1 1 2 5 H ~ .function in the resistive load case is also presented.The Bode plot is presented in Fig. 12. The transferThe co ntrol-to-output transfer function is:

It has naturally exactly the same dynam ic propertiesas IO ) .

..................................................................................... 1o o TI ,I / Phase- CPL

0

Phase -Resistive loadi L

. 2 0 9 L .................. ...................................... ................... ....................1Ph l B D h l . I Y l D Y h l O M h ? . Wh

R o b e CursorC1 - 1 .1053H. 7 .7555r rq P C 'yFig. 12. Line-to-output transfer function in CCM,Voltage Mode Control.

C. Transfer functions in DCMUsing the DCM small-signal model in Fig. 7, the

transfer functions in DCM can be calculated. Theconverter operates in DCM if R > 2 L / [T( 1-D ) ]If we define K = 2L/(RT), the conversion ratio inDCM is:

(12)2M = v /Vi l + J a 'The line-to-output transfer function is:

The modulus Ho,( l - and the position of the polesdepend on the operating point and on the type of theload. Thus, while for a resistive load:

4

for a CPL is~o, l-o)~cpL = -For a resistive load, spl is a LHP pole,

1 2 - A 4and for a CPL, spl is also a LHP pole,

1 Ms p l l c p ~ R,C 1- MThe second pole is a LHP pole:s p 2 = -,2 F s M 2D2

(17)

where F, is the switching frequency. It can be seenthat this pole is situated close to the switchingfrequencyThe difference between the modulus of the transferfunction for resistive load and for CPL is maximum atthe boundary between CCM and DCM, and lowers as

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the equivalent load resistance increases. The position ofthe poles is also different in the two cases, but mostimportant is that they are in the left-half-plane.

The Bode plots in Fig. 13 ,obtained by simu lation,confirmed these theoretical results. The higher gain atlow frequencies in the CPL case, and the differentposition of the pole s p l can be observed. The inputvoltage of the converter, the duty-cycle and the valuesof L and C are the same as in the previous subsection.However, the load resistance is increased ( R = 200C2,P = 33W ), in order to obtain the DCM o peration.

..................................... .......................................................................... .......,

r ~ l , l ~ l , ,

'.yl U/ (012 ,.* I , M Z, l Y l W1 i P- P,?,Fig. 13. Line-to-output ransfer function in DCM,Voltage Mode Control,

The control-to-output transfer function has the sameform: -

The poles are given by 16)- 18).The modulus of the transfer function is, for aresistive load:

and for a CPL:

In this case, also, the gain at low frequencies in theCPL case is higher than the gain in the resistive loadcase, as it can be seen from 20) and 21).

Iv CURRENT M OD ECONTROLIn this section, we present the open-loop line-to-output and control-to-output transfer functions whenthe controlled variable is the inductor current. We refer

to this situation as to Current Mode Con trol.A . Transfer unctions in CCM

We choose a compensating ramp mu = 0.5m2, inwhich case the DC gain of the l ine-to-output transferfunction becomes zero. For this reason, only thecontrol-to-output transfer function w ill be presented.

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The control-to-output transfer function is:

The gain is:Go,(c-o) = R x T ( l - 2 0 ) m , R , 'RXI + +-2 L Vi

or, for mu = 0.5mz ,

2 LThe pole spl is given by,

RXT(1-2D) MR,+-l + 2 L v;RXCp l = ,

or, for mu = 0.5m2,R , T ( ~- D )

2 L+-s p 1 1 m u = 0 . 5 ~ R , CThe resistance R, in (23)+(26) depends on the type

of the load. For a resistive loadX I R = > ( 2 7 )

RxIcpr.= -Re * ( 2 8 )and, for a CPL,

Comparing 25) for the two types of load, one cansee that the gain is negative in the case of a CPL. Theconverter operates in CCM if R < 2 L / [ T ( 1 - D ) ].Suppose that the load resistance R (or equivalentresistance Re in the case of a CPL) is increasedtowards the value giving the boundary between CCMand DCM . It can be seen that the gain for resistive loadis limited, while the gain for a CP L tends to infinity.

The pole spl given by (26), is a RHP pole, in theCPL case. Also, the pole moves to origin as theconverter operating point moves towards the boundarybetween CCM and DCM.

The pole sp2 is close to the switching frequency,and is given by:

2Fs--Dp2 =--The Bode plots obtained by simulations are shown

in fig. 14. The reference current I , was set to obtainthe same operating point of the converter, as thatdefined in the beginning of Section III. It can be seenthe higher DC gain of the converter with CPL. Thephase of the system starts at -180 , the system has aRHP, and also a pole close to the switching frequencycan be seen.


