Digital Control Applications in Power Electronics Lesson4

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August 2001 Lesson 4 1

Lesson 4

Implementation of Synchronous Frame

Harmonic Control for High-Performance ACPower Supplies

Lesson 4

Implementation of Synchronous Frame

Harmonic Control for High-Performance ACPower Supplies




Goal of the workGoal of the work

Investigation of selective control on output

voltage harmonics.

Motivations:

• reduction of voltage distortion in AC Power

Supplies (even with limited voltage loopbandwidth);

•

refinements aimed at an efficientimplementation in fixed-point DSP’s (here

tested on ADMC401 by Analog Devices).

Investigation of selective control on output

voltage harmonics.

Motivations:

• reduction of voltage distortion in AC Power

Supplies (even with limited voltage loopbandwidth);

•

refinements aimed at an efficientimplementation in fixed-point DSP’s (here

tested on ADMC401 by Analog Devices).




Presentation outlinePresentation outline

• Review of synchronous reference frame

harmonic regulation

• Decomposition in three-layer control

scheme

• A modified solution based on DiscreteFourier Transform

•Design guidelines for regulator parameters

• DSP implementation using ADMC401

•Experimental results

• Review of synchronous reference frame

harmonic regulation

• Decomposition in three-layer control

scheme

• A modified solution based on DiscreteFourier Transform

•Design guidelines for regulator parameters

• DSP implementation using ADMC401

•Experimental results




Current

Controller

Space

Vector Modulator

Space

Vector Modulator

Current control(optional) Current control(optional)

udc

+

_

L

C

Fi Li

+

+Ouput voltagefeed-forwardOuput voltagefeed-forward

Fi

Load

ou+

_

+

+

+

+

KsKs

conventional regulator conventional regulator

iFαααα

+

+

abc

αβαβαβαβ

+

+

Transformer

αααα

ββββ

εεεε

εεεε+

abc

αβαβαβαβ

ou*αααα

ou*ββββ

Closed-loopcontrol ofselectedharmonics

ou

+

*

iF*ββββ

Synchronous frame harmonic controlSynchronous frame harmonic control




Basic schemeBasic scheme

Outputvoltageerrors

Outputvoltageerrors

Referencecurrents

Referencecurrents

Leadingangle φ φφ φ k

Leadingangle φ φφ φ k

εεεεααααεεεε ββββ

αβαβαβαβ

αβαβαβαβ

αβαβαβαβ

αβαβαβαβ

dqk+

dqk+

dqk-

dqk-

Reg k

iFαααα∗∗∗∗+

+ +

+

θθθθk + φ+ φ+ φ+ φk

iFββββ∗∗∗∗

Reg k

Reg k

Reg k

θθθθk + φ+ φ+ φ+ φk

θθθθk

θθθθk

Regulation of positive sequence k-th harmonicRegulation of positive sequence k-th harmonic

Regulation of negative sequence k-th harmonicRegulation of negative sequence k-th harmonic




