2013 International Conference on Circuits, Power and Computing Technologies [ICCPCT-2013]

Nonlinear Modeling and Bi/ureations in Switehed Power-Faetor-Correetion Boost Regulator

Arnab Ghosh Electrical Engineering

Department National Institute o{

Technology, Durgapur, India arnab. 4u20 1 O@gmail.com

Dr. Subrata Banerjee Electrical Engineering

Department National Institute of

Technology, Durgapur, India bansub2004@rediffmail.com

Abstract-This paper describes on non linear modeling and

bifurcation phenomena in switched power-factor-correction

(PFC) boost regulator whose switching borders are varying

periodically with time. The mathematical modeling is done by

nonlinear differential equations. The system dynamics of this

switching circuit can be represented by one-dimensional (l-D) piecewise smooth maps under discrete modeling. In particular,

the dynamical behavior of the current-mode-controlled PFC boost

circuit is investigated from bifurcations viewpoint. The control of

bifurcations is also studied in this paper by varying system

parameters.

Index Terms-Switched PFC boost regulator, map based

model, bifurcation, control of bifurcation.

I. INTRODUCTION

S WITCHED dynamical systems are important for describing several engineering applications. In electrical engineering, the switched dynamical systems are observed in power

converters, chaos generators, oscillators, etc. Actually, switched dynamical system can be operated by set of switching rules [7] which are toggling among two or more dynamical states. According to mathematical point of view a switched dynamical system is defined by two or more differential equations which are related to on or off state of switch. Switching between one state to another occurs on a set of borders, which are defined on system's state space. The system with fixed borders, can be considered as autonomous system i.e. the switchings are only depended upon system states of the system, are not affected by external applied time-varying input or parameters. In some engineering applications, system switchings are not only depended upon system states but also they are controlled by external time variant driving signal, are called nonautonomous system. In most of power electronics switching circuits exhibit like nonautonomous system where switching borders are function of a periodic driving signal [14]. In this paper, we investigate the nonlinear dynamics of current-mode-controlled PFC boost circuit from bifurcations viewpoint.

The study regarding non linear behaviour of switching mode operated power converters have gained increasingly. Various kinds of non linear phenomena, such as bifurcation and chaos have been reported [3][4]. Chaos could be described as noise like, bounded oscillations with an infinite period found in some power electronic (PE) circuits has been exhibited [1]-[4] which may be responsible for the unusual high noise in some PE circuits.

978-1-4673-4922-2/13/$31.00 ©2013 IEEE 517

Dr. Pradip Kumar Saha Electrical Engineering

Department Jalpaiguri Govt. Engg.

College, Jalpaiguri, India p _ ksaha@rediffmail.com

Dr. Goutam Kumar Panda Electrical Engineering

Department Jalpaiguri Govt. Engg.

College, Jalpaiguri, India gyanda@rediffmail. com

+

k(vo - vre,)

Fig. I: Circuit diagram of power-factor-correction boost regulator.

Subharmonic and chaos of the current-mode-controlled buck converter and boost converter has been demonstrated [5]-[7]. The chaos in a current controlled boost converter first discussed by Deane [6]. Chan and Tse [10], S. Banarjee and K. Chakrabarty [I I] studied various types of routes to chaos and their dependence upon the choice of bifurcation parameters. The nonlinear dynamics in the Power Factor Correction boost converter have also been reported [12].

Border collision bifurcation [14] in switched dynamical system has drawn lot of enthusiasm recently. Many prior researchers have studied it by the method normal form. Also some researchers express the border collision as discontinuous bifurcation due to jumping of eigen values. As these methods deal with the neighbourhood of the border, they are considered as the local theory [15] of nonlinear dynamical systems. For obtaining the normal form we have to know the trace and the determinant of the jacobian matrix to the fixed points of both sides of the border. Though these have no direct relationship with the practical system. Consequently there is no general method for plotting bifurcation diagrams on the system parameter space. For resolving this problem, we follow the global behaviour of a periodic solution rather than border's neighbourhood. So we have followed here the discrete mapping system to describe this nonlinear system.

