Real-time Simulation Of The Buck Converter · Chapter 1 Introduction This report concentrates on...

Real-time Simulation Of TheBuck Converter

Janamejaya Channegowda

Power Electronics Group

Department of Electrical Engineering

Indian Institute of Science

Bangalore - 560 012

February 2014

Chapter 1

Introduction

This report concentrates on enumerating the steps involved in implementing a buck con-verter (considering non-idealities) on an FPGA platform. The focal point during theentire flow of the report will be on the conversion of differential equations of the buckconverter to discrete equations and their final implementation in real-time. The builtblock diagram files and experimental results have also been added.

Chapter 2

Switching Model

Nomenclature

• Vg = Input voltage

• Vf = Forward voltage drop of the diode

• Vc = Voltage across capacitor

• Vl = Voltage across inductor

• Vsw = Voltage drop across switch

• il = Current through inductor

• io = Current through load

• ic = Current through capacitor

• id = Current through the diode

• Rl = Resistance of inductor

• Rc = ESR of capacitor

• Ro = Load resistance

• Ts = Time step for simulating the models = 2µs

• Vb = Base voltage = 100 V

• ib = Base current = 4 A

• Zb = Base impedance = 25Ω

Development of Off-line and Real-Time Simulator for Electric vehicle /Hybrid Electric Vehicle Systems

Figure 2.1: Circuit of the buck converter being considered

Figure 2.2: Circuit of the buck converter with switch and diodes being replced by voltagesources

Figure 2.3: Circuit of the buck converter when switch is ON

2.1 Switch is ON

When Switch ”S” is ON, voltage across inductor is given by:

04/02/2014 Department of Electrical Engg, IISc, Bangalore Page 3

L · dildt

= Vg − Vsw − ilRl − Vo (2.1)

The Current through the capacitor is given by:

C · dvcdt

= il − io (2.2)

2.2 Switch is OFF

Figure 2.4: Circuit of the buck converter when switch is OFF

When Switch ”S” is OFF, voltage across inductor is given by:

L · dildt

= −Vf − ilRl − Vo (2.3)

The Current through the capacitor is given by:

C · dvcdt

= il − io (2.4)

2.3 Discrete equations obtained when switch is ON

2.3.1 Inductor current

Vl = Vg − Vsw − ilRl − Vo (2.5)

L · dildt

= Vg − Vsw − ilRl − Vo (2.6)

L · dildt

= Vg − Vsw − ilRl − (icRc + Vc) (2.7)

ic = il − io (2.8)

ic = il −VoR

ic = il −icRc + Vc

R(2.10)

ic + ic ·Rc

R= il −

(2.11)

·(il −

)(2.12)

Substituting this in the Vl expression we get :

Vl = Vg − Vsw − ilRl −[(

)·(il −

)·Rc + Vc

](2.13)

Converting the equations into dicrete quantities:

il(k + 1)− il(k)

Ts=VgL− Vsw

L− ilRl

L−[(

)·(il −

)· Rc

L+Vc(k)

](2.14)

Divide throughout by Ibase (ib) and Vbase (vb) to per-unitize the current and voltagerespectively:

ilpu(k + 1)− ilpu(k) =

Ts·Vgpu·Vb

L·ib− Ts·Vswpu·Vb

L·ib− Ts·ilpu(k)·Rl

L−[(

)(ilpu(k)− Vcpu(k)·Vb

)TsLRc

+ Vcpu(k)·Vb

]Finally:

ilpu(k+1)−ilpu(k) = a·Vgpu−a·Vswpu−a·ilpuRlpu−[(ilpu(k)− Vcpu(k)

)· b+ Vcpu(k) · a

](2.15)

Where, a =TsLZb

=2µs2mH25Ω

= (409)d = (199)h (2.16)

and, b =TsLRl

=2µs2mH2Ω

= (32)d = (20)h (2.17)

2.3.2 Capacitor voltage

From previous equations we have:

] [il −

](2.18)

Converting the equations into discrete form and per-unitizing them by dividing through-out by Ibase (ib) and Vbase (vb) we get:

Cdvcdt

] [il −

](2.19)

vc(k + 1)− vc(k)

RC + CRc

] [il −

](2.20)

vcpu(k + 1)− vcpu(k) =

RC + CRc

] [R · ilpu(k) · ib

vb− Vcpu

](2.21)

Neglecting the term CRc we get:

][Rpuilpu(k)− Vcpu(k)] (2.22)

vcpu(k + 1)− vcpu(k) =Ts

RZbZbC

[Rpuilpu(k)− Vcpu(k)] (2.23)

Finally we get:

vcpu(k + 1)− vcpu(k) =TsZbC

[ilpu(k)− Vcpu(k)

](2.24)

xc =TsZbC

= (131)d = (83)h (2.25)

2.4 Discrete equations obtained when switch is OFF

2.4.1 Inductor Current

We have:

Vl = −Vf − ilRl − Vo

L · dildt

= −Vf − ilRl − [icRc + Vc]

we have ic = il − io

ic = il −VoR

ic = il −[icRc + Vc

] [il −

]Substituting the above value of ic in Vl we get :

Vl = −Vf − ilRl −[(

)·(il −

)·Rc + Vc

]Converting the equations into discrete form and per-unitizing them by dividing through-

out by Ibase (ib) and Vbase (vb) we get:

[il(k + 1)− il(k)

]= −Vf − ilRl −

)·(il(k)− Vc(k)

