An Improved Lumped Parameter Thermal Model for Electrical Machines
G. Dajaku, D. Gerling
Institute for Electrical Drives, University of Federal Defense Munich
Werner-Heisenberg-Weg 39, D-85577 Neubiberg, Germany, tel: +49 89 6004 3708, fax: +49 89 6004 3718,
e-mail: gurakuq.dajaku@unibw.de, dieter.gerling@unibw.de
ABSTRACT – The lumped-parameter thermal method
(conventional lumped-parameter thermal method) has been
used for a long time for calculation of the temperature rises
in electric machines. In spite of being popular, generally
this method is not applied correctly for elements with
distributed heat generation. To overcome the drawbacks of
this method, in this paper, a novel thermal model for
electrical machines is developed. Using this model, the
thermal performances of a PM machine are analysed. The
accuracy of this model has been verified by comparing
with FEM calculations.
I.
INTRODUCTION
Permanent magnet synchronous machines (PMSM)
gain more and more importance for special drive
applications. Up to recent years, PMSM were known for
small drives, e.g. for servo applications. In the last years,
PMSM are increasingly applied in several areas such as
traction, automobiles, etc. Therefore, development of
different machine types requires attention to the thermal
aspects since, at the end, it is always the thermal
constraints that will determine the power rating of the
machine. To insure a successful design of the electrical
machines, it is necessary to be able to predict an accurate
temperature distribution in the most sensitive parts of the
machine to prevent the damages that can occur either by
breakdown of the stator winding insulation or by the
demagnetization of the magnets. Knowledge of thermal
behaviour in different situations can prevent overheating,
but can also improve the utilization of the system at normal
operation.
As is known, different calculation methods can be
used to analyse the thermal behaviour of electrical
machines: exact analytical calculation (“distributed loss
model”), numerical analysis, and the lumped-parameter or
nodal method (“concentrated loss model”). Compared with
other methods, the lumped-parameter thermal method is
simple and attractive which can give accurate
representation of the thermal conditions within the
machine. This method has been used for a long time by
many authors for calculation of the temperature rises in
electric machines, which solves the thermal problems by
applying thermal networks in analogy to electrical circuits.
Generally, the lumped-parameter thermal model is
composed of thermal resistances, thermal capacitances and
power losses inside the system. Such a model is based on
the hypothesis that the system, under the thermal point of
view, can be divided into several parts that are connected
to each other by means of thermal resistors and capacitors.
In the equivalent thermal network, all the heat generation
in the component is concentrated in one point. This point
represents the mean temperature of the component. In
many literatures [4]-[15], during calculation of thermal
characteristics using the lumped-parameter method, as heat
generators are taken the corresponding power losses in the
components. In spite of being popular, generally this
method is not applied correctly for elements with
distributed heat generation. From the thermal analysis
presented in [1], [2] it is shown that solving the thermal
problems with this method leads to wrong results if the
total power losses are taken as heat sources in the thermal
network (conventional lumped-parameter method). In the
past, this systematic mistake was unknown, but instrinsicly
eliminated by the fitting procedure. Based on the analysis
presented in [1], [2], the required change to the
conventional lumped-parameter thermal method to omit
this systematic mistake is simple, but decisive:
• Calculating a single lossy element, half of the
losses should be applied as heat generator source,
• Calculating a more complex system composed of
more than one lossy element, compensating
elements have to be introduced into the
lumped-parameter thermal network,
• Transient calculations can easily be developed
from the described steady state analysis.
II.
THERMAL MODELING OF ELECRICAL
MACHINES
During the thermal analysis with lumped parameter
method, the electrical machine is divided geometrically
into a number of lumped components, each component
having a bulk thermal storage and heat generation and
interconnenctions to neighbouring components through a
linear mesh of thermal impedances. The lumped
parameters are derived from entirely dimensional
information, the thermal properties of the materials used in
the design, and constant heat transfer coefficients. The
thermal circuit in steady-state condition consists of thermal
resistances and heat sources connected between the
component nodes. For transient analysis, the heat thermal
capacitances are used additionally to take into account the
change in internal energy of the body with the time.
