power_electronics_harmonic_analysis_-_ignatova (1).doc

8/11/2019 Power_Electronics_Harmonic_Analysis_-_Ignatova (1).doc

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POWER ELECTRONICS HARMONIC ANALYSIS BASED ON THE LINEARTIME PERIODIC MODELING. APPLICATIONS FOR AC/DC/AC POWER

ELECTRONIC INTERFACE.

Vanya Ignatova*, Pierre Granjon**, Seddik Bacha*

*LEG: Laboratory of Electrical Engineering of Grenoble, France**LIS: Laboratory of Images and Signals, France

SUMMARY

This paper presens an ana!"i#a! $re%&en#"'()*ain *eh)( $)r har*)ni#s *)(e!in+ an( e,a!&ai)n in

p)-er e!e#r)ni# s"se*s. The #)nsi(ere( s"se* is (es#rie( " a se )$ (i$$erenia! e%&ai)ns -hi#h are

#)n,ere( in he $re%&en#" ()*ain an( presene( in a *ari0 $)r*. In(ee( #&rrens an( ,)!a+es are

(es#rie( in er*s )$ F)&rier series an( arran+e( in a ,e#)r $)r*. The passi,e e!e*ens an( he s-i#hin+

$&n#i)ns are hen represene( " har*)ni# rans$er *ari#es. The res)!&i)n )$ he *ari0 e%&ai)ns !ea(s

) he)rei#a! i*e an( $re%&en#" e0pressi)ns )$ he s"se* ,)!a+es an( #&rrens.

This *eh)( is app!ie( ) a #!)se('!))p hree phase AC/DC/AC PWM #)n,erer. The #)nr)! !))p )$ he

#)n,erer is *)(e!e( " a((ii)na! e%&ai)ns. The spe#ra )$ he s-i#hin+ $&n#i)ns ne#essar" ) &i!( he

#)rresp)n(in+ har*)ni# rans$er *ari#es are #a!#&!ae( hr)&+h a ()&!e F)&rier series (e#)*p)sii)n.

The *ari0 e%&ai)ns are s)!,e( an( he res&!s are #)*pare( ) h)se )aine( " rea! *eas&re*ens an(

Ma!a/Si*&!in1 si*&!ai)ns.

2e" -)r(s3power system harmonics, power electronic, linear time periodic modeling, PW, control system

4. INTRODUCTION

!he wide spread "se of power electronic de#ices in power networ$s is d"e to their m"ltiple f"nctions:

compensation, protection and interface for generators% &dapting and transforming the electric energy,

they ma$e possible the insertion in the power networ$ of independent generators and renewable

so"rces of energy% 'owe#er, beca"se of their switching components, power electronic con#erters

generate c"rrent and #oltage harmonics which may ca"se meas"rements, stability and control problems%

In order to a#oid this $ind of harmonic dist"rbances, a good $nowledge on the harmonic generation

and propagation is necessary% & better "nderstanding of the harmonic transfer mechanisms co"ld ma$e

the harmonic atten"ation more efficient, optimi(ing filters and impro#ing power electronic control%

)


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!he harmonic st"dy can be effect"ated in the time domain or in the fre"ency domain% In the time

domain, c"rrents and #oltages spectra are obtained by application of Fo"rier transform% !his domain

can not gi#e an analytical harmonic sol"tion for the considered system and the relations between

harmonics can not be e+pressed%

In the fre"ency domain, se#eral methods for power networ$ harmonic analysis e+ist )-% !he simplest

consists to model the networ$ presenting power electronic de#ices by $nown so"rces of harmonic

c"rrents% &nother method presents con#erters by their .orton e"i#alent%

!hese two methods are the most often "sed in the networ$ harmonic analysis% !hey are simple, b"t not

acc"rate, beca"se they do not ta$e into acco"nt the dynamics of the switching components%

ore precise models designed for the power electronic de#ices e+ist% S"ch a model is the transfer

f"nction model, which lin$s the con#erter state #ariables by matri+ e"ations% &nother method

proposed in /- describes the con#erter by a set of nonlinear e"ations sol#ed by .ewton0s method%

