dimensionless approach to multi-parametric stability analysis of nonlinear time-periodic systems:...

IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: REGULAR PAPERS, VOL. 60, NO. 2, FEBRUARY 2013 491

Dimensionless Approach to Multi-ParametricStability Analysis of Nonlinear Time-PeriodicSystems: Theory and Its Applications to

Switching ConvertersHao Zhang, Member, IEEE, Yuan Zhang, Student Member, IEEE, and Xikui Ma

Abstract—This paper proposes a dimensionless approach toanalyze the multi-parametric stability behavior of switching con-verters, which can be characterized by a nonlinear time-periodic(NTP) system. The main objective is to analyze how multiplecircuit parameters affect the stability patterns of the derivedNTP system and to simplify the parametric complexity of suchNTP system. In contrast to previous work, the proposed methodfocuses on the parametric resultant relationships of the NTPsystem in the sense of topological equivalence, and investigatesits stability in terms of the homeomorphic NTP system. Firstly,an equivalent stability theory of NTP systems is proposed. Then,based on the equivalent theory, a normalized map is introducedand various interesting properties are derived so as to formulatethe dimensionless approach. Moreover, the approximate solutionof the NTP system in dimensionless parameter space is calculatedby using the Galerkin method, and its stability pattern is identifiedwith the help of eigenvalue analysis approach. Finally, a casestudy of one-cycle controlled Zeta PFC converter is discussedin detail to exemplify the application of the proposed method.These analytical results agree well with those ones obtained fromexperimental measurements.

Index Terms—dimensionless approach, equivalent stability,multi-parametric stability analysis, nonlinear time-periodicsystem, parametric resultant relationship, switching converter.

I. INTRODUCTION

A S A CLASS of typical piecewise smooth systems,switching converters can exhibit a great variety of non-

linear behaviors such as period-doubling bifurcation, Hopfbifurcation and chaos [1]–[17]. Due to the improperly de-signed values of some practically relevant circuit parameters,distortion behaviors can frequently occur, which will lead tothe rapid rise of device stress and the drastic degeneration ofsystem performance. Hence, the stability issue associated withbehavior boundaries in terms of some major circuit parameters

Manuscript received November 08, 2011; revised February 28, 2012; ac-cepted March 18, 2012. Date of publication October 09, 2012; date of currentversion January 24, 2013. This work was supported in part by the National Nat-ural Science Foundation of China (Grant No. 51177118, 50607015), in partby the Fundamental Research Funds for the Central Universities (Grant No.XJJ20100058), and in part by the Creative Foundation of the State Key Lab ofElectrical Insulation & Power Equipment, China (Grant No. EIPE11301). Thispaper was recommended by Associate Editor X. Li.The authors are with the State Key Laboratory of Electrical Insulation and

Power Equipment, School of Electrical Engineering, Xi’an Jiaotong University,Xi’an 710049, China (e-mail: [email protected]).Digital Object Identifier 10.1109/TCSI.2012.2215798

is very crucial to the dynamics knowledge of switching con-verters, and has become an important goal and subject of muchon-going research [4]–[8], [11]–[13], [16], [17]. However, withthe ever-increasing energy conversion requirement of powerelectronic systems, switching converters are developing towardthe trend of higher dimension and more complex structure,such as two-stage or even multistage cascade (or parallel) con-verter configuration [7], [8], [16]–[21]. Despite their gainingincreased acceptance in many applications of telecommunica-tions, aeronautics, and new IC technologies [22]–[24], theseswitching converter architectures have inevitably rise to thegreat challenge of stability analysis.Up to now, several typical approaches such as bifurcation

analysis, large- or small-signal analysis and Lyapunov func-tion method have been proposed to identify stability patternsof switching converters [11]–[17], [25]–[28]. Although theseprevious results have greatly improved the understanding ofthe system dynamical behaviors, they merely focused on singleparametric stability boundaries, which didn’t consider resultantrelationships among system parameters at all. For example, re-sistor , capacitor and line cycle were usually regarded asthree independent parameters no matter how separately or to-gether they were discussed in previous approaches. In fact, therephysically exist such resultant relationships among the three pa-rameters as . Thus, if the resultant relationship isn’t un-covered, these approaches will become too complex to enhancethe in-depth understanding of the systems dynamics, which re-sults in the fact that some useful information essential to design-oriented optimization is of great difficulty to obtain. Evidently,these parametric resultant relationships are quite indispensableto stability analysis. As far as switching converters of great com-plexity and high dimension are concerned, the system stabilityanalysis will become more complicate, which is certainly asso-ciated with these resultant relationships among multiple circuitparameters. In this sense, the actual concept ofmulti-parametricstability analysis should be based on the parametric resultant re-lationships. Nowwemay ask the following question: Does thereexist an appropriate approach to characterize the parametric re-sultant relationships? In the previous literatures on nonlinearanalysis of dc/dc converters [7], [14], a dimensionless techniquehas slightly involved the related question. However, these di-mensionless treatments failed to give a general non-dimension-alized rule, and never established the parametric resultant rela-tionships in the sense of the system stability. This reason is thatthere exists a major drawback in such technique, which is the

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492 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: REGULAR PAPERS, VOL. 60, NO. 2, FEBRUARY 2013

lack of a rigorous mathematical theory to formulate the aboveresultant relationships. Of course, this technique is not appli-cable to multi-parametric stability analysis at all.In this paper, a general dimensionless approach (DA) is sys-

tematically proposed to establish a theoretical basis for para-metric resultant relationships, and illustrate its applications tomulti-parametric stability analysis of switching converters. Theproposed method begins with a nonlinear time-periodic (NTP)system that can be regarded as a general model of practicalswitching converters. Unlike the previous studies, it presentsa rigorous mathematical theory for switching converters in thesense of topological equivalence, and investigates the stabilityof a derived NTP system by means of its corresponding dimen-sionless homeomorphic NTP system. For readability and effec-tiveness of exposition, we illustrate the method and its salientfeatures using a typical Zeta PFC converter as an example. Theanalysis is of help to us in doping out the resultant relationshipsin such NTP system on earth, and more significantly, some an-alytical results obtained here can potentially facilitate design ofswitching converters for stable operation.The paper is organized as follows. Section II gives detailed