εεεε

εεεε

ωωωωωωωω−−−−

ωωωωωωωω====

εεεε

εεεε

ββββ

αααα

++++

++++

)tkcos()tk(sin

)tk(sin)tkcos(

SS

SS

qk

dk

εεεε

εεεε

ωωωωωωωω−−−−

ωωωωωωωω====

εεεε

εεεε

ββββ

αααα

++++

++++

)tkcos()tk(sin

)tk(sin)tkcos(

SS

SS

qk

dk

φφφφ++++ωωωωφφφφ++++ωωωω

φφφφ++++ωωωω−−−−φφφφ++++ωωωω====

++++

++++

++++ββββ

++++αααα

qk

dk

kSkS

kSkS

k

k

y

y

)tkcos()tk(sin

)tk(sin)tkcos(

y

y

φφφφ++++ωωωωφφφφ++++ωωωω

φφφφ++++ωωωω−−−−φφφφ++++ωωωω====

++++

++++

++++ββββ

++++αααα

qk

dk

kSkS

kSkS

k

k

y

y

)tkcos()tk(sin

)tk(sin)tkcos(

y

y

φφφφ++++ωωωωφφφφ++++ωωωω−−−−

φφφφ++++ωωωωφφφφ++++ωωωω====

−−−−

−−−−

−−−−ββββ

−−−−αααα

qk

dk

kSkS

kSkS

k

k

y

y

)tkcos()tk(sin

)tk(sin)tkcos(

y

y

φφφφ++++ωωωωφφφφ++++ωωωω−−−−

φφφφ++++ωωωωφφφφ++++ωωωω====

−−−−

−−−−

−−−−ββββ

−−−−αααα

qk

dk

kSkS

kSkS

k

k

y

y

)tkcos()tk(sin

)tk(sin)tkcos(

y

y

εεεε

εεεε

ωωωωωωωω

ωωωω−−−−ωωωω====

εεεε

εεεε

ββββ

αααα

−−−−

−−−−

)tkcos()tk(sin

)tk(sin)tkcos(

SS

SS

qk

dk

εεεε

εεεε

ωωωωωωωω

ωωωω−−−−ωωωω====

εεεε

εεεε

ββββ

αααα

−−−−

−−−−

)tkcos()tk(sin

)tk(sin)tkcos(

SS

SS

qk

dk

Basic scheme equationsBasic scheme equations

αβαβαβαβ

dqk+

αβαβαβαβ

dqk-

αβαβαβαβ

dqk+

αβαβαβαβ

dqk-




Equivalence with stationary frame controlEquivalence with stationary frame control

Reg kAC

Reg kAC

[[[[ ]]]]

[[[[ ]]]])k js(gRe)k js(gResin j

)k js(gRe)k js(gRecos)s(gRe

skskk

skskkkAC

ωωωω++++−−−−ωωωω−−−−φφφφ++++

++++ωωωω++++++++ωωωω−−−−φφφφ==== [[[[ ]]]]

[[[[ ]]]])k js(gRe)k js(gResin j

)k js(gRe)k js(gRecos)s(gRe

skskk

skskkkAC

ωωωω++++−−−−ωωωω−−−−φφφφ++++

++++ωωωω++++++++ωωωω−−−−φφφφ====

εεεεααααεεεεββββ

αβαβαβαβ

αβαβαβαβ

αβαβαβαβ

αβαβαβαβ

dqk+

dqk+

dqk-

dqk-

Reg k

iFαααα

∗∗∗∗++

+

+

θθθθ k θθθθk + φ+ φ+ φ+ φ k


EquivalenceEquivalence

iFαααα∗∗∗∗


εεεεαααα

εεεε ββββ

Reg k

Reg k

Reg k

θθθθ θθθθ k + φ+ φ+ φ+ φ k k




• If Regk(s) is a integral regulator, the previous

equivalence implies that RegkAC(s) is a band-

pass filter centered on the kth

harmonicfrequency.

• This is true if and only if both direct and reverse

sequence controllers are implemented.

• It is possible to implement synchronous

regulators either in the dq rotating frame or in

the αβαβαβαβ stationary frame, with perfectly equivalent

performance.

• If Regk(s) is a integral regulator, the previous

equivalence implies that RegkAC(s) is a band-

pass filter centered on the kth

harmonicfrequency.

• This is true if and only if both direct and reverse

sequence controllers are implemented.

• It is possible to implement synchronous

regulators either in the dq rotating frame or in

the αβαβαβαβ stationary frame, with perfectly equivalent

performance.






• The leading angle φφφφk can be used to

compensate for internal delays, such as that of

the current controller.

• In practice, the current reference is phase

shifted to compensate for the currentcontroller delay.

• In the stationary frame implementation, based

on band-pass filters, this may or may not have

a possible equivalent (depending on the

regulator structure).

• The leading angle φφφφk can be used to

compensate for internal delays, such as that of

the current controller.

• In practice, the current reference is phase

shifted to compensate for the currentcontroller delay.

• In the stationary frame implementation, based

on band-pass filters, this may or may not have

a possible equivalent (depending on the

regulator structure).




Three-layer decompositionThree-layer decomposition

AC Power Supplies requirements

•

Fast transient response with limitedovershoot under load changes;

• Possibly fast regulation of output voltage

fundamental harmonic component;

• For distorting loads with slowly-varying

harmonics, harmonic control in somefundamental cycles (decoupling between

different controllers is needed!).

AC Power Supplies requirements

•

Fast transient response with limitedovershoot under load changes;

• Possibly fast regulation of output voltage

fundamental harmonic component;

• For distorting loads with slowly-varying

harmonics, harmonic control in somefundamental cycles (decoupling between

different controllers is needed!).




