Download Study of a Transformer Thermal System

1
Study of a Transformer Thermal System
Pedro de Carvalho Peixoto de Sousa Barros
Department of Electrical and Computer Engineering (DEEC) - Instituto Superior Técnico (IST)
Abstract—This project has as goal the study and modulation of
a dry single-phase transformer thermal system, with 1kVA of
electrical power. During this study is identified and characterized
the different zones of the transformer on which the temperature is
considered homogeneous. It is presented a lumped model as well
as the methodology to determinate its parameters. Tests are made
for different load situations, quantifying the model error and
giving particular attention to overload and short-circuit
situations, being outstanding the importance of using models in
order to foresee with precision the maximum temperature of the
transformer as well as its overload capacity. I also presented a
introductory study of the thermal behavior of the transformer
and its modulation when it finds a regimen of forced cooling
convection.
Index Terms—Transformer,
parameters, lumped parameters.
thermal
model,
distributed
I. INTRODUCTION
circuit situations in order to analyse its thermal behaviour and
to calculate its overload capacity.
Finally, I made an introductory study with the objective to
verify the thermal behaviour of the transformer when it finds a
cooling forced convection and how this regimen can increase
the transformer efficiency.
II. IDENTIFICATION OF HOMOGENEOUS ZONES
A transformer has in its constitution some kind of materials,
namely copper, iron and some types of isolation. Each one of
these materials have different thermal characteristics, as such,
normally it’s expected that the temperature in the transformer
is not the same on its different points.
To become possible to read the temperature in the different
regions of the transformer, I used type J thermocouples in a
single-phase transformer of 1kVA and nominal voltage of
240/120V as it shows in the Fig.1.
T
HE project and the overload capacity of electric machines,
namely transformers, are strongly conditioned by its
thermal performance. Due to difficulties of modulating the
thermal system, exists the conscience that the advantages of
materials are not explored integrally in the construction of
transformers, so for precaution reasons, are fixed exaggerated
safety margins for its operation.
Nowadays there are numerical calculation methods capable
to deal with complicated geometries and represent on the same
time several phenomenas (heat and mass transportation) that if
are used carefully and systematically can allow the creation of
more numerical models, namely the thermal component of the
transformer.
In this work, I have done a thermal study of the transformer,
in order to be possible to know its dynamics as well as its
hottest zone. The knowledge in identifying and studying the
hottest zone of the transformer is directly related with the lifetime reduction of electric machines as the material aging
depends on the temperature they are subjected to.
It is intended to identify several temperature rises in
different zones of the transformer, once the simplest models
estimate a homogeneous temperature over all the transformer
area.
In this study I considered a thermal lumped model, as well
as the justification and characterization of its parameters,
becoming important the realization of several tests to validate
the model and determinate the error.
A study is made for overload situations as well as short
Fig. 1. Thermocouples. Nu-Core, A-A winding, B-B winding, BC-BC
winding.
The transformer has three windings, being the internal
winding (the one next to the core) named as winding A. In the
2
external windings, named as winding B and C, the nominal
tension is 120V and when connected in series allows a
transformation ratio of 1:1.
Please note that the thermocouples from T1 to T6 are
assembled under the iron core and the T7 and T8 are out of the
core window.
The thermocouple T9 has been assembled over the upper
part of the core in order to be possible to read its temperature.
When a test is carried out in the transformer, for instance a
short circuit test, it shows that the transformer temperature is
not homogeneous, existing zones where the temperature is
higher and others where it is lower. In Fig. 2 is showed the
temperature rise for a short circuit test where it’s possible to
verify the non homogeneity in transformer temperature, and
conclude that it is lower in the iron core (T9).
55
T1
T2
T3
T4
T5
T6
T7
T8
T9
Temperature Rise (ºC)
50
45
40
35
30
25
Fig. 4. Temperature variation as the readings move away to the external
layers of the copper.
variation occurs in the isolation material, whereas in the
copper and in the iron it stays practically constant.
