Canadian Journal on Electrical and Electronics Engineering Vol. 3, No. 5, May 2012
The Effect of Harmonic Distortion on a Three phase Transformer
Losses
Hussein I. Zynal, Ala'a A. Yass
University of Mosul
linear load applied within power system, such
as arc furnace, electromotor loads, solid state
electronic devices which contain a poor power
supply, solid state devices, electronic
equipments that have control function and
leakage current at surface of polluted insulators
[2].
2-Transformer losses
Transformer losses are generally
classified into no load or core losses and load
losses [3]. This can be expressed in equation
form:
PT = PNL + PLL
(1)
Abstract-Electrical
transformers
are
designed to work at rated frequency and
sinusoidal voltage and current waves. At
present time the use of non linear loads, such
as power electronic loads are increased and
this leads to increase of power loss of
transformer. The problem of increasing the
power loss leads to several problems including
increasing the temperature of the transformer,
insulation damage and decrease the
operational life of the transformer . To avoid
these problems in case of non linear loads,
transformer should work with capacity less
than the rated capacity given by the designer.
In this research different type of losses in a
2KVA three phase transformer are studied with
linear and non linear loads, also the effect of
harmonics on transformer loss are evaluated.
The linear loads are simulated by a pure
resistance but the non linear loads are
simulated by a three phase bridge rectifier ,
also a three single phase rectifiers are
simulated as a non linear load to obtain the
effect of third harmonic . The different types of
losses and capacity of a three phase
transformer is then evaluated analytically and
simulation in MATLAB/SIMULINK and results
are compared.
PNL is the no load losses due to the
induced voltage in the core. PLL is the load loss
and consist of Pdc losses (I2Rdc) and stray losses
caused by electromagnetic fields in the
windings, core clamps, magnetic shields,
enclosure or tank walls, etc. Pdc is calculated
by measuring the dc resistance of the winding
and multiplying it by the square of the load
current. The stray losses can be further divided
into winding eddy current losses and structural
part stray losses. Winding eddy losses consist
of eddy current losses and circulating current
losses, which are all considered to be winding
eddy current losses. Other stray losses are due
to losses in structures other than windings,
such as clamps, tank or enclosure walls, etc.;
this can be expressed as [3]:
1- Introduction
Transformers are usually designed for
utilizing at the rated frequency and linear load.
Nowadays with the present of nonlinear load,
transformer leads to higher losses and
reduction of the useful life [1]. It is one of the
most important apparatus in power system
operation. It maintains supply continuity to the
consumers, so that transformer must be well
maintained to fulfill its technical life
expectation. Age of the transformer depends
on its insulation condition. Degradation of
transformer insulation can be caused by many
factors, like increasing of transformer
temperature, oxidation process of liquid or
solid insulation, improper cooling system of
transformer and short circuit current level on
the transformer loads. Those phenomena can
decrease the strength of transformer insulation
and the power quality. Other phenomenon
which affect to power transformer's operation
is harmonic. This harmonic is driven by non
PLL = P dc+ PEC + POSL
(2)
The total stray losses are determined by
subtracting I2Rdc from the load losses measured
during the impedance test and there is no test
method to distinguish the winding eddy losses
from the stray losses that occur in structural
parts.
PTSL= PLL-Pdc
(3)
2-1 Eddy Current Losses in Windings:
This type of loss is due to time variable
electromagnetic flux that covers windings.
Skin effect and proximity effect are the most
important phenomenon in creating these losses.
In transformers, in comparison to external
windings, internal windings adjacent to core
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Canadian Journal on Electrical and Electronics Engineering Vol. 3, No. 5, May 2012
Main current Eddy current
have more eddy current loss. The reason is the
high electromagnetic flux intensity near the
core that covers these windings. The winding
eddy current loss in the power frequency
Spectrum tends to be proportional to the square
of the load current and the square of frequency,
which are due to both the skin effect and
proximity effect × [4]. A portion
of the stray loss is taken to be eddy-current
loss. For dry-type transformers, the windingeddy loss is assumed to be [4]:
PEC=0.67*PTSL
(4)
POSL=PTSL-PEC
(5)
c
current
Flux
Main current
Fig (2) the proximity effect on eddy current.