Heat transfer in electrical machines is a combination
of conduction within solid and laminated components, and
convection from surfaces which are in contact with air or
other cooling fluids. This section describes the thermal
modelling procedure of electrical machines with different
rotor topologies. The axial cut of a PM machine with
surface mounted magnets in the rotor is shown in figure1.
Figure 1. The axial cut of the electrical machine
In general, the geometric complexity of an electric
machine requires a large thermal network if a solution with
a high resolution of the temperature distribution is
required. Instead of using a large, complex model, the
geometrical symmetries of the machine were used to
reduce the order of the model. The distributed thermal
properties have been lumped together to form a small
thermal network, representing the whole machine. The
following assumptions have been made during this
analysis:
• The motor has the following symmetrical
characteristics: cylindrical around the shaft and
mirror-like concerning a plane perpendicular to
the shaft in the middle of the shaft length,
• Each cylinder is thermally symmetrical in the
radial direction,
• The inner heat sources are uniformly distributed,
• The heat flux in the axial direction has been
considered only in the shaft. It has been neglected
in the rest of the machine. In this way, the thermal
radial resistance can be computed using the
equations of the hollow cylinder.
Based on the above assumptions, the electric machine
is divided geometrically into a number of components. The
number of the nodes is reduced to seven that provides
thermal access to all key elements inside the machine. One
node (1) is placed in the stator frame. Two nodes (2, 3)
were used for the stator core, which is the largest machine
part usually exposed to high temperature gradients. The
placement of one node (2) is taken to be in the stator yoke
and one in the stator teeth (3). Motor operation at high
torque is associated with large heat dissipation in the
machine windings. Two separate nodes are assigned to the
stator windings, making it possible to separately estimate
the temperatures of the coil sides embedded in the stator
core and the end windings. Therefore, in the model
presented, there are two nodes (4, 5) representing the
windings; one node (5) for the end windings and one for
the coil sides (4). It is assumed that heat dissipated from
the end winding enters the slots via the conductors. As the
end winding reaches the slot, heat starts to flow into the
stator iron core and further out to the stator frame.
Predicting the temperature of the rotor, especially of the
permanent magnets (for a PMSM), is of great importance.
The behaviour of the permanent magnet material at
increased temperature can involve risk for permanent
demagnetization. For this reason, two nodes (6, 7) were
used for the rotor. The placement of one node is taken in
the first rotor layer (bandage) (6) and the other one in the
magnets (7).
According to [1], [2] and the above modeling
concepts, each element is identified by a node in the
equivalent circuit and is associated with a thermal
capacitance, a heat source and corresponding
compensation thermal elements. The nodes are put in the
middle of each corresponding component and are coupled
to their neighbours by thermal resistances. Figure 2 shows
the improved thermal model for the i-th element.
Therefore, development of a thermal model can be
divided into four parts:
• Forming the thermal network for the electric
machine,
• Determining the thermal resistances and
capacitances,
• Determining the compensation thermal elements,
• Determining the losses and their distribution in
the machine.
All these items are important for the calculation of the
thermal performances of the electrical machines.
Figure 2. Lumped-parameter thermal model with compensation
elements for the i-th element
II.1 THERMAL RESISTANCES
It is possible to distinguish between two types of
resistances, depending on the kind of thermal exchange
between the surfaces of the concentric cylinders that they
represent: one is the conduction heat transfer inside solid
masses, and the other one is the convection heat transfer
through the surface separating a solid mass from the fluid.
The conduction thermal resistance in radial direction for a
hollow cylinder can be calculated as follows [3]:
Rconduction =
ln ( ro / ri )
2π ⋅ k ⋅ L
(1)
where, with ro , ri , L, and k are denoted the cylinder
outside radius, inside radius, the axial length, and thermal
conductivity of the material.
