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Presentation Title (can go to two or three lines) This font is Georgia
Department of Electrical and Computer Engineering
Presentation Title (can go to two or three lines) This font is Georgia
n
m
x
x
00 uxs −=> ,
00 uxs =< ,0=+= xcxs
Concept of Sliding Mode ( Second order relay system )
0 0
( , ) ,sgn( ), , , : const
x f x t uu u s s cx x u c= += − = +
Presentation Title (can go to two or three lines) This font is Georgia
Introduction - Chattering
Causes of Chattering
1. Imperfections of switching elements 2. Fast dynamics disregarded in an ideal model (unmodeled dynamics) 3. Switching frequency limitation * Discrete time system (discretization chattering)
s
0 t
Presentation Title (can go to two or three lines) This font is Georgia Sliding Mode Control
Power Converter
to be tracked. Error
0 ( , )u x t
vusvu
uus
−==
−=
0
0
)()( 0uMsMsignv >=
)(sMsignv h=
v
s
s
t
)()( 0 tutu =
v
s
s
t
∆v
Chattering (ripple) amplitude 2∆
=A
Frequency M
uMf2
)(1 20
2 −∆
=
Sliding Mode
0( , , ) ( , )x f x t u u u x t= ⇒ =
Presentation Title (can go to two or three lines) This font is Georgia Frequency Control
MaMf
21 22 −∆
= .0 constua ≈=
K
+M
-M + Input + Output
∆
Output
Input=
f * : Desired Frequency M
aMf 21 22
*
−=∆ (*)
KM2=∆ (**) (*), (**): 2
22
* 41
MaM
fK −=
Presentation Title (can go to two or three lines) This font is Georgia
SMC in Power Electronics
• DC-to-DC Converters • Three-phase Converters
Presentation Title (can go to two or three lines) This font is Georgia
Buck Boost
Models:. ........
dvdt
iC
vRC
dvdt
iC
uvRC
didt
VL
uvL
didt
VL
vL
u
o L o o L o
L g o L g o
= − = − −
= − = −
( - )
( )1
1
Control: u =
10
if switch is in position 1 if switch is in position 2
Switching Converters - Dynamics
Presentation Title (can go to two or three lines) This font is Georgia
Buck Converter – Control
( )
( ) ( )
( )
,
+
1 0; 0, 0
0 0
0
o oL ov o L
g LoLi i
refi L L i
refrefo orefo oL
o o
ref refref o Lo o
dv vi dvf v i
dt C RC dtV divdi f b uu
dtdt L Lif
i i b uif
d v vdv vi and v vdt C RC dt
dv viv vdt C
σσ σ σ
σ
µ
µ
•
= − = ⇒
= += − <
= − < > ⇒ >
−= − − + =
− + − − ( )0
refref refo o oL o o
refL L
v dvi C v v C
RC R dt
i i C
µ
σ
= ⇒ = + − +
⇒ = +
Cv
dtdv
Ci refrefLref
00 +=
Reference input )(0 tv ref
uL
Vg+= ...σ
Presentation Title (can go to two or three lines) This font is Georgia Boost Converter Control
( )
( )22
20
(1 )
(1- )
1 0
; 0, 00 0
:
1
2 2
o oL
g oL
refi L L i
refgL
eqo
refLo
g L
dv vi udt C RC
V vdi udt L L
Current controlif
i i b uif
Equivalent controlVdiLu
v dt L
d idvRC RLv RV i nonlineardt dt
σσ σ σ
σ
•
= − −
= −
<= − < > ⇒ >
− = − +
+ = − → system
,00 vv ref −=σ ,00 vv ref −=σ
)1(... uCiL −=σ
Presentation Title (can go to two or three lines) This font is Georgia
Power converter
Chattering (Ripple) Problem
f * : Switching Frequency
Amplitude of oscillation should be reduced.
Vs
switch
LRa
C R (Load)
L
Lm
T1
switch
LRa L
Lm
T2
i1
VL
iL
i2
.
..
...
..
..
∑= ki
Multiphase converter
- Triangular wave with frequency f * (offset)
- Timer (delay)
- Transformers
- Filter depends on frequency
t
Presentation Title (can go to two or three lines) This font is Georgia Chattering (Ripple) Problem
Two phases (without phase shift) Two phases (with phase shift)
Presentation Title (can go to two or three lines) This font is Georgia
Implementation of Multiphase converter
mu
mu0
mu0
m
,0ii u
mus −= )(0
ihi sMsignmus −=
- Phase in each PC depends on initial condition
- Phases in PC’s are independent
chatteringichattering mAA =max
Presentation Title (can go to two or three lines) This font is Georgia Number of Phases I
m-phase converter
Any discontinuous signal v is a periodic function with the same frequency ω. ∑∞
=
+=1
)sin(k
ikki ktgv ϕω (Fourier Series)
The phase shift between each phase and preceding is equal to T/m.
Then the k-th harmonic is the output signal (sum of the outputs).
