159-Inverters & Harmonics

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E/CT 159 first issued september 1993

Jean Noël Fiorina

Joined Merlin Gerin in 1968 as alaboratory technician in the ACS(Alimentations et ConvertisseursStatiques) department where heparticipated in the performancesetting up procedures for staticconverters. In 1977 he obtained his

ENSERG Engineering degreefollowing a 3 years evening courseand rejoined the ACS department.Starting as development engineer hewas soon afterwards entrusted withprojects. He became laterresponsible for design projects inEPS department (Electronic PowerSystem).He is in some ways the originator ofmedium and high power inverters.At present he is with the UPSDivision where, as responsible forInnovations he works on the

preparation of new UPS designs oftomorrow.

n° 159

inverters andharmonics

(case studies ofnon-linear loads)



Cahier Technique Merlin Gerin n° 159 / p.2

glossary

UPS Uninterrupted Static Power Supply - Static UPS

IEC International Electrotechnical Commission

CIGREE Conférence Internationale des Grands Réseaux Electriques et Electroniques

(International conference on hight voltage electric systems)PWM Pulse Width Modulation

D global distortion rate

Hn individual ratio of harmonics of order n

ϕ n phase angle shift of harmonic component at t = 0

In effective current of harmonic component of order n

Uref reference voltage

ν distortion factor

Vn effective voltage of harmonic component of order n

Zsn output impedance for harmonic of order n




inverters and harmonics(case studies of non-linear loads)

summary

1. Introduction p. 4

2. Characteristics of non-sinusoidal Harmonic analysis of a periodicfunction p. 4

Effective value of a non-sinusoidalalternating quantity p. 4

Distortion rate p. 4

Power factors and cos ϕ1

p. 5

Distortion factor ν p. 5

Crest factor p. 5

Relation between current distortionand voltage distortion p. 5

3. Impedances of some conventional Impedance of a transformer p. 6

Impedance of an alternator p. 7

Output impedance of an inverter p. 7

Impedance of line p. 11

4. Micro and mini-computer loads Description p. 12Influence of source impedance p. 12

Calculation of source powerfor supplying RCD type loads p. 13

5. Conclusion p. 16

Appendix 1: influence of line impedances on voltage distortions p. 16

Appendix 2: input filters in computer/micro-processor equipment p. 18

Appendix 3: bibliography p. 19

alternating quantities

sources




1. introduction

that modern inverters are excellentgenerators of sinusoidal voltage evenwhen they supply non-linear loads.This is considered quite normal asUPS are designed and very oftenutilised to supply computer/ microprocessor systems which drawnon-sinusoidal currents.

view of frequency stability as well asvoltage stability, performancessuperior to those of the mains.The only doubtful characteristic is, inthe opinion of many engineers, itsability to deliver a sinusoidal voltageregardless of the shape of the currentdrawn by the load.The aim of this «Cahier Technique» isto clarify this point and to demonstrate

Static UPS are virtually perfect electricgenerators.

They are highly reliable and, bynature, ensure (within the performancelimits of the battery) the uninterrupted

availability of electric power.

As regards electrical characteristics,the inverter (which constitutes the UPSgenerator) possesses from the point of

2. characteristics of non-sinusoidal alternating quantities

harmonic analysis of a

periodic functionAs alternating non-sinusoidal currentsand voltages are the main topic of thisstudy, it will be worth while to reviewthe electric quantities in the presence ofharmonics.

Fourier theorem states that any non-sinusoidal periodic function can berepresented by a series of termsconsisting: of a sinusoidal term at fundamentalfrequency, of sinusoidal terms whosefrequencies are whole multiples of thefundamental (harmonics), and eventually of a continuouscomponent (DC component).

The formula denoting the harmonic

analysis of a periodic function is asfollows:

y (t) = Yo + Yn 2∑n = 1

n = ∞

sin (nωt - ϕ n)

where:Yo: value of continuous componentgenerally equal to zero and consideredas such hereafter,Yn: effective value of harmonic oforder n,ω: pulsation of fundamental frequency,

ϕ n: phase shift angle of harmoniccomponent at t = 0.

effective value of a non-sinusoidal alternating

quantityApplying the general formula

Y rms = 1

T ∫ y 2 (t) dt

0

T

gives with harmonic representation:

Y rms = Yn 2∑n = 1

n = ∞

distortion rates

Harmonic rates

(as defined in IEC dictionary)

This parameter, also called harmonicdistortion or distortion factor representsthe ratio of the effective value ofharmonics (n ≥ 2) to that of thealternating quantity:

THD % = DF % = 100

Yn 2∑

n = 2

n = ∞

Yn 2∑

n = 1

n = ∞

Global rate of distortion

(as defined by CIGREE)

This parameter represents the ratio ofthe effective value of harmonics to thatof the fundamental alone:

D % = 100Yn

2

∑n = 2

n = ∞

Y 1

Note: when the distortion rate is low, asis most frequently the case for thevoltage, the two definitions lead inpractice to the same result.

For example, if:

Yn 2∑

n = 2

n = ∞

= 10 % de Y 1

The IEC expression gives:

THD = DF = 100 (0.1) 2

1 + (0.1) 2 = 9.95 %

Whereas the CIGREE expressiongives:

D % = 100 0.1

1 = 10 %

Hereafter we shall retain for the

distortion rate, the expression «D»which corresponds to a more analyticalview of the influence of harmonics on a

non-deformed wave.




crest factorAs defined by IEC, it is the ratio of crestvalue to the effective value of a periodic

quantity.

relation between currentdistortion and voltagedistortionFor a given voltage source, it is alwayspossible to define an outputimpedance, even if the latter isfrequency dependent. To the extentwhere this impedance is independent ofthe current value (linear case) it ispossible to calculate for each currentharmonic a corresponding voltage

harmonic and thus to deduce theindividual harmonic rate (percentage).The effective value of voltage harmonicof order n equals:

Un = Zsn . In

whereZsn: output impedance for harmonic nandIn: effective current of harmonic n.