Three-layer decompositionThree-layer decomposition

• The previous requirements suggest a

decomposition of the control system in three

layers:• reference tracking control (for a quick

dynamic response) loop bandwidth;

• fundamental component control high loop

gain at the fundamental frequency, with low

selectivity;

• harmonics control high loop gain at each

harmonic frequency with high selectivity.

• The previous requirements suggest a

decomposition of the control system in three

layers:• reference tracking control (for a quick

dynamic response) loop bandwidth;

• fundamental component control high loop

gain at the fundamental frequency, with low

selectivity;

• harmonics control high loop gain at each

harmonic frequency with high selectivity.




Three-Layer DecompositionThree-Layer Decomposition

x xz-1

e+jωωωω st e-j ωωωω st

x xz-1

z-Na

running DFTrunning DFT

+ ++

+

+

+

+

+

K P

proportional gainproportional gain

fundamental frequency controlfundamental frequency control

1

uoαβαβαβαβ

uoαβαβαβαβ

_

*

i Fαβαβαβαβ

currentreferences

*

K F

TswK I1

TswK I1

εεεεαβαβαβαβ

HarmonicsControl

HarmonicsControl

FundamentalComponentControl

FundamentalComponentControl

TrackingControl

TrackingControl

e-j ωωωω st e+jωωωω st

2

3






• In principle, harmonics could be controlled by

integral, rotating frame regulators.

• This solution is cumbersome, requiring a largenumber of dq transformations.

•

The AC equivalent of the integral controller is ahigh selectivity band-pass filter. This is difficult

to implement because of fixed point arithmetic.

• Even if no stability problems can be generated,the selectivity is strongly limited by rounding

errors affecting the filter coefficients.

• In principle, harmonics could be controlled by

integral, rotating frame regulators.

• This solution is cumbersome, requiring a largenumber of dq transformations.

•

The AC equivalent of the integral controller is ahigh selectivity band-pass filter. This is difficult

to implement because of fixed point arithmetic.

• Even if no stability problems can be generated,the selectivity is strongly limited by rounding

errors affecting the filter coefficients.

NotesNotes




F h- pass-band filter with unity gain and zero phase at harmonic h, with good selectivity F h- pass-band filter with unity gain and zero phase at harmonic h, with good selectivity

(((( ))))

∈∈∈∈

∈∈∈∈

∈∈∈∈∈∈∈∈ −−−−≈≈≈≈

−−−−====

ωωωω++++====

h

h

hh

Nh

h

Nh h

F

Nh h

hF

Nh2

s2

Ih

hACF1

F

KF1

FKhs

sK2)s(gRe(((( ))))

∈∈∈∈

∈∈∈∈

∈∈∈∈∈∈∈∈ −−−−≈≈≈≈

−−−−====

ωωωω++++====

h

h

hh

Nh

h

Nh h

F

Nh h

hF

Nh2

s2

Ih

hACF1

F

KF1

FKhs

sK2)s(gRe

Harmonic controlHarmonic control

sh

IhF

h

KK

ωωωωξξξξ

====sh

IhF

h

KK

ωωωωξξξξ

====

KF is defined as follows:KF is defined as follows:

KF is constant provided KIh are equal to h·KIKF is constant provided KIh are equal to h·KI




−−−−

====

−−−−

ππππ====

1N

0i

idh zih

N

2cos

N

2)z(F

−−−−

====

−−−−

ππππ====

1N

0i

idh zih

N

2cos

N

2)z(F

The above approximation is well verified by

several types of band-pass filters. A good choice,

which offers significant implementationadvantages, is represented by FIR filters based on

DFT such as:

The above approximation is well verified by

several types of band-pass filters. A good choice,

which offers significant implementationadvantages, is represented by FIR filters based on

DFT such as:

For a single harmonic frequencyFor a single harmonic frequency






−−−−

====

−−−−

ππππ====

1N

0i

idh zih

N

2cos

N

2)z(F

−−−−

====

−−−−

ππππ====

1N

0i

idh zih

N

2cos

N

2)z(F

• This equation is based on the expression of

the h-th harmonic component of a given input

signal’s DFT to derive a filter (running DFT).• The structure is that of a typical FIR filter

(linear combination of delays).

•From this standpoint N·Ts (Ts is the samplingperiod) does not necessarily represent the

period of the input signal, which can be even

non-periodic.