The temperature differences in the previous figures can be
explained appealing to Fig. 5, where it is represented the heat
flux on the zone in analysis.
20
15
10
5
5.2
5.4
5.6
5.8
6
Time (h)
Fig. 2. Temperature rise. Short-Circuit test.
In the copper windings, the temperature is lower in the
winding A, and it gradually rises as we read the temperature in
the external windings, reaching its higher reading (for the short
circuit test) in the winding B (T4 and T5) and coming back to
lower reading in the winding C.
As an example, is presented in the Figs. 3 and 4 the
temperature inside windings, as well as its variation as the
readings move away to the external layers of the copper.
Fig. 3. Temperature inside the winding in the central zone of the
transformer.
Please note that the Fig. 3 and 4 were made with simulation
programs that solve the equations for a distributed parameters
model. To simplify the simulation, it was only presented the
central zone of the transformer, where were assembled the
thermocouples from T1 to T6.
In Fig. 4, it is possible to verify that the temperature
Fig. 5. Total heat flux [W/m2]
Due to the large contact surface with the air temperature, the
iron core has a very small increase of temperature when
compared with the windings, acting as a heat exchanger. This
way it explains the fact the temperature is lower near the core.
As the reads move away from the core, the temperature of the
winding A rises, because the heat as more difficulty to flow
due to the various isolation layers.
As was already explained, the hottest temperature is verified
in winding B, what it would be expected to happen as it is an
internal winding having several isolations layers, and therefore
making the heat flux difficult through it. In addition, the
winding B connected in series with winding C, has more 18
meters of length than winding A. With more copper, for a
giving current it will appear more losses, therefore the
temperature will be higher.
In the external part of winding C, the temperature comes
back to lower values, as it is a outside zone of the winding and
the heat is easily dissipated to the air around.
In thermocouples T7 and T8, that are not assembled
underneath the iron core, when analyzing Fig. 2, we verify that
T7 as the same temperature of T5, as both are assembled
between winding B and C and near the same wire, however,
T8 has a substantially lower reading as it is assembled near the
exterior, revealing that the temperature is not constant along
the wires on the same layer, being higher on the central zone
of the winding and getting lower as it moves for the wires near
3
the exterior.
Of everything that was observed it is evidenced that the
transformer as several zones with different temperatures,
indicating clearly that the temperature is not homogeneous.
However, it becomes difficult to make the study of the
transformer considering all the different temperatures. In order
to be possible to establish a lumped model, it will be
considered zones with material homogeneity, as well as zones
where it can be found heat sources and using the Biot number
[14] is possible to establish a criterion where we can say that a
specific region of the transformer is assumed as having a
homogeneous temperature.
According with (1), it can be assumed that the temperature
is uniform if the Biot number (NB) is lower than 0,1. This
means that the thermal resistance for internal conduction is
much less than the thermal resistance for convection at the
surface of the transformer.
NB =
h× L
k
(1)
Fig. 6. Lumped model. Nu-Core, A-A winding, BC-BC winding.
each region and Pi represents the heat generation sources. The
Gi-j conductances represent the resistance to the heat exchange
between regions i and j. Once the winding A is an internal
winding, it was verified that most of the heat produced in this
winding is lost for the core and winding BC and not directly to
the air. Due to this fact, the existence of a conductance
between winding A and the air is not considered.
A. Determination of the model parameters
The thermal equations of the presented model are, in the
matricial form, represented by (3)
where
C×
V
L=
A
(2)
In (1), h is the heat transmission coefficient for convection,
L is the ratio between the volume of the body (V) and the area
of its surface (A) and k is the thermal conductivity.
Knowing the transformer geometry and the materials
properties is possible to calculate the Biot number for the
different zones of the transformer. Therefore, for the iron core,
I got a Biot Number of 6x10-3, being this value of 1,4x10-3,
1,5x10-3 and 60x10-6 for the windings A, B and C respectively.