!hese models ha#e a good acc"racy, b"t beca"se of their comple+ity they can not be applied to systems

containing m"ltiple con#erters%

For an acc"rate networ$ harmonic analysis, a simple and efficient method ta$ing into acco"nt the

harmonics ind"ced by the switching process is re"ired%

!he method proposed in this paper "ses the periodicity of the con#erter #ariables in steady state in

order to p"t them in a matri+ form in the fre"ency domain% Pre#io"s researches in this area ha#e been

already made% In 1-, the models of power electronic str"ct"res are b"ilt "sing harmonic transfer

matrices and are implemented in atlab2Sim"lin$% !his method is especially "sed for stability analysis

and for that reason data are simplified and high fre"encies are neglected% In 3-, a method "sing the

periodicity of the #ariables is presented, b"t it only gi#es a n"merical sol"tion and it is not applied in the

case of switching circ"its and networ$ analysis% 4oth pre#io"s methods do not gi#e analytical

e+pressions of the harmonics%

In this paper the presented method describes the considered system by differential e"ations, which are

then con#erted in the fre"ency domain% 4eing periodic signals, c"rrents and #oltages are described in

terms of Fo"rier series and then by #ectors of harmonics% !he passi#e elements and the switching

f"nctions are described by matrices% !he resol"tion of the matri+ e"ations gi#es time and fre"ency

e+pressions of #oltages and c"rrents%

!his paper is organi(ed as follows: Section / describes the harmonic transfer #ia the different

components of power electronic systems% !he method for harmonics assessment is described in Section

/


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1 and ill"strated with a simple e+ample% In section 3 the method is applied to a closed5loop three phase

PW &62762&6 con#erter% !he obtained res"lts are confirmed by real meas"rements and sim"lation

in Section 8%

5. HARMONIC TRANSFER 6IA PASSI6E AND SWITCHING ELEMENTS

Power electronic systems can be considered as combination of switching and passi#e components% In

this section the harmonic propagation thro"gh these elements is analysed and the necessity of their

matri+ representation is demonstrated%

When b"ilding the harmonic transfer matrices, some ass"mptions are made: the switching and the

passi#e components are s"pposed ideal, the considered system is s"pposed to be in steady state and

periodically time5#ariant%

5.4 Har*)ni# rans$er *ari0 hr)&+h)& s-i#hin+ e!e*ens

For the simple switching process presented in Fig%), the relation between ac and dc c"rrents 9:tiac and

9dci t is gi#en by:

9:9:9: tituti acdc = , )9

where 9:tiac is s"pposed iT 5periodic periodic with period of iT seconds9 and the switching

f"nction 9:tu is uT 5periodic with ui NTT = % In the following, N is an integer so that 9:tu can be

also considered as iT 5periodic%

Fig"re ): a simple switching process

!herefore, the pre#io"s signals can be decomposed in Fo"rier series as a f"nction of the same

f"ndamental fre"encyi

T

)and E% )9 finally becomes:

1


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tjk

k n

nknac

m n

tmnjmnac

m

tjmm

n

tjnnacdc

iiii euieuieueiti

=

=

=

=

+

=

=

> = is the kth

harmonic component of the iT 5periodic

signal 9:tx %

E% /9 shows that 9:tidc can be #iewed as a iT periodic signal with the following Fo"rier

coefficients:

= >-:

5 !he considered con#erter str"ct"re is described by differential e"ations% !he e"ations n"mber

depends on the n"mber of ind"ctances and capacitors in the system%5 !he differential e"ations are con#erted in the fre"ency domain and represented in a matri+ form%

6"rrents and #oltages are represented by #ectors of harmonics, passi#e elements become matrices

with diagonal str"ct"re, and the switching f"nctions become matrices with !oeplit( str"ct"re%

5 !he matri+ e"ations are sol#ed in the fre"ency domain, and the fre"ency e+pression of the

c"rrents and #oltages is obtained% !heir time e+pression can also be ded"ced by Fo"rier series%

>


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7.5 E0a*p!e

In order to be better ill"strated, the pre#io"s method is applied to the simple con#erter str"ct"re

described in Fig% / and containing both passi#e and switching elements%

Fig"re /: a simple con#erter str"ct"re!he matri+ e"ations describing the considered system are:

[ ] [ ][ ]

[ ] [ ][ ]