• This equation is based on the expression of

the h-th harmonic component of a given input

signal’s DFT to derive a filter (running DFT).• The structure is that of a typical FIR filter

(linear combination of delays).

•From this standpoint N·Ts (Ts is the samplingperiod) does not necessarily represent the

period of the input signal, which can be even

non-periodic.




−−−−

====

−−−−

ππππ====

1N

0i

idh zih

N

2cos

N

2)z(F

−−−−

====

−−−−

ππππ====

1N

0i

idh zih

N

2cos

N

2)z(F

Computing this equation in the time-domain gives:Computing this equation in the time-domain gives:


FIR filter transfer function

−−−−

====

−−−−

ππππ====1N

0i

h )ik(xihN2cos

N2)k(y

−−−−

====

−−−−

ππππ====1N

0i

h )ik(xihN2cos

N2)k(y

yh(k) is a sinusoidal signal that represents theprojection of the input signal x(k) upon the cosine

base function of order h.

yh(k) is a sinusoidal signal that represents theprojection of the input signal x(k) upon the cosine

base function of order h.

coefficients are nottime dependent!





Comparison with DFT:Comparison with DFT:

−−−−

====

ππππ====

1N

0k

N )k(xkhN

2cos)h(X

−−−−

====

ππππ====

1N

0k

N )k(xkhN

2cos)h(X

−−−−

====

−−−−

ππππ====

1N

0ih)ik(xih

N

2cos

N

2)k(y

−−−−

====

−−−−

ππππ====

1N

0ih)ik(xih

N

2cos

N

2)k(y FIR filter

hth order cosine

component in theDFT of signal x(k)

k is the time index, so in the DFT coefficients are

time dependent. The structure of the two algorithms

is exactly the same.




0 2 4 6 8 10 12 14 16 18 200

1

Harmonic orderHarmonic order


Sampledfrequency

response, asseen by signalswith periodN·T

s /h.

Sampledfrequency

response, asseen by signalswith periodN·T

s /h.

Frequency response of function Fdh

(z) for h = 3.Frequency response of function Fdh

(z) for h = 3.





−−−−

====

−−−−

ππππ====

1N

0i

i

dhzih

N

2cos

N

2)z(F

−−−−

====

−−−−

ππππ====

1N

0i

i

dhzih

N

2cos

N

2)z(F

−−−−

====

−−−−

∈∈∈∈∈∈∈∈

ππππ====

1N

0i

i

NhNh

dh zihN

2cos

N

2)z(F

kh

−−−−

====

−−−−

∈∈∈∈∈∈∈∈

ππππ====

1N

0i

i

NhNh

dh zihN

2cos

N

2)z(F

kh

For a single harmonicFor a single harmonic

No additional calculations

for more harmonics

No additional calculations

for more harmonics

For multiple frequenciesFor multiple frequencies

0 2 4 6 8 10 12 14 16 18 200

1

Harmonic order

0 2 4 6 8 10 12 14 16 18 200

1

Example: 3rdExample: 3rd

Example: 3rd, 5thExample: 3rd, 5th

Harmonic order

H i lH i l





−−−−

====

−−−−

∈∈∈∈∈∈∈∈

ππππ====

1N

0i

i

NhNh

dh zihN

2cos

N

2)z(F

kh

−−−−

====

−−−−

∈∈∈∈∈∈∈∈

ππππ====

1N

0i

i

NhNh

dh zihN

2cos

N

2)z(F

kh

• The coefficient for the i-th term can be

computed off-line according to:

• The control complexity does not depend on

the number of harmonic components taken

into account.

• The coefficient for the i-th term can be

computed off-line according to:

• The control complexity does not depend on

the number of harmonic components taken

into account.

∈∈∈∈

ππππ

kNh

ihN

2cos

∈∈∈∈

ππππ

kNh

ihN

2cos




Design CriteriaDesign Criteria

Proportional terms and fundamental frequency

control are based on specified bandwidth andphase margin:

Proportional terms and fundamental frequency

control are based on specified bandwidth andphase margin:

2c

2

s

2

cIc1I

2KK

ωωωωωωωω−−−−ωωωω====2c

2

s

2

cIc1I

2KK

ωωωωωωωω−−−−ωωωω====Equivalence with PI Equivalence with PI

Fundamental frequency controlFundamental frequency control

KIc is the integral gain of a conventionally

designed PI regulator.