Earlier when the nonlinear behaviour of the boost converter was recognized, the development of a nonlinear model of this system becomes necessary. For this problem a large signal analysis of the continuous time system is used [9] as well as discrete models [13] should have to be developed for PE circuits as mapping of the form

xn+1 = f(xn) (I)

2013 International Conference on Circuits, Power and Computing Technologies [ICCPCT-2013]

Such mappings when applied to the state phase give the state of the system at a switching instance in respect of the previous one. The dynamics between switching instant would not be established by a map based model but it could explain the nonlinearity since most of the PE circuits are linear piece wise, with nonlinear behaviour attributed only when there is switching discontinuity. However the closed form expressions for the map cannot be establish for most PE circuits, for which numerical derivation of the map is necessary, though the current mode control of PFC boost converter is an exception for such instances. A map based

20

18

16

13

12

100 150

model for the PFC boost regulator has been introduced based on idealised circuit elements.

Some literature surveys are discussed in above paragraphs. In this paper, Nonlinear Modeling and Bifurcations in Switched Power-Factor-Correction Boost (PFC) Regulator is designed by MATLAB 7.8.0, FORTRAN and the reporting of dynamics is observed by varying bifurcation parameter (only Load Resistance).

15

14

12

250

100 120 140 160 180 200 220 OulputCapacitQlVolt1!!je

(b)

Output Capacitor Voltage

(e) Fig. 2: (a) Period I phase-plane trajectory of the PFC boost regulator. Parameter values are: vin = 220sinwt, L = 30 mH, C = 120 f1.F, R = 19.0.. (b) Period 2 phase-plane trajectory of the PFC boost regulator. Parameter values are: vin = 220 sin wt, L = 30 mH, C = 120 f1.F , R = 25.0.. (c) Chaotic phase-plane trajectory of the PFC boost regulator. Parameter values are: Vin = 220 sin wt. L = 30 mH. C = 120 f1.F. R = 75.0..

A. The PFC Boost Regulator

The Power-Factor -Correction boost regulator circuit (Fig. 1) consists of an inductor L, a controlled switch SW, a power diode D and a capacitor C. These are the main components of power stage circuit. The error voltage is taken from the comparator, having a predetermined reference voltage source which senses the output voltage across the capacitor. Then it is fed to the multiplier block which multiplies rms of input voltage, inductor current iL , and error voltage to get the reference current iref. The difference of iL and iref provides

the correct timing logic for the switching driver circuit to turn on and off the Switch SW in the Boost converter. There are various ways to implement the current-mode contral of the step-up converter. Only the Constant-Frequency Control is used to contral fs i. e. switching frequency. Hence, this method ensures continuous conduction of current tlow for the full cycle of the input voltage.

There are two sets of linear differential equations depend on whether the controlled switch is on [0< t < dTsl or off [dTs< t < Tsl. The output voltage Vc and the inductor current iL are taken as state variables. When switch is closed, the current through the inductor rises and any clock pulse arriving during that period is ignored. The switch will open when inductor current reaches the reference current. After opening

518

of switch, the current falls. The switch closes again upon the arrival of the next clock pulse.

1) The State Equations during "ON" period diL Vin dt L

dvc - Vc dt RC

2) The State Equations during "OFF" period diL Vin Vc - ---dt L L

dvc R*iL - Vc dt RC

where. Vin = Vin Isin wtl

(2)

(3)

(4)

(5)

Vin = Input Voltage. L = Inductor. C = Capacitor. iL = Inductor Current. Vc = Capacitor Voltage.

ire!

i iL

c;n+2

i Vc

Tüne (sec) _ Fig. 3: Time plot of output voltage and inductor current of the boost converter with clock pulses.

2013 International Conference on Circuits, Power and Computing Technologies [ICCPCT-2013]

A few trajectories are reported in the phase-plane (the inductor current versus output voltage) is shown in Fig. 2. The nominal values of the fixed parameters are vin = 220 sin wt, L = 30 mH, C = 1 20 flF. Three cases are

shown: period 1 (for R = 19.(1), period 2 (for R = 25 .(1), and chaos (for R = 75 .(1).

