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Analysis of a controllable LC-filter for compensation of specific harmonicsH.W. Klesser, S.W.H. deH aan , J.B. Woudstra

Delft University of TechnologyPower Electronics and E lectrical MachinesS. .H.deHaan@irs.tudelfi.nl

AbstractIn this paper a recently proposed circuit tocompensate harmonics is analysed. The circuitconsists of an LC-filter which is connected in serieswith an electronic switch, which enables control ofthe system [I]. The compensator is intended to beconnected in parallel to a harmonic-generatingappliance, such as line commutated converters.The system is analysed for the third harmonic byapplying the principle of superposition. The circuit isconsidered to be excited by the undistorted gridvoltage with the so called switch voltagesuperimposed on it. After fourier-series expansion ofthe switch voltage, linear circuit theory is applied.Based on the analysis, a mathematical relationbetween the harmonic and major control and circuitparameters is derived. The obtained formula's canbe used for purpose of control and design for asystem with one zero current interval. The model hasbeen verified by comparison with experimentdyobtained results. The analysis led to followingconclusions:1) for control with a single dead time, the currentamplitude is roughly proportional tocos (ot,) , where t, is the f ~ n gnstant with respect tothe -/+ zero crossing of the phase voltage.2) . phase and amplitude cannot be controlledindependantly if one zero current interval is applied3) for a phase angle of 90" he amplitude has amaximum, which is for many applications anacceptable combination.Nomenclaturad ' delay timet,TI period of the fundamentalU, gridvoltageusw,wI frequency of the fundamental;w,,

firing instant with respect to zero crossingof grid voltage

voltage accross the switch Sresonant frequency of the LC-circuitfrequency of the third harmonic;

1. IntroductionSeveral types of power electronic converters,such thyristor controlled converters, arecharacterized by ac currents which contain manylow order harmonics with a significantamplitude. Either by their large number, such asin the input stage of electronic power supplies,or by their large power level, such as in HVDCsystems, these harmonics can cause harmonicsdistortion of th e grid voltage at the point ofcommon coupling. These harmonics may resultin extra losses and in disturbances with othercustomers. To prevent these compatibilityproblems the addition of filters is often requiredto block or reduce the harmonics. Often passiveLC filters are used which are connected inparallel to the load. Because of their lowimpedance for harmonics, these filter have thedrawback that they form a sink for harmicsbeing present in the grid, thus causing evenlarger harmonic current then would be present 'without filter. With activ e filters this problem isavoided because the current can be controlledacurately. The draw back of these.systems is thatth e care complex and contain many componentsW . 5 1Recently a new filtering system [ l] has beenproposed that consists of a passive LC-filterwith a series connected thyristors as shown infig. 1.

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Fig.1 Block diagram of he compensatorsystem0-7803-4879-6/98/$10.000 998IEEE 588

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Fig. 2 Line voltage U, and compensator cur-rent i,The filter system .is connected in parallel to theappliance that is generating the harmonics. Thethyristor enables control of the circuit in such away that the harmonic that is generated by theappliance is absorbed exactly by the filter. Theapplicability fo r the compensation of the thirdharmonics in office buildings has been provedby experiments [1,2]. In this paper anapproximating analysis of the system will bepresented for one of the modes of operation.2. . Principle of operationThe princple of operation for a 3rd harmoniccompen sator is described extensively in [l] an dwill be reviewed shortly. The value of L and Care chosen such that the resonant frequency isslightly higher than the frequency of the thirdharmonic, for instance 160 Hz for a 150Hzcompensator. Because' the resonant period isslightly shorter than the the period of the thirdharmonic, it is possible to inhibit the firing ofthyristors after a current zero crossing for a(very) short time without losing synchronismwith the fundamental. Within each period of the(50Hz) fundamental six zero crossing of thecurrent can be identified. Experiments haveshown that both phase and amplitude of thegenerated third harmonic can be controlled byinserting two delay times d, and d, (see fig. 2) attw o of six properly selected zero crossings. Inthis paper the charactistics of the circuit withonly one dead time d will be analyzed.