2-1-2The computation of proximity effect
parameter by using electromagnetic theory:
The division of eddy-current loss and other
stray losses between the windings is assumed
to be as follows [4]
The electromagnetic theory used to
computation of proximity effect in term
voltage and current by using differential forms
of Maxwell’s equations as given below [6]:
∇ × = − ∂B⁄∂t
(6)
∇ × = + ⁄
(7)
The ratio of the conduction current density (J)
to the displacement current density (∂D/∂t) is
given by the ratio σ/(jωε), which is very high
even for a poor metallic conductor at very high
frequencies (where ω is frequency in rad/sec).
Since this analysis is for the (smaller) power
frequency, therefore the displacement current
density is neglected in case eddy currents
analysis in conducting parts of the transformers
therefore [6].
∇× =
(8)
Now, let us assume that the vector field E has
component only along the x axis.
∇ E = μσ ∂E⁄∂t
(9)
Where the operator ∇ represent partial
differential
∂ E ⁄∂ x + ∂ E ⁄∂ y + ∂ E ⁄∂ z =
μσ ∂E ⁄∂t
(10)
Suppose, that Ex is a function of z only (does
not vary with x and y), then equation(10)
reduces to the ordinary differential equation
d E ⁄dz = σμ dE ⁄dt
(11)
Now eq. (11) can be presented in terms of the
proximity effect voltage induced in the
conductor by the magnetic field that penetrates
the conductor [5]
d v!" ⁄dz = σμ dv!" ⁄dt
(12)
The proximity effect voltage in terms of the
current
d v!" ⁄dz = σμ d i⁄dt (13)
After double integration of eq. (13) with
distance, assuming the current i is not a
function of distance and the flux is in one
direction, the proximity effect voltage result is
expressed as
a) 60% in the low voltage winding and 40% in
the high voltage winding for all transformers
having a maximum current rating of less than
1000 A (regardless of turns ratio).
b) 60% in the low voltage winding and 40% in
the high voltage winding for all transformers
having a turns ratio of 4:1 or less.
c) 70% in the low voltage; winding and 30% in
the high voltage winding for all transformers
having a turns ratio greater than 4:1 and also
having one or more windings with a maximum
self cooled current rating greater than 1000 A
2-1-1 Proximity effect
The proximity effect contribution to the
winding eddy current loss is defined as
follows. Consider Fig.2. The HV winding
produces a flux density due to a changing
current. The LV winding and core cut the flux
density. The flux density that cuts the LV
winding induces an emf that produces
circulating or eddy currents. This effect is
called the proximity effect, which is caused by
a current-carrying conductor, or magnetic
fields that induce eddy currents in other
conductors in close proximity to the other
current carrying conductor or magnetic fields.
These eddy currents will dissipate power, PEC,
and contribute to the electrical loss in the
windings in addition to those caused by normal
dc losses [5].
.
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Canadian Journal on Electrical and Electronics Engineering Vol. 3, No. 5, May 2012
V!" = σμ d i⁄dt (14)
or in terms of the winding eddy current, ipe [5]
%&' ()*) =
7'8 =
, - . ⁄,/ -
0,- .123 ⁄,/ - 0
4123 9:;21
<- =
4123 56<>-
×
=
, - . ⁄,/ 4123 56-
the square of the rms current and the square of
the frequency [4]
X-
PTU = PTUVW ∑GQNR
n XYGQS
(15)
6
(20)
To obtain the true value of eddy current loss it
must be multiplying by harmonic loss factor
(F[\ ) when the transformer supplying
nonlinear
load
(16)
2-2 Other Stray Losses in Transformers:
Each metallic conductor linked by the
electromagnetic flux experiences an internally
induced voltage that causes eddy currents to
flow in that ferromagnetic material. The eddy
currents produce losses that are dissipated in
the form of heat, producing an additional
temperature rise in the metallic parts over its
surroundings. The eddy current losses outside
the windings are the other stray losses. The
other stray losses in the core, clamps and
structural parts will increase at a rate
proportional to the square of the load current
but not at a rate proportional to the square of
the frequency as in eddy current winding
losses [4].
3-Effect of harmonic on transformer losses
3-1 Effect of Voltage Harmonics
According to Faraday’s law the terminal
voltage determines the transformer flux level
[5]
<@
? <> = v(t)
(17)
Transferring this equation into the frequency
domain shows the relation between the voltage
harmonics and the flux components can be
written as
Nj(nw)φFG = VG
(18)
The flux magnitude is proportional to the
voltage harmonic and inversely proportional to
the harmonic order n. Furthermore, within
most power systems the harmonic distortion of
the system voltage is well below 5% and the
magnitudes of the voltage harmonics
components are small compared to the
fundamental component. This is determined by
the low internal impedance of most supply
systems carrying harmonics. Therefore
neglecting the effect of harmonic voltage and
considering the no load losses caused by the
fundamental voltage component will only give
rise to an insignificant error [7].
