# Simulation environment for designing electric drives in commercial vehicles, heavy duty machines and ships

**When electric drive systems are designed for commercial vehicles, the variety of operating conditions presents special challenges. Statements frequently have to be made on the achievable driving performance and electrical energy consumption based on a designed but not yet mass-produced electric machine and its associated inverter. With this objective in mind, a simulation environment was developed that contains a route calculation in the time domain, integrates the motor currents in discrete time steps, and calculates the losses and temperatures with the efficiency of the motor and inverter. This paper focuses on the analytical calculation of the critical winding temperature of electric machines. The simulation tool only requires a few main data and is therefore also suitable for continuously calculating the tem-peratures during driving operation. In addition to data on power losses, it also pro-vides a statement on whether the electric machine and inverter are sufficiently di-mensioned for the required task.**

## 1. Challenges in project planning for drives

When commercial vehicles are equipped with electric drives, special requirements often have to be met that can differ greatly from those of passenger cars and are more varied overall; commercial vehicles have a very wide range of vehicle masses. There can be considerable differences in mass when a vehicle is empty or loaded, carrying a trailer, or operating an attachment. The routes to be traveled can either be unfamiliar and require assumptions, or they can be relatively precisely defined, as in the case of public buses or factory transport vehicles.

A particular challenge in designing commercial vehicle drives is the correct dimen-sioning of the inverter and motor. Cost, weight, and installation space advantages are reasons for choosing smaller aggregates. At the same time, the short-term overload capability compared to the continuously possible performance should be systematically used. However, the requirements for operational safety, performance, and service life of the drive must always be met.

## 1. Challenges in project planning for drives

When commercial vehicles are equipped with electric drives, special requirements often have to be met that can differ greatly from those of passenger cars and are more varied overall; commercial vehicles have a very wide range of vehicle masses. There can be considerable differences in mass when a vehicle is empty or loaded, carrying a trailer, or operating an attachment. The routes to be traveled can either be unfamiliar and require assumptions, or they can be relatively precisely defined, as in the case of public buses or factory transport vehicles.

A particular challenge in designing commercial vehicle drives is the correct dimen-sioning of the inverter and motor. Cost, weight, and installation space advantages are reasons for choosing smaller aggregates. At the same time, the short-term overload capability compared to the continuously possible performance should be systematically used. However, the requirements for operational safety, performance, and service life of the drive must always be met.

## 2. The simulation environment

## 2. The simulation environment

### 2.1 Objectives and areas of application

Aradex AG has created a simulation environment that makes it faster and safer to plan electric drives in commercial vehicles.

The electrical and thermal behavior of the drive train (fig. 1) can be simulated taking into account the data of the vehicle, the route profile, and other operating conditions. The simulation serves several objectives:

- Calculation of the efficiency of the motor and inverter in driving operation
- Determination of the energy consumption and thus the range
- Determination of the temperatures occurring in the drive (motor, inverter)
- Determination of the driving performance for a given route profile

### 2.1 Objectives and areas of application

Aradex AG has created a simulation environment that makes it faster and safer to plan electric drives in commercial vehicles.

The electrical and thermal behavior of the drive train (fig. 1) can be simulated taking into account the data of the vehicle, the route profile, and other operating conditions. The simulation serves several objectives:

- Calculation of the efficiency of the motor and inverter in driving operation
- Determination of the energy consumption and thus the range
- Determination of the temperatures occurring in the drive (motor, inverter)
- Determination of the driving performance for a given route profile

### 2.2 Input data for the vehicle, the route, and the environmental conditions

The driving cycles known from the passenger car sector are essentially defined by value pairs of speed and time and are therefore usually inadequate for commercial vehicles with their wide range of applications. Here it is necessary to simulate driving on real routes which are defined, for example, by their geodata, i.e., a sufficient number of points with a geographical longitude, latitude, and altitude above sea level, and the respective assigned target or maximum speed values. The actual driving speed can only be determined in the simulation calculation, where various data can be taken into account: slopes of the route, the speed and sign of wind, characteristics of the road surface, additional forces (e.g. a snow blade), or the power of auxiliary aggregates.

### 2.2 Input data for the vehicle, the route, and the environmental conditions

The driving cycles known from the passenger car sector are essentially defined by value pairs of speed and time and are therefore usually inadequate for commercial vehicles with their wide range of applications. Here it is necessary to simulate driving on real routes which are defined, for example, by their geodata, i.e., a sufficient number of points with a geographical longitude, latitude, and altitude above sea level, and the respective assigned target or maximum speed values. The actual driving speed can only be determined in the simulation calculation, where various data can be taken into account: slopes of the route, the speed and sign of wind, characteristics of the road surface, additional forces (e.g. a snow blade), or the power of auxiliary aggregates.

