estudio de corrientes en neutros

8/12/2019 Estudio de Corrientes en Neutros

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Analysis of the neutral conductor current in a three phase supplied network with

non-linear single phase loads

J. Desmet(*), I. Sweertvaegher(*), G. Vanalme(*), K. Stockman(*), R. Belmans(**)

(*)

Hogeschool West-Vlaanderen, dept. P.I.H., Graaf Karel de Goedelaan 5, B-8500 Kortrijk, Belgium(**)K.U.Leuven, dept. ESAT/ELEN, Kardinaal Mercierlaan 94, B-3001 Leuven, Belgium

Abstract-This paper describes what factors (i.e. load and

supply) have an important effect on the neutral conductor

current. It is shown that an asymmetry up to 10 or an

unbalance of 10% in the power supply has only a minor effect

on the rms-value of the neutral conductor current. An

unbalance in load conditions increases the neutral conductor

current. Harmonics in the power supply voltage highly affects

the rms-value of the neutral conductor current.

I. INTRODUCTION

Nowadays non linear loads (compact fluorescent lamps,

computers, variable speed drives,), mostly used with the

aim of rational energy use, are very common. These loads,

producing harmonic currents, yield high neutral conductor

currents [1,2]. In this paper the influence of power supply

asymmetry and unbalance and load unbalance on the neutralconductor current is investigated. Also the sensitivity of the

neutral conductor current to harmonics in the power supply

voltage is studied. In order to have a better insight into the

experimental results, some theoretical considerations are

supplied first.

II. THEORETICAL CONSIDERATIONS

A. Assumptions

A three phase supplied network with neutral conductor is

considered. The load phase currents are assumed to be steady

state periodic signals only containing odd harmonics.

B. Derivation of the harmonics in the neutral conductor

current from the phase currents

Symmetric and balanced network

Using the Fourier transform, the phase currents in a

symmetric and balanced network can be written. The neutralconductor current is given by the summation of the three

phase currents. The same reference is used for the phase

angles in these equations.

( ) ( ) ( ) ( ) ...5sin3sinsin 553311 ++++++= tItItItIU (1)

( ) ...3

25sin

3

23sin

3

2sin 553311 +

+

+

+

+

+=

tItItItIV (2)

( ) ...3

45sin

3

43sin

3

4sin 553311 +

+

+

+

+

+=

tItItItIW (3)

( ) ( ) ...03sin*30 33 ++++= tItIN (4)

Notice that the first order harmonics (i = 6k+1, with i theorder of the harmonic and k = 0,1,2...) in the phase currents

are forming a direct system, the third order harmonics (i =

6k+3) are forming a homopolar system and the fifth order

harmonics (i = 6k+5) an inverse system. Consequently, the

neutral conductor current only consists of third order

harmonics.

Arbitrary networkUsing the Fortescue transform [3], an arbitrary

(asymmetric and unbalanced) system can be written as the

summation of a direct, an inverse and a homopolar system.

In (5) the Fortescue transform is applied to the harmonics of

order i in the phase currents.

=

iinv

id

ih

iW

iV

iU

I

I

I

aa

aa

I

I

I

,

,

,

2

2

,

,

,

1

1

111

(5.a)

=

iW

iV

iU

iinv

id

ih

I

I

I

aa

aa

I

I

I

,

,

,

2

2

,

,

,

1

1

111

3

1(5.b)

with

=

3

2exp

ja

As the sum of the direct components and also the sum of

the inverse components equals zero (1+a+a2=0), only the sum

of the homopolar components results in a neutral conductor

current.

( ) ( ) ihihiinvidiN IIIaaIaaI ,,,2,2, 3311 =++++++= (6)

The neutral conductor current only consists of the

homopolar components of the phase currents. In case of a

symmetric and balanced network, these homopolar

components correspond to the third order harmonics.