If a temperature exchange due to the convection exists,
the heat-transfer rate is related to the overall temperature
difference between the wall and fluid and the convection
area A. The convection resistance is defined as [3]:
Rconvection =
1
h⋅ A
(2)
The quantity h is called the convection heat transfer
coefficient. An analytical calculation of h may be made for
some systems. For a complex situation it must be
determined experimentally. The heat transfer coefficient in
many literatures is called as film conductance because of
its relation to the conduction process in this stationary
layer of fluid at the wall surface.
During thermal modelling of the electric machines,
the mechanical contacts are important, for example,
between the winding and teeth as well as between the
stator core and frame. A poor mechanical contact has gas
pockets which decreases the heat transfer across the joint.
The temperature drop between two contact surfaces is the
result of a thermal contact resistance [3],
Rcontact =
1
hcont ⋅ A
(3)
where, hcont is called the contact coefficient and A is the
contact surface area. The value of the contact coefficient,
hcont , depends on different factors such as: contact surface,
thickness of the void space, thermal conductivity of the
fluid which fills the void space, pressure etc. The contact
coefficient for different materials and surface types is
difficult to be calculated, and has to be determined
experimentally.
II.2 THERMAL CAPACITANCES
In time dependent problems, a representation of the
stored thermal energy in the system is introduced as
thermal capacitances. Each node is assigned with a thermal
capacitance from the node to ambient. The thermal
capacitance of an element is derived from geometrical and
material data of the machine,
Cthi = mi ⋅ ci
(4)
where mi is the mass and ci is the specific heat capacity
of the i-th element.
this effect appears in the slot-teeth region. The heat
generation in the stator slot leads to the wrong temperature
in the teeth region, and afterwards it is reflected to all
components of the electric machines. Therefore, to
compensate this temperature error, additional compesation
temperature elements between the midpoint of the stator
teeth and its corresponding end points have to be used.
II.4 HEAT SOURCES
The heat generation due to losses in the different
machine components was introduced with current
generator parallel connected to the thermal components of
the machine parts that generate these losses. The losses of
an electric machine consist of: stator iron losses, stator
copper losses, rotor losses and friction losses. The power
losses can be calculated analytically or using FE methods.
For each operating condition, the calculated losses of the
electric machine have to be used as the thermal model
inputs. Because the parameters of the electrical machines
(winding resistance R and the magnet remanence Brem) are
temperature dependent parameters, there is a strong
interaction between the electromagnetic and thermal
analysis, i.e. the losses are critically dependent on the
temperature and vice versa.
III. A SIMPLIFIED THERMAL NETWORK
FOR ELECTRICAL MACHINES
Based on the modifications and improvements
proposed in [1], [2], in this paper, a novel lumpedparameter thermal model (lumped-parameter model with
compensation thermal elements) for the thermal analysis of
electrical machines with different rotor topologies is
developed, figure 3. This model consists of a seven-node
network that captures all key machine temperatures. For
PM machines with different rotor topologies the only
resistances that will be different in the thermal model
presented in figure 3 are Rth5 , Rth 6 , and Rth10 .
II.3 COMPENSATION THERMAL ELEMENTS
According to [2], some compensation thermal
elements have to be used in the improved thermal network
to compensate the temperature error obtained from the
conventional lumped-parameter method loaded with the
total power losses. If the nodes of the thermal network are
applied to the middle of each corresponding machine
component (as is performed during the following thermal
analyis of the PM machine), the expression for the
compensation elements for this modeling case is [2]:
tcomp ,i =
PLosses ,i Rth ,i
⋅
4
2
(5)
In analogy to the basic reason for the error of the
conventional lumped-parameter method, where the heat
generation of the regarded element leads to the gradient
temperature error inside the own element, in some cases
the heat generation of one element leads to the temperature
error in the neighbouring component. In electric machines
Figure 3. A novel lumped-parameter model for electrical
machines.