The individual rate of harmonics oforder n for this voltage is equivalent to:

Hn = Un

U 1

whereU1 = effective value of fundamentalvoltage.

The global distortion rate of voltage isthus obtained by means ofexpression:

D % = 100

Un 2∑

n = 2

n - ∞

U 1

and also:

D % = 100 Hn

2

∑n = 2

n - ∞

The input impedance for variousharmonic frequencies plays thereforean important role in bringing about theonset of voltage distortion. The higherthis input impedance, the greater will bethe voltage distortion rate for a givennon-sinusoidal current.

Individual harmonic rate

This parameter represents the ratio ofthe effective value of a harmonic oforder n to that of the alternating

quantity (according to IEC dictionary) orto that of the fundamental alone(according to CIGREE),

according to definition in IECdictionary:

Hn % = 100 Yn

Yn 2∑

n = 1

n = ∞

and according to CIGREE definition:

Hn % = 100 Yn

Y1

This latter definition will be retained insubsequent reasoning.

power factors and cos ϕ1

According to IEC, the power factor isthe ratio of the effective power P to theapparent power S:

λ =P

S

This power factor should not beconfused with the phase shift anglefactor (cos ϕ1) which represents thecosine of angle formed by the phase

elements of fundamental componentsof voltage and current:

λ 1 = cos

ϕ 1 = P 1

S 1

where:P1 = effective power of fundamentalcomponentS1 = apparent power of fundamentalcomponent.

distortion factor νAccording to standard specification

IEC 146-1-1, this factor enables todefine the relation between powerfactor λ and cos ϕ1:

ν = λcos ϕ 1

Where voltages and currents areperfectly sinusoidal the distortion factorequals 1 and cos ϕ1 is equal to thepower factor.




of UccR is of the order of 1 % toseveral % (this latter value becomingcorrespondingly smaller as the powerrating of transformer increases).

In practice, as regards harmonics,since only the inductance impedance isfrequency dependent, it is theinductance alone which determines thebehaviour/performance of thetransformer. in three phase transformers, it isnecessary to take into account the

different possible connection types ofprimary and secondary windings, asthese exert an influence on the sourceimpedance for some harmonics (inparticular, third harmonic andmultiples of 3).

In fact, in the case of a transformerwhich supplies to each of its secondarywindings distorted and balancedcurrents comprising harmonics of order3 and multiples of 3, say 3 k, andconsidering that these currents arebalanced, it is thus possible to write foreach of these phases:

I1 3 k = I sin 3 k ωt

I2 3 k = I sin 3 k ( ωt - 2π3

)

I3 3 k = I sin 3 k ( ωt - 4π3

)

or

I1 3 k = I sin 3 k ωt

I2 3 k = I sin (3 k ωt - k 2π)

I3 3 k = I sin (3 k ωt - k 4π)

These equations show that thethree currents are in phase. It is thisphenomenon which leads one toobserve in the neutral conductor ofsome wiring installations (neon tubesfor example) the presence of muchhigher currents than originallyanticipated.

3. impedances of some conventional sources

Very often the impedance Zs (at 50 Hz)of a generator is given as percentage ofnominal impedance of load Zc:

Zs % = 100 Zs

Zc

For the nominal current, the voltagedrop across this impedance represents,therefore, as percentage in relation tothe nominal voltage, the value of thissource impedance:

Zs . InUn

% = 100 Zs . InUn

where Zc . In = Un

Zs . In

Un % = 100 Zs . In

Zc . In = 100 Zs

Zc

impedance of atransformerFigure 1 represents an equivalent circuitdiagram of a single phase transformerseen from secondary winding.

The transformer impedance consists ofan inductance L in series with aresistance R. An equivalent value of therelative impedance is given by thetransformer short-circuit voltage Ucc.

Indeed, by definition, the short-circuitvoltage is the voltage that must beapplied across a winding in order toinduce a nominal current in the otherwinding also under short-circuit,

Ucc % = 100 Ucc

Un

Ucc % = 100

Zs . In

Un = 100

Zs

Zc = Zs %

This short-circuit voltage is made up oftwo terms: UccR et Uccx (see fig. 2).

in distribution transformers or generalpurpose transformers with ratingssuperior to 1 kVA, the value of Uccxranges from 4 - 6 %, whereas the value

The behaviour of a transformer towardsthese harmonics is therefore dependenton the homopolar impedance Zh of thetransformer (refer to «CahierTechnique» n° 18 «Analyse desréseaux triphasés en régime perturbé àl'aide des composantes symétriques»).

Two types of secondary windings aresuitable for not amplifying or reducingharmonic distortions:

star connected secondary with«distributed» neutral

fig. 1: equivalent circuit diagram of a single

phase transformer seen from secondary

winding.

fig. 2: Kapp triangle of a transformer (values

referred to secondary).

e

L R

A B

Ucc Uccx = L I2nω

Ucc = R I2n I2cc = I2nR




When primary windings are delta orstar connected with the neutral pointconnected to the source neutral(see fig. 3), the harmonic impedances

of order 3 and multiples are neitherencouraged nor discouraged(Zh = Zd).The transformer behaves as threesingle phase transformers.

ZIGZAG connected secondaryThese connections ensure minimumdistortion in secondary - in fact, in thiscase, the harmonic currents of order3 k do not circulate in the transformerprimary, and the impedance Zs is nolonger dependent on secondarywindings. The inductance is thus very

low: Uccx ≈ 1 %, and the resistance isreduced roughly by half whencompared with the resistance of deltastar connected transformer of samerating.Figure 4 and the following calculationexplain why currents of pulsatingfrequency 3 kω are not found in thetransformer primary (homopolarcurrent equal to zero).

