real-time simulation of the buck converter · chapter 1 introduction this report concentrates on...
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Real-time Simulation Of TheBuck Converter
By
Janamejaya Channegowda
Power Electronics Group
Department of Electrical Engineering
Indian Institute of Science
Bangalore - 560 012
India
February 2014
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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.
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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
2
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+
−
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:
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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)
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ic = il − io (2.8)
ic = il −VoR
(2.9)
ic = il −icRc + Vc
R(2.10)
ic + ic ·Rc
R= il −
VcR
(2.11)
ic =R
R +Rc
·(il −
VcR
)(2.12)
Substituting this in the Vl expression we get :
Vl = Vg − Vsw − ilRl −[(
R
R +Rc
)·(il −
VcR
)·Rc + Vc
](2.13)
Converting the equations into dicrete quantities:
il(k + 1)− il(k)
Ts=VgL− Vsw
L− ilRl
L−[(
R
R +Rc
)·(il −
Vc(k)
R
)· Rc
L+Vc(k)
L
](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−[(
RR+Rc
)(ilpu(k)− Vcpu(k)·Vb
ibR
)TsLRc
+ Vcpu(k)·Vb
ib·L
]Finally:
ilpu(k+1)−ilpu(k) = a·Vgpu−a·Vswpu−a·ilpuRlpu−[(ilpu(k)− Vcpu(k)
Rpu
)· 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)
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2.3.2 Capacitor voltage
From previous equations we have:
ic =
[R
R +Rc
] [il −
VcR
](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
=
[R
R +Rc
] [il −
VcR
](2.19)
vc(k + 1)− vc(k)
Ts=
[R
RC + CRc
] [il −
VcR
](2.20)
vcpu(k + 1)− vcpu(k) =
[Ts
RC + CRc
] [R · ilpu(k) · ib
vb− Vcpu
](2.21)
Neglecting the term CRc we get:
vcpu(k + 1)− vcpu(k) =
[TsRC
][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)
Rpu
](2.24)
here:
xc =TsZbC
= (131)d = (83)h (2.25)
2.4 Discrete equations obtained when switch is OFF
2.4.1 Inductor Current
We have:
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Vl = −Vf − ilRl − Vo
L · dildt
= −Vf − ilRl − [icRc + Vc]
we have ic = il − io
ic = il −VoR
ic = il −[icRc + Vc
R
]ic =
[R
R +Rc
] [il −
VcR
]Substituting the above value of ic in Vl we get :
Vl = −Vf − ilRl −[(
R
R +Rc
)·(il −
VcR
)·Rc + Vc
]Converting the equations into discrete form and per-unitizing them by dividing through-
out by Ibase (ib) and Vbase (vb) we get:
L
[il(k + 1)− il(k)
Ts
]= −Vf − ilRl −
[(R
R +Rc
)·(il(k)− Vc(k)
R
)·Rc + Vc
]il(k + 1)− il(k) = −Ts
LVf − il(k)Rl
TsL−[(
TsL
)(RRc
R +Rc
)(il(k)− Vc(k)
R
)]− Vc(k)
TsL
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)
TsLRl
−
[(TsL
1.67
)(ilpu(k)− Vcpu(k)Vb
ibR
)]− TsVcpu(k)Vb
Lib
ilpu(k + 1)− ilpu(k) = −TsLZb
Vfpu − ilpu(k)TsLRl
−
[(TsL
1.67
)(ilpu(k)− Vcpu(k)
RZb
)]− Vcpu(k)
TsLZb
Finally:
ilpu(k + 1)− ilpu(k) = −AVfpu −Bilpu(k)−[Cilpu(k)− CVcpu(k)R−1
pu
]−AVcpu(k) (2.26)
Where:
A =TsLZb
=2µ2m25Ω
= (409)d = (199)h (2.27)
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B =TsLRl
=2µ2m2Ω
= (32)d = (20)h (2.28)
C =TsL
1.67
=2µ2m1.67
= (27)d = (1B)h (2.29)
2.4.2 Capacitor Voltage
From previous equations we have:
ic =
[R
R +Rc
] [il −
VcR
](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
=
[R
R +Rc
] [il −
VcR
](2.31)
vc(k + 1)− vc(k)
Ts=
[R
RC + CRc
] [il −
VcR
](2.32)
vcpu(k + 1)− vcpu(k) =
[Ts
RC + CRc
] [R · ilpu(k) · ib
vb− Vcpu
](2.33)
Neglecting the term CRc we get:
vcpu(k + 1)− vcpu(k) =
[TsRC
][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)
Rpu
](2.36)
here:
xc =TsZbC
= (131)d = (83)h (2.37)
2.5 Block diagram implementation of switching model
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Fig
ure
2.5:
Con
stan
tsuse
din
the
blo
ckdia
gram
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Fig
ure
2.6:
DA
Cin
terf
ace
par
tI
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Fig
ure
2.7:
DA
Cin
terf
ace
par
tII
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Fig
ure
2.8:
DA
Cin
terf
ace
par
tII
I
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Fig
ure
2.9:
The
induct
orcu
rren
tis
pre
vente
dfr
omb
ecom
ing
neg
ativ
e
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Fig
ure
2.10
:T
he
enab
lesi
gnal
sar
ege
ner
ated
ata
freq
uen
cyof
500k
Hz
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Fig
ure
2.11
:T
he
freq
uen
cyb
eing
gener
ated
at50
kH
z
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Fig
ure
2.12
:C
alcu
lati
onof
induct
orcu
rren
tpar