As the Biot number is lower than 0,1 we can assume a
temperature homogeneity in each one of the considered zones.
With experimental tests, it was verified that the windings B
and C can be considered as one only zone of homogeneous
temperature.
Please note that in all zones it was considered the hottest
point.
III. LUMPED MODEL
One of the goals of this work is the determination of a
lumped model on which will be possible its use in a simple
way with the lowest error.
For that, it was considered each of the transformer regions
with a current source (that corresponds to the heat generation
on that material), a capacitor (to represent the heat
accumulation) and a resistor (to represent the temperature
variation in the transformer due to the heat flux).
Fig. 6 shows the different transformer zones, being ∆θi the
variation of temperature of zone i, Ci the thermal capacity of
d∆θ
+ G × ∆θ = P
dt
(3)
In which C represents the matrix of the thermal capacities,
∆θ the vector of temperature rise for each region of the
transformer, G the matrix of thermal conductances and P the
vector of the heat source for each zone where it is verified the
heat generation.
From (3), it is possible to verify that in a steady state
regimen we get (4),
G × ∆θ = P ⇔ ∆θ = G −1 × P ⇔ ∆θ = R × P
(4)
being R the matrix of the thermal capacities.
To solve (3), it is necessary to determine the model
parameters. Vector P is an entrance value, being possible to
represent it depending on the transformer load, whereas the ∆θ
vector represents the temperature variation for each specific
part of the transformer when related to a specific load. This
way, there are two variables to be calculated: the matrix of
thermal capacities C, and the matrix of thermal resistances R
(the matrix G is obtained by inverting the matrix R).
1) Thermal capacities calculation
Knowing the transformer geometry and its materials, it is
possible to characterize with some accuracy the thermal
capacity.
Cth = ρVC p = mC p
So it is possible to write the matrix C as follows:
(5)
4
0
CA
0
0  6253 ,63
0  =  0
C BC   0
0
485 ,70
0



609 ,18 
0
0
(6)
60
2) Thermal resistance calculation
The calculation of the thermal resistances values is slightly
more complicated than the thermal capacities. Once the
transformer geometry is complex, it becomes easier to
calculate the thermal resistances values with experimental
tests. Observing the steady state equation (4) it is shown that
heating each different parts of the transformer and measuring
the temperature variation in all the other parts, it is possible to
calculate the thermal resistances values.
To heat each zone of the transformer, it was made a test
with direct current, it means that knowing the current value, it
is possible to calculate the heat generated in each zone (power
losses). This type of test is valid when is intended to heat the
windings.
Once is not possible to heat only the iron core (it is
impossible to heat it without heating also the windings), to
verify the iron core heating, it was made an open circuit test
where was monitored the total electric power on the
transformer and the current in the winding A.
Therefore, with the mentioned tests, it becomes possible to
calculate the matrix R.
Then, the thermal resistances matrix is as follows:
 R Nu
R =  R Nu −A
R Nu −BC
R Nu −A
RA
R A −BC
R Nu −BC  0,65 0,66 0,54
R A−BC  = 0,66 2,33 1,42 
R BC  0,54 1,42 2,19 
(7)
where the main diagonal line represents all the resistances
connected to a specific zone (Nu, A the BC) and outside the
diagonal are represented the thermal resistances between each
two zones of the transformer.
Inverting the matrix R it is possible to obtain the thermal
conductances matrix G.
R
G Nu − Ar + G Nu − A + G Nu −BC

− G Nu − A

− G Nu − BC

−1
=G =
− G Nu − A
G Nu − A + G A − BC
the steady state. Afterwards, was made a simulation with the
previously presented model in order to validate it.
− G Nu − BC
− G A −BC



+ G Nu − BC 
Temperature Rise (ºC)
C Nu
C =  0
 0
TNu-Simul.
TA-Simul.