For a turn ratio

N 2

N 1

,

the current circulating for instance in

the primary winding 1 equals:

N 2

N 1

(i 1 - i 3)

with

i 1 = I 1 3 k = I sin 3 k ωt

i 3 = I 3 3 k = I sin 3 k (ωt - 4π3

)

= I sin (3 k ωt - 4π)

this gives

N 2

N 1 (i 1 - i 3) = 0

The ZIGZAG connected secondarywinding acts therefore as an attenuatorto harmonics of order 3 k. This type oftransformer is often used as an outputtransformer for classic inverters of highrating.

As a general rule, the other types ofconnection are to be avoided, inparticular those that do not allow theneutral to be «distributed» in the

secondary; in fact for these Zh = ∞.

impedance of an alternatorAn alternator can also be representedby a voltage source in series with aninductance and a resistance.

However, this inductance assumes verydifferent values according to the speedof current variations to which it isrelated.

During such current variation, theequivalent reactance passes

progressively from a value called sub-transient to its synchronous value via atransient value. These different valuesreflect the variation of the alternatormagnetic flux.

As regards current harmonics, only thesub-transient reactance is to beconsidered in any phenomenon lastingless than 10 ms.This reactance, referred to as«longitudinal sub-transient reactance»is denoted as X''d.

For an alternator of current production,this reactance amounts to 15 - 20 %.In traditional machines but of designoptimised in this respect, a value of12 % can be achieved.Finally, in special machines, someconstructors claim values decreasing to6 %.

In conclusion, it is worth recalling that,except in very particular cases, thealternator output impedance isconsiderably greater than that of atransformer; consequently, the sameapplies to the voltage distortion rate inthe presence of distorted currents.

output impedance of aninverterThe impedance of an inverter isessentially dependent on the outputimpedance of its filter and on the typeof regulation adopted.

Principe of an inverter

An inverter comprises first of all a

converter referred to as «mutator» i.e.

switching device which converts the DC

voltage supplied by a rectifier or a DCbattery into AC voltage.

fig. 4: transformer with ZIGZAG connected

secondary and attenuation of harmonics of

order 3 k.

fig. 3: winding connections of three-phase

transformers which have a homopolar

impedance Zh equal to a direct impedance

Zd.

primary

A

B

C

a

b

c

n

secondary

a

b

c

n

A

B

C

N

IIN

N(I - I )

1

2

N1

N1

N1

N2

N2

N2

N2

N2

N2

1 3 3 1

I2

I3




In a single phase unit, there are

two ways of achieving this conversion:

half-bridge converter (see fig. 5a),

full-bridge converter (see fig. 5b).

The square wave voltage appearing

between A and B is then filtered so asto obtain in the output of the unit a

sinusoidal voltage wave with a low

distortion rate.

In practice, the switching device

(mutator) produces several positive

and negative pulses (see fig. 6) which

makes it possible to reduce the size of

the filter and to have a faster acting

voltage regulator.

By modulating the relative time

intervals corresponding to conductionand non-conduction periods, it is

possible to «spread» the voltage duringthe period in such a way as to make

the conduction time of the switching

device practically proportional to the

instantaneous value of the

fundamental.

This principle is called PWM (Pulse

Width Modulation) - (MLI in french).

The filter inserted behind the switching

device (mutator) is generally of the L

and C type (see fig. 7).

The inverter is therefore a voltagesource with the filter impedance in

series.

Voltage V is the voltage measured at

no load, and the impedance consisting

of L and C elements in parallel is the

impedance measured when terminals

A and B are short-circuited (obtained

by applying Thevenin theorem;

see fig. 8).

Classic inverters

When the commutation frequency is

low, regulation can:

cope with variations of current drawnby user equipment,

compensate for voltage variation of

DC battery (or rectifier),

have, however, difficulties in dealing

permanently with variations of current

due to harmonics generated during half

cycle.

In these inverters, the output

impedance is equal to the impedance

of their filter. They can, therefore, be

described as classic inverters since

fig. 8: equivalent circuit diagram of an

inverter seen from its output.

fig. 6: output voltage of switching unit

(mutator) with 5 pulses per half period.

fig. 5a: principle of switching unit (mutator)

half-bridge converter.fig. 5b: principle of mutator full-bridge

converter.

V

closed

open

open

closed

T/2 T

t

+ E

V

T/2 T

t

+ E

V

t

+ E

- E

0

+ E

II

B A II

1

2

3

4

B

I3I3

I4 I4

AB

A

openI1 closedI2

closedI1

openI2

T/2 T

t

+ E

t

+ E/2

- E/2

loadB A

+ E

+ E/2

0

VA

VAB

openI2

closedI1

closedI2

openI1

I1

I2

V

t

fundamental

AB

L

CV

fig. 7: output filter of an inverter.

A

B

mutator Cuser

(receiver)

L




In general, it is then useful to show thewhole output circuit part(switching unit + filter) in the form of aseries impedance Z1 together with a

parallel impedance Z2 (see fig. 11).

By applying Thevenin theorem, it ispossible to transform the circuitdiagram into that shown in figure n° 12.

V’m = voltage measured at no loadthus:

V'm = Vm . Z 2

Z 1 +Z 2

operationally they function in the same

way as the early design units (due to

the limited capacity of semi-conductors

to operate at high frequencies).

The output impedance of these

inverters is therefore frequency

dependent and can be represented bythe diagram used in figure 9.

thus at low frequencies the

impedance of the filter is nearly equalto Lω. at high frequencies the filter

impedance differs little from1

Cω .

at resonant frequency

Fo = 1

2 π

LC

the impedance of the filter assumes a

high value that can attain, in terms of

magnitude, the value of the nominal

load impedance of the equipment

(Zs = 100 % Zc).

