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2.1 Buck converter
2.1.1 Operation modes 2.1.2 Voltage transfer function
2.1.3 Current modes (CCM, DCM)
2.1.4 Capacitor current 2.2 Boost converter
2.2.1 Operation modes
2.2.2 Voltage transfer function 2.3 Buck-Boost converter
2.4 Comparison between topologies
2.5 Simulation of SMPS 2.5.1 The simulations problem
2.5.2 Basics of average model of SMPS 2.5.3 Example: Boost average model simulations
BUCK, BOOST, BUCK-BOOST, DCM
Prof. S. Ben-Yaakov , DC-DC Converters [2- 2]
Buck Converter Constant Switching
Frequency
t
ON ON ON
t
ON ON ON
control
switch
t on
t off
TS
s
s T
f =
D or D T
t on s
on = →
D 1 D
T
t off s
off = →−
Switch frequency:
Duty Cycle:
S
Vin D
L
C
R
control
Prof. S. Ben-Yaakov , DC-DC Converters [2- 3]
Operation modes
On
Off
At steady state I (^) a=Ib
S
V (^) in D
L
C
R
S
Vin D
L
C
R
V (^) L
I (^) L
t (^) s
t
V (^) in-V (^) o
-V (^) o
I (^) a Ib t
Self commutation
V L
I (^) L
t (^) s
t
Vin -Vo
I (^) a t
Commutation
In this case
Inductor current waveform at steady state
L
V (^) in −Vo
ton
t
I (^) L
t (^) off
L
Vo −
∆ I
S
Vin D
C
R
ton
t (^) off
Buck
Prof. S. Ben-Yaakov , DC-DC Converters [2- 5]
Voltage transfer function
The ∆ I method
Left triangle
on
in ot
L
V V
I ⋅
Right triangle
off
o (^) t
L
V
∆I = ⋅
off
o on
in o t L
V
t L
V V
on s
on
on off
on
in
o (^) D
T
t
t t
t
V
V
= (^) Independent of L!
L
V (^) in −Vo
t (^) on
t
I L
t (^) off
L
Vo −
∆ I
Prof. S. Ben-Yaakov , DC-DC Converters [2- 6]
-V o
V L
t off t
V (^) in-V (^) o
t on
Ts
At steady state, over one switching cycle:
VL = 0 ;
on in
o D V
V
S ++ S−= 0 ⇒ =
S (^) +=(Vin −Vo)⋅ton ;
S (^) −=(−Vo )⋅toff ;
S
Vin D
C
R
ton
toff
V (^) o
V (^) L
Voltage transfer function
The average voltage method
t (^) on
t
I L
t ' off T s
I pk
Voltage transfer function (DCM)
The ∆ I method
off
o on
in o pk t L
V t L
V V I = ′
−
out
in out on off V
(V V )D D
−
⋅ +
− = T T(D D ) L
V V
2
1
T
1 I (^) on S on off
in o
S
AV
R
V I (^) AV = o
) V
V V T D( 1 L
V V
2
1 I o
in o on on
in o AV
− ⋅ +
−
2 Sin o
2 R( Vin −Vo)DonTV= 2 LV
= + − 1 RDT
8 L 1 4 L
RDT
V
V
s
2 on
s
2 on
in
o
Prof. S. Ben-Yaakov , DC-DC Converters [2- 11]
Boundary of CCM and DCM
t (^) on
t t (^) off
Ts
L
Vo − L
V (^) in −Vo
IL
L 2
L (^) min
I (^) av
z For CCM L > L (^) min
z In Buck (^) off pk av min
o t I 2 I L
V
s
off
avs
o off min 2 f
RD
2 If
VD
L = =
Prof. S. Ben-Yaakov , DC-DC Converters [2- 12]
Example
A BUCK converter has a following characteristics:
Output voltage: Output current:
Input voltage: Frequency:
Current mode: CCM
Find:
Vo = 5 V I (^) out =Iav= 10 A
Vin = 10 V fs = 100 kHz
L min
1. 2 H
2 If
VD
L
D 1 D 0. 5
CCM
D 0. 5
V
V
5 avs
o off min
on off on in
o
= μ ⋅ ⋅
I
L
t
I
av
t
I
av
I R
I C
t
AC
DC
Capacitor current
Capacitor current
S
Vin D
L
I C^ R L I^ C (^) I control R
V (^) o
C L R
I =I −I
z Assumption:
V 0 has small ripple
Prof. S. Ben-Yaakov , DC-DC Converters [2- 14]
BOOST Step-Up
z V (^) o > V (^) in Why ??