Using the above expressions, the model parameters
can be derived from the entire dimensional information of
the electric machine, the thermal properties of the material
used in the design, and constant heat transfer coefficients.
In the following Tables the thermal resistances and the
compensation elements of the thermal network are
explained briefly.
magnets in the rotor. It has a stator outer diameter of 280
mm and an active length of 68 mm. The motor is designed
for a wide operation speed range (0-6000 rpm). Figure 4
shows the cross-section of the studied machine.
TABLE I: Thermal resistances
Meaning of the thermal resistances
Rth 0
Rth1
Rth 2
Rth 3
R′th 3
Rth 4
Rth 5
Rth 6
Rth 7
Rth8
Rth 9
Rth10
Forced convection thermal resistance between
stator frame and cooling channels
Equivalent thermal resistance between the stator
yoke midpoint and stator frame
Equivalent thermal resistance between the stator
teeth midpoint and stator yoke midpoint
Equivalent thermal resistance between the coil side
and the stator teeth midpoint
Equivalent thermal resistance between the coil side
and the stator yoke midpoint
Equivalent thermal resistance between the coil side
and the end-winding node
Equivalent thermal resistance between the stator
teeth midpoint and the first layer midpoint
Equivalent thermal resistance between the first
layer midpoint and the second layer midpoint
Equivalent thermal resistance between the second
layer midpoint and the stator frame
Equivalent thermal resistance between the endwinding node and the stator frame
Equivalent thermal resistance between the endwinding node and the second layer midpoint
Equivalent thermal resistance between the second
layer midpoint and the ambient
Figure 4. Geometry of the analysed PM machine.
The stator of the PM machine is mounted along its
outer periphery in an aluminium cylinder, which is cooled
by forced water flowing through serial-circumferentially
cooling channels. Furthermore, water-cooling of the stator
frame was chosen to achieve a compact motor
construction, and by making the cooling channels spiral
shaped, the temperature differences along the frame
surface can be kept small. Table III shows the calculated
steady-state temperatures for different operation conditions
when the stator frame temperature is 90 ˚C.
TABLE II: Compensation thermal elements
Meaning of the thermal compensation elements
tY -Fr
Stator yoke – frame
tY -T
Stator yoke – teeth
Stator yoke compensation
thermal elements due to PYoke
tY -Cs
Stator yoke – coil side
losses
tT-Y
Stator teeth – yoke
Stator teeth compensation
thermal elements due to PTeeth
tT-Ag
Stator teeth – air gap
TABLE III: Steady-state temperature
500
Speed [rpm]
3000
6000
[˚C]
90
90
90
T2 -Yoke
[˚C]
104.07
105.56
103.08
T3 -Teeth
[˚C]
140.03
145.28
136.27
[˚C]
195.83
199.67
161.53
T1 -Frame
T4 -Coil side
T5 -End winding [˚C]
203.1
206.4
164.85
tTeeth
losses
Stator teeth compensation thermal elements due to
PCoil -side losses
T6 -Layer 1
[˚C]
122.12
163.93
181.8
tCs-Y
Coil side – stator yoke
T7 -Rotor
[˚C]
121.69
163.9
183.18
tCs-T
Coil side – stator teeth
tCs-Ew
Coil side – end
winding
End winding – coil
side
S
tEw-Cs
tL1-Ag
Layer 1 – stator teeth
tL1-Rc
Layer 1 – rotor core
tRc-L1
Stator coil side compensation
thermal elements due to PCoil -side
losses
IV.1 SENSITIVY ANALYSIS
End-winding comp. thermal
elements due to PEnd -wind losses
First layer compensation
thermal elements due to PLayer1
losses
Rotor core – first layer Second layer compensation
thermal elements due to PLayer 2
losses (magnets)
IV. SIMULATION RESULTS
In the following analysis, the thermal model given in
the figure 3 is implemented to calculate the thermal
performances of a PM machine with surface mounted
It must be denoted here, that the above calculations
are made for an ideal case. The equivalent thermal
conductivity and the thickness of the insulation layer in the
stator slot is taken to be keq = 0.25 W/(m˚C) and xeq = 0.5
mm, respectively. In addition, the frame temperature is
taken to be constant for different operation conditions of
the PM machine. The effect of these parameters on the
thermal performances of the electric machine is discussed
below.