Working out (6), Kirchoffs law yields:

( ) iWiViUiWiViUihiN IIIIIIII ,,,,,,,,3

1*33 ++=++== (7)

Assuming iUiUiUj

eII ,,,= , iViViV

jeII ,,,= , iWiWiW

jeII ,,,= , then

iNI , is given by:

( )( )iWiWiViViUiU

iWiWiViViUiUiN

IIIj

IIII

,,,,,,

,,,,,,,

sinsinsin

coscoscos

+++

++=(8)

From the above equation, the amplitude IN,iand the phase

angle N,iof the ithharmonic in the neutral conductor current


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can be calculated. The amplitude IN,iof the ith

harmonic inthe neutral conductor current is:

( ) ( )2,,,,,,2

,,,,,,, sinsinsincoscoscos iWiWiViViUiUiWiWiViViUiUiN IIIIIII +++++=

(9)

with: IN,i: amplitude of the ith

harmonic in the neutral

conductor currentIU,i, IV,i, IW,i: amplitude of the i

thcurrent harmonic in

respectively phase U, V and W

U,i, V,i, W,i: phase angle of the ith

currentharmonic in respectively phase U, V and W

The phase angle N,i of the ith

harmonic in the neutralconductor current is:

( )( )

=

iN

iNiN

I

Iarctg

,

,,

Re

Im (10)

If the harmonics (amplitudes and phase angles) in the

phase currents are known, the harmonic content of the neutral

conductor current can be calculated using (9) and (10).

C. Rms-ratio of neutral conductor and phase currents for a

symmetric and balanced network

For a symmetric and balanced network, the rms-ratio of

neutral conductor and phase current increases with increasing

third order harmonics and with decreasing first and fifth order

harmonics in the phase current (11). The neutral conductorcurrent never can be more than 3 times the phase current.

The maximum ratio is hypothetically possible if the third

order harmonics in the phase current are infinite in

comparison to the part of first and fifth order harmonics in

the phase current (e.g. with a third order load).

( )

( ) ( ) ( )

+++

+

++=

256

236

216

2363

kkk

k

phase

N

III

I

I

I

(11)

with: IN: rms-value of the total neutral conductor currentIphase: rms-value of the total phase currentI6k+1, I6k+3, I6k+5: rms-value of a first order, resp. third

and fifth order harmonic in the phase current with

order 6k+1, resp. 6k+3 and 6k+5 (k=0,1,2,...)

Consider the particular case in which the phase currents are

consisting of odd harmonics I2n+1with I2n+1= qn*I1(0 q 1,

n = 1,2,...) or I3= q*I1, I5= q*I1, I7= q3*I1, I9= q

4*I1,...

The rms-value of the phase current is:1

21

642

1

1*...1 I

q

IqqqIphase

=++++= (12)

The rms-value of the neutral conductor current equals:

16

11482

1

*3*...*3 I

q

qIqqqIN

=+++= (13)

The rms-ratio of the neutral conductor current and the phase

current is:

( )( ) 424222

6

2

1

3

11

13

1

13

qq

q

qqq

qq

q

qq

I

I

phase

N

++=

++

=

= (14)

The maximum rms-ratio of the neutral conductor current andthe phase current is obtained when q=1 (all the harmonics in

the phase current have the same weight) and equals 3 .

In [4] it is mentioned that the neutral conductor current can

reach 1.73 times the phase current.

III. EXPERIMENTS

A. Test configuration

Power SourceLOAD

A

ch1

A

ch3

A

ch2

V

ch1

V

ch2

V

ch3

U

N

W

V

Power analyzer

Fig. 1. Measurement set-up

Using a programmable power source, an arbitrary voltage

waveform is generated, independently for each phase. Eachphase is loaded by a variable number of compact fluorescent

lamps 15 W / 220-240 V. For different set-ups of the power

source and load conditions, the phase and neutral conductor

currents are measured and analysed. Measurements are done

using a high performance power analyser.

