1120 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMSI: REGULAR PAPERS, VOL. 54, NO. 5, MAY 2007

Bifurcation Analysis of PWM-1 Voltage-Mode-Controlled Buck Converter Using

the Exact Discrete ModelSomnath Maity, Divyendu Tripathy, Tapas Kumar Bhattacharya, and Soumitro Banerjee

AbstractNonlinear phenomena in power electronic circuitsare generally studied through discrete-time maps. However, thereexist very few circuit configurations (like, for example, the cur-rent-mode-controlled dcdc converters or current programmedH-bridge inverter) for which the map can be obtained in closedform. In this paper, we show that, in a voltage-mode-controlleddcdc converter, if the switching is governed by pulse-widthmodulation of the first kind (PWM-1), an explicit form of thestroboscopic map can be obtained. The resulting discrete-timestate space is piecewise smooth, divided into five regions, eachwith a different functional form. We then analyze the bifurcationbehavior using the explicit map and demonstrate the differenttypes of border collision bifurcations that may occur in this systemas a fixed point moves from one region to another. This includesthe very interesting case of a direct transition from periodicity toquasi-periodicity through the route of border collision bifurca-tion. Mode-locking periodic windows are also obtained at certainranges of the parameters. The two-parameter bifurcation dia-gram is presented, showing the domains of existence of differentoscillatory modes in the system parameter plane.

Index TermsBifurcation, border collision bifurcation, dcdcconverter, pulsewidth modulation of first kind (PWM-1), quasi-periodicity.

I. INTRODUCTION

NONLINEAR phenomena in power electronic circuits havebeen investigated extensively over the past decade, andmany reviews are now available on the subject [1], [2]. Mostinvestigations have been directed toward the dc-dc converters,which are the simplest and widely used systems exhibiting awealth of nonlinear phenomena including subharmonic oscil-lations, quasi-periodicity, and chaos [3][5]. Out of the avail-able logics for the control of switching, voltage-mode control[6][10] and current-mode control [5], [11] have been investi-gated extensively.

In most of the above investigations, sampled-data models ormaps of the converters have been derived, and the bifurcationstructures have been investigated with the discrete models.However, it was found that, except for the current-mode-con-trolled dcdc converters [12], [13] and the pulsewidth modu-

Manuscript received May 9, 2006; revised September 18, 2006. This workwas supported in part by the Department of Atomic Energy, Government ofIndia, under Project 2003/37/11/BRNS. This paper was recommended by As-sociate Editor G. Chen.

The authors are with the Department of Electrical Engineering, IndianInstitute of Technology, Kharagpur 721302, India (e-mail: smaity@ee.iitkgp.ernet.in; tapas@ee.iitkgp.ernet.in; soumitro@ee.iitkgp.ernet.in).

Digital Object Identifier 10.1109/TCSI.2007.895526

lation (PWM) current-programmed H-bridge inverter [14], themap cannot be derived in closed form for most of the powerelectronics circuits. In such cases, the map has to be obtainednumerically [15]. For example, in the popular voltage-modecontrolwhere the switching control is exercised by PWMacontrol voltage suitably obtained as a linear combination ofthe state variables is compared with a periodic ramp waveformto generate the switching signal. The discrete-time modelingof such a system yields transcendental equations, thus makingit impossible to obtain the map in closed form [15], thoughin some studies the map obtained under some simplifyingassumptions have been used [16], [17].

There exists one variant of this control scheme, called thepulsewidth modulation of type-1 (PWM-1) [18][20], in whichthe value of the control voltage at the beginning of the rampcycle is compared with the ramp waveform to generate theswitching signal. This method of switching control is becomingpopular because it is necessary to sample the control signalonly once per clock cycle [21], which makes it suitable indigital control and VLSI implementation. Moreover, it auto-matically prevents the undesirable multiple switchings withina clock cycle. In this paper, we show that, for such systems, itis possible to obtain the sampled data model in closed form,which makes the investigation of the dynamics and stabilityquite straightforward.