Then Z=0 for all harmonics except for k=m, 2m, …
Presentation Title (can go to two or three lines) This font is Georgia
Mathematical Aspects IISliding Mode Existence Conditions
0)( ,0)( 21 =′>′ TsTs0)"( ,0)"( 21 <= TsTs
1 s1=03
s2=0
0)0(0)0(
2
1
>=
ss
2
Scalar Control: 0lim0lim00
and ><−→+→ss
ss
s=0Vector Control
. 2 2
212
211
ssignssignsssignssigns
−−=+−=
c a
d b
0=)(xs j
0=)(xsi
Presentation Title (can go to two or three lines) This font is Georgia
Presentation Title (can go to two or three lines)This font is GeorgiaDesign Principle (Two-phase PC)
The relay function v2 of the second phase depends on the two errors
12*2 sss −=
02)()(
)(*21
*2
11
orMsMsignsMsignsasMsigns
hh
h
±⇒−=
+−=
Period
Phase Shift
MT ∆=
2
42T
M=
∆=φ
0=a
∆−
=22
2aM
MT
Presentation Title (can go to two or three lines) This font is Georgia Simulation of Phase Shift
Phase shift can be controlled if 12 ∆=∆ α )0( =a
Phase shift 42T
M<
∆=αφ0≠a
1<α
1v
2v
Presentation Title (can go to two or three lines) This font is Georgia Phase Shift
Phase shift mT
M=
∆=
2αφ
Period 22
2aM
MT−∆
=
)(4
22
2
aMmM−
=α
Presentation Title (can go to two or three lines) This font is Georgia Number of Phases II
Switching sequence should be preserved
for any α: point 3 should be before point 4.
aMM +∆
<∆
2α
−<<
−>⇒
mMaaor
aMMm 212
maxmax
Finally,
<<+
−<
−=Maaif
aMM
mMaif
aMmM
max
22
2
2
21)(
4
α
=
Presentation Title (can go to two or three lines) This font is Georgia Master-slave Mode
For k=1/α the equations are identical to those of the first version.
Implementation: All phases are with the same hysteresis.
Presentation Title (can go to two or three lines) This font is Georgia Simulation I
Power converter
),,2,1( mk =
Two-phase Converter )4( VVref =
Presentation Title (can go to two or three lines) This font is Georgia Simulation I
Frequency
50 Hz
Presentation Title (can go to two or three lines) This font is Georgia Simulation I
0007.00127.00013.0 2 −+−= refrefh VVK
Frequency Control (50Hz) 4-phase Converter )4.2( VVref =
Presentation Title (can go to two or three lines) This font is Georgia Simulation I
4-phase Converter )4.2( VVref =(Unmodified)
4-phase Converter )4.2( VVref =(Modified)
Presentation Title (can go to two or three lines) This font is Georgia Simulation I
4-phase Converter )2.7( VVref =4-phase Converter )8.4( VVref =
Presentation Title (can go to two or three lines) This font is Georgia Simulation I
6-phase Converter )4.2( VVref =6-phase Converter )714.1( VVref =
Presentation Title (can go to two or three lines) This font is Georgia Simulation I
4-phase Converter time varying Vref
6-phase Converter time varying Vref
Presentation Title (can go to two or three lines) This font is Georgia Simulation I
6-phase Converter time varying Vref
4-phase Converter time varying Vref
Presentation Title (can go to two or three lines) This font is Georgia Simulation II
IKOR Power Converter 4-phase Converter )3( VVref =
Presentation Title (can go to two or three lines) This font is Georgia Simulation II
4-phase Converter )8( VVref =4-phase Converter )6( VVref =
Presentation Title (can go to two or three lines) This font is Georgia
Simulation Results (4-phase) Vref (reference voltage) = 1.5 V
L (inductance) = 50*10-9 H Ra (internal resistance) = 0.3*10-3 Ω
C (capacitance) = 1*10-3 F RL (load resistance) = 0.001 Ω
Vs (source Voltage) = 12 V
Vref (reference voltage) = 1.5 V
Presentation Title (can go to two or three lines) This font is Georgia
• using two more additional subsystems 2 *1 , / 3K a K K= − =
Master-Slave Version
Presentation Title (can go to two or three lines) This font is Georgia
Simulation Result (8-phase) Vref (reference voltage) = 1.5 V L (inductance) = 50*10-9 H Ra (internal resistance) = 0.3*10-3 Ω C (capacitance) = 1*10-3 F RL (load resistance) = 0.001 Ω Vs (source Voltage) = 12 V
8 phases
Presentation Title (can go to two or three lines) This font is Georgia
Presentation Title (can go to two or three lines)This font is Georgia
Multiphase SMC –Unmodeled Dynamics
Equivalent width of hysteresis
- The frequency of oscillation in first phase is determined by unmodeled dynamics.
- The equivalent width of hysteresis is applied for the other phases.
Presentation Title (can go to two or three lines) This font is Georgia