3-2 Effect of Current harmonics
In most power systems, current harmonics are
of significance. These harmonic current
components cause additional losses in the
windings and other structural parts [7].
a- Current harmonic effect on I2R loss
If the rms value of the load current is increased
due to harmonic components, the I2R loss will
be increased accordingly
PI = R<K × IMNO
= R <K ∑GQNR
IGMNO
(19)
GQS
b- Current harmonic effect on PEC
The eddy current losses generated by the
electromagnetic flux are assumed to vary with
F[\V"<<] =
-
^
∑Y_`ab
G- Y
Y_6
^6
^Y
∑Y_`ab
Y_6
^6
(21)
c- Current harmonic effect on other stray
losses:
The other stray losses are assumed to vary with
the square of the rms load current and the
harmonic frequency to the power of 0.8 :
cOd = PcOdW ∑GQNR
ne.g
GQS
XY
X6
(22)
To obtain the true value of other stray loss it
must be multiplied by harmonic loss factor
([\VhiW ) when transformer supplying
nonlinear load [4]
[\VhiW =
-
^
∑Y_`ab
Gj.k Y
Y_6
-
^Y
∑Y_`ab
Y_6
-
^6
(23)
^6
4- Recommended procedures for evaluating
the load capability of transformers under
nonlinear loads (containing harmonics)
The equation that applies to linear load
conditions is [4]:
P\\W ()*) = 1 + PTU (pu) + Poh\ (pu)
(24)
PLL-R:is the loss at rated load condition with
linear load.
As the effect of harmonic on losses of
transformer evaluated in pervious sections, a
general equation for calculating of losses when
transformer supplying a harmonic load can be
defined as fallow:
P\\ ()*) = I (pu)[1 + F[\ × PTU (pu) +
F[\VhiW Poh\ (pu)
(25)
The permissible transformers current is
expressed as
INR (pu) = q
rsst (!u)
[Svwxs ×ryz (!u)vwxs2{|t r}{s(!u) ]
(26)
5- Theoretical calculation
The transformer used in this paper has the
parameter as given in table (1):
Table (1) transformer parameter
KVA V1 V2 I1R
I2R
Rdc1 Rdc2
2000 380 137 3.03 4.86 1.45 0.5
In this paragraph the losses of the transformer
are calculated using equations given in
previous sections:
1- losses with linear load
a- Omic losses computation
The omic loss (Pdc) calculated using equation
(19), where Inrms equal rated current with linear
load .
Pdc= 75.24 W.
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Canadian Journal on Electrical and Electronics Engineering Vol. 3, No. 5, May 2012
b- Total stray losses computation
using
equation (3) .
PTSL = 81.5 – 75.24 = 6.26 W (where PLL-R
obtained from short circuit test).
To separate the total stray loss to eddy current
loss and other stray loss equations (4) and (5)
are used:
PEC = 0.67 * 6.26 = 4.194 W
POSL = 0.33 * 6.26 = 2.065 W
and to divided the other stray losses and
winding
nding eddy current losses between low
voltage
and high voltage
windings
assumptions given in article 2.1 are used [4].
PEC-LV = 0.6 * 4.194 = 2.5164 W
PEC-HV = 0.4 * 4.194 = 1.677 W
POSL-LV = 0.6 * 2.065 = 1.239 W
POSL-HV = 0.4 * 2.065 = 0.826 W
2- Losses with nonlinear loads
To calculate the losses theoretically it is
assumed that the secondary current is a square
wave which contain harmonic orders as given
in table (2)
Table (2) harmonic magnitude of secondary
current
Harmon
ic order
Seconda
ry
current
A
1
5
7
11
13
17
19
23
4.8
6
1.0
5
0.4
7
0.3
5
0.2
3
0.1
6
0.1
2
0.0
8
a- Omic loss (Pdc ).
By using equation (19)) the omic loss (Pdc) is
calculated as:
Pdc = 3*(3.22 * 1.45 + 5.0172 * 0.5) = 82.29 W.
b- Eddy current losses
By using equation (20) the eddy current loss is
calculated as:
PEC = 4.113 * 1.0618 = 4.36 W.
To obtain the true value of eddy current loss it
must be multiplied by harmonic loss factor
which is evaluated by using equation (21)
(2 .