In practice, frequency Fo is therefore

chosen so as not to correspond to the

possible frequency of current

harmonic, i.e. 210 Hz (harmonic

currents of order 4 are non-existent or

are of very small amplitude).

This being the case, various ingenious

ways have been devised byconstructors in an effort to reduce the

output impedance.

For example:

additional filters,

special connection circuits for the

transformer inserted behind the

three-phase switching device

(mutator).

At first sight, classic inverters have a

behaviour towards harmonic currents

comparable to that of well designed

alternators and therefore less

satisfactory than that of transformers.

Inverters with PWM and appropriateregulation

When the switching frequency of theswitching unit (mutator) is high (atleast several kHz) and the regulationsystem allows rapid interventionthrough the modification to pulsewidths during the same period, it isnaturally possible to maintain theinverter output voltage within itsdistortion limits even when dealing withhighly distorted currents.

The block diagram of such inverter,shown in figure 10 is as follows:

The output voltage Vs is constantlycompared with a reference voltageUref which is sinusoidal and has a verylow distortion rate (< 1 %).

The voltage difference ε is thenprocessed by a correction circuit oftransfer function C (p) whose aim it isto ensure the performances and thestability of control circuit systems.The resulting voltage issued from thiscorrection circuit is then amplified bythe switching unit (mutator) itself andits ancillary control circuit with anamplification gain A.

The voltage Vm supplied by the

switching unit is shaped by the filterconsisting of L and C elements beforebecoming the output voltage Vs.

In practice, one should take intoaccount: the impedance of the transformer, ifpresent in the circuit, in order to obtainthe total value of inductance (often theinductance is integrated within thetransformer. That is why it does notappear in circuit diagrams), the output impedance of theswitching unit which according todesigns, is not necessarily negligible.

fig. 12: transformed equivalent circuit

diagram of switching unit seen from output.

fig. 11: equivalent circuit diagram of a

switching unit seen from output.

fig. 10: block diagram of a PWM inverter.

Fo F

Lω

Cω

1

Zs

ZcC

ε Vm L Vs I

-

UrefC (p) A

+

V'm Vs

Zs

Vm Z Vs

Z 1

2

fig. 9: variation of output impedance in a

classic inverter with frequency.




Everything happens as if the outputimpedance of the inverter were dividedby 1 + µ (p) . H (p).To throw further light on thisimpedance, it is convenient to carry outadditional calculations.

In the band-pass of regulation, theproduct µ (p) . H (p) being ≥ 1,calculations are as follows:

1 + µ (p) . H (p) ≈ µ (p) . H (p)

Z's ≈ Zs

µ (p) . H (p)

since

Zs = Z 1 . Z 2

Z 1 + Z 2

and

H (p) =Z

2Z 1 + Z 2

thus

Z's ≈ Z 1 . Z 2

Z 1 + Z 2

. 1

µ (p) . Z 1 + Z 2

Z 2

thus

Z's ≈ Z 1

µ (p)

This means that in the band-pass ofregulation, the output impedance ofinverter is equal to the seriesimpedance of the filter for the wholeoutput circuit divided by theamplification gain of the action chain.

Beyond the band-pass of regulation,the output impedance becomes againthe impedance of filter which by thenbecomes the impedance of a capacitoroffering a low impedance at highfrequencies. Hence the shape of thecurve of output impedance in functionof frequency (see fig. 14).

fig. 13: transformed block diagram of a PWM inverter.

fig. 14: comparison of output impedances

between classic inverter and PWM inverter

in function of frequency.

With PWM inverters, the output impe-dance remains very low up to highfrequencies and the output voltage dis-tortion due to circulating currents, evenhighly distorted currents, is negligible.

Limitation of current

The semi-conductors utilized inswitching units can deliver a maximumcurrent, above which their performancecan no longer be guaranteed. It istherefore advisable to limit the currentto this maximum value in order toensure reliability of performance.

As soon as the current drawn by theload exceeds the maximum value setfor the inverter, the latter becomes agenerator of constant current until the

current value required by the loaddrops below the fixed threshold limit.

Under these conditions, the outputvoltage does not follow the shape of thereference voltage and remains distortedas long as the load current exceeds thethreshold limit.

This voltage distortion is all the moresignificant, the longer the durationstage above the threshold limit.

Zc

ε V Zs Vs IVref

µ H (p)(p)

1

Zs

classic inverter

PWM inverterF

Zs = measured in output by short-circuiting V'm, thus:

Zs = Z 1 . Z 2

Z 1 + Z 2

Ratio Z 2

Z 1 + Z 2

is the transfer function

of the filter, say H (p):

thus H (p) = Z 2

Z 1 + Z 2

To simplify still further, it is convenientto replace the product C (p) x A byµ (p) which represents the transferfunction of action chain.The block diagram becomes thatshown in figure 13, where Zs = output

impedance in the absence of regulationas is the case in classic inverters.

When a current is drawn by the load, avoltage drop appears at the terminals ofthe output impedance Zs, such that:V1 - Vs = ZsI

Developing still further:

V1 = ε . µ (p) . H (p)

ε = Vref - Vs

V1 = (Vref - Vs) . µ (p) . H (p)

V1 = Vs + ZsI

Vs + ZsI = (Vref - Vs) . µ (p) . H (p)

thus:

Vs 1 + µ (p) . H (p) =

V ref µ (p) . H (p) - ZsI

thus:

Vs = V ref.µ (p) . H (p)

1 +µ (p) . H (p)

- ZsI

1 +µ (p) . H (p)

The first term represents the resultobtained for a conventional controlsystem with no disturbance present.