definition of a nonlinear time-periodic system for practicalswitching converters. Section III establishes the framework onthe equivalent stability theory of homeomorphic NTP systems.In Section IV, based on the proposed equivalent stability theory,a systematic representation of the DA for multi-parametric sta-bility analysis is discussed in terms of dimensionless parametersets. In Section V, stability patterns of the Zeta PFC converterare analyzed as an example to exemplify the proposed method.In Section VI, experimental results and power unbalance anal-ysis are presented for verification purposes. In Section VII,some interesting behavior boundaries of the converter areillustrated. Finally, some remarkable conclusions are arrivedat in Section VIII. The paper also contains three Appendixes,which address the proofs of the equivalent stability theorem(A), and provide details of coefficient matrices (B) as well asconstraint equations (C) for the Zeta PFC converter.

II. DEFINITION OF NTP SYSTEMS FOR SWITCHINGCONVERTERS

If the piece-wise smooth feature of switching converters isreplaced by their overall smooth trait after a standard averagingapproach [2], then their low-frequency dynamical behaviors canbe described by the following differential equation:

(1)

where is a nonlinear continuous smooth function definedin with an initial conditionand represent the time interval and the variable domain, re-spectively.Moreover, if the right-hand side function of (1) satisfies

for , where is the least period, thenwe define such system as a NTP system.It is worth noting that these converter systems are modeled by

the low-frequency model as shown in (1), which is, of course,not applicable to the fast-scale dynamics at the time scale ofswitching frequency. In this paper, we focus on low-frequency

dynamics of the switching converters with periodic force inputsor outputs.Note that the solution in (1) is usually a non-zero one.

Thus, for the convenience of later discussion, we introduce thefollowing transformation:

(2)

where is the perturbed solution of . Substituting (2)into (1), becomes the equilibrium point for the followingsystem:

(3)

Since

(4)

(5)

Subtracting each side of (5) from the corresponding side of(4), one obtains

(6)

combining (3) with (6), we get the transformed system asfollows:

(7)

where for . Hence, (7) is alsoa NTP system, which holds the same period of (1). For brevity,

denotes the perturbed solution of (7) with an initialcondition .the stability properties of such as (uniform) stability and

(uniform) asymptotical stability are fully equivalent to those ofthe equilibrium point , which can be accurately expressedby the Lyapunov definitions [29]. Moreover, the following the-orem on the relation between the local stability of a NTP systemand its corresponding linearized system is relevant to our sub-sequent study. For conciseness, we refer the readers to [30] fora detailed proof.Theorem 2.1: Assume that is sufficiently smooth (i.e., at

least ) for the NTP system (1), and• let be its transformed zero solution systemdefined in (7);

• let be the Jacobianmatrix ofwith respect to , evaluated at the solution .

Suppose that there exists a common period betweenand , and then for the linear equation with the followingperiodic coefficients:

(8)

if is an asymptotically stable equilibrium point of (8), thesolution of (1) is asymptotically stable.Remark 2.1: The least period of and are denoted

by and , respectively. As indicated in [31], don’t needto be the same as . However, the two least periods shouldbe commensurable, namely, satisfy , whereand are the least positive integers if available. Thereby, we

ZHANG et al.: DIMENSIONLESS APPROACH TO MULTI-PARAMETRIC STABILITY ANALYSIS 493

define as the common period for and .In particular, if is equal to , one obtains .

III. THEOREMS ON EQUIVALENT STABILITY OF NTP SYSTEMS

A. Time Reparameterization

In general, the stability of the NTP system (1) depends on itsstructural parameters. For the sake of clarity, the NTP system(1) is rewritten as the following parameter-dependent system:

(9)

where is a time-invariant circuit parameter vector,is a parameter set defined in and is of perioddefined in . The unperturbed solution of (9) is periodic,satisfying for .As is mentioned above, there exists a common period for

the NTP system , which is generally related to the externalforcing period of the system. However, such periodic featurefails to be explicitly expressed in (9), and accordingly the con-nection between the forcing period and the natural periods1 ofthe system cannot be easily obtained. Note that the stability isa systematic property with time approaching infinity. Thus, forthe NTP system , the information relevant to transient andtime- dependent behaviors can be omitted, while only the in-formation on the stability of cycles needs to be preserved. it isnecessary to introduce a time reparameterization process for theNTP system [32]. Suppose that is a smooth positivenumber with respect to and there exists an invertible map

(10)

where and . Then, is transformed intothe following time-parameterized system:

(11)

where denotes the derivative of with respect to . Obvi-ously, the periods of and satisfiesand .

B. Theorems on Equivalent Stability

In this subsection, we consider the stability properties of NTPsystem in the sense of topological equivalence. Note thathomeomorphic spaces possess the same topological propertyand there is certainly no intrinsic difference among homeomor-phic spaces [32], [33]. However, in terms of generality and con-venience, there can be a significant difference. By using a properhomeomorphism, a physical problem of high parametric com-plexity may be homeomorphic to an easier and more generalone with reduced parametric complexity so that its topologicalproperties are more easily determined.There are, however, two basic questions to be appropriately

answered. One is whether the stability of a NTP system isequivalent to that of its homeomorphic NTP system; the otheris whether there exists a proper homeomorphism retaining theequivalent stability. Based on the two questions and the above

1Roughly speaking, a natural period is the intrinsic time scale of a givensystem related to its own physical parameters, describing the time process for astudied phenomenon.

time reparametrization, we propose the following two theo-rems on the equivalent stability of NTP systems as sufficientconditions. Detailed proofs are given in Appendix A.Theorem 3.1: Consider the NTP system . Suppose that

is stable in the neighborhood of its cycle when . Ifthere exist two invertible maps and satisfying that• possesses a smooth positive function such that

where ;• is an invertible linear map as suchthat where is the corresponding matrix.and are the two domains defined in .

Let be the product map of and with the form of

(12)

Then, the following three statements hold.1) The mapped system is a NTP systemwith a parameter vector , and is the vector function from

to .2) For the NTP systems and , their cycles are stable iffthey are uniformly stable.

