162_differential mode current harmonics forecast for dcm boost rectifiers design

7/27/2019 162_Differential Mode Current Harmonics Forecast for DCM Boost Rectifiers Design

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Differential mode current harmonics forecast for DCM boost rectifiers design Jean Christophe CREBIER

EPE '99 - Lausanne P. 1

Differential mode current harmonics forecast for DCM boost rectifiers

design

Jean Christophe CREBIER, Marc BRUNELLO, Jean Paul FERRIEUX

Laboratoire d’Electrotechnique de Grenoble (LEG)INPG/UJF-CNRS UMR 5529

ENSIEG, BP 46, 38402 St Martin d’Hères cedex,GRENOBLE, FRANCE

Phone 33/(0)4.76.82.62.99 Fax 33/(0)4.76.82.63.00E-mail: [email protected]

Keywords:

EMC/EMI, harmonics, high frequency power converters, modelling, power factor correction, power quality, switched mode power supplies.

Abstract:

This new method allows to forecast differential mode current harmonics of discontinuous currentmode (DCM) rectifiers from the fundamental up to the switching frequency multiples. It is a lot faster than time domain simulations and it gives exact theoretical predictions. This allows designoptimization because it takes into account control variables with respect to operation point. It isapplicable to any kind of boost derived converters. Comparisons with experiments and simulationsemphasize its interesting characteristics.

Introduction:

Since low frequency harmonic standards appeared IEC1000-3-2 [1], lots of power factor correction(PFC) rectifiers have been designed and developed. Some of them operating in discontinuous currentmode (DCM) are simple to control, very efficient and reach high power density. Boost derivedrectifiers [2,3] represent a large part of them. They provide two essential tasks: active input current

shaping and output voltage control. However these converters generate high frequency harmonicswhich must be studied and reduced in order to comply with corresponding standards [4]. Besides,

depending on control strategy possibilities, it is necessary to look at the low order harmonics in order to design correctly the converter in order to comply with corresponding standards.Existing methods have been developed and provide good results [5,6]. A new method faster thanclassical time domain simulations is proposed here. It allows to forecast in the frequency domain PFCrectifier input current harmonics from the fundamental up to switching frequency harmonic multiples.This allows to find optimal operation point, and to characterize high frequency filter with respect to

required specifications (power flow, output power) and standards. The method is presented anddescribed. It is applied to the single-phase single switch boost rectifier operating in DCM for twodifferent control strategies. Circuit oriented simulation and experiments validate theoretical predictions.

The method:

The main idea is to get a theoretical time domain representation of differential mode current created bythe converter over the line period (with respect to operation point and control variables) and to use

mathematical tools to get a frequency domain representation of this current from the fundamental up toswitching frequency multiples.

Description.

The idea is to consider the input current waveform over a line period. DCM hard-switched converters

draw high frequency waveforms (see figure 2), which can easily be mathematically described.





Knowing the switching events (duty cycle "α" and switching frequency "Fo" for example) as well asswitching period current waveform and maximum (see figure 3 for example) a simple model based onthe Laplace formulation can be derived [7]. All these switching period mathematical formulations arethen summed up over a line period. This gives the complete input current pattern in a mathematicalform (eq.1). Using the relation linking the Laplace transform to the Fourier transform, a frequencydomain representation of this current is derived. This allows to get current harmonics from thefundamental up to switching frequency multiples.Playing with variable parameters (inductor, output voltage, duty cycles, frequency…) optimization can be performed and/or ideal operation point can easily be derived.

Application area.

This kind of method can be applied anytime input current can easily be characterized (boost or buck derived converters). Any kind of control strategy can be used as long as switching events and inputcurrent waveform of each switching period can be theoretically forecasted. This mainly means that:- Control strategy is well known.- Discontinuous shape is well known (meaning simple propagation paths).- Evolution of high frequency patterns has a line period reproducibility.

However, it is not possible to take into account common mode disturbances.

The above method is going to be applied to the single-phase single switch boost rectifier [3]. Twodifferent operation modes will be considered. First operation with a simple control strategy "constantswitching frequency and duty cycle over a line period" will be studied. Then it will be applied to asliding mode control [8] considering an operation at the limit between continuous current mode(CCM) and discontinuous current mode (DCM).

Application I: Constant switching frequency and duty cycle operation

Presentation.