IV.1.1
COOLING EFFECT
Generally, for the water-cooled electric machine in the
stator frame, the total heat generated in the electric
machine flows throughout the components of the electric
machine to the frame and then by the convection the heat is
transferred away to the coolant. In the figure 3, the heat
transfer from the stator frame to the coolant is presented
with a convective thermal resistance Rth 0 . This parameter
depends on many factors, such as geometry of the cooling
surface, cooling effectivity etc., and requires a careful
consideration because of its position in the main heat flow
path of the stator losses to the ambient. To show the
influence of the stator frame convection heat transfer
coefficient in the thermal performances of the PM
machine, the temperature is calculated for the different
stator frame cooling conditions. The following calculations
are done for the case when the coolant temperature is 90
ºC, and the PM machine operates at 500 rpm. The other
parameters of the thermal model are taken to be the same
as in the previous calculation. Table IV shows the
temperature variation in the PM machine for different
convection heat transfer coefficients.
⎣⎡W /(m ⋅ C) ⎤⎦
h
Rth 0
o
[˚C/W]
the air-gaps between the conductors.
The equivalent thermal conductivity is difficult to be
determined analytically. For this parameter, a thermal test,
proposed in [4], with a stator DC supply is simply
requested. According to the experimental results made for
an induction motor [15], the equivalent thermal
conductivity can be considered in the range of 0.06 to 0.09
W/(m˚C).
In the above thermal analysis, an insulation material
with thermal conductivity keq = 0.25 W/(m˚C) and
thickness xeq = 0.5 mm is taken during the simulation. To
show the influence of the equivalent insulation layer on the
thermal performances of the electric machine, the
following calculations are made for different values of keq .
For h = 12000 W /(m 2 ⋅ o C) stator frame convection
coefficient, the steady-state temperatures of the electric
machine at 500 rpm are presented in the Table V.
TABLE V: Influence of slot insulation material
TABLE IV: Influence of the stator frame cooling
2
-
0.25
0.1
0.06
[˚C/W]
0.0148
0.0360
0.0597
[˚C/W]
0.0654
0.1574
0.2597
1000
6000
12000
keq
[W/(m˚C)]
0.0128
0.0023
0.0012
Rth 3
T1 -Frame
[˚C]
156.8
102.13
96.33
Rth′ 3
T2 -Yoke
[˚C]
170.69
116.16
110.38
T1 -Frame
T3 -Teeth
[˚C]
206.06
152.02
146.29
T4 -Coil side
[˚C]
262
207.84
202.12
[˚C]
96.33
96.33
96.33
T2 -Yoke
[˚C]
110.38
110.21
110
T3 -Teeth
[˚C]
146.29
148.45
148.57
[˚C]
202.12
291.95
387.98
T5 -End winding [˚C]
269.17
215.1
209.36
T4 -Coil side
T6 -Layer 1
[˚C]
159.78
128.96
125.69
T5 -End winding [˚C]
209.36
298.63
394.05
T7 -Rotor
[˚C]
T6 -Layer 1
[˚C]
125.69
129.21
131.91
T7 -Rotor
[˚C]
125.21
128.73
131.48
158.82
128.42
125.21
Table IV shows that the frame temperature (and as a
consequence the temperatures at the other parts of the PM
machine) varies with the variation of the heat transfer
coefficient. Increasing the cooling intensity (using an
effective cooling method), clearly decreases the
temperature in all components of the machine.