FHL_EC = 3.849.
PEC = 16.781 W
c- other stray loss
By using equation (22)) the other stray loss is
calculated as.
POSL = 2.026 * 1.0618 = 2.152 W .
To obtain the true value of other stray loss it
must be multiplied by harmonic loss factor
evaluated by equation (23) .
FHL-OSL = 1.175 .
POSL = 2.53 W.
al calculated transformer losses
The analytical
under linear and non linear loads are tabulated
in table (3) below.
Table (3) analytical calculated losses value
Type
of
losses
Loss
under
linear
load
(W)
Harmonic
factor
40
75.36
Loss
under
non
liner
load
(W)
40
82.29
Corrected
losses
under
non
linear
load (W)
40
82.29
Iron
Pdc
PEC
4.113
4.36
3.849
16.781
POSL
Total
losses
2.025
121.499
2.152
1.202
2.53
141.601
By using equations (24), (25)) and ((26) the
transformer capability under non linear loads is
calculated as:
IMax = 0.926 * 4.86 = 4.50 A.
VA = 0.926 * 2000 = 1852.
6- Simulation Result
In this article the three – phase 2KVA
transformer is simulated with linear and non
linear loads using matlab/ simulink. The eddy
current loss is represented as a dependent
voltage source, its voltage depend upon the
second derivative of the load current and other
stray losses represented as a resistance in series
with the leakage inductance and dc resistance.
The non linear load is a three – phase
uncontrolled rectifier with resistive and high
inductive loads ( with and without 5th harmonic
filter ). Fig (3) show the simulation circuit.
Figure (3) simulation circuit
By using matlab P.S.B the losses are calculated
and the results are tabulated as given in table
(4). Fig(4) showss the percentage loss compared
to linear load plotted against % rated load.
The effect of harmonics on omic loss with
different load values is given in table (5) and
fig (5) shows the variation of omic loss with
different load values.
Canadian Journal on Electrical and Electronics Engineering Vol. 3, No. 5, May 2012
Figure (4) shows the increase of percentage
loss compared with linear load
Figure (5) the omic loss for different type
of loads
Table (4) total losses of transformer and change of losses compare to linear load
Total loss
Linear
load(W)
Total loss under Non linear load(W)
%
Rated
load
Resistive
Load
rectifier
with
resistive
load
% change
of loss
compare
with linear
load
rectifier
with
resistive
load and
filter
% change
of loss
compare
with linear
load
rectifier
resistive
with
inductive
load
% change
of loss
compare
with linear
load
controlled
rectifier with
resistive load
% change
of loss
compare
with linear
load
FL
122.9
139.2
13.26
129.1
5.044
138.7
12.85
141
14.72
0.75
FL
87.59
98.31
12.23
92.25
5.32
98.13
12.02
99.16
13.20
0.67
FL
73.36
81.45
11.02
77.19
5.22
81.38
10.93
82.03
11.81
0.5 FL
61.53
67.2
9.215
64.51
4.84
67.18
9.182
67.55
9.78
0.375
FL
52.17
55.72
6.8
54.36
4.19
55.72
6.80
55.93
7.20
0.25
FL
45.38
47.2
4.01
46.85
3.23
47.21
4.03
47.3
4.23
Canadian Journal on Electrical and Electronics Engineering Vol. 3, No. 5, May 2012
Table(5)the omic loss for different type of loads
%Rated
load
Omic loss
with Linear
load(W)
Resistive
load
Omic loss with Non linear load (W)
Rectifier
with resistive
load
rectifier with
resistive load
and5th
harmonic filter
FL
77.29
82.41
0.75 FL
44.78
48.48
0.625 FL
31.65
34.6
0.5 FL
20.69
22.9
0.375 FL
11.98
13.49
0.25 FL
5.538
6.485
The eddy current loss for different type of
loads are given in table (6) for different rated
load and figure (6) shows the variation of eddy
current loss with different rated load
78.3
46.03
32.9
21.87
13.03
6.476
Rectifier
with
resistive and
inductive
load
82.05
48.3
34.48
22.83
13.46
6.473
controlled
rectifier with
resistive load
α=30
82.7
48.63
34.7
22.96
13.53
6.5
Table (6) the effect of harmonics on eddy current loss with different load values
%Rated load
Eddy loss with
Linear load(w)
Resistive load
FL
0.75 FL
0.625 FL
0.5 FL
0.375 FL
0.25 FL
3.782
1.885
1.15
0.563
0.131
0.038
Eddy loss with Non linear load (w)
rectifier with
resistive load
14.12
8.215
5.693
3.524
1.769
0.501
Figure (6) change of eddy current loss with
percentage of rated load
The other stray loss for different type of loads
is given in table(7)for different load values. Fig
(7) shows the variation of other stray loss with
different load values.