Here, a simple setup is considered, the desired task being to get an idea of differential mode input

current harmonics from the fundamental up the switching frequency multiples. The converter operatesin discontinuous current mode, working at constant switching frequency and constant duty cycle. Thisoperation mode allows to get correct but not perfect power factor correction and is used as example because it corresponds to a simple case.The operation point is set at:

Vout=650V Vin=230Veff Pout=1kW L b=600µH α=0.49 Fo=20kHz

Vout

Lb

CoutVin

Boost chopper

Diode rectifier

Capacitor filter

AC source 230V

50Hz

Load

F

A

B

D K E

Iin

Fig.1: Converter topology.

0 0.005 0.01 0.015 0.02400

200

0

200

400

R f

Sf

Iin(A)

t s

Fig.2: Line current waveform for a reduced

switching frequency.

Time domain simulations have been realized using Simplorer software [9]. Below are given a timedomain and a frequency domain representations of the input current for the above operation point.





-15

-10

-5

0

5

10

15

0 0.005 0.01 0.015 0.02

t (s)

A m p l i t u d e ( A

1.E-04

1.E-03

1.E-02

1.E-01

1.E+00

1.E+01

10 100 1000 10000 100000 1000000

Frequence(Hz)


Fig.3: Input current time and frequency domain representations for DCM boost rectifier operating atconstant switching frequency and duty cycle.

As it can be seen in the frequency domain representation, this operation mode does not remove totallylow frequency harmonics. This is often the case in three phase application where control strategiesremain basic. Therefore, it can be interesting to forecast what will be low order harmonics and to

optimize converter specific values (L b, Fo,…) in order to comply with corresponding standards.Besides, the frequency domain representation clearly shows high frequency harmonics. Here again atheoretical forecast might help in filter design and optimization. These two aspects can be deducedusing the following model.

Modelling.

An example of line current Iin patterns is shown Fig.2. A focused time domain representation of eachswitching period is given below Fig.4. Each of them can be described by a sum of causal ramps thatmust be correctly defined with respect to operation point and converter components.In case of boost rectifier operating in DCM with constant frequency and duty cycle over the line period, it is easy to define all of them.

To

α.To

βk .ToILb(A)

t

βk+1.To

ILb(k)

Fig.4: Switching period input current patterns

For each switching period, three causal ramps are required. Their sum gives a high frequency time

domain mathematical representation of the input current.

)To).k (t(u.t.To.

I

)To).k (t(u.t).To.

I

To.

I()To.k t(u.t.

To.

I)k (F

k

k

)k (Lb

k

)k (Lb)k (Lb)k (Lb

h

β+α+−β

∆+

α+−β

∆+

α

∆−−

α

∆=

(eq.1)

where k ∈[0, m] m being the modulation factor m=ωo/ωr

with To.L

)k (VI in

Lb(k) α=∆ (eq.2)

and outin

in

k V)k (V

)k (V

. −α=β (eq.3)

and Vin(k)=Vmax.sin(2.π.k/m) (eq.4)

t

Iin Iin (zoom)





The following step consists in a summation of Fh(k) functions over a line period Tr:

∑ω

ω

=

=r

o

1k

hl )k (FF (eq.5)

Then, using the Laplace/Fourier transform mathematical tools, it is simple to get a mathematicalexpression of the frequency domain representation of the input current. This is given by the Laplacetransform over a line period (eq.6):

∑ω

ω

=

−β+α−α−

β+

β+

α−

α=

r o

1k

p.To.k p.To).(

k

p.To.

k

2

k e.e.To.

1e).

To.

1

To.

1(

To.

1.

p

axIm) p(L k (eq.6)

To get the Fourier transform and therefore harmonics values, it is necessary to do : p=j.n.ωr

where n represents the harmonic number from the fundamental up to the switching frequencymultiples.

Theoretical results and comparisons.

Having now the mathematical form of the frequency components of the input current, low order harmonics as well as high frequency harmonics can be studied easily. The plot below (Fig.5) presentsa comparison between two frequency representations of low order harmonics of the studied converter input current. One comes from the above mathematical model whereas the other one comes from aSimplorer time domain simulation. Both of them are also compared to corresponding standardIEC1000-3-2. Both theoretical results match very well allowing to validate our model in thisfrequency range.

0 100 200 300 400 500 600 700 8000.001

0.01

0.1

1

10

Normn

Simun

..C2n

2Fr

.50n

IEC 1000-3-2

Simplorer

Method

Fr(Hz)

Harmonics (A)

Fig.5: Low frequency harmonics comparison between both method results and standards.

Having this frequency representation, it is simple to get best possibilities of the converter with respect

to its characteristics (Vin, Vout, α…) and standards. It is particularly interesting in three-phase caseswhere output voltage must be lowered as much as possible.