B. Neutral conductor current for a symmetric and balanced

network and sinusoidal power supply voltages

Set-up parametersPower source: A sinusoidal voltage with rms-value of

220 V is generated on each phase. The voltage signal on

phase U is taken as reference, the voltage signals on phase V

and W are leading with respectively 120 and 240.Load: Each phase is loaded by 5 compact fluorescent

lamps 15 W / 220-240 V.

Results of measurement

In Fig. 2 two graphs are given, representing the harmonic

contents of phase and neutral conductor currents. The phase

currents contain harmonics of first, third and fifth order,while the neutral conductor current mainly contains third

order harmonics. Notice that the third order harmonics in theneutral conductor current are three times as high as the

corresponding harmonics in the phase currents, as

theoretically expected (4). The small part of first and fifthorder harmonics in the neutral conductor current is caused by

the fact that the lamps are not completely identical. The load

is not perfectly symmetric and balanced. The small

unbalance in the load can be seen in the small differences

between the phase currents (Fig. 2.a).

The rms-ratio of the neutral conductor current and thephase currents is 1.7.


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phase cu r ren ts

0

50

100

150

200

250

300

350

400

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39

order ha rmon ic in phase cu r ren t

rms-valueharm

onic[mA]

phase U

phase V

phase W

(a)

neu t ra l conduc to r cu r ren t

0

10 0

20 0

30 0

40 0

50 0

60 0

70 0

80 0

90 0

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39

o rd e r h a rmo n i c i n n e u t ra l c o n d u c to r c u r re n t

rms-valueharmonic

[m

A]

(b)

Fig. 2. Phase (a) and neutral conductor (b) currents in case of a symmetric

and balanced power supply and a symmetric and balanced load (5 compact

fluorescent lamps in each phase)

C. Influence of asymmetry or unbalance in the power supply

on the neutral conductor current

Set-up parameters

Power source: A sinusoidal voltage is generated on each

phase. An overview of the used rms-values and phase angles

of the power supply voltages is given in Table I.Load: Each phase is loaded by 5 compact fluorescent

lamps 15 W / 220-240 V.

TABLE I

POWER SOURCE PARAMETERS

phase U phase V phase WCharacteristicspower supply

Vrms[V]

[]

Vrms[V]

[]

Vrms[V]

[]

Symmetric,

balanced220 0 220 120 220 240

Symmetric,

unbalanced220 0 200 120 240 240

Asymmetric,

balanced220 0 220 115 220 250

Asymmetric,

unbalanced220 0 200 115 240 250


In Table II a summary of the measured rms-values of the

total neutral conductor current INand the third harmonic IN,3in the neutral conductor current is given for the different

power source set-ups according to Table I and for a load

consisting of 5 compact fluorescent lamps in each phase. The

deviations of the rms-values for an asymmetric and/orunbalanced power supply from the values for a symmetric

and balanced supply are also mentioned in the table.

TABLE II

TOTAL CURRENT AND THIRD HARMONIC IN THE NEUTRAL CONDUCTOR

FOR DIFFERENT POWER SUPPLY PROPERTIES

Power supply properties (see Table I)

symmetric

andbalanced

(reference)

symmetric

andunbalanced

asymmetric

andbalanced

asymmetric

andunbalanced

rms-value

IN867.9 mA 860.0 mA 843.2 mA 839.1 mA

deviation

from ref.- -0.91% -2.85% -3.32%

rms-value

IN,3833.2 mA 818.9 mA 790.7 mA 797.6 mA

deviationfrom ref.

- -1.72% -5.10% -4.27%

load condition: 5 lamps in each phase

Notice that the neutral conductor current, mainly

consisting of the third harmonic, has the highest rms-valuefor a symmetric and balanced power supply. Only in this

case the phase angles of the third harmonics in the phase

currents are the same and the amplitude of the third harmonic

in the neutral conductor current is the sum of the amplitudesof the third harmonics in the phase currents (9). In the other

cases the amplitude of the third harmonic in the neutralconductor current is less than the sum of the amplitudes of

the third harmonics in the phase currents.

