Average Model of Boost Converter, including Parasitics ...[1], the Boost converter with typical...

International Journal on Power Engineering and Energy (IJPEE) Vol. (7) – No. (2)ISSN Print (2314 – 7318) and Online (2314 – 730X) April 2016

Reference Number: W15-P-0020 636

Haytham Abdelgawad and Vijay SoodFaculty of Engineering and Applied Science, University of Ontario Institute of Technology (UOIT),

Oshawa, [email protected]

Abstract—This paper presents the principles of state-spaceaverage modeling of DC-DC Boost converters operating inthe Discontinuous Conduction Mode (DCM) which occursin converters due to low load current operation. Theswitching ripples in inductor current or capacitor voltagecause polarity reversal of the applied switch current orvoltage and thus a zero current mode is reached. Modernconverters are often designed to operate in DCM for allloads due to their higher efficiency. In this paper, thereduced-order & full-order average models for the Boostconverter are derived. Also, the output transfer function ofthe Boost converter is calculated which can be furtherutilized for designing a robust controller. Furthermore, theeffects of parasitic elements and losses are included basedon the state-space averaging technique. The open-looptransfer functions of the proposed models are derived andthe behavior of the Boost converter is verified by analysingtransient step responses.

Index Terms—Boost converters, Discontinuousconduction mode, Full-order model, Parasitic realization,Reduced-order model, State-space average modeling.

NOMENCLATURE

TS Total time period for one cycleton ON-time periodd1 Duty cycle

d1Ts ON-time periodd2Ts OFF-time periodipk Peak value of inductor current

Average value of inductor currentx state vector

A1 state (or system) matrix during ON-stateB1 input matrix during ON-stateA2 state (or system) matrix during OFF-stateB2 input matrix during OFF-stateA3 state (or system) matrix during DCMB3 input matrix during DCMnL No. of inductors in the converter circuitK Correction matrix

Small perturbation value of inductor currentSmall perturbation value of capacitor voltageSmall perturbation value of input voltage

Small perturbation value of duty cycleEquilibrium point value of inductor currentEquilibrium point value of capacitor voltageEquilibrium point value of input voltageEquilibrium point value of duty cycle

M Output to input voltage ratio

I. INTRODUCTION

MONGST the different topologies of DC-DC converters[1], the Boost converter with typical efficiencies of 70-

95% [2], is the one where the output voltage is always greaterthan its input voltage. By comparing the Boost converter withthe Buck or Buck-Boost converters, it is found that the designof the Boost converter is more difficult since the Boostconverter is considered as a non-minimum phase system andalso has a zero root in the right half of the s-plane. In otherwords, as a result of the duty cycle, which is the control inputfor this converter and appears in the current and voltageequations, the solution of the state-space equations of thisconverter is more difficult [3].

The structure of DC-DC converters consists of linear (i.e.resistor R, inductor L and capacitor C) and nonlinear (i.e.switch) components. Since these converters can becharacterized as nonlinear and time-variant systems, then thesmall-signal model of the state-space average model isrequired to design a linear controller. References [4-6] showexamples of the small-signal analysis and design of a linearcontroller in the frequency domain for DC-DC converters.

Consideration of all of the system parameters (such asconduction resistances, switch conduction voltages, switchingtimes, and inductor- and capacitor-resistances) in themodelling procedure is an essential step towards the design ofa robust controller. In [7-9] Basso, Tomescue and Towaticonsidered the inductor- and capacitor-resistances and theoutput current in the modelling procedure of the Boostconverter. In [10], Ben-Yaakov considered the capacitorresistance and the output current in the modelling procedure ofthe Boost converter. In [11, 12], a Pulse Width Modulated(PWM) converter with ideal system parameters, both incontinuous and discontinuous conduction modes, is modelledand the effect of conduction resistances is included for thesame model. In [13], the average model of the PWM converter

Average Model of Boost Converter, includingParasitics, operating in Discontinuous

Conduction Mode (DCM)

A

mailto:[email protected]



is demonstrated by considering the conduction resistances andtheir voltage drop in DCM.