3. AnalysisIn the cyclic stable state the the thyristor is firedperiodically at times t,+kT, that aresynchronized with the period of the fundamental(see fig 3b). As a result of the firing a resonantcurrent (about 150Hz) will start to flow in thecircuit. T he thyristor switch opens after the nextcurrent zero crossing of the resonant harmoniccurrent and the diode will take over the currentin opposite direction.

b

U,+I$R

Fig. 3 Model to calculate the current generatedby the compensator currentThe analysis of the circuit can be split up inthree steps.Step 1: The voltage acting on the LCR-circuit isconsidered as being the superpostion of the gridvoltage u,(t) and the voltage accross the switchus(t), where. the following mesh equation appliesto the circuit as shown in fig. 3:

u l ( t ) - u s ( t ) - u J t ) - i R - L - i = 0.tIf it is assumed that the tim e interval of the openinterval d is short in comparison to thefundamen tal period, than the switch vo ltage canbe considered constant du ring the open interval.To have a zero current during the open intervalthe inductor voltage should be zero during thisinterval. Th is mean s that the switch voltage u,(t)during the op en interval is equal to:

U& ) = - u c ( t , - d ) + u , ( t ) when S open= o when S closedBecause the grid voltage u,(t) is a continuoussinoid, the response of the circuit is easy tocalculate. For the response we can write in the

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time domain:i,(t) = ~,(~(ino,)~sin(no,(t)argF(ino ,) )(1)Here F(jo) is the frequency response of thefilter: F(jwj=i(jo)/u(joj .To calculate its response to u,(t), it is written asa series of rectangular pulses with width du,(t) =E=.-uc(tl-4 - O, sino,(t, -d/2,) .(2)' p ( t - t , +d2-kT1)

with p(t) = 1 for ltld/2.The series of pulses can be expanded in afourier series:

U&) = a , + x ; , a"COS(nol(r-t ,+d2))For the filter current, being the response t o U&),we can write:

(3)For the ratio of the harmonics of the series ofpulses we can write:a,, 3 s i n n q d I 2a3 n sin 3o,d /2_ - (4)

Step 2: During the zero-current interval the sumof the response to us an d p is zero:for kT ,+il d

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model predicts slightly lower values than themeasurements. This is p robably rooted in the thefact that the sum mation in eq. (7) is carried ou tover 5 terms. Further a difference in dampingmay ne a cause. This shows that the presentedmodel and method of calculation can be usedfor design and form the base for control of thecircuit.

c0 2W IC0 WO 8W l W O ( t ~ r )d ,

td l < d2 d , =d , d p d ZFig. 4 Amplitude of the third harmonic currentasa function of dl and d? The markers give thetheoretical amplitude fo r d l =O andd2 = 1.4 ms

5. ConclusionThe analysis method leads to approximate,explicit exp ressions for all low order harmonicsas a function of the circuit parameters L, C andR (as reflected inF(io)) and the controlparameter d. The expression can be used fordesign and control of the compensator-circuit

ReferencesJ.B. Woudstra, S.W.H. de Haan, .C. vander SluijsThird harmonic compensator applicablefor large office buildings, Eur. PowerElectronics Conf. EPE 1997, Trondheim,pp. 4.917-4.922J.B. Woudstra,S.W.H.e HaanActive filter for compensation of specificharmonics; experimental results,submitted PEDE S98G. Kamth, N. Mohan, V. AlbertsonHardware implementation of a NovelReduced VA Rating filter for Nonlinearloads in 3-phase, 4-wire Systems, IEEE-ApEC'95, pp. 984-989.H. AkagiNew trends in Active Filters, Proc.European Power Elctronics Conf.,EPE '95, Sevilla, pp. 0.017-0.026.R.M. Duke, S.D. RoundThe Steady State Performance of acontrolled current active filter, IEEETrans. Power Electronics,. Vo1.8, "3,Apr. 1996, pp140-146,

with one zero current interval d. Thecompensator circuit with one zero-current-interval can be u sed in applications wh ere phaseand amplitude need not be controlledindependently. This is amongst other the casewith harmonic producing loads like dioderectifiers with capacitive outpu t filter.The method of analysis can be extended tooperation modes with more zero-current-intervals.

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