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....................... ............................ ~ .................................~.~~...~ CPL Moduluso j Buck, control-to-output, CCMCurrent Mode ConLTol

.............~ . . ~................................. ~ .~~~~~~ ...............................bIj RM I

Current Mode Control.The poles o f the line-to-output transfer function arethe same as for the control-to-output transfer function,but the DC gain is theoretically zero for the particular

case of the compensating ramp ma = OSm .B. Transfer unctions inDCM

The control-to-output transfer function is:

An equivalent resistance is defined as:Rk = k . R , ,

where, for ma = 0.5m2 ,( 1 - M ) ( 2 - M )k = M+4(1- M ) ( 1 - 2 M )

The resistance R, in (31) depends on the type ofRxlR = R , and, for a CPL,~ x I c p L -Re .Then, the gain is:

the load. For a resistive load

The pole spl is given by:( 3 3 )

(34)The sp2 is close to the switching frequency .The Bod e plot is given in Fig. 15.

P=27W (R =lO8Q) b

I \ Phase,001

s2 1 1 1wz Y P , l l P I I I , Mi I am1 I I H Z I W H Z t 0 D X l 1 owlI r e l Y e n r yFig. 15 Control-to-output ransfer function, DCM,Current Mode Control

V . CONCLUSIONIn this paper a dynamic model of a Buck converterfeeding a Constant Power Load is presented. Theresults show that by using Voltage Mode Control, theopen loop is unstable in CCM and stable in DCM, andby using C urrent Mode Co ntrol, the system is unstable

in both conduction modes. This will pose additionalrestrictions on the applied control solution as com paredto the normal situation where a Buck converter isfeeding a purely resistive load.The authors are aware that modeling the d istributedsystem by two cascaded converters is a crudeapproximation of the real application. This research hasits origin in trying to model a complete distributedpower supply system. However, a good understandingof the behavior of a single unit (converter + CPL), andespecially of its dominant dynamic characteristics,must be achieved first. After this step, the completesystem can be modeled b y these single units.In the literature there are presented various waysthat could be applied for stabilizing the unit analyzed inthis paper. However, based on our experience, usingdirectly these single units in a large scale system, willnot necessarily produce satisfactory results. Furtherresearch will be don e to fi id a suitable control strategyto solve the overall problem.

REFERENCES[11 Kislovski, A., Olsson, E., C onstant-Pow er Rectif iers forConstant-Power Telecom Loads, Proc. INTELEC,Vancouver, Canada, 1994, pp. 630-634.[2j Suntio, T., Vallittu, P., Laurinen,T., Ikonen, M., Designof an AC/DC Power Supply for Telecom Applications,Proc. Finnish Workshop on Power and Industrial Electronics1997, FINPIE/97, August 26, 1997, Finland, pp. 85-92.[3] Middlebrook, R.D., Cuk, S., A General UnifiedApproach to Modeling Switching-Converter Power Stages,Proc. PESC 1976, Cleveland pp. 18-3 4.[4] VorpCrian, V. Sim plified Analysis of PWM ConvertersUsing Model of PWM Switch. Part 1: ContinuousConduction Mode, IEEE Trans. on Aerospace andElectronic System s, Vol. 26, No. 3, May 1990, pp. 490-496.[5] VorpCrian, V ., Simp lified Analysis of PWM ConvertersUsing Model of PWM Switch. Part 11: DiscontinuousConduction Mode, IEEE Trans. on Aerospace andElectronic System s, Vol. 26, No. 3, May 1 990, pp. 497-505 .[6] Rodriguez, F.D., Chen, J.E., A Refined NonlinearAveraged Model for Constant Frequency Current ModeControlled PWM Converters, IEEE Trans. on PowerElectronics, Vol. 6, No. 4, October 1991, pp. 656-664.[7] Liu, Y., Sen, P., Large-Sign al Modeling of H ystereticCurrent-Programmed Converters, IEEE Trans. on PowerElectronics, Vol. 11,No. 3, May 1996, pp. 423-430.[SI Erickson, R.W., Fundamentals of Power Electronics,Chapman & Hall, 1997.[9j M ohan, N., Undeland, T., Robbins, W., PowerElectronics. Converters, Applications and Design, JohnWiley & Sons, Inc., 1995.[lo] Kislovski A., Red1 R., Sokal N., Dynamic Analysis ofSwitching-Mode DC/DC Converters, Design AutomationInc., Massachusetts, 1991.

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