TBC-Simul.
TNu
TA
TBC
50
40
30
20
10
0
0
2
4
6
8
10
Time (h)
Fig. 7. Steady-State test.
From Fig. 7, it is verified that the obtained results for the
simulation approaches the experimentally values. It is also
observed that the transformer shows a maximum variation of
temperature, for the operation at nominal power rate, about
58ºC for the copper windings and 28ºC in the iron core.
As it would be expected, it is evidenced that the thermal
behaviour of the windings is different from the thermal
behaviour of the iron core, being its time constants about 3600
(1 hour) and 8200 seconds (2 hours and 18 minutes),
respectively.
For this test, it is evidenced that the winding A losses are
about 10,4W being 13,9W for the winding BC, noticing that
this difference is due to the longer length of the winding BC.
Through a test open circuit test (carried out for the model
parameters calculation) it was verified that the iron core losses
are about 20W.
TABLE I
MAXIMUM ERROR
8,0ºC
|Absolute
Error|
1,5ºC
Relative
Error %
18,8
41ºC
3ºC
7,3
41ºC
3ºC
7,3
Temperature
Simulated
Experimental
TNu
9,5ºC
TA
44ºC
TBC
44ºC
IV. EXPERIMENTAL RESULTS AND SIMULATIONS
On table I is presented the maximum error for this test,
verifying that it occurs during the transient state. It’s noticed
that the relative error in the core is high during this test,
however, it is verified that it occurs at extremely low
temperature with a small absolute error. On the other hand, the
relative error in windings is lesser, but occurring an increase of
the absolute error. After reaching the temperature steady state
the error decreases to values lower than 3,5% in the core and
3,8% in the windings.
A. Steady State operation
For this test, the transformer was placed supplying energy to
a resistive load at nominal power rated, reading the
temperature in various parts of the transformer until it reaches
B. Overloads and Short circuits
In this chapter it is intended to study which is the maximum
overload capacity of the transformer, doing tests and
simulations, in order to assure the normal operation of the
− G A − BC
G BC−Ar + G A − BC
 2,23 − 0,49 − 0,23
= − 0,49 0,82 − 0,41
 − 0,23 − 0,41 0,78 
(8)
5
1) Overloads
In Fig. 8 it is presented the experimental and computational
result obtained for a test where the transformer operates at
nominal power rate, afterwards operating at a temporary
overload, increasing the current to a value 10% higher than its
nominal current.
Afterwards it decreases the current until it reaches 90% of
its nominal value.
70
TNu-Simul.
TA-Simul.
TBC-Simul.
TNu
TA
TBC
Temperature Rise (ºC)
60
50
40
As an example, the laboratory transformer, as it was already
referred, at rated power it produces power losses of 20W in the
iron core, being 10,4W and 13,9W in the windings A and BC,
respectively. Once this transformer has class H isolation, it’s
possible a maximum temperature variation of 125ºC with an
ambient temperature of 40ºC and 15ºC error margin for the
hottest point, it means that this point can reach 180ºC.
Fig. 9 represents a simulation where initially the transformer
operates at rated power, later on increasing 2,5 times the
winding losses it produces losses of 26W in the winding A and
34,8W in winding BC. The losses in iron core can be
considered constant once these only depend on the supplied
voltage value.
120
Temperature Rise (ºC)
transformer and avoiding failures.
Regarding the short circuits, this study becomes important
mainly for the project of transformer circuit breakers and its
time actuation reaction, which has to be fast enough to avoid
major failures.
TNu-Simul.
TA-Simul.
TBC-Simul.
100
80
60
40
30
20
20
1
2
3
4
Time (h)
5
6
7
Of course that when it reaches an overload state increasing
the 10% the current, the windings verify an increase on losses
of 20% and a temperature rise.
As it is possible to see in table II, the error of the simulation
values, for the hottest point, is about 2% for the windings and
3% for the iron core.