IV.1.2 INFLUENCE OF THE SLOT INSULATION
MATERIAL
Various modelling strategies have been developed by
many authors to model the heat transfer and temperature
distribution within a winding. One of them is the
composite thermal conductivity method [4] in which the
calculations of the thermal resistance perpendicular to the
winding conductors were made by considering the
insulated conductors and impregnation material as
homogeneous body of slot material. The slot region is
modeled with an homogeneous material (copper) at the slot
center, insulated with an equivalent insulation layer. This
insulation layer is characterised by an equivalent thermal
conductivity keq , which takes into account the thermal
Table V shows that the equivalent thermal
conductivity of the insulation layer has a significant effect
on the stator windings temperature. To improve the
thermal characteristics of the electric machine, especially
in the stator windings, the slot insulation material with
high thermal conductivity must be used.
From the above analysis it is shown that the stator
frame temperature depends on the efficiency of the cooling
method and the generated power losses in the electric
machine. The variation of the convection heat transfer
coefficient in the stator frame alters the temperature rise in
all components of the electric machine. Otherwise, the end
winding temperature is of great importance since it is the
part in the machine, which reaches the highest temperature.
As is shown in Table V, Rth3 and Rth′ 3 affect the predicted
end winding temperature significantly. Focusing on
regions with high temperature gradients is especially
important here. In this case, the slot insulation and air
pockets around the stator windings should be brought to
attention.
conductivity of:
-
the slot insulation layer,
air-gap between the slot insulation and the
laminations,
the insulation varnish of the windings, and
IV.2 COMPARISON WITH FE METHODS
In this section the proposed thermal model is
validated by comparison with FE methods. Figure 5 shows
the temperature distribution inside the PM machine. The
FE simulations are performed under the following
conditions: 500 rpm operating speed, 90 °C coolant
temperature, h = 12000 W /(m2 ⋅ o C) (coolant convection
coefficient), and keq = 0.25 W/(m˚C) (thermal conductivity
of the slot insulation layer).
machine. The influence of the equivalent slot insulation
material and the cooling efficiency on the thermal
performances of the studied PM machine is investigated.
Furthermore, a comparison with FEM is performed. It is
shown that the calculation results obtained with the
improved lumped-parameter method are in good
agreement with the ANSYS results.
VI.
Figure 5: Temperature distribution inside the PM machine.
Under the same simulation conditions, Table VI compares
the results obtained from the improved lumped-parameter
thermal method, conventional lumped-parameter thermal
method, and from ANSYS thermal analysis.
TABLE VI: Comparison of results.
Temperature of the PM
machine at 500 rpm
T1 -Frame
[˚C]
ANSYS
2D
97.26
Improved
LPM
96.33
Conventional
LPM
96.33
110.7
110.38
110.36
T2 -Yoke
[˚C]
T3 -Teeth
[˚C]
145.6
146.29
152
[˚C]
202.49
202.12
207.2
T4 -Coil side
T5 -End winding [˚C]
--
209.36
222.35
125.69
125.78
125.21
125.32
T6 -Layer 1
[˚C]
124.6
T7 -Magnets
[˚C]
124.01
As is shown in the Table VI, the calculation results
obtained with the improved lumped-parameter method are
in good agreement with the ANSYS results. Comparing the
conventional lumped-parameter method and the ANSYS
results, it is shown that in some sections of the studied
electric machine such as in the stator teeth, coil side, and
end winding, there is a relatively large discrepancy
between the results. Based on the expression for the
compensation thermal elements presented in the section
II.3, the deviation of the results obtained with conventional
lumped-parameter thermal method from an accurate
calculation method (FEM, improved lumped-parameter
method) depends on the operation conditions of the studied
machine (power losses), thermal characteristics of the
materials, and the operation temperature (thermal
conductivity of the materials vary with the temperature).
V.
CONCLUSION
In this paper, an improved lumped-parameter thermal
model (lumped-parameter method with compensation
thermal elements) for the thermal performances of
electrical machines is developed and analysed. This model
has been applied to predict the thermal behaviour of a PM
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