Here, the disturbance is introduced bymeans of current I circulating in theinternal impedance Zs.In the absence of regulation, the termdenoting the disturbance would haveassumed a value of ZsI.With regulation, this disturbance islimited to:

ZsI

1 + µ (p) . H (p)




Influence of line impedance on

voltage distortion

Since the line impedance is additionalto the source impedance, it has the

effect of increasing the distortion rate ofthe voltage in installations drawingdistorted currents.

Figure 16 shows an example where anuser installation U2 draws a highlydistorted current. When this occurs, thedistortion rate measured at its inputterminals is D2; however, because ofthe impedance divider consisting of Zsand ZL2, a distortion rate D is measuredat the output terminals of the source Dbeing smaller than D2.

Consequently, to minimise the influence

of receiver installations which generateharmonic currents in other «receivers»,it is recommended to supply the receiverinstallations through a special line.Readers interested in further details canrefer to appendix 1.

Such cases are met essentially whensingle-phase loads consisting of acapacitor in front of a rectifier giving ahigh crest factor. The latter is usually of

the order of 3 (crest value ≈ 3 times theeffective value of current) whereas for a

pure sine wave it is only 2 .

The performance of the PWM inverterfor this type of load is examined inchapter 4.

impedance of lineThere is always a length of cable ofgreater or lesser importance betweenthe voltage source and each userinstallation.

Value of line impedanceThe line impedance consists essentiallyof an inductance L in series with aresistance R (see fig. 15). The value ofthe inductance is hardly dependent onthe section of conductors and is usuallyassumed to be 0.1 Ω /km (at 50 Hz)which is roughly equivalent to0.3 µH/m.The value of the resistance isdependent on the section of the cableand is taken as r = 20 Ω /km for 1 mm2

section.

For example, a cable of 16 mm2 section

exhibits a resistance of 1.25 Ω /km anda reactance of only 0.1 Ω /km.

As a first approximation, it will bepossible to represent a cable by itsresistance only in the case of small andmedium size power rating installationswhere the use of small sectionconductors is quite common.

Note: for harmonic frequencies, it mightbe necessary to take into account theskin effect.

In this respect, one must remember thatin a copper conductor, the equivalentconduction thickness, referred to asskin thickness, is given by the formula:

a (mm) = 66

F (HZ)

Thus, at 50 Hz the skin thickness is9.3 mm, whereas at 1 kHz it is reducedto 2.1mm.

The skin effect must therefore be takeninto account for large sectionconductors which generally carryharmonic currents of high order.

fig. 17: output impedance of different sources in function of frequency.

fig. 16: supply to polluter receiver (U 2 ) by

means of special line.

fig. 15: equivalent circuit diagram of line.

In conclusionFigure 17 below shows the variationof output impedances of varioussources of same power rating with

frequency.

e

Zs

source

DZL

ZL D

U

U

1

2 2

2

1

L R

0

50

100

150

50 250 500 750

ratio of output impedance

to nominal load impedance

classicinverter

PWM inverter

F (Hz)

Zs

Zc

%

alternator X"d = 12 %

transformer Uccx = 4 %




It is clearly apparent that the PWMinverter exhibits by far the lowestoutput impedance. To better clarify thispoint, figure 18 shows three sources,

each with the same impedance at150 Hz.It is thus obvious that the impedance ofa classic transformer as well as theimpedance of the supply line, mustboth be taken into account whendistorted currents are to be supplied toa load.

The PWM Inverter is by far the bestgenerator on the market as regards itsability to minimise the voltage harmonicdistortion. It is 5 to 6 times better thana transformer of the same rating.

Figure 20 shows the voltages andcurrents obtained with a relatively lowsource impedance consisting of aninductance and a resistance such thattheir short-circuit voltages referred to

the load power are respectivelyUccx = 2 % and UccR = 2 %.

It must be pointed out the distortionrate of the voltage v in the rectifierinput is already important as it reaches7.5 % even despite a low sourceimpedance.The current i starts flowing as soon asthe voltage e becomes higher than Ubut its rate of rise is limited by thesource inductance.

This inductance extends the time ofcurrent circulation when voltage e

becomes again smaller than v.It is therefore essentially the value ofthe source inductance whichdetermines the shape of current i.

It is apparent that the current is highlydistorted compared with a perfect sinewave and, in addition, slightly out ofphase with respect to the sourcevoltage. In this example, the powerfactor is equal to 0.72.

descriptionThese single-phase loads, as manyother types of electronic equipment, aresupplied by means of switched-modepower supplies.

Thus, a load of RCD type (Resistances,Capacitors, Diodes) has been retainedin Standard Specification NF C 42-810to characterise inverters of rating below3 kVA.

A load of RCD type consists of a Graetzfull-bridge converter and preceded by acapacitor. The latter acts as an energystorage reservoir in order to supplycurrent to the load between twosuccessive peaks of the rectifiedvoltage.

The supply source is represented by a

voltage e and an output impedanceZs.In the examples cited in this chapter,the time constant of discharge of thecapacitor through the resistance is fixedat 125 ms (see fig. 19).

Current i starts flowing when voltage eexceeds the DC voltage U andcirculates for a relatively short time torecharge the capacitor to its nominalvoltage.

influence of sourceimpedanceIn the previous example it is shown thatthe load cannot be considered as a

generator of harmonic current, but onthe contrary, that the current is highlydependent on the source impedance.

Figure 21 shows the variation of currenti and voltage v in the rectifier inputwhen the source impedance changesfrom Uccx = 0.25 % to Uccx = 8 %

4. micro and mini-computer loads

fig. 19: basic circuit diagram of micro and

mini-computer type load.

e

Zs i

v C R U

electric cable transformer PWM inverter

=

S = 12 kVAS = 60 kVA

S = 6 mm2

I = 30 m

fig. 18: sources exhibiting same impedance at 150 Hz.