TNu
4
6
Time (h)
8
10
12
Fig. 9. Maximum Overload.
Fig. 8. Overload test.
Temperature
2
TABLE II
MAXIMUM ERROR: HOT-SPOT
|Absolute
Simulated Experimental
Error|
30ºC
31ºC
1ºC
Relative
Error %
3,1
TA
66,1ºC
67,3ºC
1,2ºC
1,8
TBC
65,1ºC
64,2ºC
0,9ºC
1,4
To study temporary overloads situations, a model that
calculates which is the temperature variation with a small error
shows particularly useful, it means, for a given operation state
of the transformer, it is possible to calculate with some
accuracy the maximum temperature of the hottest point, if it
operates under overload, as well as which is the maximum
time that the transformer can support overload situations
without isolation damages.
This model can also be useful for the thermal design of the
transformer as knowing the maximum temperature reached it
becomes possible to manufacture smaller transformers and this
way avoiding the usual over sizing of electric machines.
As it is presented, the model foresees that the winding
temperature variation reaches the limit value, which is the
125ºC.
Assuming that the windings resistance does not change with
the temperature, it is possible to calculate the theoretical
current value that flows trough them for the considered
situation, and it is verified that the values reaches 6,6 amperes.
Once the nominal current of the transformer is 4,16
amperes, a current of 6,6 amperes corresponds to an increase
of 58% in the nominal current, reveals that the laboratory
transformer is clearly oversized.
In the previous simulation, it was presented an extreme
situation of overload, on which the transformer finds an
operating state during a period of time long enough to reach a
steady state.
However, it is possible to put the transformer operating with
higher values than the ones mentioned previously if the
operating time doesn’t exceed the period of time necessary to
reach its maximum value of 125ºC.
The Fig. 10 presents some simulations for different current
values. In this simulation, it is assumed as beginning point the
temperature variation in steady state, making afterwards a
current change multiplying it by 1.58, 1.7, 2 and 2.5.
The situation where the nominal current is multiplied by
1.58 was already presented, being the maximum overload
situation. As we increase the current value, it is verified that
the temperature reaches quickly the maximum value.
6
Temperature Rise (ºC)
100
80
60
40
20
0 t1 t2
1
t3
2
Time (h)
3
4
5
Fig. 10. Overload Time.
For the situations considered in this simulation, it is also
verified that it is possible to operate the transformer with a
current 70% higher than the nominal value during a period of
time about 80 minutes (t3), decreasing this period to 25
minutes if the current is double of its nominal value (t2). If the
current reaches 2.5 times the nominal value, the period of time
that the transformer can operate overloaded without any
isolation damages decreases to 12 minutes (t1).
With this simulation, it is also intended to enhance the
importance of the thermal behaviour of the iron core, it means,
the iron core working as a heat exchanger how it contributes
for the windings cooling operating under overload conditions.
Observing again Fig. 10, it is verified that for operating
states of soft overloads ( near rated values) the iron core
constant thermal time has an important role on the global
transformer cooling, it means, the heat generated by the
transformer is slowly dissipated by the iron core making the
temperature of the hottest point to raise also gradually.
As the overload becomes more violent (extremely higher
than the rated conditions) the heat generated by the windings
reaches extremely high values, producing a quickly
temperature increase.
The heat generated in the windings needs time to spread into
the iron core, being then dissipated by this one. As the
windings heat increase time is sufficiently lower than the
period of time necessary to spread and dissipate the heat
trough the core, the cooper temperature will increase quickly
that the core temperature.
Take for instance the situation where the current is 2.5 times
the nominal current. The temperature in the winding BC takes
approximately 12 minutes (t1) to reach its maximum value. At
that exactly time, it is verified that the iron core temperature
practically doesn’t change.
2) Short circuits
The use of a proper thermal model allow us to calculate
efficiently the period of time that the transformer can operate
with short circuit, knowing which is the maximum overload
protection actuation time.