V^ S
in
L D
C R
V
X
V
o
Prof. S. Ben-Yaakov , DC-DC Converters [2- 15]
ON V L =V in
OFF V L =V in -V o
V (^) in
L
C R
Vo
V (^) in
L
C R
V (^) o
Operation modes
VL
I (^) L
t (^) s
t
V in
I (^) a t
V (^) L
I (^) L
t (^) s
t
V (^) in
V (^) in-V (^) o
I (^) a Ib t
Boost
Comparison between basic topologies CCM
S
V (^) in
L D
C R
V (^) o
S Vin D L
C R
V (^) o
S V (^) in D
L C R
V (^) o
S
L D
Basic Cell
L a
b
c
Switched inductor
Prof. S. Ben-Yaakov , DC-DC Converters [2- 20]
Iin
t
Iin
t
Iin
t
Io
t
Io
t
Io
t
Source current Load current
Buck
Boost
Buck Boost
Continues current -> Low ripple component
Discontinues current -> High ripple component
Input and Output Currents
Prof. S. Ben-Yaakov , DC-DC Converters [2- 21]
Modulator Control
D Ve
Vin
Assembly
Switched
o
V
−
The simulation problem
•The problematic part : Switched Assembly
- Rest of the circuit continuous - SPICE compatible
- Only possible simulation :
Time domain (cycle-by-cycle) -Transient
- The objective : translate the
Switched Assembly into an equivalent
circuit which is SPICE compatible
Modulator Control
D Ve
Vin
Assembly
Switched Vo
−
The simulation problem
Prof. S. Ben-Yaakov , DC-DC Converters [2- 23]
−
−
−
b d^ c
a
C (^) f
RLoad
Vout
Vin IL
Ib (^) IC
Vout (^) Vout
RLoad RLoad Cf Cf
L
a d (^) c
b
IL IC Ib Vin Vin
b on
T (^) L
Ib IL
IC
d
c
L
Buck Boost
Buck −Boost
T on
−
Average Simulation of PWM Converters
Prof. S. Ben-Yaakov , DC-DC Converters [2- 24]
T (^) on - switch conduction time
T (^) off - diode conduction time
T (^) DCM - no current time (in DCM)
b L^ a
c
b Ton
TDCM
Toff
L
c
a
The Switched Inductor Model
b
c
Ib
Ic
a a
G
Gb
Gc
Ia
b
c
Ia =IL a
I (^) b =IL⋅D on
Ic =IL⋅D off
G (^) a , G (^) b ,C (^) c - current
dependent sources
c L off
b L on
a L
G I D
G I D
G I
Toward a continuous model
Prof. S. Ben-Yaakov , DC-DC Converters [2- 29]
L
DerivingI
V L
t
IL
IL
V
VL
L
V
IL
L
I
L
V
dt
dI
L
V
dt
dI (^) L L L L = ⇒ =
Average inductor current
Prof. S. Ben-Yaakov , DC-DC Converters [2- 30]
b
c
a L
V(a,b)
V(a,c)
VL
V( a,b)
V ( a,c)
Ton (^) Toff
Ts
on off
S
on off L
V(a,b)D V(a,c)D
T
V(a,b)T V(a,c)T V
Average inductor current
b
c
a
Ga
Gb
Gc
L
rL
IL
EL L
V
Topology independent!