KIc is the integral gain of a conventionally

designed PI regulator.

H i t lH i t l





• Amplification of frequencies close to the

harmonics is an undesired effect of the filter

superposition.

• Introduction of the leading angle φφφφk is possible

by means of positive feedback, provided that

the angle is proportional to the harmonicfrequency.

• Internal delays can be compensated.

• Amplification of frequencies close to the

harmonics is an undesired effect of the filter

superposition.

• Introduction of the leading angle φφφφk is possible

by means of positive feedback, provided that

the angle is proportional to the harmonicfrequency.

• Internal delays can be compensated.

−−−−

====

−−−−

∈∈∈∈∈∈∈∈

++++

ππππ====

1N

0i

i

Nk

a

Nh

dh z)Ni(kN

2cos

N

2)z(F

kh

−−−−

====

−−−−

∈∈∈∈∈∈∈∈

++++

ππππ====

1N

0i

i

Nk

a

Nh

dh z)Ni(kN

2cos

N

2)z(F

kh





Specification

on the response time (npk )

Specification

on the response time (npk )Spk

Ik Tn

2.2

K ====


npk is the desired number of fundamental cyclesfor the dynamic response. It must be high enough

to provide de-coupling with the fundamental

frequency control. The equation is derived byapproximated relations between gain and settling

time of 2nd order band-pass filters.

npk is the desired number of fundamental cyclesfor the dynamic response. It must be high enough

to provide de-coupling with the fundamental

frequency control. The equation is derived byapproximated relations between gain and settling

time of 2nd order band-pass filters.

CC






gain of DFT filtersgain of DFT filters s

Ik

F 32.0

K

K ωωωω≈≈≈≈ s

Ik

F 32.0

K

K ωωωω≈≈≈≈

This relation is based on the approximation of thesingle DFT filter with a conventional second order

band-pass filter.

By trial and errors the 0.32 coefficient can bedetermined as the one minimizing the “distance”

between the two frequency responses.

This relation is based on the approximation of thesingle DFT filter with a conventional second order

band-pass filter.

By trial and errors the 0.32 coefficient can bedetermined as the one minimizing the “distance”

between the two frequency responses.



DSP I l iDSP I l i




DSP ImplementationDSP Implementation

Main features of DSP-based controller ADMC401:

• 16-bit fixed point DSP based on ADSP 2171 core.

• Fast arithmetic unit (38.5ns cycle).

• High-performance peripherals (double-update

PWM modulators, flash A/D 12 bit converters,

etc..).

• Suited for single-chip high-performance motion

control applications.

Main features of DSP-based controller ADMC401:

• 16-bit fixed point DSP based on ADSP 2171 core.

• Fast arithmetic unit (38.5ns cycle).

• High-performance peripherals (double-update

PWM modulators, flash A/D 12 bit converters,

etc..).

• Suited for single-chip high-performance motion

control applications.

C t l PC t l PInitialization

Routines




Control Program

Flow-Chart

Control Program

Flow-Chart

Routines

Wait

Interrupt?

No

Yes

Reference Generation

[ s ]µµµµ

0

transformationsαβαβαβαβerrors calculations

Voltage control:proportional +

fundamental frequency

Harmonic control

deadbeat currentcontrol

SVM

Dead-time compensation

PWMSYNC

Timing starting

from ADC conversion

2 µµµµs

~3 µµµµs

~3 µµµµs

~18.4 µµµµsN=200

~ 6.4 µµµµsN=50

~2.7 µµµµs

~5 µµµµs

~36 µµµµs ~24 µµµµs

• Implementation on

ADMC401

• The control program

is written in assembly

language.• The use of DFT based

filters greatly

simplifies theimplementation.

• Execution times are

short.

• Implementation on

ADMC401

• The control program

is written in assembly

language.• The use of DFT based

filters greatly

simplifies theimplementation.

• Execution times are

short.