3) The (uniformly) stability of in the neighborhood ofis equivalent to the (uniformly) stability of original systemin the neighborhood of .

Note that Theorem 3.1 deals with the Lyapunov stability of. However, for practical applications, it is always desirable

that the operating point of switching converters would even-tually converge to the original steady state once the perturba-tion factors vanish, i.e., the switching converters are asymptoti-cally stable. the following asymptotic equivalent stability theoryshould be proposed.Theorem 3.2: Consider the NTP system . Suppose that

is asymptotically stable in the neighborhood of its cyclewhen . If the product map is of the same definitionshown in Theorem 3.1, then besides the first statement 1) pro-posed in Theorem 3.1, the following two statements hold.1) For the NTP systems and , their cycles are asymptot-ically stable iff they are uniformly asymptotically stable.

2) The (uniformly) asymptotic stability of in the neighbor-hood of is equivalent to the (uniformly) asymptoticstability of in the neighborhood of .

Here Theorem 3.1 and Theorem 3.2 are referred as the equiv-alent stability theorem for the NTP systems. It should be high-lighted that Theorems 3.1 and 3.2 give the sufficient conditionsfor the equivalent stability properties of NTP systems. Namely,the product map can be selected as a combined invertible mapwith and . According to the proofs, a significant feature ofis the boundedness of its submap . However, it is not true

for every homeomorphism in general2. the first question pro-posed earlier in this subsection will not be fully established. Anecessary and sufficient condition may possibly exist for certainbounded operators, which is not the focus of the interest herein.Note that the stability behavior of depends on the param-

eters assigned in . By using the proper map , a param-eter-dependent map can be induced between the original pa-rameter set and the mapped parameter seti.e., (where and can be equal or not),

2For instance, the tangent homeomorphism belongs to an unbounded operatordefined in a bounded region due to its unbounded image with

.


which maps the cycles of at onto the cycles of at, preserving the direction of time. Moreover, if

the dimension between and satisfies , then thedimension of original parameter space of will be reduced,which may simplify the parametric complexity of the originalNTP system. As will be shown shortly, the main function of theproposed theorems lie in this.

C. Parametric Dimensionality Reduction

In the following subsection, some discussions on how tochoose the invertible matrix in the map are provided inorder to guarantee a simplified parameter space with reduceddimension. For clarity, the parameter vectors and are writtenas follows:

(13)

(14)

where and .Since the coefficients in (9) contains the elements and, a feasible way to reduce the number of the elements inis to represent by using certain combinations of andsuch as their sums or products, i.e.,

or . Moreover, according to the Buckinghamtheorem [35], if the dynamical behavior of the NTP systemdepends upon dimensional variables3, then by taking

a suitable transformation to remove their units in order to yieldsome dimensionless variables, the system can be reduced to only

dimensionless variables, where the reduction numberdepends upon the problem complexity

that equals to the number of its fundamental dimensions. Notethat and are the dimensional circuit parameters of theswitching system, which are embodied as coefficients in (9).Thus, a compact form of can be selected as an invertible diag-onal matrix, which transforms the dimensional state vector

into the dimensionless state vectoras follows:

(15)

where and the invertible matrix is written as

(16)

Consequently, the key point of selecting is to choose aproper which should ensure that the circuit state variablecan be transformed into the dimensionless state variable ,

and is usually a combination of and as aforemen-tioned.

IV. THE DA FOR MULTI-PARAMETRIC STABILITY ANALYSIS

A. Representation of the DA

As discussed in Section III, by removing units from the NTPsystem with physical quantities to yield its dimensionless vari-ables, one can get an equivalent NTP system of reduced param-eter dimension. an apparent purpose of the DA is to reduce the

3The variables here are extensional notions, which include the system’s statevariables and the circuit parameters. For the NTP system (9) studied in thispaper, they consist of the state vector and the parameter vector .

number and complexity of variables by shaping them into di-mensionless forms. However, a more significant role lies in re-vealing some fundamental properties of the complex system andgiving new insights into the resultant relationships among var-ious parameters, which affect its dynamical behaviors.A famous example to illustrate the above role is the dimen-

sionless Reynolds number in fluid dynamics [35], whichis used to characterize the fluid in a pipe being either laminarflow or turbulent flow. Note that Reynolds number involves acombination of various factors into a dimensionless variable

, and subsequently analyze the given physical phenom-enon in an equivalent manner. Inspired by this dimensionlesstreatment, it becomes possible to investigate the parameter-de-pendent NTP system in its dimensionless space. This formu-lation, together with the equivalent stability theory, brings aboutthe complete establishment of DA as follows.Firstly, the time reparameterization function in the product

map should represent the characteristic period, wherein thefundamental dynamics of the system are captured. Since we aremainly concern with the low-frequency dynamics of switchingconverters, can be selected as the reciprocal of the commonperiod in the system under study, i.e., .Secondly, we choose the invertible matrix as shown in (16).

Then, by using the map , we get the dimensionless variablein the product space . Based on Theorems 3.1 and

3.2, the original dimensional NTP system is transformed intothe following dimensionless NTP system:

(17)

where is the dimensionless parameter vector. The commonperiod of and equals to 1 here.Finally, we define the product map and

the induced parameter-dependent map as the nor-malized map and the parameter normalized map, respectively.As highlighted earlier, the dimension of original variable spacecontaining state variables and circuit parameters can be reducednow, and its reduction number equals to the number ofthe fundamental dimensions of the NTP system.

B. Some Properties of

So far, we have derived the dimensionless NTP system. Notethat is an invertible square matrix and is a smooth pos-itive number. Referring to Theorems 3.1 and 3.2, the (asymp-totic) stability of the dimensionless system near is fullyequivalent to the stability properties of the dimensional systemnear . one can perform the multi-parametric stability

analysis in the dimensionless NTP system, which not only re-tains equivalent stability properties but helps to establish theparametric resultant relationships on the system stability. Sev-eral straightforward properties of the map , regarding to theparametric resultant relationships, can be naturally held as fol-lows and are stated without proof.1) For such that is (asymptotically) stable, thenthe dimensionless parameter vector canyield to be (asymptotically) stable.