11. THE MAP-BASED MODEL

A. System Formulation {or Moving Borders The Switched dynamical systems is considered with moving

borders and it is mathematically expressed as Consider two dynamical systems 51 and 52'

dx 51 : -= g(x, (1 ) , x E Rn (6) dt 52: �: = hex, (2) , X E Rn (7)

Where, 01 and 02 are the system parameters with appropriate dimensions. Next, B is defined as a border

B = {x ERn: a (x, t) = O} (8) Where a(.) is a time-varying periodic function of period T, i.e.,

a(x, t + T ) = a(x, t) Vx E B Vt ER (9) This border B thus divides the state space into two parts, namely, Mland M2, which are given by

Ml = {(x,t) E Rn X R: a(x,t)?: 0 (10) M2 = {(x, t) E Rn X R : a(x, t) ::; O} (11)

The solution of the system in Ml and M2 is governed by the state equation corresponding to 51 and 52, respectively, as given in (6)&(7). Suppose the solutions in Ml and M2are given by

x(t) = ({J(t, X01 ) x(t) = ljJ(t, X02)

Where XOl and X02 respectively.

x(O) = XOl (x, t) E Ml (12) x(O) = X02 (x, t) E M2 (13)

are initial points located in Ml and M2,

B. System Formulation for PFC Boost Regulator

The mapping from one switching instant to the next in lR2 space formed by Ve and iL . Following assumptions are considered.

I) The value of inductor and switching frequency are so chosen to avoid any discontinuity in conduction.

2) The value of the capacitance is also chosen that the value of output voltage ve always exceed that of the input voItage at any part of the cycle.

The behavior of state variables along with dock pulses is shown in Fig. 3. We take the c10sure of control switch as t = O. At this time let the initial conditions be iL = iL,nand Ve = ve,n-

The switch turns off when the current iL in the inductor reaches reference current ire!' The on-time tn can be

calculated

(n dt = � riret di o Vin lL,n L C' . ) tn = v:- lre! -lL,n Ln (14)

The final value of ve(tn) = ve ne-tn!RC (15) and ve(tn) should not be less than vin if the model is to remain valid.

When switch is off, substituting iL and diddt from (5) into (4) we get a second-order differential equation of ve in the form

(16)

519

Where,

ß - L+RC

ß 1

ß Vin 1 - RLC ' 2 = LC' 3 = LC'

If the dock period is T, then tn/T will count the dock pulses in the on period. The remainder part of it multiplied by T will give the time of switching off after the last dock pulses.

The solution of the homogeneous equation has three different functional forms depending on the roots of the characteristic equation

Al = -�1 + J ß} -ß2

,12 = -�1 + J ß} -ß2

2 Case 1: ßl > ß 2 : In this case, the two roots are real and

4 distinct, and the solution of (16) is

ve(t) = KleA1t + K2eA2t + Vin (17) Putting the initial condition for off condition, we get

K = v e-tn!RC - v:- -K 1 e,n Ln 2

K = _1 _ [-v:- ,1 - iret + V e-tn!RC {A + .!...}] 2 Al -A2 Ln 1 C e,n 1 RC Let the output voltage at next switch on instant be ve,n+l' the current ie,n+l' These are obtained as

v = K eA1tJ, + K eA2tJ, + v:- (18) e,n+l 1 2 m i = K1 [1 + Reit ]eA1tJ, + K2

[1 + RCA ]eA2tJ, + Vin (19) L,n+l R 1 R 2 R Z

Case 2: ßl = ßz : In this case, the solution of (16) is 4

ve(t) = (Kl + K2t)e-Cßd2)t + Vin (20) Where,

K = v e-tn!RC - v:-1 e,n m . {RCßl } _ lretR+ -2--1 K1-Vin K 2 - ----'--'--'''-----'---RC and the mapping from one switching instance to another is given by

v = (K + K t')e-Cßd2)tJ, + v:- (21) e,n+l 1 2 n m iL,n+l = � [K + K t' + RC {K -K1ßl _ K2ß1tJ,

}] e-Cßd2)tJ, + Vin (22) R 1 2 n 2 2 2 R Z

Case 3: �l < ß2 : In this case, the oscillatory solution of

the linear nonhomogeneous equation (16) is ve(t) = [Kl coswt + K2 sinwt]eC-ß1t!2) + Vin (23)

Where

W= Jß2-ß}

And the initial conditions give K = v e-tn!RC - v:-1 e,n m . RCßIKI K - lretR-Kl +-z--Vin

2 - wRC The mapping is given by

Ve,n+l = e( -ßltJ,!2) [Kl cos wt� + K2 sin wt�]

i = e-Cßd2)tJ, [K1 + � + RC {WK -K1ßl}] cos wt' + L,n+l R R 2 2 n

(24)

e-Cßd2)tJ, [:2 + * + RC {WKl -K2t1}] sin wt� + V�n (25)

Thus, Ve,n+l and iL,n+l are given explicitly as functions of ve,n and iL,n in all the three cases.