~




Uccx crest power current spectrum global distortion

factor factor rate of voltage

% H3 H5 H7 H9 H11 H13

0.25 2.7 0.64 87 64 38 15 1 7 2.8

0.5 2.63 0.65 85 60 33 11 4 7 3.5

1 2.51 0.68 81 52 24 6 7 6 5.4

2 2.35 0.72 76 42 14 7 6 3 7.5

4 2.19 0.75 69 29 8 8 4 4 11.2

6 2.1 0.77 63 21 8 6 3 3 14.2

8 2 0.78 59 17 8 5 3 2 16.8

whilst the resistive part has beenarbitrarily fixed at UccR = 2 % .

Table in figure 22 brings to light, forthese different impedances, thevariation of the different characteristicparameters relating to current andvoltage; when the source impedanceincreases, the power factor improveswhereas the distortion rate (see page 4)of the voltage in the input of userinstallation increases.

It is the value of the distortion ratewhich determines the choice of thesource. A distortion rate of 5 % is oftenthe limiting value admissible for receiverinstalla-tions that can be either pollutersor polluted.

Curves in figure 23 page 14 show thevariation or the global distortion rate ofvoltage in the input of the rectifier infunction of two parameters: when the short-circuit voltage of thesource varies from 0 to 8 %, for 3 values of resistive short-circuitvoltage (UccR = 0, UccR = 2 % andUccR = 4 %).

They also show that, in practice, it is theinductive short-circuit voltage thatdetermines the voltage distortion rateexcept when this short-circuit voltage islower than 1 %.

calculation of source powerfor supplying RCD typeloadsKnowing the active power absorbed bythe rectifier (Pr), it is essential tochoose correctly the power source (Ps)that must supply it.

fig. 22: variation of principle characteristic parameters of current and voltage for a computer type load supplied from a source of impedance Ucc R

constant and equal to 2 % for values of Uccx varying from 0.25 % to 8 %.

Hn % = 100 I N

I 1

λ = P

S

v

u

i

e

t

fig. 20: currents and voltages of a computer type load of 1 kW with source such that:

Uccx = 2 % and Ucc R = 2 %.

fig. 21: variation of current and voltage at the computer type load input when the short-circuit

voltage Uccx changes from 0.25 % to 8 % while the short-circuit voltage Ucc R remains

constant and equal to 2 %.

I crest

I rms

0,25 %0,5 %

1 %2 %4 %8 %

Uccx

t




order of 1 % (in accordance withfigure 23), for a distortion rate of 10 %, ashort-circuit voltage of the order of 3 %

must be retained.

For a transformer

if Uccx = 4 %

for D = 5 % a power ratio of:

Ps

Pr = 4 %

1 % = 4 is sufficient,

for D = 10 % the power ratio wouldbe:

Ps

Pr = 4 %

3 % = 1.33

but in this case, a value at least equal

to 1.43 would be required by the powerfactor.

if Uccx = 6 %

for D = 5 % a power ratio of:

Ps

Pr = 6 %

1 % = 6 is necessary ,

for D = 10 %, a power ratio of 2 isrequired.

Note: for a transformer, it is often

necessary to take a much higher powerratio considering that distortions can

already be present in the network.

A distortion rate of 3 % due solely to

the working of the rectifiers, leads oneto retain a inductive short-circuit

voltage of 0.45 % (in accordance withfigure 23) which amounts to multiplying

by 2.2 the power ratings of

transformers to obtain a distortion rate

of 5 %.

For an alternator

As distortion rates of 5 % and 10 %

lead to inductive short-circuit voltagesof 1 % and 3 % respectively, powerratios of alternator to rectifier are

therefore equal to respectively:

Uccx

1 % and Uccx

3 % .

If Uccx = 18 %, it will be necessary:

for D = 5 %

to have a power ratio of:

Ps

Pr = 18 .

for D = 10 %

to have a power ratio of:

Ps

Pr = 18 %

3 % = 6.

For an inverter

classic inverter

As it was explained in our discussion

on source impedances, this type of

inverter of single phase mode exhibits

an impedance comparable to that of an

alternator of good design (with Uccx of

the order of 12 %).

As generally the output distortion of an

inverter must be limited at 5 %, then it

is desirable to retain a power ratio ofthe order of 12.

Inverters of the classic type are

available today mostly in three phaseversion. In these, always assuming a

distortion rate of 5 %, the power ratio is

7 when operated with a transformer

whose secondary is connected in

ZIGZAG.

PWM inverter with appropriate

regulation

(reminder: its impedance is at least

five times lower than that of a

transformer for which the power rating

must be multiplied by 4).

In our development hereafter, theimpedance of the supply line isneglected (or integrated into Ucc ofsource).

A first indication is provided by thepower factor:

λ =P

S

This power factor is dependent on thetotal short-circuit voltage upstream ofrectifier but can be given a meanvalue of the order of 0.7.

Having established this first criterion,the power of the source musttherefore be at least equal to theactive power absorbed by the rectifiermultiplied by

1

0.7 or 1.43.

The second criterion is related to adistortion rate that would beacceptable: if a distortion rate of 5 % isenvisaged, it is possible to retain aninductive short-circuit voltage of the

fig. 23: variation of voltage distortion rate at input of microprocessor type load with respect to

Uccx and several values Ucc R of the source.

0 1 2 3 4 5 6 7 8

5

10

15

Uccx %

U = 0

U = 2 %U = 4 %

voltage distortion

rate as % CCR

CCR

CCR




As long as the current drawn by theload exhibits a crest value lower thanthe limiting threshold value for theequipment, the distortion rate remains

very low and inferior to 5 %. As soon asthe threshold limit is reached, thevoltage supplied by the inverterbecomes distorted (sine wave becomesaffected by crest flattening) and thevoltage distortion rate increases.