2) Assume that there exists a dimensionless parameter vectorsuch that is (asymptotically) stable. Then, it

follows that for such that would makebe (asymptotically) stable.


3) Considers the stability boundaries of andof . If there exists a critical parameter vector

, then . Moreover, if there exists, then it follows that such that

for .Note that can establish the relationship between the dimen-

sionless stability boundary and its dimensional boundary .Thereby, the parametric resultant relationships among all groupsof circuit parameters affecting the system stability can be visual-ized by plotting a few behavior boundaries in dimensionless pa-rameter space. However, boundary margins are often expressedas 3-D surfaces or 2-D curves. For effectiveness of exposition,we suppose the parameter vectors and can be divided intoseveral subvectors, such as and , wherethere is no coupling relation between and . Thus, we get theparameter normalized submap ,where and are defined in (18)

(18)

the original parameter sets (or ) and (or ) can bedivided into the following independent subsets (or ) and

(or ), which satisfy

(19)

(20)

Hence, the above properties of the map are also suitable forits submap . As such, the multi-parametric stability analysisbased on the parametric resultant relationships can be carriedout by means of several families of dimensionless parameters.

C. Stability of Dimensionless Periodic Solutions

In this subsection, we employ the Galerkin method to get theapproximate solution of (9), and investigate its stability patternsvia an eigenvalue analysis approach.1) Dimensionless Periodic Solutions: Note that the state

variables of a switching converter generally consist of thevoltage across capacitors and the current through induc-tors in its power and control stages. Thus, the state vectorof (9) can be decomposed into where

and are the powerstate vector and the control state vector, respectively. Likewise,the corresponding dimensionless state vector of (17) can

be represented as , whereand . For convenience, we construct thefollowing diagonal matrix as . Multiplying

by both side of (17) simultaneously, we obtain

(21)

Then, we add the first rows of (21) on each side and thefollowing equation is readily acquired:

(22)

where denotes the th element of and is thecorresponding th row of the . At first glance, thereare at least state variables in (22), which seems difficult toget their periodic solutions. However, there lie often some extraconstraints among these variables based onsome fundamental circuit theories and feedback control featuresof the system. Suppose that there are constraint equationsfor the system (9) with state variables of the following form:

...(23)

Putting (23) into (22), the state variables in (22) can be thenreduced to an unique and independent variable, which is denotedby . the multivariable NTP system is simplified into a single-variable nonlinear differential equation which only containsas follows:

(24)

where is a differential operator. Since is a continuous peri-odic vector of period 1, (24) can be solved in the time interval[0, 1]. It is well-known that the solution of a given NTP systemcan be formulated via the Galerkin method [36]. The key idea ofGalerkin method is based on the fact that any function in afunction space can be uniquely expressed as a linear combi-nation of the linear independent basis function . Thus, anapproximate solution of is written as

(25)

where is the coefficient for the th linear independent basisfunction. Then, the residual error satisfying the differentialequation (24) with terms of the sum (25) is defined as

(26)

According to [36], the residual error should be orthogonal toeach basis function as follows:

(27)


Due to the orthogonality of the trigonometric function set, wecan simply choose as

(28)

By substituting (26) into (27), one obtains

(29)where .To utilize the approach, all we need to do is to solve these

algebraic equations for the coefficients in (29).According to [36], the Galerkin approximation wouldconverge to the exact solution of if is sufficiently large.For of order , it is feasible to neglect high order basisfunctions beyond the th if a required accuracy is satisfied.Here we simply denote the Galerkin approximation asif no confusion arises. Then, the other solutions of canbe easily calculated by putting into (23). As such, the numer-ical solution is obtained.2) Stability of Periodic Solutions: So far, we have got the pe-

riodic solution of the NTP system (17). According to Theorem2.1, the stability of can be investigated by its correspondinglinearization system. Suppose in (17) satisfies ,then the asymptotic stability of can be equivalently char-acterized by its linear-time periodic equation as follows:

(30)

where denotes the Jacobian matrix at . In general,an eigenvalue analysis approach [37] can be one of the mostpowerful tools to analyze the stability properties of (30). Theapproach plays an important role in investigating the relationbetween the stability patterns and the eigenvalues of the mon-odromy matrix for (30). Since detailed derivations of a mon-odromy matrix for a given linear periodic equation has beenproposed in our earlier work [12], we omit the explicit form ofthe matrix here and only give the results of the calculated eigen-values as follows:

(31)

where the eigenvalues obtained in (31) can provide enough in-formation to analyze the stability of the cycle and to iden-tify its bifurcation patterns when the parameter varied. As in-dicated in [38], if , for all , the solu-tion is asymptotically stable. However, if , for some(at least one), then the associated solution is unstable. Besides,the unstable behaviors of the system depend on the manner inwhich the eigenvalues leave the unit cycle. Specifically, if a pairof complex conjugate eigenvalues across the unit circle, then aNeimark-Sacker bifurcation pattern occurs.

V. APPLICATION OF THE DA TO ZETA PFC CONVERTER

A. Zeta PFC Converter and Its Mathematical Model

1) One-Cycle Controlled Zeta PFC Converter: The one-cycle controlled (OCC) Zeta PFC converter as shown in Fig. 1contains the power stage, the output voltage controller and the

Fig. 1. One-cycle controlled Zeta PFC converter. The parameter values areV–70 V, mH–1.8 mH, mH, F,F, V, k

k –7 k , k , nF, kHz.

OCC PWM modulator. Since detailed circuit configuration andoperation have been illustrated in [39], [40], we only give thefollowing remarks on its operation. Firstly, the system is re-stricted to CCM operation owing to the fact that almost all ofpractical medium and high power supplies usually work in thisoperation. Secondly, we focus on the case that once reachesduring one switching cycle, the integrator will be reset to the

normal operation. Here, we denote as ,where and represent the rms value and the line voltagefrequency, respectively.Note that the Zeta PFC converter (without the voltage con-

troller) can be modeled as

(32)

where equals toand equals to . Note that the coefficientmatrix can be easily obtained from Kirchoff’slaws, and the voltage controller can be formulated by

(33)

where and. Here

and denote the dc gain and time constant of thefeedback network, respectively. By combining (32) and (33),the closed-loop model of the OCC Zeta PFC converter is for-mulated as follows:

(34)

where . The coefficient matrices

and stand for

and respec-tively, and their detailed expressions are given in Appendix B.