[ ][ ] [ ] [ ][ ]

=

=

=

dcacac

dcac

dcdc

VUVIL

IUI

VCI

, where ))9

[ ]C and [ ]L are the capacitor and the ind"ctor diagonal matrices,

[ ]U is the switching f"nction matri+ with a !oeplit( str"ct"re,

[ ]dcV , [ ]acV , [ ]dcI et [ ]acI are #ectors containing the harmonics of the corresponding state

#ariables%

& problem which may occ"r in this case is the non in#ersibility of the matrices corresponding to the

ind"ctor and the capacitor when the dc component harmonic of rang ;9 is ta$en into acco"nt%

Fort"nately, the in#ersion of these matrices can be a#oided by simple mathematical perm"tations% !he

sol"tion of the matri+ e"ations a#oiding the in#ersion of the capacitor and ind"ctor matrices is in this

case:

?


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[ ][ ][ ] [ ]( ) [ ]

[ ] [ ][ ][ ][ ] [ ][ ]

=

=

+=

dcdc

dcac

acdc

VCIVCUI

VUCULV )

-,

% )/9

It can be noted that this method directly leads to an analytical sol"tion of the harmonics of the different

electrical "antities% For this reason, this fre"ency domain method can be considered as more acc"rate

and rapid than the time domain one, where time wa#eforms of the state #ariables are first obtained, and

then corresponding spectra are ded"ced% &nother ad#antage of this method is that the harmonics

analytical e+pression can be "sed to increase the efficiency in harmonics red"ction and elimination%

8. APPLICATION OF THE METHOD TO AN AC/DC/AC CON6ERTER

In order to completely ill"strate the pre#io"s method, the model of a closed5loop &62762&6 PW

con#erter is elaborated% !he considered str"ct"re is chosen beca"se of its comple+ity and its wide

spread "se as power interface% !he considered system is composed of two con#erters ha#ing the same

str"ct"re see Fig% 19, so that the application of the method is presented only for the &6276 con#erter%

!he method can be analogically applied to the whole con#erter str"ct"re by "sing similar e"ations for

the second con#erter%

Fig"re 1: &62762&6 three phase con#erter "sed as power interface

8.4 M)(e!in+ he AC/DC PWM #)n,erer

!he method is first applied to the &6276 con#erter described in Fig%3, where the 762&6 con#erter of

Fig%1 has been replaced by a resistor%

@

R1

R2

R3

L3

L2

L1

Lc

Lb

La

C

Rc

Ra

Rb

i1

i2

i3

ia

ib

ic

Vdc

SYSTEM

1

SYSTEM

2

R1

R2

R3

L3

L2

L1

Lc

Lb

La

C

Rc

Ra

Rb

i1

i2

i3

ia

ib

ic

Vdc

SYSTEM

1

SYSTEM

2


9/18

Fig"re 3: &6276 con#erter

4y s"pposing switching components, passi#e elements, and networ$ #oltage as ideal, the con#erter can

be described by the set of following differential e"ations:

( )

( )

( )

( )

++=

=

=

=

R

tVtitutitutitu

dt

tdVC

tVtutututiRtV

dt

tdiL

tVtutututiRtV

dt

tdiL

tVtutututiRtV

dt

tdiL

dcdc

dc

dc

dc

9:9:9:9:9:9:9:

/

)9:>

9:9:9:9:/9:9:

9:

>

9:9:9:9:/9:9:

9:

>

9:9:9:9:/9:9:

9:

11//))

/)11111

1

1)/////

/

1/)))))

)

,

)19

where 9:tui is the switching f"nction of the thi con#erter leg:

=

)

)9:tui % )39

In steady state, these e"ations can be con#erted in the fre"ency5domain and presented in the

following matri+ form:

A

R1

R2

R3

L1

L2

L3

E2

E3

E1

C RV

dc

i1

i2

i3

idc

R1

R2

R3

L1

L2

L3

E2

E3

E1

C RV

dc

i1

i2

i3

idc


10/18

[ ][ ] [ ] [ ][ ] [ ] [ ] [ ]( )[ ]

[ ][ ] [ ] [ ][ ] [ ] [ ] [ ]( )[ ]

[ ][ ] [ ] [ ] [ ] [ ] [ ] [ ]( )[ ]

