EEE-1206-016-Harmonic-Distortion-Three-phase-Transformer-Losses.pdf

Canadian Journal on Electrical and Electronics Engineering Vol. 3, No. 5, May 2012

255

The Effect of Harmonic Distortion on a Three phase Transformer

Losses

Hussein I. Zynal, Ala'a A. Yass

University of Mosul

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.

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

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)

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]:

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


256

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 winding-

eddy loss is assumed to be [4]:

PEC=0.67*PTSL (4)

POSL=PTSL-PEC (5)

The division of eddy-current loss and other

stray losses between the windings is assumed

to be as follows [4]

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


having a turns ratio of 4:1 or less.

c) 70% in the low voltage; winding and 30% in


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].

.

Fig (2) the proximity effect on eddy current.

2-1-2The computation of proximity effect

parameter by using electromagnetic theory:

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

current

Flux

c

Eddy current Main current

Main current


257

V!" = σμ d�i dt�⁄ (14)

or in terms of the winding eddy current, ipe [5]

%&'()*) = ,-. ,/-⁄0,-.123 ,/-⁄ 0 = ,-. ,/-⁄

412356- (15)

7'8 = 4123 9:;214123 56-

× <- =<>- (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 ∑ IGMNO�GQNR�GQS (19)

b- Current harmonic effect on PEC

The eddy current losses generated by the

electromagnetic flux are assumed to vary with

the square of the rms current and the square of

the frequency [4]

PTU = PTUVW ∑ n�GQNR�GQSXY-X6-

(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

F[\V"<<] = ∑ G-^Y-^6-

Y_àbY_6∑ ^Y-

^6-Y_àbY_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 ∑ ne.g XY-X6-

GQNR�GQS (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 = ∑ Gj.k^Y-^6-

Y_àbY_6∑ ^Y-

^6-Y_àbY_6

(23)

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[\VhiWPoh\(pu) (25)

The permissible transformers current is

expressed as

INR�(pu) = q rsst(!u)[Svwxs×ryz(!u)vwxs2{|tr}{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.


b- Total stray losses computation using

equation (3) .

PTSL = 81.5 – 75.24 = 6.26 W (where P

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 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 1713 11 7 5 1 Harmon

ic order

0.1

6

0.2

3

0.3

5

0.4

7

1.0

5

4.8

6

Seconda

ry

current

A

a- Omic loss (Pdc ).

By using equation (19) the omic loss (P

calculated as:

Pdc = 3*(3.22 * 1.45 + 5.017

2 * 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.



which is evaluated by using equation (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 .



evaluated by equation (23) .

FHL-OSL = 1.175 .

POSL = 2.53 W.

The analytical calculated transformer losses

under linear and non linear loads are tabulated

in table (3) below.

Table (3) analytical calculated losses value


Total stray losses computation using

(where PLL-R

To separate the total stray loss to eddy current

loss and other stray loss equations (4) and (5)

and to divided the other stray losses and

nding eddy current losses between low

voltage and high voltage windings

assumptions given in article 2.1 are used [4].

To calculate the losses theoretically it is

assumed that the secondary current is a square

wave which contain harmonic orders as given

Table (2) harmonic magnitude of secondary

23 19 17

0.0

8

0.1

2

0.1

6

) the omic loss (Pdc) is

* 0.5) = 82.29 W.

current loss is



which is evaluated by using equation (21) .

) the other stray loss is



al calculated transformer losses

under linear and non linear loads are tabulated

Table (3) analytical calculated losses value

Type

of

losses

Loss

under

linear

load

(W)

Loss

under

non

liner

load

(W)

Harmonic

factor

Iron 40 40

Pdc 75.36 82.29

PEC 4.113 4.36 3.849

POSL 2.025 2.152 1.202

Total

losses

121.499

By using equations (24), (25) and (

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

uncontrolled rectifier with resistive and high

inductive loads ( with and without 5

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) shows 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.


Harmonic Corrected

losses

under

non

linear

load (W)

40

82.29

16.781

2.53

141.601

) and (26) the

transformer capability under non linear loads is

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

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

s 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


Figure (4) shows the increase of percentage

loss compared with linear load

Total loss under Non linear load(W) Total loss

Linear

load(W)

% change

of loss

compare

with linear

load

controlled

rectifier with

resistive load

% change

of loss

compare

with linear

load

rectifier

resistive

with

inductive

load

% change

of loss

compare

with linear

load

rectifier

with

resistive

load and

filter

% change

of loss

compare

with linear

load

rectifier

with

resistive

load

Resistive

Load

%

Rated

load

14.72 141 12.85 138.7 5.044 129.1 13.26 139.2 122.9 FL

13.20 99.16 12.02 98.13 5.32 92.25 12.23 98.31 87.59 0.75

FL

11.81 82.03 10.93 81.38 5.22 77.19 11.02 81.45 73.36 0.67

FL

9.78 67.55 9.182 67.18 4.84 64.51 9.215 67.2 61.53 0.5 FL

7.20 55.93 6.80 55.72 4.19 54.36 6.8 55.72 52.17 0.375

FL

4.23 47.3 4.03 47.21 3.23 46.85 4.01 47.2 45.38 0.25

FL

Figure (5) the omic loss for different type

of loads

Table (4) total losses of transformer and change of losses compare to linear load


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

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.

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

Omic loss with Non linear load (W) Omic loss

with Linear

load(W)

controlled

rectifier with

resistive load

α=30

Rectifier

with

resistive and

inductive

load

rectifier with

resistive load

and5th

harmonic filter

Rectifier

with resistive

load

Resistive

load

%Rated

load

82.7 82.05 78.3 82.41 77.29 FL

48.63 48.3 46.03 48.48 44.78 0.75 FL

34.7 34.48 32.9 34.6 31.65 0.625 FL

22.96 22.83 21.87 22.9 20.69 0.5 FL

13.53 13.46 13.03 13.49 11.98 0.375 FL

6.5 6.473 6.476 6.485 5.538 0.25 FL

Eddy loss with Non linear load (w) Eddy loss with

Linear load(w)

controlled

rectifier with

resistive load

α=30

rectifier with

resistive and

inductive

rectifier with

resistive load

and5th

harmonic

filter

rectifier with

resistive load

Resistive load %Rated load

15.52 13.98 8.426 14.12 3.782 FL

8.919 8.219 4.829 8.215 1.885 0.75 FL

6.161 5.733 3.295 5.693 1.15 0.625 FL

3.811 3.569 1.981 3.524 0.563 0.5 FL

1.951 1.808 0.930 1.769 0.131 0.375 FL

0.586 0.521 0.177 0.501 0.038 0.25 FL

Table(5)the omic loss for different type of loads

Table (6) the effect of harmonics on eddy current loss with different load values


261

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.

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. Analytical simulation practical

Total

Losses

(W)

141.6 139.2 146.7

VA

rating

1852 1878 1818

Imax (A) 4.5 4.56 4.423

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

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 19-

23,2009, Harbin, China.

[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

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[6] S.V. Kulkarni, S. A. Khaparde,

”Transformer Engineering Design and

Practice", Indian Institute of Technology,

Bombay Mumbai, India, Marcel Dekker. Inc.

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EEE-1206-016-Harmonic-Distortion-Three-phase-Transformer-Losses.pdf

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