Fig. 3 shows two graphs representing the harmonics in the

phase and the neutral conductor currents for a symmetric and

unbalanced power supply. As a result of the unbalance in the

power supply, the harmonic contents of the phase currents aredifferent (Fig. 3.a) and the first and fifth order harmonics

show the highest differences, so the neutral conductor current

contains more first order and fifth order harmonics than in the

case of a balanced power supply (Figs 3.b and 2.b). The third

order harmonics in the neutral conductor have slightlydecreased in comparison with a balanced power supply. This

can be attributed to the differences (caused by the power

supply unbalance) in the phase angles of the third order

harmonics in the phase currents. The rms-value of a

harmonic in the neutral conductor current is not only

depending on the rms-values of the corresponding harmonics

in the phase currents, but also on their phase angles (9).


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phase cu r ren ts

0

50

100

150

200

250

300

350

400

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39

order ha rmon ic in phase cu r ren t

rms-valueharm

onic[mA]

phase U

phase V

phase W

(a)


0

100

200

300

400

500

600

700

800

900

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39

order ha rmon ic in neu t ra l conduc to r cu r ren t

rm

s-valueharm

onic[m

A]

(b)

Fig. 3. Phase (a) and neutral conductor (b) currents in case of a symmetric

and unbalanced power supply and a symmetric and balanced load (5 compactfluorescent lamps in each phase)

Fig. 4 shows clearly that an asymmetry in the power

supply increases the first order and fifth order harmonics in

the neutral conductor current and decreases the third order

harmonics. A power supply asymmetry hardly affects the

third harmonic.

I n f l uence o f asymmet ry i n the power supp l y on the neu t ra l

c o n d u c t o r c u r r e n t

0

100

200

300

400

500

600

700

800

900

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39

order ha rmon ic in the neu t ra l condu c to r cu r ren t

rms-valueharm

onic[m

A]

symm. supply

asymm. supply

Fig. 4. Neutral conductor current in case of a balanced and (a)symmetricpower supply and a symmetric and balanced load (5 compact fluorescent

lamps in each phase)

From the above results it is seen that an unbalance or

asymmetry in the power supply leads to an increase of the

first and fifth order harmonics and a decrease of the third

order harmonics in the neutral conductor current. However,

the change is the smallest for the third harmonic, while it is

the determining factor in the neutral conductor current.

Finally, it is concluded that the rms-value of the total neutralconductor current is only slightly affected by an asymmetry

or unbalance in the power supply (Table II).

D. Influence of load unbalance on the neutral conductor

current

Set-up parameters

Power source: A sinusoidal voltage is generated in eachphase, with rms-value and phase angle as in Table I.

Load: Each phase is loaded by a number of compact

fluorescent lamps 15 W / 220-240 V. Measurements were

done for the following load conditions, considering a constant

three phase power: 6 lamps in phase U, 6 lamps in phase V, 0 lamps in

phase W (unbalanced load)

6, 4 and 2 lamps in phase U, V and W respectively(unbalanced load)

4 lamps in each phase (balanced load)


Table III summarises the measured rms-values of the

neutral conductor current for different power supply (Table I)

and load conditions. Again an asymmetry or unbalance in the

power supply has only a minor effect on the rms-value of the

neutral conductor current. The load conditions, on the other

hand, have a high influence on the neutral conductor current.The neutral conductor current increases with increasing load

unbalance. Consequently, the lowest neutral conductor

current is obtained for a balanced load.

TABLE III

RMS-VALUES OF THE NEUTRAL CONDUCTOR CURRENT FOR DIFFERENTPOWER SUPPLY AND LOAD CONDITIONS

Power supply properties (see Table I)Load

conditions:

number oflamps in

each phase

symmetricand

balanced

symmetricand

unbalanced

asymmetricand

balanced

asymmetricand

unbalanced

phase U: 4phase V: 4

phase W: 4

675.9 mA(reference)

666.3 mA(-1.42%)

655.2 mA(-3.06%)

654.3 mA(-3.20%)

phase U: 6

phase V: 4phase W: 2

751.0 mA

(+11.11%)