[ ][ ] [ ][ ] [ ][ ] [ ][ ]( ) [ ] [ ]

++=

=

=

=

dcdc

dc

dc

dc

VRIUIUIUVC

VUUUIRVIL

VUUUIRVIL

VUUUIRVIL

)11//))

/)111111

1)//////

1/))))))

/

)

/>

)

/>)

/>

)

%

)89

!he state #ariables are represented by a set of #ectors of harmonics, and the system parameters by

matrices as described in the pre#io"s section:

[ ] 1,/,)%%%%%%-/);)/

== kIIIIII T

kkkkkk

[ ]Tdcdcdcdcdcdc VVVVVV %%%%%%- /);)/ =

9:-

9:-

1,/,)9:-

1,/,)9:-

RdiagR

HCjdiagC

kHRjdiagR

kHLjdiagL

kk

kk

=

=

==

==

,

)>9

where -,H is a #ector containing the ran$s of the harmonics:

[ ]%%%/);)/%%%- =H %

)?9

atrices -, )U , -, /U and -, 1U contain the Fo"rier coefficients of the different switching f"nctions:

[ ][ ]%%%%%%-

%%%%%%-

%%%%%%-

/1)1;1)1/11

//)/;/)////

/)));)))/))

uuuuutoeplitzU

uuuuutoeplitzU

uuuuutoeplitzU

=

=

=

,

)@9

!hese Fo"rier coefficients are obtained by "sing Fo"rier series decomposition in the case of periodic

switching f"nction for e+ample f"ll5wa#e con#erters9% In the case of PW con#erters, the switching

f"nction is not e+actly periodic, b"t it can be represented as a two5dimensional f"nction of the career

and the reference wa#eforms, which are periodic% !herefore, the Fo"rier coefficients can be obtained

);


11/18

thro"gh a do"ble Fo"rier series decomposition% For e+ample, the switching f"nction of a nat"rally

sampled PW is gi#en by ?-:

[ ] [ ]

+++

++++=

=

= 1/9):cos

//sin)39

1/9):cos:9:

)

itntmMmJnmm

itMtu ccnm n

i

, )A9

where

M is the magnit"de of the mod"lating signal,

andc c are the carrier p"lsation and phase,

and are the f"ndamental p"lsation and phase,

( )nJ is the 4essel f"nction of order n%

4y s"pposing that the career fre"ency c is an integer m"ltiple of the f"ndamental fre"ency , the

Fo"rier coefficients of : 9iu t can be easily determined from E% )A9%

8.5 C)nr)! *)(e!in+

In open loop, the magnit"de M and phase of the mod"lating signal "sed for the calc"lation of the

Fo"rier coefficients of the switching f"nction are $nown and constant% In closed loop these two

parameters are "sed to control the magnit"des of the con#erter state #ariables, "s"ally the ac c"rrent%

For that reason they are not fi+ed, b"t depend on the real and the desired reference9 #al"es of the

controlled state #ariables%

In this section the impact of the control system is ta$en into acco"nt by calc"lating the phase and

magnit"de of the mod"lating signal% In the proposed algorithm, the con#erter E% )19 are first sol#ed

for the f"ndamental fre"ency, and the f"ndamental of the switching f"nction is fo"nd by replacing the

con#erter state #ariables by their reference #al"es% !he mod"lating signal parameters are fo"nd "sing

the fact that the mod"lating signal is the f"ndamental of the switching f"nction% 4y $nowing M and

the real switching f"nction can be calc"lated and the method for harmonics estimation presented in

Section 1 can be applied% !he described algorithm is gi#en in details in this section%

!he con#erter state #ariables and switching f"nctions are s"pposed symmetrical, only the f"ndamental

component is ta$en into acco"nt:

))


12/18

=

=

=

1

/

1

1

/

/

)

j

j

iei

iei

ii

=

=

=

1

/

1

1

/

/

)

j

j

VeV

VeV

VV

=

=

=

1

/

1

1

/

/

)

j

j

ueu

ueu

uu

/;9

!he passi#e elements in the three phase are considered as e"al:

k

k

RRRR

LLLL

===

===

1/)

1/)/)9

4y "sing /;9 and /)9, only one phase of the con#erter can be considered% !hen, E% )19 become:

=

=

R

Vui

dt

dVC

iRV

uVdt

diL

dcdc

kdc

k

/

1

>

//9

!he con#erter #ariables and switching f"nction are transformed in the d; frame in order to ma$e them

appear as constant:

+=

+=

+=

d

d

d

juuu

jVVV

jiii

/19

E"ations //9 transformed in the d; frame become:

)/


13/18

( )

+=

=

+=

R

Viuiu

dt

dVC

VuiRiLdtdiL

VuViRiLdt

diL

dcdd

dc

dckdk

k

dcddkkd

/

1

/)

/

)

/39

!he magnit"des of the state #ariables are constant in the d; frame, so that their deri#ati#es are e"al

to (ero:

;=

dt

did;=

dt

di ;=

dt

dVdc

/89

4y s"pposing the ac c"rrent e"al to its reference #al"e the PI controllers are ideal9, the d and

components of the switching f"nctions can be fo"nd%

( ) ( )( )

( )

( )

=

=

+=

!e"kd!e"k

dc

d!e"k!e"k

dc

d

!e"kd!e"k!e"d!e"k!e"kd!e"dc

iRiLV

u

ViRiLV

u

iRiLiViRiLiRV

/

/

/

1

%

/>9

From the obtained #al"es of du and u , the f"ndamental magnit"de and phase of the switching

f"nction are then calc"lated:

=

+=

d

d

u

u

uuM

arctan;

//

/?9

)1


14/18

8.7 App!i#ai)n )$ he *eh)( ) he -h)!e #)n,erer sr&re

!he resistor of Fig%3 is replaced by 762&6 con#erter% Similar e"ations are "sed to describe the whole

system% !he con#erter str"ct"re is connected to the grid and a resistor is "sed as load% !he PW

fre"ency is /$'(% !he obtained res"lts are compared with those obtained by meas"rements and

sim"lation, and are presented in the following section%

9. SIMULATION AND PRACTICAL RESULTS

!he res"lts obtained from the theoretical method, the atlab2Sim"lin$ sim"lation, and the e+perimental

bench are compared in this section%

9.4 The)rei#a! *eh)(

!he matri+ e"ations describing the con#erter and its control system are implemented% !he switching

f"nctions and the $nown state #ariables as the inp"t #oltage are decomposed in Fo"rier series and the

corresponding harmonic transfer matrices and #ectors are b"ilt% !he resol"tion of the matri+ system

e"ations leads to the fre"ency e+pression of the con#erter state #ariables, and the corresponding time

wa#eforms can be e#ent"ally determined by in#erse Fo"rier transform% !he calc"lation time depends on

the n"mber of harmonics considered in the different signals%

9.5 Ma!a/Si*&!in1 si*&!ai)n

& model of the con#erter based on its differential e"ations is implemented "nder atlab2Sim"lin$% !he

obtained res"lts are in the time domain and a Fo"rier transformation is "sed to obtain the c"rrents and

#oltages spectra%

9.7 E0peri*ena! en#h

!he e+perimental bench is presented in Fig% 8 and its str"ct"re in Fig% >% !he networ$ #oltage is

adapted thro"gh a"totransformers%

)3


15/18

Fig"re 8: E+perimental bench Fig"re >: E+perimental bench description

9.8 Res&!s

!heoretical res"lts are compared to those obtained by meas"rements and atlab2Sim"lin$ sim"lations%

In Fig% ?, the spectr"m of the ac c"rrent from the networ$ side is shown between )8;; and >8;; '(

aro"nd PW harmonics9% In the three cases, the harmonics are sit"ated at the same fre"encies and

ha#e almost the same magnit"des% !he small differences are d"e to the ass"mptions "sed in o"r method,

the sim"lation errors, and the dist"rbances in the real system non5ideal components, noises, etc%9% !he

res"lts obtained for the dc #oltage and the ac c"rrent from the load side are "ite similar%

Fig"re ?: &6 c"rrent spectr"m from the networ$ sideB theoretical, sim"lation, and e+perimental res"lts

:. CONCLUSION

& new analytical fre"ency5domain method for harmonics modeling and e#al"ation in power electronic systems

has been presented in this paper% !he considered system is described by a set of differential e"ations, which are