This paper presents DC and small-signal circuit models ofthe DC-DC Boost converters operating in DCM. The parasiticcomponents are included in these models to improve themodelling accuracy. The DC model is used to extract theequations for the DC voltage transfer function and theefficiency. The small-signal model is used to extract theequations for the open-loop small-signal transfer functions,such as control to output transfer function, input to outputtransfer function, and input impedance.

Finally, the various transfer functions (Control, OutputImpedance, etc.), various Bode diagrams and transient stepresponses have been plotted and compared with ideal cases.

II. BOOST REGULATOR STATE EQUATIONS FOR ON-OFF TIME SWITCHING

For modelling with the state-space technique, the desiredstate variables for any electric circuit are the energy storingelements (i.e. inductor current and capacitance voltage). Tostart applying the state-space technique to any complicatedcircuit, it must first be converted into a piece-wise linearizedcircuit(s) in which circuit laws can be applied.

In the Boost converter (Fig.1), there are two operatingmodes; the ON- and OFF-modes. The ON-time is defined byd1Ts, and the OFF-time is defined by d2Ts = (1-d1)Ts, where Ts

is the total time period for one cycle and d1 is the duty cyclewhich is defined as the ratio of the ON-time period to the totalswitching time period for one cycle i.e. d1 = ton / Ts.

The main switch S is turned ON and OFF by a pulse with aduty cycle equal to d1. Therefore, the piece-wise linearizedequivalent circuits of the system in ON- and OFF-modes withduration d1Ts and (1-d1)Ts are represented by Figs. 2 and 3respectively.

Fig.1 Boost converter circuit

Using iL and vC as the two state variables (x = [iL vC]T), andby writing the KVL and KCL for the loops of Figs. 2 and 3,the state-space equations can be derived:

Fig.2 Equivalent circuit of Boost converter during ON-state

During the ON-state, the state-space (ss) form will be:

During the OFF-state, the ss form will be:

Fig.3 Equivalent circuit of Boost converter during OFF-state

Fig.4 shows the steady-state waveforms for the ContinuousConduction Mode (CCM) where the inductor current flowscontinuously [iL(t) > 0].



Fig.4 Steady-state waveforms for CCM

In Discontinuous Conduction Mode (DCM), in addition totwo modes as in Continuous Conduction Mode (CCM), thereis a third mode of operation in which capacitor voltage orinductor current is zero. For DCM operation, during firstinterval (i.e. ON-period) the switch is turned on and inductorcurrent rises and reaches a peak when the switch is about toturn off, and resets to zero at the end of the OFF-period.During DCM, in ss form:

Fig.5 Inductor current waveform of DC-DC Converter in DCM

Thus,‘ON’ Mode For t ϵ [0, d1TS](7)‘OFF’ Mode For t ϵ [d1TS, (d1+d2) TS](8)‘DC’ Mode For t ϵ [(d1+d2) TS, TS](9)Where,d1Ts = ON-period timed2Ts = OFF-period timeTs = Total time period for one cycleipk = Peak value of inductor current

= Average value of inductor currentvin = Input voltage

The modelling method for DCM operation comprises ofthree steps:a) Averaging;b) Inductor current analysis;c) Duty-ratio constraint.

State-space averaging techniques are employed to get a setof equations that describe the system over one switchingperiod. After applying an averaging technique to equations(7)-(9), the following expression can be found:

The above equation can be written as , where,

&

.

For state-space averaging technique in DCM, only thematrix parameters are averaged and not the state variables.Equation (10) will hold when true average of every statevariable is used.