.MODULE/RAM/SEG=USER_PM1 DFT_FILTER200;

.CONST ORD_N=200;

.VAR/RAM/PM/CIRC Filt_Coef[ORD_N];

#include "dft.dat” /* coefficients

initialization */

.ENTRY DFT200;

DFT200:

i5=^Filt_Coef; /* use dedicated */l5=%Filt_Coef; /* circular registers */

m5=1; /* i,l,m (0-7) */

l0=%Filt_Coef; /* i0 data pointer (same lenght) */

.MODULE/RAM/SEG=USER_PM1 DFT_FILTER200;

.CONST ORD_N=200;

.VAR/RAM/PM/CIRC Filt_Coef[ORD_N];

#include "dft.dat” /* coefficients

initialization */

.ENTRY DFT200;

DFT200:

i5=^Filt_Coef; /* use dedicated */l5=%Filt_Coef; /* circular registers */

m5=1; /* i,l,m (0-7) */

l0=%Filt_Coef; /* i0 data pointer (same lenght) */

DSP ImplementationProgram sample






m1=1;

mr = 0, mx0 = dm(i0,m1), my0 = pm(i5,m5);

cntr=%Filt_coef-1;

do calc0 until ce;

calc0: mr = mr + mx0 * my0 (ss),mx0 = dm(i0,m1), my0 = pm(i5,m5);

mr = mr + mx0 * my0 (rnd);

if mv sat mr;

rts;

.ENDMOD;

m1=1;

mr = 0, mx0 = dm(i0,m1), my0 = pm(i5,m5);

cntr=%Filt_coef-1;

do calc0 until ce;

calc0: mr = mr + mx0 * my0 (ss),mx0 = dm(i0,m1), my0 = pm(i5,m5);

mr = mr + mx0 * my0 (rnd);

if mv sat mr;

rts;

.ENDMOD;








Experimental SetupExperimental Setup

Prototype ratings:

•DC-link voltage 300V

• Filter Inductance 1 mH

•Output Filter Capacitor 120µµµµF

• Switching frequency 10kHz

•Selected frequencies: 3rd, 5th,7th,9th,11th

Experimental ResultsExperimental Results




Experimental ResultsThree-phase diode rectifiers - Proposed solution

Experimental ResultsThree-phase diode rectifiers - Proposed solution

(a) Ouput Voltage (40V/div)(a) Ouput Voltage (40V/div)

time (5ms/div)time (5ms/div) frequency (50Hz/div)frequency (50Hz/div)

(a)

(b) Ouput current (10A/div)(b) Ouput current (10A/div)

c) Output voltage

spectrum (10dB/div)

c) Output voltage

spectrum (10dB/div)

(c)(c)(b)

22

33 44 55 9966

77 88






time (5ms/div)time (5ms/div) frequency (50Hz/div)frequency (50Hz/div)

(a)(a)


(c)(c)

(b)(b)

22

33 44

55

9966

77

88

Experimental ResultsThree-phase diode rectifiers - PI controller

Experimental ResultsThree-phase diode rectifiers - PI controller

c) Output voltage

spectrum (10dB/div)

c) Output voltage

spectrum (10dB/div)





pSingle-phase diode rectifier

pSingle-phase diode rectifier


time (5ms/div)time (5ms/div)

(a)(a)


(b)(b)

(a)(a)

(b)(b)

Proposed solutionProposed solution Conventional PIConventional PI






pDistorting Load Turn-on

pDistorting Load Turn-on


(a)(a)

(b)(b)(b)

(a) Ouput Voltage (40V/div)(a) Ouput Voltage (40V/div) (b) Ouput current (10A/div)(b) Ouput current (10A/div)





pLinear Load Step-Changes

pLinear Load Step-Changes

Turn-off Turn-off Turn-onTurn-on

time (5ms/div)time (5ms/div) time (5ms/div)time (5ms/div)

(a) Ouput Voltage and its reference (40V/div)(a) Ouput Voltage and its reference (40V/div)


(a)(a)

(b)(b)

(a)(a)

(b)(b)

R fR f




ReferenceReference

P. Mattavelli, S. Fasolo: “Implementation of Synchronous

Frame Harmonic Control for High-Performance AC Power

Supplies”, IEEE IAS Annual Meeting 2000, Rome, Italy, 8-12 October, 2000, pp. 1988-1995.

P. Mattavelli, S. Fasolo: “Implementation of Synchronous

Frame Harmonic Control for High-Performance AC Power

Supplies”, IEEE IAS Annual Meeting 2000, Rome, Italy, 8-12 October, 2000, pp. 1988-1995.

Digital Control Applications in Power Electronics Lesson4

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