rectifier with
resistive load
and5th
harmonic
filter
8.426
4.829
3.295
1.981
0.930
0.177
rectifier with
resistive and
inductive
controlled
rectifier with
resistive load
α=30
13.98
8.219
5.733
3.569
1.808
0.521
15.52
8.919
6.161
3.811
1.951
0.586
Figure (7) change of other stray loss with
percentage of rated load
7- Practical Result
A three phase 2KVA 380/137volt star – delta
transformer is connected in laboratory with
linear and non linear loads. The nonlinear
loads are a three phase bridge rectifier with
resistive and high inductive loads. By using (3
Canadian Journal on Electrical and Electronics Engineering Vol. 3, No. 5, May 2012
Phase Power Quality (3945-B powepad) the
input, output powers were measured.
Also the current waveform were recorded and
analyzed. Fig (8) shows the practical
waveform of the current for high inductive
load.
28% at full load compared with linear load
case.
Reference
[1] D.M. Said, K.M. Nor, “Simulation of the
Impact of Harmonics on Distribution
Transformers”, 2nd IEEE International
Conference on Power and Energy (PECon 08),
December 1-3, 2008, Johor Baharu, Malaysia.
[2] Sumaryadi, Harry Gumilang, Achmad
Susilo, “Effect of Power System Harmonic
on Degradation process of Transformer
Insulation System”, Proceedings of the 9th
International Conference on Properties and
Applications of Dielectric Materials, July 1923,2009, Harbin, China.
Figure(8) current waveform for rectifier
with high inductive load.
The result obtained as
The total power loss with linear load is equal
to 121.34 W.
The total power loss with non linear load is
equal to 146.7 W.
The maximum permissible secondary current
with non linear load is 4.423 A.
The maximum permissible VA with non linear
load 1818 .
Table (8) gives the comparison between
analytical simulation and practical results for
non linear load.
Table (8) comparison between analytical
simulation and practical results.
Total
Losses
(W)
VA
rating
Imax (A)
Analytical
141.6
simulation
139.2
practical
146.7
1852
1878
1818
4.5
4.56
4.423
[3] Asaad A. Elmoudi, " Evaluation of Power
System Harmonic Effects on Transformers
Hot Spot Calculation and Loss of Life
Estimation", Ph. D. Thesis, Helsinki
University of Technology, 2006.
[4] IEEE Std C57.110-1998, “IEEE
Recommended Practice for Establishing
Transformer Capability When Supplying
Nonsinusoidal Load Currents”.
[5] S. B. Sadati, A. Tahani, B.Darvishi, M.
Dargahi, H.yousefi, “ Comparison of
Distribution
Transformer Losses and
Capacity under Linear and Harmonic
loads”, 2nd IEEE International Conference on
Power and Energy (PECon 08), December 1-3,
2008, Johor Baharu, Malaysia .
[6] S.V. Kulkarni, S. A. Khaparde,
”Transformer Engineering Design and
Practice", Indian Institute of Technology,
Bombay Mumbai, India, Marcel Dekker. Inc.
2004 .(Book)
[7] A. Elmoudi, M. Lehtonen, Hasse Nordman,
“Effect of Harmonics on Transformers Loss
of life”, Conference Record of the 2006 IEEE
International Symposium on Electrical
Insulation.
8-Conclusion
In this paper the effect of current harmonics
upon transformer losses based on(IEEE
standard c57-110) have been analyzed and
evaluated. The equivalent KVA and maximum
current ratings of a three – phase transformer
for supplying harmonic loads are evaluated.
The analytical simulation and experimental
results shows that losses increase with increase
of total harmonic distortion of the transformer
current and rated capacity decreases. when
transformer supplying non linear load the
percentage increase of losses at full load was
13.26% compared with case of linear load, but
when 5th harmonic filter is connected the
percentage increase of the losses reduced to
5.04%. The percentage increase of omic loss
was 31.4% while the increase of eddy current
loss was 63.4% and increase of other stray loss
was 5.2% compared with the case of linear
loads. When the transformer loaded with three
single phase rectifier ( the secondary current
contains third harmonic as well as the other
harmonics) the percentage increase of loss was
261