In Fig. 11 is presented a short circuit situation in the
secondary winding of the transformer, where the current is 30
times higher the nominal current.
120
Temperature Rise (ºC)
TNu - 1.58*In
TBC - 1.58*In
TNu - 1.7*In
TBC - 1.7*In
TNu - 2*In
TBC - 2*In
TNu - 2.5*In
TBC - 2.5*In
120
TNu
TA
TBC
100
80
60
40
20
-1
-0.5
0
0.5
1
1.5
Time (s)
2
2.5
3
3.5
Fig. 11. Short-Circuit.
In the figure above it is possible to observe that the iron
core does not have time to heat, reason why its behaviour can
be disgard, considering just the windings behaviour.
To project the overload protections, it is verified that the
transformer can remain in operation under short circuit during
2.75 seconds from which can occur failure. Therefore, the
protection actuation time must be lower than the previously
referred time.
C. Forced Convection
In the previous chapters, it was presented a study for the
transformer thermal behaviour during a natural convection
situation. In this chapter it is intended to present an
introductory study for a transformer thermal behaviour when it
operates under forced cooling convection.
As it is presented, the hottest point of the transformer, under
nominal conditions, is located inside the copper windings.
Cooling under forced convection has as goal the
temperature reduction in that point.
As already referred, it was observed that the iron core acts
as a heat exchanger. Having verified this, a fan was used
pointing directly to iron core and this was increasing,
efficiently, the heat exchange.
To calculate the model parameters, it was proceeded as
previously referred, doing tests with direct current in order to
heat each winding separately and a open circuit test in order to
heat only the core.
This way, it was possible to calculate the model parameters
when the transformer operates under forced cooling
convection, being the thermal resistance matrix given by (9):
0,23 0,25 0,21
R = 0,25 1,82 1,05 
 0,21 1,05 1,84 
(9)
Inverting the matrix is then possible to obtain the thermal
conductances matrix.
7
 5,20 − 0,55 − 0,28
R = G =  − 0,55 0,88 − 0,44
− 0,28 − 0,44 0,83 
Comparing the values obtained for matrix G with forced
convection (10) and for the matrix G with natural convection
(8), it is observed that almost all the values are similar to the
exception to the value obtained for the core conductance.
This fact was expected, once only cooling the core with
forced convection, increases the thermal conductance between
the core and the air, GNu-ar.
As the iron core is the only part of the transformer where the
forced convection acts, all the other conductance values don’t
have significant variations.
With the intention of comparing the behaviour of the
lumped parameters model under forced convection with the
real temperature evolution, a test has been made where the
transformer, that was initially at ambient temperature, placing
it later on operating under nominal conditions until the
temperature rises up to the steady state.
Fig. 12 presents the experimental results and the results
obtained with the simulation. Comparing the results obtained
with forced convection with the ones obtained with natural
convection, it is verified a global temperature reduction in the
transformer giving the indication that the position chosen for
the fan has been efficient.
45
Temperature Rise (ºC)
40
35
TNu-Simul.
TA-Simul.
TBC-Simul.
TNu
TA
TBC
30
25
TABLE III
MAXIMUM ERROR: FORCED COOLING CONVECTION
|Absolute Relative
Temperature Simulated Experimental
Error|
Error %
TNu
5,8ºC
4,5ºC
1,3ºC
28,8
(10)
−1
20
28,5ºC
25,8ºC
2,7ºC
10,5
29,1ºC
26,5ºC
2,6ºC
9,8
From table III, where it is possible to read the model error,
we can verify an increase of it when comparing with the same
test but with natural convection. This error increase is due to
the fact that we are not modelling correctly the forced
convection, it means, representing the complex phenomena as
the convection, which involves mass transportation, just trough
a resistance, what could not be sufficient and therefore
increasing the error value.
V. CONCLUSION
This report presented a lumped model for a dry single-phase
transformer, considering it with several zones with
homogeneous temperature. The model revealed accuracy, with
small error when calculating the transformer temperatures.
