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Presentation Title (can go to two or three lines) This font is Georgia

Department of Electrical and Computer Engineering


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= += − = +


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 −=


SMC in Power Electronics

• DC-to-DC Converters • Three-phase Converters


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


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


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)


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, …


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 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 =


Frequency

50 Hz


0007.00127.00013.0 2 −+−= refrefh VVK

Frequency Control (50Hz) 4-phase Converter )4.2( VVref =


4-phase Converter )4.2( VVref =(Unmodified)

4-phase Converter )4.2( VVref =(Modified)


4-phase Converter )2.7( VVref =4-phase Converter )8.4( VVref =


6-phase Converter )4.2( VVref =6-phase Converter )714.1( VVref =


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 =


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


• using two more additional subsystems 2 *1 , / 3K a K K= − =

Master-Slave Version


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

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.

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