E (^) L =V(a,b)⋅Don+V(a,c)⋅D off
Gc =IL⋅D off
Gb =IL⋅D on
Ga =I L
b
c
a
L on
T
Toff
The Generalized Switched Inductor Model
(GSIM)
Prof. S. Ben-Yaakov , DC-DC Converters [2- 32]
- The formal approach
b
c
a
Ga Gb
Gc
Co Ro
Vin
Vo
EL
I L
L
V(a,b)
V( a,c) rL
E (^) L =[V 0 −Vin]⋅Don+[ 0 −V 0 ]⋅Doff
Ga =I(L) Gb=I(L)⋅Don Gc=I(L)⋅D off
Example Implementation in Buck Topology
S
Vin D
L V (^) o
Ro
Co
b
c
a
Prof. S. Ben-Yaakov , DC-DC Converters [2- 33]
- The intuitive approach - by inspection
L
Co
in Ro
V
Vo
IL
Ein
Gb
S L
Co o
R
Vin
D
Vo
Polarity: (voltage and current
sources) selected by inspection
Ein −Vo→VL
Ein =Vin⋅D on
Gb =IL⋅D on
Implementation in Buck Topology
V in
Rdson (^) b
c
a
G
Gb
Gc
Co
Ro
rc
a
L
rL
IL
EL L
V
E (^) L =(Va−Vb)⋅Don+(Va−Vc)⋅Doff
Gc =IL⋅D off
Gb =IL⋅D on
Ga =I L
Modified Average Model
Prof. S. Ben-Yaakov , DC-DC Converters [2- 38]
IL and D (^) on are time dependent variables {IL (t), D (^) on (t) }
D (^) on is not an electrical variable
ILDon
b
G L^ L
I
Making the model SPICE compatible
Prof. S. Ben-Yaakov , DC-DC Converters [2- 39]
D (^) on is coded into voltage
Source
Don
Name ofnode:"Don"
V (D ) I(L)
on
∗ L
Gvalue
In SPICE environment
Running SPICE simulation
DC (steady state points) - as is
TRAN (time domain) - as is
AC ( small signal) - as is
- Linearization is done by simulator!
Simulation
Prof. S. Ben-Yaakov , DC-DC Converters [2- 41]
I L
b
c
a L^
Ton
Toff
Ton Toff
Toff
Ts
t
L ILpk
I
on s
s on off 1 D T
T T
D' = −
T' (^) off =Ts−T on
Discontinuous Model (DCM)
Prof. S. Ben-Yaakov , DC-DC Converters [2- 42]
1.The average inductor current in DCM
VL (^) V(a,b)
V(a,c)
Ts
Ton Toff
T'off
t
b
c
a L
V(a,b)
V(a,c)
Ton
VL =V(a,b)Don+V(a,c)DoffasinCCM
Combining CCM / DCM
b
c
a
L
b
c
a
Ga
Gb
Gc
Ga ≡IL
on off
L on b D D
ID
G
on off
L off c D D
ID
G
E (^) L =V(a,b)⋅Don+V(a,c)⋅D off
= − − on
on
L s off on D V(a,b)D
2 ILf D min( 1 D),
L
rL
IL
EL VL
The combined DCM / CCM mode
Prof. S. Ben-Yaakov , DC-DC Converters [2- 47]
Example: Boost average model simulation
Rsw {Rsw}
EDoff
min(2I(Lmain)Lmain/(Tsv(a,b)V(Don))-V(Don),1-V(Don))
etable
OUT+ OUT-
IN+ IN-
Resr {Resr}
Gc
V(Doff)*I(Lmain)/(V(Don)+V(Doff))
GVALUE
OUT+
OUT- IN+
IN-
PARAMETERS: LMAIN = 75u COUT = 220u RLOAD = 10
Doff
Gb
V(Don)*I(Lmain)/(V(Don)+V(Doff))
GVALUE
OUT+
OUT- IN+
IN-
0
Lmain {Lmain}
RLoad {RLoad}
Dbreak
Dmain
VDon {VDon}
Rinductor
{Rinductor}
EL
(V(Don)V(a,b)+V(Doff)V(a,c))
EVALUE
OUT+ OUT-
IN+ IN-
1
0
PARAMETERS: FS = 100k TS = {1/fs}
b
Vin_DC
{Vin_DC}
a Cout {Cout}
PARAMETERS: RESR = 0. RINDUCTOR = 0. RSW = 0.
PARAMETERS: VIN_DC = 10v VDON = 0.
c out
Ga
I(Lmain)
GVALUE
OUT+ OUT-
IN+ IN-
Don
S
L
Co o R Vin
D Vo
Prof. S. Ben-Yaakov , DC-DC Converters [2- 48]
Example: Boost average model simulation
Example: Boost average model simulation
S
L
Co o R Vin
D Vo
Prof. S. Ben-Yaakov , DC-DC Converters [2- 50]
Example: Boost average model simulation
S
L
Co o R Vin
D Vo
Prof. S. Ben-Yaakov , DC-DC Converters [2- 51]
Boost: Response to step of input voltage
Ti me
3 0 ms 3 5 ms 4 0 ms 4 5 ms 5 0 ms V( o u t )
1 8 V
1 9 V
2 0 V
2 1 V
SEL>>
V( a )
9 V
1 0 V
1 1 V
1 2 V
(average model simulation)
Vin
Vout