)8


16/18

con#erted in the fre"ency domain and presented in a matri+ form% !he resol"tion of these matri+ e"ations leads

to theoretical e+pressions of the different #oltages and c"rrents% It can be noted that thismethod is designed for

power systems with periodically switching components, and leads to an analytical e+pression of the

different electrical "antities, which is one of its main ad#antages% Indeed, this allows to determine the

infl"ence of the system parameters control strategy, passi#e elements, etc%9 on the harmonic contents

of the con#erter state #ariables% It can be s"ccessf"lly applied for power "ality assesment, harmonic

filters optimisation and con#erter control design%

;. LIST OF SYMBOLS

aci , acI &lternati#e c"rrent

dci , acI 7irect c"rrent

L Ind"ctance

M agnit"de of the mod"lating signal

R Cesistor

C 6apacitor

u , U Switching f"nction of one leg

v Doltage

ac

V &lternati#e #oltage

dcV 7irect #oltage

F"ndamental p"lsation

c 6arrier p"lsation

F"ndamental phase

c 6arrier phase

,no%3, ct% /;;)B p%?A)5@;;

/- 4ath"rst5G, Smith54, Watson5., &rrillaga5:


17/18

1- ollerstedt5E%, 4ernhardsson54%: )npierre%granonlis%inpg%fr9 was born in France in )A?)% 'e recei#ed

the %S% in electrical engineering from the Jni#ersity 6enter of Science and

!echnology, 6lermont5Ferrand, France, in )AA3 and the Ph%7% degree from the

.ational Polytechnic Instit"te of Grenoble I.PG9, France in /;;;% 'e oined the

Laboratory of Images and Signals LIS9 at I.PG in /;;), where he holds a position

as assistant professor% 'is general interests co#er signal processing theory s"ch as nonlinear signals and

filters higher order statistics, Dolterra filters9, non stationary signals and filters cyclostationarity,

LP!D filters9 and acti#e control% 'is c"rrent research is mainly foc"sed on signal processingapplications in electrical engineering s"ch as fa"lt diagnosis in electrical machines and power networ$s%

Se((i1 Ba#ha seddi$%bachaleg%ensieg%inpg%fr9 recei#ed his Engineer and aster

from .ational Polytechnic Instit"te of &lgiers respecti#ely in )A@/ and )AA;% 'e

oined the Laboratory of Electrical Engineering of Grenoble LEG9 and recei#ed his

)?
http://erlgre.ujf-grenoble.fr/webspirs/doLS.ws?ss=Mollerstedt-E+in+AUhttp://erlgre.ujf-grenoble.fr/webspirs/doLS.ws?ss=Bernhardsson-B+in+AUhttp://erlgre.ujf-grenoble.fr/webspirs/doLS.ws?ss=2000-IEEE-Power-Engineering-Society-Winter-Meeting-Conference-Proceedings-Cat-No00CH+in+SOhttp://erlgre.ujf-grenoble.fr/webspirs/doLS.ws?ss=2000-IEEE-Power-Engineering-Society-Winter-Meeting-Conference-Proceedings-Cat-No00CH+in+SOhttp://erlgre.ujf-grenoble.fr/webspirs/doLS.ws?ss=Mollerstedt-E+in+AUhttp://erlgre.ujf-grenoble.fr/webspirs/doLS.ws?ss=Bernhardsson-B+in+AUhttp://erlgre.ujf-grenoble.fr/webspirs/doLS.ws?ss=2000-IEEE-Power-Engineering-Society-Winter-Meeting-Conference-Proceedings-Cat-No00CH+in+SOhttp://erlgre.ujf-grenoble.fr/webspirs/doLS.ws?ss=2000-IEEE-Power-Engineering-Society-Winter-Meeting-Conference-Proceedings-Cat-No00CH+in+SO


18/18

Ph7 and '7C respecti#ely in )AA1 and )AA@% 'e is presently manager of Power System Gro"p of

LEG and Professor at the Jni#ersity oseph Fo"rier of Grenoble% 'is main fields of interest are power

electronic systems, modeling and control, power "ality, renewable energy integration%

)@

power_electronics_harmonic_analysis_-_ignatova (1).doc

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