As far as high frequency harmonics are considered, the forecast method results are given Fig.6. Theyare compared with time domain simulation results. It can be seen that both match very well. Thisdemonstrates the validity of the method as a computer-forecast method of differential mode EMI inDCM PFC rectifiers. The computer time necessary to calculate the most important harmonics up to1MHz is about 10 seconds on Mathcad software. Considering a circuit-oriented simulation enoughaccurate (100ns steps), the time computer is much longer, around 2mn. Besides FFT is then required.





1.E-04

1.E-03

1.E-02

1.E-01

1.E+00

1.E+01

1.E+04 1.E+05 1.E+06

F(Hz)

H a r m

o n i c s ( A )

Forecast method

Simulation results.Simplorer

Fig.6: High frequency range comparison between both method results.

These theoretical results and comparisons clearly demonstrate the interest of this new forecast method

compared with classical time domain simulations. The mathematical form of the model allowsoptimization routine that might be useful providing in addition time gains.

To conclude with this first application, a comparison with practical results is now proposed.

Practical results and comparisons.Experiments have been realized in order to check theoretical forecast with reality. Chosen operation point and spectral representations are given below:

Vout=385V Vin=127Veff Pout=500W L b=300µH α=0.49 Fo=23kHz

1.E-04

1.E-03

1.E-02

1.E-01

1.E+00

1.E+01

1.E+01 1.E+02 1.E+03 1.E+04 1.E+05 1.E+06

Frequency (Hz)

A

m p l i t u d e ( A )

Fig.7: Comparison between theoretical and practical spectral representation of input current.

As it can be seen, theoretical and practical results match correctly. Differences are due to two knownand explainable origins.

The converter is operating in open loop. Therefore, if supply voltage is disturbed, then input currentwill also be. Fig.8.left presents a spectral representation of supply voltage. It clearly appears that lowfrequency harmonics are due to this practical effect that has not been taken into account.

In the upper frequency first switching frequency harmonic multiples match very well. Around 300kHzdifferences appear. This is due to model approximation. In fact, during non conduction phase, ringingeffect might appears. This is underlined this the plot Fig.8.right showing high frequency input currentwaveforms.

Theoretical resultsPractical results





1.E-04

1.E-03

1.E-02

1.E-01

1.E+00

1.E+01

1.E+02

1.E+03

1.E+01 1.E+02 1.E+03 1.E+04 1.E+05

Frequency (Hz)

A

m p l i t u d e ( V )

.

-2

0

2

4

6

8

0.E+00 5.E-05 1.E-04

T (s)

A m p l i t u d e ( A )

Fig.8: left-. Spectral representation of supply voltage.right- A high frequency input current pattern

It has to be mentioned that these effects can be theoretically taken into account. For that, it is requiredto consider a supply voltage with additional harmonics and an input current taking into account the

above described ringing effect.

1.E-04

1.E-03

1.E-02

1.E-01

1.E+00

1.E+01

1.E+01 1.E+02 1.E+03 1.E+04 1.E+05 1.E+06

Frequency (Hz)


Specific forecast assumptions:- Voltage power supply:

V1 = 180Vmax

V1 = 10Vmax

V1 = 2Vmax

Input current resonance:A = 0.3Amax

Fres = 320kHzD = 10

5

Corresponding Laplace transform:

2res2

res

)D p(.A) p(Res

ω++

ω=

Fig.9: Spectral representation comparison with modified forecasted results.

These results clearly validate the theoretical forecast and the interest of the method. It is now going to be applied to another operation mode in order to show that this mathematical model can be widelyapplied.

Application II: Constant duty cycle and variable switching frequency

operation:

Presentation.

Here is proposed another application based on the same structure but with a different control strategy.

The rectifier is operating in this case at the limit of CCM and DCM. This allows to realize a highquality PFC (with no low order harmonic in theory) and besides it induces reduced high frequency

harmonics due to sliding mode control [8]. It is studied here as a complementary example to show thatthe forecast method is general.

The operation point is set at:

Vout=650V Vin=230Veff Pout=1kW L b=660µH Ton=25µsTime domain simulations have been realized using Simplorer software. Below are given a timedomain and a frequency domain representations of the input current for the above operation point.





-15

-10

-5

0

5

10

15

0 0.005 0.01 0.015 0.02

t (s)


1.E-04

1.E-03

1.E-02

1.E-01

1.E+00

1.E+01

10 100 1000 10000 100000 1000000

Frequence(Hz)


Fig.10: Input current representations for DCM boost rectifier operating at constant duty cycle and

variable switching frequency.