2013 International Conference on Circuits, Power and Computing Technologies [ICCPCT-2013]

III. BlFURCATIONS DIAGRAMS

The nonlinear modelling of the current-control power factor correction boost regulator can be investigated in details from the bifurcation view point. One can define periodicity state variables as the repetitive behaviour of output waveform as sheen in phase space.

The map-based model provides a fast and easy way of obtaining the bifurcation diagrams. We consider to iterate the map with taking initial condition(O, 0), and eliminate first 500 consecutive values for neglecting the transient.

It may be noted that the dock period is much less than the time constant RC. Here the Load Resistance is taken as a major bifurcation parameter.

Bifurcation diagrams are obtained from FORTRAN and ORIGIN 5.0 software. The data files are generated after executing the FORTRAN programme of State Equations of Switched Power Factor Correction Boost Regulator. This data files are plotted by ORIGIN 5.0. Results are given below for taking major system parameter i.e. Load Resistance as a Bifurcation Parameter.

A. Load Resistance (R) as the Bifurcation Parameter

�80 ,-------------------, <fJ a. �70

>-060 Z w :::l 050 w it I :::.:::40 I o S 030

!;: § 20

a. � 10

-�

100 200 300 400 500 600 700 800 900 1000

RESISTANCE (OHM)

(a)

'""'1800 ,------------------,

� o 1600 G [) 1400 Z � 1200 8 er 1000 " � 800

� 600

o ::1 400

� 200 <fJ "-C

1 00 200 300 400 500 600 700 800 900 1000

RESISTANCE (OHM)

(b)

1800 ,-------------------,

W'600

�. � 1400 A�_.; / G �:/ , � 1200

d 1) ------ ' . � � ,

r:: r� /A o 600

� 0.. 400

100 200 300 400 500 600 700 800 900 1000

RESISTANCE (OHM)

(C) Fig. 4: Bifurcation diagram of the boost converter with Resistance as parameter: (a) stroboscopic sampling of inductor current, (b) stroboscopic sampling of capacitor voltage, and (c) switch-on sampling of peak capacitor voltage.

The bifurcation diagram of the boost converter with load resistance as a bifurcation parameter is shown in Fig. 4 (a) & (b) which shows bifurcation diagram for stroboscopic sampling of state variables and Fig. 4(c) which shows bifurcation diagram for the switch-on sampling of peak output capacitor. Resistance is varied from I to 1000n with step of 0.5n with other parameters fixed atvin = 220 sin wt Volt, L = 30 mH, C = 120 flF. According to the above diagrams the inductor current and capacitor voltage are almost same i.e. in phase as this PFC circuit is forced to provide current and voltage same phase.

Bifurcation and chaos are observed when load resistance is increased. This bifurcation diagrams are like staircase with increasing the load resistance. Period I at 19 n. Period 1

520

bifurcates to period 3 at 19.2 n and extend up to 19.7 n. Period 4 is shown at 27. A finite chaotic (i. e. periodicity is greater than five) zone occur at 54.76 n and this zone has still extended with increasing the bifurcation parameter (i.e. load resistance ).

The bifurcation diagram of the switch-on sampling of peak output capacitor voltage with load resistance as parameter (which is varied from I to 1000 n) is shown in Fig 4( c). Period 1 behaviour is observed from 1 to 52.93 n. Period 1 bifurcates to period 2 at 52.93 n and extend up to 113.19 n. At 113.19 n periods 2 bifurcated to period 3 and continue up to 175.08 n and then system operates with period 4. Ajump is observed at 417.54 n. Then the system operates with finite periodicity which is continued up to 1000 n.

2013 International Conference on Circuits, Power and Computing Technologies [ICCPCT-2013]

IV. CONTROL OF BIFURCATIONS (LOAD RESISTANCE AS BIFURCATION PARAMETER)

�80�---------------------------------, <fJ CL �70

>-060 Z w :::l 050 w it ,,40 o S 030

!;: §20

CL �10

100 200 300 400 500 600 700 800 900 1000

RESISTANCE (OHM)

(a)

�20 �---------------------------------,

g: �18

>-016

d] �

i

14 �." � 12 i\�

S �\i 010 .\ � :� § 8 :\ Cl... I: � 6 ;\

i\ 100 200 300 400 500 600 700 800 900 1000

RESISTANCE (OHM)

(b)

Fig. 5: Bifurcation diagram of the boost converter with Resistance as parameter: stroboscopic sampling of inductor current (a) before controlling bifurcation diagram (b) after controlling bifurcation the bifurcation diagram.