Experience shows that, in order toavoid a voltage distortion exceeding5 %, it is necessary to set the thresholdlimit for current at 1.5 times the crestvalue of the nominal effective current ofthe inverter,

thus I limit = 1.5 2 I rms.

The corresponding crest factor ofcurrent is then equal to

1.5 2 that is 2.12.

Figure 24 shows the variation(evolution) of voltage and current in a5.2 kVA inverter with a threshold limitset at:

5,000

220 . 1.5 . 2 = 48 A .

A voltage distortion rate of 5 % isreached here for an apparent power of5.2 kVA, that is slightly greater than5 kVA which is its design parameter forrating.

The power factor of the RCD load is, inthis case very close to 0.8 (0.79) andconsequently the inverter does notneed to be over-dimensioned in orderto supply this type of load (except whenthe distance between inverter andloads is relatively significant, this beinghowever, true for all sources).

In the example shown in figure 24, a5 kVA inverter is capable of supplying a4 kW rectifier with a distortion rate

inferior to 5 %.

Thus P inverter = PR

0.8 = 1.25 PR

It is worth noting that the fact of limitingthe current improves the power factor.

In the preceding paragraph dealing withtransformer, it was noticed that, with apower factor of rectifier amounting to0.7, it was necessary, even in theabsence of constraints on the distortionrate, to choose a transformer whosepower rating was at least equal to1.43 PR.

The PWM inverter appears therefore tobe the ideal source of voltage forsupplying not only loads of RCD typebut also all receiver equipment which

are generators of harmonic currents(non-linear loads).

In the preceding section, we havediscussed the case of inverters and

single phase loads; the samereasoning can indeed be applied tothree phase equipment providing theequipment is fitted with independent

regulation in each phase (this isgenerally the case with this type ofequipment).

fig. 24: variation of output voltage of 5 kVA inverter with threshold limit set at 48 A.

Urms : 220 V

Irms : 20 A

power factor : 0.69

crest factor : 2.4

distortion rate : 3 %

apparent power : 4.4 kVA

active power : 3 kW

48A

48A

48A

Urms : 220 V

Irms : 11 A

power factor : 0.61

crest factor : 3.6

distortion rate : 2.7 %


active power : 1.5 kW

Urms : 220 V

Irms : 24 A

power factor : 0.79

crest factor : 2




Urms : 220 VIrms : 29 A

power factor : 0.82

crest factor : 1.64







first assuming that Z2 = 0 (load veryclose to transformer).

Curves in figure 23 will giveD = 4.6 % = D2.

it is necessary now to calculate D etD2 with a line 100 m/10 mm2

(i.e. 100 m long and a section 10 mm2):

thus short-circuit impedances of theline referred to PR:

U'2ccx = l ω .

PR

Un 2 . 100

U'2ccR = R . PR

Un 2 . 10

thus with:

lω = 0.1 . 100

1,000 = 10 m Ω

r = 20 . 100

1,000 . 1

10 = 0.2 Ω

characteristics of transformer:50 kVA (with Uccx = 4 % etUccR = 2 %).

It is necessary to calculate the

impedances of the inductive short-

circuit and resistive short-circuit of the

transformer but referred to the activepower of micro-computers, thus:

U'1ccx = U1ccx . PR

Ps

U'1ccR = U1ccR . PR

Ps

thus

U'1ccx = 4 % . 10

50 = 0.8 %

U'1ccR = 2 % . 10

50 = 0.4 %

5. conclusion

them to supply a low distortion voltageto receivers that are generators ofharmonic currents (non-linear loads).

market for supplying electronic andmicro-processor loads. The high speedresponse of their regulation systemsendows them with a very low«harmonic impedance»; thus enables

Static inverters equipped with PWM arenearly perfect sources of voltage.Besides their qualities as regardsvoltage stability and frequency stability,they are the best generators on the

appendix 1: influence of line impedances on voltage distortions

The end of paragraph 3 stresses thefact that it is desirable to supply«receivers» that are generators ofharmonic currents by means of speciallines.

This is true for loads of RCD type, butalso for all «receivers» utilising powerelectronics such as rectifiers, batterychargers, speed controllers etc.

The use of a special line provides

isolation of harmonics throughimpedance (see fig. 25).

For a «clean» receiver

The distortion rate D1 is practicallyidentical to D, and this is all the moretrue as the impedance of line Z1 issmall compared with that ofreceiver Zp.

For a non-linear receiver

D2 will be all the more lower as thesum Z2 + Zs will remain low, in otherwords as the non-linear «receiver» willhave a low power rating in relation to

its supply.The following example shows moreclearly the influence of Z2 on D and D2.

Let’s consider a set of micro-computerabsorbing 10 kW at 230 V that is beingsupplied by a cable conductor 100 mlong connected to a transformer.

characteristics of cable:

section: 10 mm2,

Lω = 0.1 Ω /km at 50 Hz,

r = 20 Ω /km for a 1 mm2 section.fig. 25: power supply through a specific line a receiver generator of harmonic currents.

e

Zs

D

D

Zp

Ls Rs

U1ccx U1

Z

Z

I r

Dclean

receiver

non-linear

receiver

1

2

1

2

CCR




fig. 26a: voltage distortion rates due to Uccx for various values of Ucc R .