748.9 mA

(+10.80%)

735.4 mA

(+8.80%)

751.6 mA

(+11.20%)

phase U: 6

phase V: 6phase W: 0

854.7 mA

(+26.45%)

852.5 mA

(+26.13%)

847.8 mA

(+25.43%)

856.1 mA

(+26.66%)


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0

100

200

300

400

500

600

700

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39

order ha rmon ic in the neu t ra l condu c to r cu r ren t

rm

s-valueha

rm

onic[mA]

4 lamps in each phase

phase U, V, W loade d by resp. 6, 4, 2 lamps

phase U, V, W loade d by resp. 6, 6, 0 lamps

Fig. 5. Neutral conductor current in case of a balanced and symmetric power

supply and for different load conditions

Fig. 5 shows the harmonic content of the neutral conductor

current for different load conditions in case of a symmetric

and balanced power supply. The third order harmonics are

not depending on the load conditions considering the

constraint of the constant three phase power. The first and

fifth order harmonics are nearly zero for a balanced load and

they increase with increasing load unbalance.

E. Sensitivity of the neutral conductor current to harmonics

in the power supply voltage

Set-up parameters

Power source: The power supply is symmetric and

balanced (with set-up parameters as in Table I), but the power

supply voltage contains only one odd harmonic (with order

3,5,...,21) in addition to the fundamental. The amplitude of

the voltage harmonic (relative to the fundamental) variesfrom 1% to 5%; the phase angle is 0, 90 or 180 (values

seen from the harmonic) referred to the voltage fundamental.

Load: Each phase is loaded by 5 compact fluorescentlamps 15 W / 220-240 V.

Measurement results

Fig. 6 shows the influence of a third harmonic (2%) in the

power supply voltage on the phase currents and the neutral

conductor current. In the phase currents (Fig. 6.a) the fifth

harmonic is more influenced than the third harmonic

(changes of respectively 10% and 2.5%). The change of the

fifth harmonic in the phase currents has no effect on the

neutral conductor current (Fig. 6.b). Consequently, the sameconclusions can be drawn as for the set-up of a symmetric

and balanced network considered in B, where the first and

fifth order harmonics are zero in the neutral conductor. Only

the changes of the third order harmonics are determining the

neutral conductor current.

Table IV gives, for different harmonic contents of the

power supply voltage, the deviations (%) of the rms-value of

the neutral conductor current from the reference value in case

of a sinusoidal voltage. From this table it is seen that in

general the changes of the rms-values of the neutral

conductor current are higher for voltage harmonics of higher

order and for increasing amplitude of the harmonic. The rms-value of the neutral conductor current is very sensitive to the

presence of harmonics with high order in the power supply

voltage.

TABLE IV

SENSITIVITY OF THE RMS-VALUE OF THE NEUTRAL CONDUCTOR CURRENT TO

THE HARMONICS IN THE POWER SUPPLY VOLTAGE

Deviations (%) of the rms-value of the neutral

conductor current form the reference value

(sinusoidal power supply voltage)

phase angle harmonicharmonic content

power supply

voltage0 180 90

average

(absolute

values)