Furthermore, since the OCC block is a built-in PWM modu-lator, the duty cycle can be dictated as follows: [39], [40]

(35)

we get the following general form:

(36)

Substituting (36) into (34), we obtain the mathematical modelof the system as follows:

(37)

Note that the Zeta PFC converter is a parameter-dependentNTP system as shown in (9), where its common period equalsto the periods of and , i.e., . Forbrevity, we omit the average overbar for the correspondingstate variables in the following discussions.2) Dimensionless Mathematical Model: Here we apply the

DA derived in Section IV to get the dimensionless mathematicalmodel of (37). First, we choose as , i.e., andselect as . Then, bytaking upon (36), we obtain

(38)

Likewise, by applying to (37), we get the following dimen-sionless NTP system (after rearranging):

(39)

where and. The explicit form of (39) is shown in (40),

shown at the bottom of the page, where the dimensionlessparameters are specifically expressed as

(41)

where and in (13) and (14) aredenoted as follows, respectively:

(42)

(43)

the parametric resultant relationships affecting the stabilityof the Zeta PFC converter are revealed by the equations shownin (41) after the DA, and the dimension reduction equals to thenumber of the fundamental dimensions of the NTP system. Infact, the terms of in (41) can beregarded as the natural periods of the system, which will bediscussed later.Although is a dimensional parameter, it is often fixed as a

constant in practical applications. Hence, we omit in anduse the simplified one in the rest of the discussions. Thus, theparameter subvectors and defined in Section IV can bewritten as

(44)

where the parameter subsets andsatisfy (19) and (20).Moreover, the constraint equations among and

can be easily obtained as follows, and their detailed derivationare given in Appendix III:

(45)

By combining (22), (38) with (45), we get the explicit formof (24) as shown in

(46)

(40)


Fig. 2. Comparison of the dimensionless solution waveforms by usingnumerical simulation performed in MATLAB/SIMULINK (solid line) and theGalerkin Approach of the DA model (dashed line). (a) . (b) .

Due to the limit of length, we cannot provide the detailed formsof its partial derivation and only give the result of forthe converter as shown in (47).

(47)

B. Eigenvalue Analysis Using the DA

1) Periodic Solutions by the Galerkin Approach: Here, thedimensionless parameters are listed as follows:

and, which the corresponding dimensional param-

eters are V, k and mH. Here wechoose because the accuracy of the approximate solu-tion from the Galerkin approach can be ensured adequately inmost cases. Thereby, the dimensionless solutions and arecalculated as follows

(48)

(49)

From (48) and (49), it follows that these analytical resultsagree very well with those ones obtained from the numericalsimulation, as shown in Figs. 2(a) and (b).2) Eigenvalue Loci: Now, we use the derived monodromy

matrix to investigate the possible bifurcation pattern of suchNTP system. Specifically, we investigate the movement of theeigenvalues as the dimensionless parameters andare varied, which correspond to the variation of and. Table I shows the changing trend of the five eigenvalues as

and are varied, respectively. We clearly observe thatthe loci of a pair of complex conjugate eigenvalues andbegin to cross the unit circle as increases to 44.3733 (i.e.,

arrives at 5.348 k when V, mH),moves away from 0.041391 (i.e., removes from 60 V as

Fig. 3. Movement of the eigenvalues with the increase of , where Fig. 3(a)shows the trend of the loci of eigenvalues as is increased from 34.9842 to44.4329 and Fig. 3(b) depicts the detailed loci corresponding to Table I(A).

TABLE IEIGENVALUES FOR DIFFERENT VALUES OF DIMENSIONLESS PARAMETERS

AND IN THE ZETA PFC CONVERTER WITH OCC (A) WITHAND , (B) WITH

AND , (C) WITH AND

k mH) and reaches(i.e., reaches 1.77855 mH when V, k ),while other eigenvalues and still stay inside the unitcycle. Thus, this implies Neimark-Sacker bifurcation occurs. Tomake the movement of the eigenvalues more intuitive, we plottheir loci as is increased (see Fig. 3). Here, we omit the othertwo figures for and to save space since they are similarto Fig. 3.

VI. EXPERIMENTAL RESULTS AND POWER UNBALANCEANALYSIS

A. Experimental Observation

To further validate these results, experimental tests of theOCC Zeta PFC converter were carried out. Experimental wave-forms are presented in Fig. 4, which show that oscillatory insta-bility occurs as the parameter values of and are smalleror the value of is larger than one in normal operation. Theseresults are in good agreement with the theoretical predictions


Fig. 4. Experimental waveforms of the one-cycle controlled Zeta PFC con-verter ( : 1 A/div; : 50 V/div; time: 5 ms/div). (a) Stable operation for

k V, mH. (b) Oscillatory instability fork V, mH, which manifests as medium-fre-

quency oscillation of . (c) Stable operation for V, kmH. (d) Oscillatory instability for mH, V,

k , which manifests as medium-frequency oscillation of .

presented in Table I. Fig. 4(a) shows the measured waveforms ofthe output voltage and the rectified input current for kwhen V, mH, where the system is stable for

W. Fig. 4(b) shows these measured waveforms asdecreases to 5.3 k and the system becomes unstable, whereincreases to 115 W. As the input voltage is increased from60 V to 70 V when k mH, the measuredwaveforms (see Fig. 4(c)) become normal again compared tothose shown in Fig. 4(b), and increases to 125 W. However,the oscillatory instability appears again as increases to 1.8mH when k V (see Fig. 4(d)), andremains near at 125 W.