From Fig.5, it can be deduced that the average value ofinductor current is:

Consider the circuit behavior when the switch is ‘OFF’; thecurrent which is delivered to the capacitor does not necessarilyhave the same value as the average inductor current. Since theinductor current changes slowly with time, the capacitorequation can be solved employing the ‘conservation ofenergy’ principle, and after that the averaging step isperformed. The total amount of charge which the capacitorobtains from the inductor during switching cycle is:

Thus, average charging current would be of value:

When a capacitor is connected to a resistive load, then the netaverage charging current which is delivered to the capacitor isgiven by:

Hence, the average capacitor current will be given by:



Note here that the above expression differs from the KCLexpression of capacitor which is obtained through state-spaceaveraging. From (10), we can define the state-space-average(SSA) charging current as the inductor current’s averagemultiplied by the duty ratio for which the inductor is chargingthe capacitor. From (11), the SSA charging current can beexpressed as:

This expression is different from the actual charging current in(13). It can be implied that a ‘charge conservation’ law isviolated in unmodified SSA as the averaging step is done on acomplete model thus leading to mismatching of responseswith averaged response of DC-DC converters. Thus (10) ismodified by dividing the inductor current by the factor(d1+d2). The basic method is to rearrange the x, thus x =[iL,vC]T, where sub-vector iL contains all (nL) inductor currentsof the converter and defined by a matrix K, as below:

With this correction matrix, the average modified modelbecomes

Where,

The SSA model for the above equation is

Since there is only one inductor, the x is of dimension two, thecorrection matrix K is simply given by

Thus, the modified average model of Boost converter in DCMis given below:

III. REDUCED ORDER AVERAGE MODEL FOR BOOSTSWITCHING REGULATOR

To complete the average model represented in (18), a dutyratio constraint is defined showing the dependency of d2 onother variables. Usually, in conventional SSA techniques, theinductor voltage balance equation is used in defining the duty-ratio constraint.

For the Boost converter topology, utilizing the volt-secondbalance over the switching cycle,

For time T1 = d1TS,

For time T2 = d2TS,

By removing Imax from above equations,

Substituting d2 in (21), this will result in

From these calculations for a Boost converter, it can be seenthat inductor current dynamics disappear, thus resulting in adegenerate model. Since inductor current is not present in thestate variable in this reduced-order model, it must be replaced



by expressing it as an algebraic function of other variables, sothat inductor dynamics are removed.

For a Boost converter, the peak of the inductor current isgiven by:

The average value of inductor current is given by:

Substituting by (22), the above relationship can be written as

Substituting (27) in (24), this will result in the conventionalaverage model for a Boost converter in DCM. In this model,the dependency on the average inductor current is removed.

Now, applying standard linearization techniques andapplying small perturbations as follows to (28):

Separating terms of and then converting itto state-space form,

Which is in the form of:

Since,

Where,

Hence, two transfer functions can be found as follows:

And

IV. FULL ORDER AVERAGE MODEL FOR BOOSTSWITCHING REGULATOR

A reduced-order model can correctly only predict DC andlow frequency behavior of PWM converters. However, at highfrequencies, it is unable to capture the dynamics of theconverter. Hence, a full-order model is also desired.

The full-order derivation starts from a modified averagedmodel represented by (18). This model differs from thereduced-order one in terms of duty ratio constraint. From (25),the following relationship is obtained:

Substituting this into (11), this will result in a duty constraint

This constraint is different from the earlier one which isderived for a reduced-order model showing that it enforcescorrect average charging of output capacitor. Putting d2 into(21), the following relations are derived:



Equating (35) and (36) to zero and finding the solutionsfor , the DC-operating point can be obtained. Letthe scalar value of M be the output to input voltage ratio.Thus,

Now applying standard linearization techniques andapplying small perturbations as follows to (35) and (36):

Then,

And

Now, the small signal model can be derived with thefollowing equation:

Where,

Since,

Where,

Hence, the following transfer functions can be formulatedfrom the small signal model:

V. PARASITIC REALIZATION IN THE BOOSTSWITCHING REGULATOR

Due to the difficulties faced during the modelling procedureof the Boost converter, parasitic elements such as conductionvoltages, conduction resistances, inductor resistances andequivalent series resistances (ESR) of capacitors have beenignored. The idea of simply considering ideal/losslesscomponents and leaving out parasitic elements simplifies themodel development procedure and allows to understand thefundamental behaviour of the switching converter system.However, the effects of parasitic elements and losses areimportant for improving model accuracy, studying efficiencyand dynamic performance of the system. The problem withincluding the parasitic elements is that they lead to nonlinearcurrent and voltage waveforms and hence result incomplications in the modelling procedure.