With the considered model, it was possible to foresee the
maximum temperature of the hottest point for situations of
overload and short circuits. With the presented tests, it was
intended to point out the importance of using models of
temperature forecast for a transformer in order to be possible
to calculate the overload capacities and the protections time
actuation.
Placing the transformer in a situation of forced cooling
convection, an increase of the model error is verified, this way
concluding that modelling complex phenomena as convection
requires another kind of study.
15
REFERENCES
10
[1]
5
0
0
TA
TBC
[2]
1
2
3
Time (h)
4
5
6
Fig. 12. Forced cooling convection.
During this test it is observed that the maximum temperature
variation in the windings is about 43ºC and without forced
convection the temperature variation increases to 58ºC.
Regarding the core, it is verified a reduction to 12ºC compared
with the 28ºC with natural convection. In general, with forced
convection it is verified a reduction of about 15ºC in the
transformer temperature.
This temperature reduction reveals particularly importance
during overload situations, once it makes possible that the
transformer operates with overload during more time. For
failure situations, as short circuits, the position chosen for the
fan can not be the best one, as it just affects the core thermal
behaviour.
[3]
[4]
[5]
[6]
[7]
[8]
Çengel, Yunus A., Heat Transfer – A Practical Approach, International
Edition, McGraw-Hill, 1998.
Dente, António, Modelo Térmico das Máquinas Eléctricas, Lisboa,
Instituto Superior Técnico, 2002.
Analog Devices, Monolithic Thermocouple Amplifiers with Cold
Junction Compensation, Rev C, Norwood, Analog Devices, 1999.
Lindsay, J. F., Temperature Rise of na Oil-Filled Transformer with
Varying Load, Vol. PAS-103, Montreal, IEEE Transactions on Power
Apparatus, 1984.
Glen, Swift, Molinski, Tom S., Lehn, Waldemar, A Fundamental
Approach to Transformer Thermal Modeling – Part I: Theory and
Equivalent Circuit, Vol. 16, No.2, IEEE Transactions on Power
Delivery, 2001.
Tang, W. H., Wu, Q. H., Richerdson, Z. J., Equivalent Heat Circuit
Based Power Transformer Thermal Model, Vol 149, No. 2, IEE Proc.Electr. Power Appl, 2002.
Tang, W. H., Wu, Q. H., Richerdson, Z. J., A Simplified Transformer
Thermal Model Based on Thermal-Electric Analogy, Vol 19, No. 3,
IEEE Transactions on Power Delivery, 2004
Alvares, Marcelo Carvalho, Samesima, Milton Itsuo, Delaiba, António
Carlos, Análise do Comportamento Térmico de Transformadores
Suprindo Cargas não Lineares Utilizando Modelos Térmicos,
Uberlândia, Universidade Federal da Uberlândia, 1999.
8
[9]
[10]
[11]
[12]
[13]
[14]
[15]
[16]
Resende, Maria José, Thermal Ageing of Distribution Transformers Due
to Load and Ambient Temperature Variability, Lisboa, Instituto
Superior Técnico , 1998.
International Standard - Power transformers - Loading Guide for OilImmersed Power Transformers (IEC 354:1991).
RS Data Sheet, Type J/K/N/T Welded Tip Glass Fiber, 2006.
RS Data Sheet, Thermocouples, 2005.
Susa, Dejan, Dynamic Thermal Modeling of Power Transformers,
Helsinki, Helsinki University of Technology, 2005.
Rizzoni, Giorgio, Thermal Systems – Module 6, Ohio, The Ohio State
University, October 2005.
Montsinger, V. M., Loading Transformers by Temperature, AIEE
Trans., Vol. 49, 1930.
Dakin T. W., Electrical Insulation Deterioration Treated as a Chemical
Rate Phenomena, AIEE Trans., Vol. 67, 1948.