B. Power Unbalance Analysis

As discussed in [20], the Zeta PFC converter belongs to theType I-IIA configuration, in which the power flows between andwithin the power stage are cataloged into the high-frequencypower flow and the low-frequency power flow. Figs. 5(a) and(b) show the equivalent high-frequency power flow diagramsfor the Zeta PFC converter under CCM operation. Firstly, whenthe equivalent synchronized switches for converters 1 and 2 turnon, the inductor of converter 1 absorbs energy from the input,while transfers energy through of converter 2 into theload with low-frequency buffering capacitor . Then, after theswitches turn off, starts to release the energy absorbed inthe previous on-time interval to , which will retransfer thatenergy to converter 2 as shown in Fig. 5(b). Also, releasesenergy to and .the system operation can be regarded as the process of con-

stantly absorbing and releasing energy in the form of chargingand discharging for inductors and capacitors within high-fre-quency cycles, which results in an overall low-frequency powerflow. It is clear that if the values of and are rel-atively small, the dynamical response of energy delivery willbe rather fast. One can see shortly that the transferring speed ofenergy around the high-frequency, i.e., the power flow response

Fig. 5. Equivalent high-frequency power flow diagrams for the Zeta powerstage under CCM operation. The switches of the converter 1 and 2 are synchro-nized by the same switching sequence. (a) Power flows as the switches turn on.(b) Power flows as the switches turn off.

(PFR) of the power stage does affect the system stability. Gen-erally, the low-frequency power flow lies in two different man-ners. Firstly, if the power demand of the load is met by rapid PFRthrough a well-designed power stage which accomplishes therapid PFR requirement, then the high-frequency power flow or-derly occurs. Thus, the system has stable low-frequency powerflow, and its instantaneous power balance relation among theinput power , the output power andthe buffered power in satisfy

(50)

However, if the dynamical responses of the inductors and ca-pacitors around the high-frequency fail to meet the rapid PFRrequirement, the buffering capacitor will lose the role of ab-sorbing (filling) the instantaneous power surplus (deficit). Then,the power balance relation (50) will be untenable, which leadsto instantaneous power unbalance. The residual powerin and equals

(51)

which causes the oscillatory surge of the system with an un-stable low-frequency power flow.The above discussions lead to an interesting idea of the power

unbalance analysis, i.e., the instantaneous power unbalance re-sults in an oscillatory manner of the system and such oscilla-tory dynamics tend to attenuate the power unbalance scenario.Specifically, when decreases from 7 k to 5.3 k , the outputpower demand is increased from 80 W to 115 W. If (50) can bemet by the rapid PFR, the system is stable. Unfortunately, sincethe selected parameters and fail to meet such require-ment, the power unbalance mode shown in (51) occurs with alarge magnitude of oscillations depicted in Fig. 4(b). However,as increases from 60 V to 70 V, the PFR is accelerated andthe power unbalance mode ceases as shown in Fig. 4(c). More-over, if the PFR slows down by the increase of , the powerunbalance case also occurs due to an unmatched power require-ment, which leads to the oscillation shown in Fig. 4(d).From the above power unbalance analysis, the aforemen-

tioned parametric resultant relationships can be regarded as acompromised effect on the power balance of the system, whichis subjected to major circuit parameters in the power stage, thevoltage controller and the input/reference voltages.

C. Dimensionless Distributed Discharge Time Constants

Note that and represent the discharge (or charge)time constants of the well-known and fundamental


circuits. Now, if the time constants and are rela-tively small, the discharge (or charge) process will be faster thanthat of large time constants. That is to say, the PFR of the in-ductor and the capacitor will be rapider during the chargeor discharge process. Inspired by such concept, we define thedimensionless variables and shown in (41) as di-mensionless distributed discharge time constants , i.e.,

and as follows:

(52)

As aforementioned, the system stability is associated with thecircuit parameters in the power stage, the voltage controller andthe input/reference voltages, which result in a resultant effect onthe power balance of the system. In fact, the can be usedto describe such effect in the power stage. Moreover, the di-mensionless parameters and for the voltage controllerand the dimensionless parameter for the input/referencevoltages can also be applied to formulate this effect. We drawsome interim conclusions on the multi-parametric stability ofthe system according to the above discussion.• The increase of corresponding to the increase ofwill reduce the discharge (or charge) process in the powerstage, which results in a slower PFR. The power unbalancecase will arise because of the unmatched power require-ment, which leads to oscillatory instability of the system.

• The decease of corresponding to the increase of willcause an increasing demand of the output power. If theincreasing power demand fails to bemet by the power stagewith a given , then the power unbalance case occurs,which results in the oscillatory instability.

• The increase of corresponding to the decrease ofwill enlarge the discrepancy of power supply and demand.When the relatively “increasing” power demand, com-pared with the decreasing power supply, fails to be metby a given power stage, the oscillatory instabilityemerges due to the power unbalance mode.

VII. FURTHER APPLICATION

In this section, we will apply the method developed in thispaper to visualize the qualitative behaviors of the system bothin the dimensionless and dimensional parameter spaces, andpresent the boundary surface and curves of stable region andoscillatory unstable region in terms of practically relevant pa-rameters. Specifically, theoretical dimensionless boundaries areshown in Figs. 6(a) and 7(a), which is obtained by using theanalytical method derived in Section V. Using the parameternormalized map, we get the circuit parameter boundary sur-face as shown in Figs. 6(b), 7(b) and 8. Also, we take a fewcross sections from the boundary surface as indicative boundarycurves shown in Figs. 6(c)–(d) and 7(c)–(d). Experimental dataare plotted along with the analytical results for verification pur-pose. To emphasize the parametric resultant relationships on thestability types of the system, the following general discussionsare made.Firstly, the oscillatory instability is prone to occur for rela-

tively large values of or small values of or . In general,a relatively small can increase the PFR and enlarge

Fig. 6. Stability boundaries for the Zeta PFC converter with OCC plotted in theparameter space and . (a) Boundary surface underdifferent and for and

. (b) Boundary surface plotted in under differentfor F, k and V. (C) Cross

section curves for F under different . (d) Cross section curves formH under different .