tI
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Fig
ure
2.13
:C
alcu
lati
onof
induct
orcu
rren
tpar
tII
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Fig
ure
2.14
:B
lock
sin
volv
edin
inte
grat
ion
ofin
duct
orcu
rren
tan
dca
pac
itor
volt
age
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Fig
ure
2.15
:B
lock
show
ing
the
sele
ctio
nof
on/o
ffin
stan
ces
ofth
ein
duct
orcu
rren
tb
eing
sele
cted
bas
edon
swit
chin
gfr
equen
cy
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Fig
ure
2.16
:B
lock
show
ing
the
calc
ula
tion
ofin
duct
orcu
rren
tduri
ng
the
offp
erio
d
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Fig
ure
2.17
:B
lock
show
ing
the
calc
ula
tion
ofou
tput
volt
age
duri
ng
on/o
ffp
erio
dof
the
swit
ch
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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)
DVg
] [R
Rl +R
](3.1)
Il =VoR
(3.2)
Discontinuous conduction mode
Vo =DVg
D +D2
[1− VfD2
DVg
](3.3)
Il =VoR
(3.4)
K =2L
RTswhere Ts = Switching frequency (3.5)
Kcri = 1−D (3.6)
and finally:
D2 =−D +
√D2 + 4K
2(3.7)
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3.1 Block diagram implementation of average model
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Fig
ure
3.1:
The
const
ants
invo
lved
inth
eblo
ckdia
gram
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Fig
ure
3.2:
Cal
cula
tion
ofD
2par
tI
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Fig
ure
3.3:
Cal
cula
tion
ofD
2par
tII
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Fig
ure
3.4:
Clo
cks
that
are
use
din
the
DA
Cin
terf
ace
par
tI
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Fig
ure
3.5:
DA
Cin
terf
ace
par
tII
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Fig
ure
3.6:
Gen
erat
ion
ofen
able
puls
esat
500k
Hz
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Fig
ure
3.7:
Sel
ecti
onof
induto
rcu
rren
tin
CC
M
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Fig
ure
3.8:
Sel
ecti
onof
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Development of Off-line and Real-Time Simulator for Electric vehicle /Hybrid Electric Vehicle Systems
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Development of Off-line and Real-Time Simulator for Electric vehicle /Hybrid Electric Vehicle Systems
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Development of Off-line and Real-Time Simulator for Electric vehicle /Hybrid Electric Vehicle Systems
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Chapter 4
Real-time simulation results
Switching Model
Figure 4.1: The start-up transient of the inductor current of the switching model
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Development of Off-line and Real-Time Simulator for Electric vehicle /Hybrid Electric Vehicle Systems
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
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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
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Development of Off-line and Real-Time Simulator for Electric vehicle /Hybrid Electric Vehicle Systems
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
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Development of Off-line and Real-Time Simulator for Electric vehicle /Hybrid Electric Vehicle Systems
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
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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
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Development of Off-line and Real-Time Simulator for Electric vehicle /Hybrid Electric Vehicle Systems
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
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Development of Off-line and Real-Time Simulator for Electric vehicle /Hybrid Electric Vehicle Systems
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
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Development of Off-line and Real-Time Simulator for Electric vehicle /Hybrid Electric Vehicle Systems
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
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Development of Off-line and Real-Time Simulator for Electric vehicle /Hybrid Electric Vehicle Systems
Figure 4.18: The transient of the output voltage of the average model when input voltagechanges from 40 V to 60 V
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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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