8/12/2019 Estudio de Corrientes en Neutros
1/6
1 of 6
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
8/12/2019 Estudio de Corrientes en Neutros
2/6
2 of 6
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
q
qqq
q
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.
8/12/2019 Estudio de Corrientes en Neutros
3/6
3 of 6
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
Results of measurement
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).
8/12/2019 Estudio de Corrientes en Neutros
4/6
4 of 6
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
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)
Results of measurement
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%)
8/12/2019 Estudio de Corrientes en Neutros
5/6
5 of 6
I n f l uence o f l oad unba lance on the neu t ra l conduc to r cu r ren t
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
8/12/2019 Estudio de Corrientes en Neutros
6/6
6 of 6
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
voltage with third harmonic, 2%, 180
(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
neu t ra l conduc to r cu r ren 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 neu t ra l conduc to r cu r ren t
rms-valueharm
onic[m
A]
sinusoidal voltage
voltage with third harmonic, 2%, 0
voltage with third harmonic, 2%, 180
(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.