Fig. 7. Stability boundaries for the Zeta PFC converter with OCC plotted inthe parameter space and . (a) Boundary surface for

.and . (b) Boundary surface for V andmH under different . (C) Cross section curves for

ms under different . (d) Cross section curves for under different.

the stable operation region of the system. However, the oppo-site scenarios for the small and , which enlargethe unstable margin, can be attributed to the insufficient energydelivery and buffering in the power stage.Secondly, the variation of the load resistance plays

a twofold role in the change trend of the. Specifically, as decreases, the

and increase, while the


Fig. 8. Stability boundaries for the Zeta PFC converter with OCC plotted inthe dimensional parameter space for mH, kunder different . Here, the critical bifurcation points of forand equals to 0.04386 and 0.04135, respectively.

and also decrease. In fact, the increase of can en-large the stable operation boundary, and the increase oftogether with the decrease of and will reduce the stableboundary. the system stability will be a trade-off between theabove two opposite trends. As a result, the impact of loadresistance on the system stability is not significant as shownin Figs. 6(c), (d) and 7. For this reason, extensive behaviorboundary as being a variable has not been analyzed herein.Thirdly, the input/reference voltages have a significant ef-

fect on the system stability, and the oscillatory manner of thesystem can be eliminated by increasing the input voltage or de-creasing the reference voltage. a power balance mode can be ac-complished by increasing the input voltage to improve the inputpower supply, or by decreasing the reference voltage to reducethe output power demand.Finally, the oscillatory instability here tends to occur for rela-

tively small as shown in Fig. 8. Since the output power de-mand arises as becomes smaller, the power unbalance caseis more prone to emerging under a poor power matching degree,such as a lower input voltage. In fact, a relatively large andcorresponding to small and will aggravate the power

unbalancemode drastically. However, nomatter what the valuesof is, the oscillatory boundary region is not almost affected.This reason is that its corresponding dimensionless parameterhas little effect on the system power delivery.

VIII. CONCLUSION

A general methodology for investigating parametric resultantrelationships of parameter-dependant NTP systems in the home-omorphic space has been proposed systematically, which leadsto the DA for multi-parametric stability analysis of switchingconverters. We start with the concept of NTP systems whichformulates the general mathematical model of switching con-verters. Then, we prove that under a proper map, the stabilityanalysis of the original NTP system can be simplified to thatof the homeomorphic NTP system with a lower parameter di-mension, but possesses equivalent stability properties. Subse-quently, the DA is carried out based on the derived map, and aspecific example of the Zeta PFC converter is finally given tovalidate the proposed method.

It is shown that the dimension of circuit parameter space canbe reduced by using the DA, and the parametric resultant re-lationships are carried out by means of influencing the powerunbalance modes of the system, which can be analyzed by sev-eral families of dimensionless parameters. In contrast to pre-vious works, the parametric resultant relationships of the NTPsystem have been fully considered in the sense of topologicalequivalence. It has been highlighted that the proposed method iscapable of both simplifying the system parametric complexityand revealing how these resultant relationships affect the sta-bility patterns.Another important aspect of our study here is to identify the

behavior boundaries bymeans of major circuit parameters, fromwhich plenty of useful information can be used to give somedesign-oriented guidelines for stable operation. Compared withprevious results, multi-parametric stability behavior boundariesin the form of 2-D curve or 3-D surf are displayed as one lineor surface rather than one point in a mapped parameter space.That is to say, the proposed method can provide a family of cir-cuit parameters rather than a set of ones so that optimal designof circuit parameters becomes possible. Although the approachis applied to the Zeta PFC converter as an example, the method-ology of using homeomorphic space map and dimensionless pa-rameter set is applicable to the multi-parametric stability anal-ysis of other switching converter circuits, such as the two-stagePFC converter studied in [13].

APPENDIX APROOFS

For the proofs, we need two auxiliary results. One is con-cerned with the equivalent relation between stability and uni-form stability for the zero solution of (7), and the other is re-lated to the equivalent relation between its asymptotic stabilityand uniformly asymptotic stability.Lemma 1 ([34], Theorem 1.8.9): For the NTP system (7), the

equilibrium of (7) is stable iff it is uniformly stable.Lemma 2 ([34], Theorem 1.8.11): For the NTP system (7), its

zero solution is asymptotically stable iff it is uniformlyasymptotically stable.Remark L.1: Lemma 1 implies that if the NTP system (7) is

stable over and , then for( is independent of ), as long as

, the perturbed solution of (7) with the initial statewill satisfy for . Thus, the (uniform)stability of the zero solution can be fully characterized bythe case of ordering . Similar to Lemma 1, the (uniformly)asymptotic stability of the zero solution can also bethoroughly investigated by ordering for Lemma 2.

A. Proof of Theorem 3.1

For notational simplicity, the parameter vector in (9) isomitted. Proof of each statement in Theorem 3.1 is given as fol-lows.1) As discussed in Section III, the periodic feature of the orig-inal system still lies in the map , i.e.,and . By applying the map to theNTP system (11), we obtain the mapped system whosevector field satisfies for .


Note that and. This implies that the mapped system is also

a NTP system.2) The zero solution systems for and for aredescribed by the following expressions, respectively:

(53)

(54)

where andfor . As mentioned in Section II, the (uniform)stability of and would be fully equivalent to

and respectively. Based on Lemma 1, the cyclesof and are stable iff they are uniformly stable.

3) Note that the proof of the third statement 3) in Theorem3.1 can be reduced to proving the (uniform) equivalent sta-bility of and of the corresponding NTPsystem and . For notational simplicity, de-notes the (uniform) equivalent stability between and ,and does the same relation between and .Let be a transition system after the map with theform of

(55)

Since is a NTP system, is also be a NTP system.Hence, the proof of can be decomposed to prove

and .: Note that is a NTP system. Considering

Lemma 1, the equilibrium of (53) is stable, if andonly if it is uniformly stable. Following the Remark L.1one can get for such that

(56)

Let be zero, thus the statement (56) can be completelyequivalent for such that

(57)

Obviously, for , it follows thatsuch that

(58)

Note that is a NTP system. Using Lemma 1, one canobtain for such that

(59)

Hence, the (uniform) stability of of is estab-lished, and vice versa, i.e., .