From a hybrid system point of view, the system consistsof three subsystems, and each one is chosen according to someswitching condition. There are five possible sequences of sub-systems that can be followed within a clock period. This makesthe discrete-time state space divided into five compartments,separated by four borderlines. We derive the exact functionalforms representing the map in each compartment and the equa-tions representing the borderlines.

We then use the map thus derived to investigate the possibletypes of sudden transition in dynamic behavior that may becaused when a fixed point collides with one of the borderlines.Out of the possible cases, one is particularly interestingwherea border collision results in the birth of a quasi-periodic orbit di-rectly from a periodic orbit. Note that the occurrence of quasi-periodicity in voltage-mode-controlled converters have been re-ported earlier [16], [22], but the transition was reported to occurthrough NeimarkSacker bifurcation that occurs when complexconjugate eigenvalues continuously move out of the unit circle.In contrast, in the phenomenon we report in this paper, eigen-values jump discontinuously across the unit circle. This type ofbifurcation has been shown to occur in more complicated con-

1549-8328/$25.00 2007 IEEE

MAITY et al.: BIFURCATION ANALYSIS OF PWM-1 BUCK CONVERTER USING EXACT DISCRETE MODEL 1121

Fig. 1. PWM-1 voltage-mode-controlled buck converter.

verter configurations, for example, the multilevel dcdc con-verter [23], [24]. In this paper, we show that border collisionbifurcation resulting in the birth of a torus can also occur ina simple voltage-mode-controlled dcdc converter. Moreover,while in the multilevel converter such a bifurcation occurs whenthe control voltage crosses the border between the levels, weshow that in a normal single-level PWM it can happen when aconverter normally operating in DCM moves to CCM. We alsoreport the bifurcation sequences involving parameter ranges ofquasi-periodicity and mode-locked periodic behavior.

II. MODELING OF PWM-I VOLTAGE-MODE-CONTROLLEDDCDC BUCK CONVERTER

We consider a voltage-mode-controlled buck converter asshown in Fig. 1. It consists of a controlled switch (MOSFET),an uncontrolled switch (diode), an inductor , a capacitor

, and a load resistance . The switching of the MOSFET iscontrolled by the PWM-1 feedback logic.

This is achieved by obtaining a control voltage as a linearcombination of the output capacitor voltage and a referencesignal in the form

(1)

where is the gain of the error amplifier and is the factorof reduction of the output voltage . An externally generatedsawtooth voltage , of timeperiod and upper and lower threshold voltages and ,respectively, is used to determine the switching instants. Here,

denotes the fractional part of . Intype-1 PWM, the value of the control voltage at the beginningof each ramp is compared with .1 For the th clock cycle,the switch is turned on at the beginning of the clock period if

and is turned off when . If, the switch remains off, and, if ,

the switch remains on throughout the clock period. The inductorcurrent ramps up during the ON time and falls during the OFF

1Note the difference with type-2 PWM, in which the switching instant is de-termined through a continuous comparison of v and v .

time. If the inductor current reaches zero value before the nextclock cycle, the operation is said to be in discontinuous conduc-tion mode (DCM), else it is in the continuous conduction mode(CCM).

There are three subsystems described by three sets of differ-ential equations, as follows:

: the equations for the ON state

(2)

: the equations for the OFF state

(3)

: the equations for the discontinuous conduction state

(4)

where

and and are the current through the inductor and the voltageacross the capacitor, respectively. Here, switching occurs when-ever the solution of each subsystem reaches the border functionspecifically defined for that subsystem. The border function be-tween the subsystems and is given by

where is the value of the capacitor voltage at the th clockinstant. When hits the border , switching occurs, and, sub-sequently, the evolution of is governed by the subsystem .Two borders exist for the subsystem . One is the clock signalfor resetting the switch that moves the system from back to

, and the other is at , which moves the system fromto . These two borders can be described by

while in , the system has only one border given by

1122 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMSI: REGULAR PAPERS, VOL. 54, NO. 5, MAY 2007

Fig. 2. Switching flow diagram of PWM-1 voltage-mode-controlled buck con-verter.

Fig. 3. Possible evolutions of control voltage and inductor current betweentwo clock instants where: (a) control voltage v (nT ) V ; (b) V