A Boost converter circuit, with parasitics included, willlook like as shown in Fig.6.



Fig.6 Boost Converter Circuit with parasitic

During the ON-state:

During the OFF-state:

During DCM period:

Now, applying the averaging technique, this will result in:

So, the SSA model for the above equation is

Since there is only one inductor, the x is of dimension two, thecorrection matrix K is simply given by

With this correction matrix, the average modified modelbecomes

Hence, the modified average model of Boost converter inDCM will look like:

In the above state-space equation, replacing d2 by the dutyconstraint (34), this will result in



And

Now applying standard linearization techniques andapplying small perturbations to (51) and (52), this will resultin the small-signal model as:

Where,

Where,

Since,

Where,

Hence, the following transfer functions can be formulatedfrom the small signal model:

Since,

This is in the form of:

Since,



Hence, two transfer functions can be found as follows:

VI. SIMULATION RESULTS

The simulations were done by using the following modelparameters: L= 18μH, C= 4.7μF, Vin= 5V, D= 0.7, RC= 0.1 Ω,Rd= 0.15 Ω, RL= 0.8 Ω, Rsw= 0.17 Ω and FS= 350 kHz. Thesimulations were performed for two different cases.

Figures 7 and 8 show the Bode diagrams for the transferfunction representing the relation between the output voltageand duty ratio (d) (i.e. Control transfer function) for ideal andnon-ideal DC-DC Boost converters in DCM, respectively.

Fig.7 Bode diagram for ideal DC-DC Boost converter in DCM

Fig.8 Bode diagram for non-ideal DC-DC Boost converter in DCM

As seen from Fig.7 and Fig.8, there is a remarkabledecrease in the open loop gain as the load resistance decreases.When the load resistance goes below a certain range, the openloop gain goes under 0 dB which makes the converterineffective without an external feedback controller.

Figures 9 and 10 show the transient step responses for idealand non-ideal DC-DC Boost converters in DCM respectively.As indicated in Fig.9 and Fig.10, these figures were plotted byusing two different load values to see the transient behavior asa function of the load value. As seen from Fig.9 and Fig.10,the steady-state behavior is a function of the load value. Thisis the important reason behind why the closed-loop control isessential, where the duty cycle (the only controllableparameter in the circuit) must be controlled and adapted inorder to preserve the required steady-state DC voltage fordifferent load values. Moreover, by comparing Fig.9 andFig.10 for the same values of the load resistance, it is easy toobserve that the step response for the ideal DC-DC Boostconverter is looking identical to the step response for the non-ideal DC-DC Boost converter. This indicates that the ideal andnon-ideal transfer functions should be approximately equal;which, in turn, reveals that the loss elements found in the non-ideal DC-DC Boost converter circuit have little effect on thedynamic response of the DC-DC Boost converter operating inDCM.

Fig.9 Transient step response for ideal DC-DC Boost converter in DCM



Fig.10 Transient step response for non-ideal DC-DC Boost converter inDCM

VII. CONCLUSION

First, the various aspects of average modelling of DC-DCBoost converter operating in DCM are studied. Basically, themodelling procedure consists of three steps:

1. Averaging the matrix parameters and selection of thecorrection matrix (K) depending on the number ofinductor currents of the converter.

2. Conversion of state-space equations into differentialequations for inductor current and capacitor voltage.

3. Defining a duty ratio constraint so that the expressionconsists of only one duty ratio.

Second, the reduced- and full-order average models havebeen derived. It was found that the reduced-order model canestimate the behavior in the low frequency range but the full-order model, since dynamics of inductor are present, is moreprecise.

Finally, various parasitic components have been taken intoconsideration and a full-order model is developed. The systemdynamic behavior for the DC-DC Boost converter with idealcomponents and DC-DC Boost converter with non-idealcomponents operating in DCM are compared via Bode plotsand transient step responses under different values of the loadresistance in order to help in designing a robust controller.