: Note that the zero solution of of is(uniform) stable. Then, for the essential condition part of(59), one obtains

(60)

where is the matrix defined in Theorem 3.1. Accordingto the comparability of matrix norm, we get

(61)

Since , then we obtain

(62)

Therefore, substituting (61) and (62) into (61), we obtain

(63)

Denote as , then can be written as .For conciseness, also denote as . Hence,it follows that for suchthat

(64)

Therefore, the (uniform) stability of the zero solutionof is established, and vice versa, i.e., . Fi-

nally, the statement 3) of Theorem 3.1 is valid.

B. Proof of Theorem 3.2

According to the proof of Theorem 3.1, we find that its firststatement addressing the periodic character of can be directlyapplied to Theorem 3.2. Hence, we only need to give the proofsfor other two statements of Theorem 3.2.1) Reviewing the proof of the Statement 2) of Theorem 3.1,we find that the zero solution systems and are twoNTP systems. By applying Lemma 2, it follows that thecycles of and are asymptotically stable, if and onlyif they are uniformly asymptotically stable.

2) Note that the proof of the Statement 2) of Theorem 3.2 canbe similarly performed as that of the Statement 3) in The-orem 3.1. For clearance, we redefine the (uniform) asymp-totic stability equivalence between and as ,and the same relation for and as . Letbe the same transition system defined in the previous Proof.Hence, the proof of can also be decomposed toprove and .

: Note that is a NTP system. Then consid-ering Lemma 1, the zero solution of (53) is asymp-totically stable, if and only if it is uniformly asymptoti-cally stable. As mentioned in Remark L.1, one gets for

such that

(65)

Let be zero, thus (65) is completely equivalent forsuch that

(66)

For , it follows that forand such that

(67)


Using Lemma 2, one obtains for andsuch that

(68)

Hence, the (uniform) asymptotic stability of of isestablished, and vice versa, i.e., .

: Note that of is (uniform) asymptoticallystable. Then, for the essential condition part of (68), one obtainsthe same argument as (60). Also, the same form of (61) canbe get for asymptotical stability. Following the same notationsfor and can be written as ,which is denoted by . Hence, it follows that for

such that

(69)

the (uniform) asymptotic stability of the equilibrium ofis established, and vice versa, i.e., . This concludes

the proof of Statement 2) of Theorem 3.2.

APPENDIX B

The coefficient matrices , and for theone-cycle controlled Zeta PFC converter are written as follows:

(70)

(71)

(72)

APPENDIX C

According to [39], [40], the voltage across the capacitorand , i.e., and can be written as

(73)

Substituting (35) into (73), one obtains

(74)

Consider the Kirchhoff’s current law for the node in Fig. 1,the relation between and can be obtained as

(75)

For the OCC controller with a voltage follower, can beexpressed as follows

(76)

applying the map to (73), (74), (75) and (76), we willreadily get the constraint equations among and asshown in (45).

ACKNOWLEDGMENT

The authors would like to thank the Associate Editor andthe anonymous reviewers for their helpful suggestions and con-structive comments which have led to significant improvementin the presentation of this paper. They also would like to thankProf. C. K. Tse (Hong Kong Polytechnic University), Dr. Y. Ma(UPMC, France) and Dr. P. Chen (EPFL, Switzerland), for theirvaluable discussions and insightful feedback.

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Hao Zhang (M’06) was born in Shaanxi Province,China, in 1973. He received the B.E. degree fromXi’an University of Science & Technology, Xi’an,China, in 1996, the M.Sc. degree from Xi’an Univer-sity of Technology, Xi’an, China, in 2002, and thePh.D. degree in electrical engineering (with honors)from Xi’an Jiaotong University, Xi’an, China, in2005.From December 2004 to June 2005, he was a Re-

search Assistant with Hong Kong Polytechnic Uni-versity, Hong Kong. Since 2007, he has been an As-

sociate Professor, School of Electrical Engineering, Xi’an Jiaotong University.During the academic year 2010–2011, he was a Visiting Professor of the Centerfor Power Electronics Systems (CPES), Virginia Tech. His research interestsare in complex behaviors of distributed power systems, and power electronicsinterfaces in micro-grid systems.

Yuan Zhang (S’11) was born in Chaohu, AnhuiProvince, China, in 1989. He received the B.E.degree (hons.) in electrical engineering in 2010 fromXi’an Jiaotong University, Xi’an, China, where heis currently pursuing the M.E. degree in the areas ofmodeling, control and design of practical switchingpower converters, grid-connected inverters of powerelectronics interfaces in micro-grids.Mr. Zhang was the recipient of many top schol-

arships and awards during his undergraduate andgraduate studies. His B.E. dissertation also won the

Distinguished Thesis Award from Xi’an Jiaotong University in 2010. He is amember of IEEE Power & Energy Society and Chinese Physics Society.

Xikui Ma was born in Shaanxi, China, in 1958. Hereceived the B.E. and M.Sc. degrees, both in elec-trical engineering, from Xi’an Jiaotong University,China, in 1982 and 1985, respectively.In 1985, he joined the Faculty of Electrical

Engineering, Xi’an Jiaotong University, where heis currently the Chair Professor of the Electromag-netic Fields and Microwave Techniques ResearchGroup. During the academic year 1994–1995, hewas a Visiting Scientist at the Power Devices andSystems Research Group, Department of Electrical

Engineering and Computer, University of Toronto, Toronto, ON, Canada. Hisresearch interests include electromagnetic field theory and its applications,analytical and numerical methods in solving electromagnetic problems, mod-eling of magnetic components, chaotic dynamics and its applications in powerelectronics, and the applications of digital control in power electronics. He hasbeen actively involved in more than 25 research and development projects. Heis also the author or coauthor of more than 190 technical papers, and also theauthor of five books in electromagnetic fields, including Electromagnetic FieldTheory and Its Applications (Xi’an: Xi’an Jiaotong University Press, 2000).Prof. Ma received the Best Teacher Award from Xi’an Jiaotong University in

1999.

dimensionless approach to multi-parametric stability analysis of nonlinear time-periodic systems:...

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