The system behaves different periodic nature with changing the bifurcation parameter, after bifurcation control the system

�1800 ,_----------------------------------,

� o 1600 ;,. G 1400 z � 1200 o W tt 100J G 800 S o � 600

o ::1 400 CL � 200

,," 100 200 300 400 500 600 700 800 900 1000

RESISTANCE (OHM)

(a)

initially operates with period 2, and then it bifurcates to period 1. Ultimate the system goes to stabilize zone.

�28° I-:::;�

�==============� � 260 �240 ;r � 220 (f a 200 � 180 � 160 S 140 o !;;: 120 o 100 W cl 80 " (7j 60

\\ '�-------�

:::,. (j 40 ����������������� 100 200 300 400 500 600 700 800 900 1000

RESISTANCE (OHM)

(b)

Fig. 6: Bifurcation diagram of the boost converter with Resistance as parameter: stroboscopic sampling of output capacitor voltage (a) before controlling bifurcation diagram (b) after controlling bifurcation the bifiucation diagram.

Before bifurcation control the output voltage increases with load resistance and system can operate 1 periodicity to several periodicities but after bifurcation control we get few

1800,_----------------------------------,

,.....,,1600

� � 1400 � 1200 ':i 01000 > f-ii 800 f-Ö 600

� CL 400

1 00 200 300 400 500 600 700 800 900 1000

RESISTANCE (OHM)

constant voltage level zones where system can operate up to period 4 i. e. no chaotic zone will be occurred.

�2 ,_------------------------------_, U) 291 ':i 0290 G w 289 � 288 eS 287 > f- 286 :::l � 285 0284 " � 283

282

281

280 -l,I,,"""'""'''""''"'''�'"''"''""''''''''""''''''''"''''�'"''"''""''''''"''''" 100 200 300 400 500 600 700 800 900 1000

RESISTANCE (OHM)

W 00 Fig. 7: Bifurcation diagram of the boost converter with Resistance as parameter: switch-on sampling of peak capacitor voltage (a) before controlling bifiucation diagram (b) after controlling bifurcation the bifurcation diagram.

521

2013 International Conference on Circuits, Power and Computing Technologies [ICCPCT-2013]

The periodicity of switch-on sampling of peak capacitor voltage is obtaining both sides. In Fig. 7(a) system periodicity and voltage increase with load resistance but in Fig. 7(b) the

V. CONCLUSION

The BifUfcations in Switched Power-Factor-Correction

(PFC) Boost Regulator with current-mode control has been

investigated depending on the nonlinear model. Results highlight that the proposed model of practical PFC regulator,

simulation results and phase-portrait diagrams. The phase­

plane-trajectory CUfves and bifUfcation diagrams are observed

by varying value of load resistance (R). The value of load

resistance (R) is increased or decreased; the phase-portrait of

output capacitor voltage (vc) and inductor current Ud is

going to period I to period 11 to chaotic-mode. Chaos

phenomena are shown by multiple loops on phase-plane

diagram. The most important point of all the case studies, if

the entire system is operated in chaotic-mode the output capacitor voltage ripples has been minimized by increasing the

chaotic-region. The diagrams of bifurcation control are also

reported in this paper. We can control entire system in oUf

desired region according to oUf demand.

VI. ACKNOWLEDGMENT

We are also grateful to Ministry of Human Resource, Govt. of India for providing the financial support of this project work.

REFERENCES

[1] O. Dranga, C.K. Tsc and H. H. C. Iu and I. A. Nagy, "Bifurcation

Behaviour 0/ A Power-Factor-Correction Converter" in Int. Journal 0/

Bifurcation and Chaos.

[2] J. R. Wood, "Chaos: A real phenomenon in power electronics, " in App!.

Power Electron. Conj, Baltimore, MD, 1989. [3] 1. H. B. Deane and D. C. Hamill, "Instability, subharmonics, and chaos

in power electronic circuits," IEEE Trans. Power Electron., vol. 5, pp.260-268, 1990.

[4] D. C. Hamill, "Power electronics: Afield rich in nonlinear dynamics,"

in Nonlinear Dynamics 0/ Electronic Systems, Dublin, Ireland, 1995. [5] J. H. B. Deane and D. C. Hamill, "Chaotic behavior in current-mode

controlled dc-clc converter, " Electron. Lett., vol. 27, no. 13, pp. 1172-1173, 1991.