REFERENCES

[1] N.Mohan, T. M. Undeland, and W. P. Robbins, “PowerElectronics, Converters, Applications, and Design,” JohnWiley & Sons, 2003.

[2] R. Erickson, “DC-DC Converter,” Article in WileyEncyclopedia of Electrical and Electronics Engineering.

[3] V. I. Utkin, “Sliding Mode Control Design Principles andApplications to Electric Drives,” IEEE Trans. OnIndustrial Applications, Vol. 40, pp. 23-36.

[4] J.H. Su, J.J. Chen, and D. S. Wu, “Learning FeedbackController Design of Switching Converters viaMATLAB/SIMULINK,” IEEE Trans. On Education,Vol.45, pp. 307-315, 2002.

[5] J R. B. Ridley, “A New Continuous-Time Model forCurrent –Mode Control,” IEEE Trans. On PowerElectronics, Vol. 6, No. 2, PP. 271-280, 1991.

[6] P. Li, and B. Lehman, “A Design Method for ParallelingCurrent Mode Controlled DC-DC Converters,” IEEE

Trans. On Power Electronics, Vol. 19, PP. 748-756, May2004.

[7] C. P. Basso, “Switch-Mode Power Supply SPICECookbook,” ISBN: 0071375090, McGraw-Hill, NewYork, 2001.

[8] B. Tamescu, “On the Use of Fuzzy Logic to controlParalleled DC-DC Converters,” PHD Thesis, Blackbury,Virginia Polytechnic Institute and State University,October 2001.

[9] A. Towati, “Dynamic Control Design of Switched ModePower Converters,” Doctoral Thesis, Helsinki Universityof Technology, 2008.

[10]R. Naim, G. Weiss, and S. Ben-Yaakov, “H∞ ControlApplied to boost Power Converters,” IEEE Trans. OnPower Electronics, vol. 12, no. 4, pp. 677-683, July 1997.

[11]V. Vorperian, “Simplified Analysis of PWM ConvertersUsing the Model of the PWM Switch, Parts I (CCM) andII (DCM),” Trans. On Aerospace and Electronics systems,vol. 26, no. 3 May 1990.

[12]V. Vorperian, “Fast analytical techniques for Electricaland Electronics Circuits,” Cambridge University Press,2004, ISBN 0-521- 62442-8.

[13] C. M. Ivan, D. Lascu, and V. Popescu, “A New AveragedSwitch Model Including Conduction Losses PWMConverters Operating in Discontinuous Inductor CurrentMode,” SER. ElEC ENERG. Vol. 19, No. 2, PP 219-230,August 2006.

Haytham Abdelgawad (S’15) was born inCairo, Egypt, in 1978. He received theB.Sc. (Eng.) and M.Sc. degrees in electricalengineering from Helwan University,Cairo, Egypt, in 2000 and 2009,respectively. He is currently pursuing thePh.D. degree in electrical engineering at theFaculty of Engineering and AppliedScience, University of Ontario Institute of

Technology (UOIT), Oshawa, ON, Canada.His research interests include power converter topologies

and their control aspects, Maximum Power Point Tracking ofphotovoltaic systems, Artificial Intelligence and OptimizationAlgorithms and the integration of renewable energy systemsinto the smart grid.

Vijay Sood (SM’79–F’06) received thePh.D. degree from the University ofBradford, Bradford, U.K., in 1977.

He is currently an Associate Professorand the OPG Design Co-chair at UOIT,Oshawa, Ontario. He has extensiveexperience in the simulation of HVDC-FACTS systems and their controllers. Hehas authored two text books on HVDC

Transmission. His research focuses on the monitoring, control,and protection of power systems and on the integration ofrenewable energy systems into the smart grid.

Dr. Sood is a Registered Professional Engineer in theprovince of Ontario. He is a Fellow of the IEEE, EngineeringInstitute of Canada and the Canadian Academy of



Engineering. He served previously as a Director of the IEEECanadian Foundation and is a former Editor of the IEEETransactions on Power Delivery, and Co-editor of the IEEECanadian Journal of Electrical and Computer Engineering.

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