The above frequency representation of the input current clearly shows PFC quality of this operationmode. Besides, HF harmonics are lowered compared with the previous operation principle (applicationI). This emphasize interest of this control strategy and its theoretical study which is going to be

performed now.

Modelling.

The mathematical model is derived in a similar way as before.

The high frequency mathematical representation of the input current (Fig.10) is given by equation(eq.7):

)TonToff Tont(u.t.Toff

I

)TonTont(u.t).Toff

I

Ton

I()Tont(u.t.

Ton

I)k (F

k k

k

)k (Lb

k

k

)k (Lb)k (Lb

k

)k (Lb

h

−−−∆

+

−−∆

+∆

−−∆

=

(eq.7)

with ∑−

=

+=1k

1i

ik Toff TonTon (eq.8)

And .TonL

)k (VI in

Lb(k) =∆ (eq.9)

Andoutin

ink

V)k (V

)k (V.TonToff

−= (eq.10)

These values are calculated up to final time equal to 20ms:

∑= +==

k

1iiToff Ton02.0Tr (eq.11)

Ton

Toff k ILb(A)

t

Toff k+1

Ton

ILb(k)

Fig.11: High frequency input current patterns.

t

Iin Iin (zoom)





Once the sum over a line period is made, the final mathematical form (equ.12) is derived usingLaplace tools.

∑=

−+−−

++−=

nb

1k

p.Ton p).Toff Ton(

k

p.Ton

k 2

k k k e.e.Toff

1e).

Toff

1

Ton

1(

Ton

1.

p

axIm) p(L (eq.12)

Where "nb" represent the number of high frequency switching patterns over a line period.

To get the Fourier transform and therefore harmonics values, it is necessary to do : p=j.n.ωr

where "n" represents the harmonic number from the fundamental up to the switching frequencymultiples.

Results and comparisons.

Results and comparisons between forecasted results coming from the frequency model and the time

domain simulations are given below Fig.12 (It is for the above operation point).

1.E-04

1.E-03

1.E-02

1.E-01

1.E+00

1.E+01

1.E+01 1.E+02 1.E+03 1.E+04 1.E+05 1.E+06

Frequency (Hz)


Fig.12: The two theoretical frequency representations of the input current in sliding mode operation.

As it can be seen in this second example, both results match very well in low and high frequencyranges. This allows to conclude that the frequency model is suitable for this kind of operation mode(sliding mode) whenever it is possible to theoretically forecast high frequency input current waveform(shape and switching event) over a line period.

Conclusion:

A new powerful method has been presented. It allows to forecast all input current harmonics from the

fundamental up to switching frequency multiples. It can be used as an optimization tool in order tohelp in converter design (comply with standards, filter design, optimization of converter characteristics…). It is very easy to derive the mathematical model from known input current patterns.

It can be implemented very easily on any kind of software and can be used in an optimization routine.Finally, it is faster (in a ratio of 100) than classical time domain simulations and besides results aredirectly in the frequency domain.

References:

1. International electrotechnical commission IEC1000-3-2, "Limits for harmonic current emissions (equipment

input current < 16A per phase)", August 1995.

2. A.R. Prasad, P.D. Ziogas, S. Mantias, "An active power factor correction technique for 3 phase diode

rectifier", PESC 89, pp58-66.





3. O. Garcia, J.A. Cobos, R. Priet and J. Uceda, "Single switch AC/DC power factor correction converter valid

for both three phase and single phase application", EPE'97, pp1.188-1.193.

4. ISM550011 "Limits and methods of measurement of radio disturbance characteristics of industrial,scientific and medical (ISM) radio-frequency equipment", July 1991.

5. M. A.E. Andersen "Fast prediction of differential mode noise input filter requirements for flyback and boost

unity power factor converters", EPE 97, p2.806-2.809

6. F.S. Dos Reis, J. Sebastian, J.Uceda "Determination of power factor preregulators conducted EMI", EPE 95,

pp3.259-3.264

7. R. Scheich, J. Roudet, J. Bigot "Common mode RFI of a HF power converter: Phenomenon, its modeling and

its measurement" EPE 93, pp164-169

8. J.S. Lai, Daoshen Chen "Design consideration for power factor correction boost converter operating at the

boundary of continuous conduction mode and discontinuous conduction mode", APEC 93, pp267-273.

9. B. Knorr, U. Knorr, L. Zacharias, H. Puder, "SIMPLORER, reference manual – version 3.3", SIMEC GmbH

& Co KG, 1998.

10. J.C. Crebier, "Study of conducted disturbances in boost derived rectifiers", PhD dissertation from INPG, May

1999

162_differential mode current harmonics forecast for dcm boost rectifiers design

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