3rd harm. 1% -0.77 1.21 -0.86 0.952% -2.32 2.39 -2.25 2.32

3% -4.16 3.17 -3.21 3.51

4% -6.30 4.05 -4.07 4.815% -8.49 5.02 -5.12 6.21

5th harm. 1% 0.84 -0.78 3.67 1.76

2% 1.20 -1.90 6.52 3.21

3% 1.29 -3.73 9.23 5.504% 2.43 -6.46 12.22 7.04

5% 3.78 -9.73 14.49 9.33

7th harm. 1% 0.85 -0.60 -1.76 1.07

2% 0.89 -1.29 -2.88 1.69

3% 0.79 -1.84 -3.53 2.05

4% 0.62 -2.12 -3.99 2.245% 1.27 -1.68 -3.88 2.28

9th harm. 1% 0.56 -1.53 -0.07 0.72

2% 0.91 -1.01 0.77 0.903% 1.08 -0.90 1.22 1.07

4% 2.59 2.13 2.51 2.41

5% 4.65 6.30 3.98 4.98

11th harm. 1% -2.59 2.67 0.21 1.82

2% -5.94 4.79 1.45 4.06

3% -7.01 7.27 3.90 6.06

4% -6.49 10.12 7.52 8.045% -2.90 13.22 11.79 9.30

13th harm. 1% 1.84 -1.39 0.28 1.17

2% 3.60 -1.36 0.59 1.853% 6.18 0.01 2.01 2.71

4% 10.51 2.59 4.43 5.84

5% 15.13 5.63 7.33 9.36

15th harm. 1% -1.08 -0.17 0.54 0.60

2% 2.21 1.50 2.42 2.04

3% 6.49 3.71 4.76 4.99

4% 11.11 7.27 8.60 8.995% 15.84 12.10 13.02 13.65

17th harm. 1% 1.57 0.37 -1.82 1.25

2% 2.92 2.24 -2.68 2.613% 5.39 6.15 -1.31 4.29

4% 9.17 11.31 2.21 7.57

5% 13.61 17.41 7.27 12.76

19th harm. 1% 0.39 1.47 2.19 1.35

2% 1.89 3.24 4.36 3.16

3% 5.21 6.75 8.38 6.784% 9.60 11.29 13.60 11.50

5% 15.00 15.98 19.66 16.88

21st harm. 1% 0.01 2.17 1.45 1.21

2% 1.87 4.87 4.75 3.833% 6.30 8.98 9.75 8.34

4% 12.33 13.57 15.46 13.79

5% 19.35 18.97 21.14 19.82


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I n f l uence o f a th i rd ha rmon ic i n the power supp l y vo l tage on the

phase cu r ren ts

0

50

100

150

200

250

300

350

400

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39

order ha rmon ic in the phase cu r ren t

rms-valueharm

onic[mA]

sinusoidal voltage

voltage with third harmonic, 2%, 0


(a)

I n f l uence o f a th i rd ha rmon ic i n the power supp l y vo l tage on the


0

100

200

300

400

500

600

700

800

900

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39

order ha rmon ic neu t ra l conduc to r cu r ren t

rms-valueharm

onic[m

A]

sinusoidal voltage



(b)

Fig. 6. Phase (a) and neutral conductor (b) currents for different harmonic

contents of the power supply voltage. The power supply and load aresymmetric and balanced

IV. CONCLUSIONS

It is shown that an asymmetry up to 10 or an unbalance of

10% in the power supply has only a minor effect on the rms-

value of the neutral conductor current. An unbalance in load

conditions increases the neutral conductor current.

Harmonics in the power supply voltage highly affects the

rms-value of the neutral conductor current.These conclusions can be drawn for all equipment with

similar current fingerprints as those of the tested compact

fluorescent lamps (e.g. computers).

ACKNOWLEDGMENT

The authors wish to thank the Flemish Government for

granting the project Studie van de nadelige gevolgen van hetgrootschalig gebruik van verlichting en office-equipment in

nutsgebouwen (IWT-HOBU).

REFERENCES

[1] A.-C. Liew, Excessive neutral currents in three-phase fluorescentlighting circuits,IEEE Transactions on Industry Applications, Vol. 25,

No. 4, pp. 776-782, July/August 1989.

[2] T.M. Gruzs, A survey of neutral currents in three-phase computerpower systems,IEEE Transactions on Industry Applications, Vol. 26,

No. 4, pp. 719-725, July/August 1990.

[3] C.L. Fortescue, Method of symmetrical co-ordinates applied to thesolution of polyphase networks, Transactions of the American Institute

of Electrical Engineers, Vol. 37, Part II, pp. 1027-1140, June 1918.

[4] L. Van der Veken, Safety and inspection perspective, EuropeanCopper Institute: Workshop on economic cost of poor power quality,

Brussels, Belgium, 8 June 2000.

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