[6] 1. H. B. Deane, "Chaos in a current-mode controlled boost dc-dc

converter, " IEEE Trans. Circuits Syst. I, vol. 39, pp. 680-683, 1992. [7] C. K. Tse, "Flip biforcation and chaos in three-state boost switching

regulators," IEEE Trans. Circuits Syst. I, vol. 41, pp. 16-23, 1994. [8] K. Chakrabarty, G. Podder, and S. Bane�jee, "Bifurcation behavior 0/

buck converter," IEEE Trans. Power Electron., vol. 11, pp. 439-447,1995.

[9] F. D. Tan and R. S. Ramshaw, "Instabilities 0/ a boost converter system

under large parameter variations, " IEEE Trans. Power Electron., vol. 4, no. 4, pp. 442-449, 1989.

[10] W. C. Y. Chan, C. K. Tse, "Study o/ bifurcation in current- programmed

boost dc-dc converters: From quasi-periodicity to period doubling, "

IEEE Trans Circuits syst. I, vo1.44, pp. 1129-1142, Dec. 1997. [lI] S. Banarjee, K. Chakrabarty, "Nonlinear modeling and bifurcations in

the boost converter, " IEEE Trans. Power Electron., vol. 13, pp. 252-260, 1998.

522

system periodicity (fOUf) and output voltage are almost constant with increasing load resistance.

[12] M. Orabi, T. Ninomiya, "Nonlinear dynamics 0/ power /actor correction

converter." IEEE Trans. Industrial Electronics, vol. 50, pp.1116-1125, 2003.

[13] D. C. Hamill and J. H. B. Deane, "Modeling 0/ chaotic dc-dc converters

by iterated nonlinear mappings, " IEEE Trans. Power Electron., vol. 7, no. 1, pp. 25-36, 1992.

[14] S. Bane�jee and G. Verghese, Eds., "Non linear Phenomena in Power

Electronics: Attractors, Bifurcations, Chaos, and Nonlinear Contro!. "

New York: IEEE Press, 2001. [15] S. H. Stroratz, "Nonlinear Dynamics and Chaos". Reading, MA:

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BIBLIOGRAPHY

Arnab Ghosh received B.Tech (Electrical) from JIS College of Engineering, Kalyani, M.Tech (Electrical) Specialization: Power Electronics & Drives from Jalpaiguri Govt. Engineering College, Jalpaiguri.

He is currently working as a Research Scholar in electrical engineering department to National Institute of Technology, Durgapur, West Bengal. His area of interest is to study non linear dynamics of power electronic circuits.

Dr. Subrata Banerjee was born in 1968. He received B.E. (Electrical) in 1989, M.E. (Electrical Machines) in 1994 and PhD in Electrical Engineering from Indian Institute of Technology, Kharagpur in the year 2005.

He is presently working as an Associate Professor in electrical engineering in National Institute of Technology, Durgapur, India. He has published a numbers of research papers in NationalIInternational Conference

Records/Journals. His research interest includes Control Systems, Electrical Machnies and Power Electronics. He is a Iife member of Systems Society of India, Institute of Engineers (India) and member of IEEE(USA). His Biographical inclusion is in Marquis Who's Who 2007 and IBC FOREMOST Engineers of world-2008. He has worked many research and consultancy projects. Dr. Banerjee has received some academic awards. He has visited different countries in IEEE conferences and seminars.

Dr. Pradip Kumar Saha received BE (Electrical) trom B.E.College, Shibpore. M.Tech«Electrical) Specialization: Machine Drives & Power Electronics trom IIT­Kharagpur. PhD from University of North Bengal. FIE, MISTE, Certified Energy Auditor.

He is currently a Professor and Head, Department of Electrical Engineering, Jalpaiguri Government Engineering College, Jalpaiguri,WB-735102. His research interests

include chaotic dynamics in drives and power electronics.

Dr. Goutam Kumar Panda received BE (Electrical ) from 1.G.E. College, Jalpaiguri, M.E.E( Electrical) Specialization: Electrical Machines & Drives from Jadavpur University. PhD from University of North Bengal. FIE, MISTE, Certified Energy Auditor.

He is currently Professor, Department of Electrical Engineering, Jalpaiguri Government Engineering College,

Jalpaiguri, WB-7351 02. His research interests include chaotic dynamics in drives and power electronics.