)·Rc + Vc

]il(k + 1)− il(k) = −Ts

LVf − il(k)Rl

TsL−[(

)(il(k)− Vc(k)

)]− Vc(k)

Noting that RRc

R+Rcratio is always 0.59, for example 100·0.6

100+0.6, we take it’s inverse (1.67)

in all the calculations of constants.

ilpu(k + 1)− ilpu(k) = −TsLVfpu

Vbib− ilpu(k)

)(ilpu(k)− Vcpu(k)Vb

)]− TsVcpu(k)Vb

ilpu(k + 1)− ilpu(k) = −TsLZb

Vfpu − ilpu(k)TsLRl

)(ilpu(k)− Vcpu(k)

)]− Vcpu(k)

Finally:

ilpu(k + 1)− ilpu(k) = −AVfpu −Bilpu(k)−[Cilpu(k)− CVcpu(k)R−1

]−AVcpu(k) (2.26)

Where:

A =TsLZb

=2µ2m25Ω

= (409)d = (199)h (2.27)

B =TsLRl

=2µ2m2Ω

= (32)d = (20)h (2.28)

C =TsL

=2µ2m1.67

= (27)d = (1B)h (2.29)

2.4.2 Capacitor Voltage

From previous equations we have:

] [il −

](2.30)

Converting the equations into discrete form and per-unitizing them by dividing through-out by Ibase (ib) and Vbase (vb) we get:

Cdvcdt

] [il −

](2.31)

vc(k + 1)− vc(k)

RC + CRc

] [il −

](2.32)

RC + CRc

] [R · ilpu(k) · ib

vb− Vcpu

](2.33)

Neglecting the term CRc we get:

][Rpuilpu(k)− Vcpu(k)] (2.34)

vcpu(k + 1)− vcpu(k) =Ts

RZbZbC

[Rpuilpu(k)− Vcpu(k)] (2.35)

Finally we get:

vcpu(k + 1)− vcpu(k) =TsZbC

[ilpu(k)− Vcpu(k)

](2.36)

xc =TsZbC

= (131)d = (83)h (2.37)

2.5 Block diagram implementation of switching model

induct

Chapter 3

Average Model

The various formulae for calculating inductor current and output voltage during CCMand DCM are given below:

Continuous conduction mode

Vo = VinD

[1− Vf (1−D)

](3.1)

Il =VoR

Discontinuous conduction mode

Vo =DVg

[1− VfD2

](3.3)

Il =VoR

RTswhere Ts = Switching frequency (3.5)

Kcri = 1−D (3.6)

and finally:

D2 =−D +

√D2 + 4K

2(3.7)

3.1 Block diagram implementation of average model

induto

Chapter 4

Real-time simulation results

Switching Model

Figure 4.1: The start-up transient of the inductor current of the switching model

Figure 4.2: The transient of the inductor current of the switching model when load changesfrom 1000 ohm to 100 ohm

Figure 4.3: The transient of the inductor current of the switching model when duty ratiochanges from 0.2 to 0.5

Figure 4.4: The transient of the inductor current of the switching model when inputvoltage changes from 40 V to 60 V

Figure 4.5: The start-up transient of the output voltage of the switching model

Figure 4.6: The transient of the output voltage of the switching model when load changesfrom 1000 ohm to 100 ohm

Figure 4.7: The transient of the output voltage of the switching model when load changesfrom 100 ohm to 1000 ohm

Figure 4.8: The transient of the output voltage of the switching model when duty ratiochanges from 0.2 to 0.5

Figure 4.9: The transient of the output voltage of the switching model when input voltagechanges from 40 V to 60 V

Average Model

Figure 4.10: The start-up transient of the inductor current of the average model

Figure 4.11: The transient of the inductor current of the average model when load changesfrom 1000 ohm to 100 ohm

Figure 4.12: The transient of the inductor current of the average model when duty ratiochanges from 0.2 to 0.5

Figure 4.13: The transient of the inductor current of the average model when input voltagechanges from 40 V to 60 V

Figure 4.14: The start-up transient of the output voltage of the average model

Figure 4.15: The transient of the output voltage of the average model when load changesfrom 1000 ohm to 100 ohm

Figure 4.16: The transient of the output voltage of the average model when load changesfrom 100 ohm to 1000 ohm

Figure 4.17: The transient of the output voltage of the average model when duty ratiochanges from 0.2 to 0.5

Figure 4.18: The transient of the output voltage of the average model when input voltagechanges from 40 V to 60 V

References

[1] A VHDL Primer, 3rd Edition by Jayaram Bhasker

[2] Digital Electronics - A Practical Approach with VHDL, 9th Edition by William Kleitz

[3] FPGA board document by Venugopal.S

[4] FPGA DOCUMENT-2 by N. Praveen Kumar

[5] FPGA Board Document, Ver.1.3, Revision 2, February 20, 2007 by Parag AnandRajne, Jayalakshmi and Ravi Krishna

[6] Digital Systems Design with FPGA, course material by Kuruvilla Varghese

[7] Dynamic Performance of Switched Mode Power Converters in Simulation of powerelectronic circuits, course material by M.B.Patil, V.T.Ranganathan and V. Rama-narayanan. Narosa, New Delhi, 2009.

[8] Course material on Switched mode power conversion by V.Ramanarayanan.

[9] Fundamentals of power electronics by Robert.W.Erickson and Dragan Maksimovic.

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