U'2ccx = 10 . 10 -3 . 10 4

(230) 2 . 100

= 0.19 %

U'2ccR = 0.2 . 10 4

(230) 2 . 100 = 3.8 %

total short-circuit impedances:

U'ccx = 0.8 % + 0.19 % = 0.99 %

U'ccR = 0.4 % + 3.8 % = 4.2 %

thus

U'ccx = U'1ccx + U'2ccx

U'ccR = U'1ccR + U'2ccR

the voltage distortion rates D’L and

D’R related to «impedances» ofinductive short-circuits and resistive

short-circuits.

These values are obtained from curves

in figure 26a and figure 26b and are

respectively

D’L = 3.9 %,

D’R = 3.9 %.

distortion rate at input of personal

computers:

D 2 = (3.9 ) 2 + (3.9 %) 2 = 5.52 %.

voltage distortion rates DL and DR atthe source:

D L = D' L . U'1ccx

U'ccx

D R = D' R . U'1ccR

U'1ccR

thus:

D L = 3.9 % . 0.8

0.99 = 3.15 %

D R = 3.9 % . 0.4

4.2 = 0.37 %.

voltage distortion rate D at the source

D = D L 2 + D R

2

D = (3.15 %) 2 + (0.37 %) 2 = 3.17 %.

in this example, the supply line

causes D and D2 to change as follows

D from 4.6 % to 3.17 %,

and D2 from 4.6 % to 5.52 %.fig. 26b: distortion rates due to Ucc R for various values of Uccx.

0 1 2 3 4

5

10

Uccx %

distortion rate

due to UccxU = 0CCR

U = 2 %CCR

U = 4 %CCR

0 1 2 3 4

1

2

3

%

Uccx = 0

Uccx = 1 %

Uccx = 2 %

Uccx = 3 %

Uccx = 4 %

distortion rate

due to UCCR

UCCR

%)2




fig. 28: basic circuit diagram of an antiparasitic interference filter.

fig. 27: basic circuit diagram switched mode

power supply to RCD load.

appendix 2: input filters in computer/micro-processor equipment

Their purpose is to stop thepropagation of disturbances caused byswitched mode power supplies towardsother equipment installations that couldbe adversely affected.Conversely, they help attenuate somedisturbances present in the networkwhich are likely to alter the functioningof electronic and data informationequipment.

The question is to know if these filtersattenuate harmonic currents generated

by RCD loads.

Interference rejection in network

Switched mode power supplies operateat high frequencies in an effort toreduce the size and weight oftransformers.In figure 27, the load resistance R is thebasic circuit shown in figure 19 isreplaced by a transformer and its load.In this circuit, the line current remainsidentical because of the presence ofcapacitor C.

To achieve silent operation, the

switching frequency is always high andin any case in excess of 20 kHz.

The commutation times of a transistor(change from non-conducting toconducting state and vice versa) arevery brief and do not, in some cases,exceed a few tens of nano seconds.

These high frequency commutations(switching) do generate HF interferencethat is propagated by conduction andradiation. This gives rise to thepresence of parasitic interference alongthe line upstream of the switchingdevice, that is in the mains (on this

subject, it is recommended to refer to«Cahier Technique» n° 149«Electromagnetic Compatibility»).

In order to limit the circulation of theseHF currents, constructors of datainformation processing equipmentinsert filters upstream of the switchedmode power supply unit; a typicalcircuit of such filters is shown infigure 28.

These filters reduce disturbances: of common mode which affect in thesame way both conductors with respectto earth,

of differential mode which are presentbetween the two conductors.

Inductance L offers a high impedanceto currents of common mode butpractically none to those of differentialmode as its windings are wound inopposition.

Disturbances of common mode areconducted to earth by capacitors C1

and blocked by inductance L.

Disturbances of differential mode are

attenuated by capacitors CA and CRwhich, at high frequency, offer a lowimpedance between the conductors.

Protection of switched mode powersupply

The filter inserted between the ACmains and the RCD supply ensures asecond function: it protects the RCDsupply from impulse type over-voltagesand from HF interference of differentialand common mode which are presentin the mains.

Leakages to earth

The presence of capacitors C1 causesa leakage current at 50 Hz to f low toearth.

Design standards generally specifyvalues of leakage current not to beexceeded (a few milliamperes forequipment connected to a mains point).For example, standard specificationIEC 950 relating to data processingequipment recommends that theseleakage currents should be kept below3.5 mA for equipment connected to amains point.

In fact, currents of the order of 1 to2 mA have been measured by UTE.If a line supplies several electronic anddata processing equipment, the sum ofthe leakage currents can trip the highlysensitive differential residual currentdevice (30 mA) inserted in the line.

Filtering of harmonics

The filters inserted between the mainsand the RCD supply operate efficientlyin the frequency band-pass ranging

from 10 kHz to 100 MHz.Unfortunately, they are of no useagainst harmonic currents injected intothe mains network.This is due to the fact that harmoniccurrents produced by RCD supplies areof relatively low frequency: 1 kHzcorresponds in fact to a harmonic oforder 20 in relation to a fundamental at50 Hz!

C

+ 12 V

0 V

mains

supplyRCD supplyCC

C

C

L

Imc Imd

Imc Imd

R A

1

1, ,@ @ , ,@ @ , ,@ @ , ,@ @ , ,@ @ , ,@ @




appendix 3: bibliography

Normes

IEC 146-1-1Semi-conductor converters. Generalrequirements and line commutatedconvectors - part 1-1: specificationsbasic requirements.

IEC 950Safety of information technologyequipment including electrical businessequipment.(NF C 77-210, modification 1incorporated).

NF C 42-810Alimentations sans interruption, depuissance nominale inférieure à 3 kVA.

Cahiers Techniques Merlin Gerin

«Analyse des réseaux triphasés enrégime perturbé à l'aide descomposantes symétriques»Cahier Technique n° 18- by B. De METZ-NOBLAT.

«EMC: electromagnetic compatibility»Cahier Technique n° 149- by F. VAILLANT.



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