### 2.3 Route simulation in the time domain

A driver model with a look-ahead function is used to start braking in time to meet the speed requirements from the route file. In this driver model, the agility of the driving style, i.e., the acceleration and deceleration values, can also be limited.

The target acceleration is used to calculate the required current strength that the inverter is to supply to the motor in the drive train model, which includes the gearbox, electric machine, and inverter. This calculation follows the procedure of the ARADEX inverter, which calculates and controls the optimum current in regard to strength and angle for each operating point of the motor, so that the total current strengths and thus the losses are minimal. The simulation tool dynamically takes into account the possible limitations of the current strength due to a high temperature in the motor or inverter as well as definable current limits of the battery.

The motor torque is determined from the controlled current in conjunction with the applied speed. This is used to calculate the actual acceleration in the vehicle mod-el. This is integrated over time so that the new speed is calculated as the result.

The repetition frequency of the calculation can be adapted to the simulation task.

### 2.3 Route simulation in the time domain

A driver model with a look-ahead function is used to start braking in time to meet the speed requirements from the route file. In this driver model, the agility of the driving style, i.e., the acceleration and deceleration values, can also be limited.

The target acceleration is used to calculate the required current strength that the inverter is to supply to the motor in the drive train model, which includes the gearbox, electric machine, and inverter. This calculation follows the procedure of the ARADEX inverter, which calculates and controls the optimum current in regard to strength and angle for each operating point of the motor, so that the total current strengths and thus the losses are minimal. The simulation tool dynamically takes into account the possible limitations of the current strength due to a high temperature in the motor or inverter as well as definable current limits of the battery.

The motor torque is determined from the controlled current in conjunction with the applied speed. This is used to calculate the actual acceleration in the vehicle mod-el. This is integrated over time so that the new speed is calculated as the result.

The repetition frequency of the calculation can be adapted to the simulation task.

### 2.4 Efficiencies and temperature calculations

Losses in the inverter occur at diodes and IGBTs. Simulation tools from the manu-facturers are used to calculate these losses. The heat sink temperature is calculated from the coolant flow rate and the coolant input temperature according to the design; the heat sink temperature and the power loss produce the temperature of the chips. This temperature limits the current that the inverter can supply.

The temperatures that result from the losses limit the torque and power for the elec-tric machine as well. In principle, electric machines are very well suited for the oper-ating requirements of vehicles, as these machines can be strongly overloaded for a short time beyond their continuous output limit. The enamel of the copper conductors in the coils, especially in the so-called winding heads (fig. 2), i.e., the loops of the winding wire that extend beyond the laminated core and are cooled the worst in typical PM machines, is critical with regard to overheating. Another limitation is caused by the properties of the permanent magnets in the rotor.

*Fig. 2: Winding head of a permanent magnet-excited electric machine for commercial vehicles with 300/600 kW continuous / peak power*

It is therefore important to limit the amount and duration of the current load to ensure that the temperature limit is not exceeded at any point on the machine at any time. However, this should not result in a hard switch-off of the current. The power should instead be reduced so that the driver can recognize it and adapt his or her driving style accordingly.

Constant temperature measurement leads to high construction costs and is not pos-sible at all during the project planning phase. This is why simulation is so important here. The converted power of vehicle drives can vary greatly over time. The simula-tion task therefore requires the calculation of the transient, i.e., time-dependent, temperature behavior.

Because it is to be used in project planning, the method described here must do without CAD data from the electric machine. In the area of thermodynamics, it there-fore works with analytical equations derived from basic equations of heat conduction. Therefore, suitable simplifications are necessary to reduce the complexity and still consider the essential effects.

- The rotor and rotor shaft are considered as one body with a homogeneous tem-perature distribution.
- The heat input to be assigned to the stator at an imaginary cylinder surface is as-sumed at a stator diameter close to the air gap. Temperature differences between the copper wires of the coils and the teeth between them made of electrical steel, the coil cores, are neglected. Averaged values for temperature, heat conduction, and heat capacity are assumed for both of them together. These assumptions are supported by an FEM temperature model of a single tooth of the coil that was studied as an example.
- Heat radiation and convection on the external surface of the housing are negligible in comparison to the heat transfer via the cooling medium.

#### 2.4.1 Stationary temperature behavior

The equations for calculating the stationary temperature behavior, i.e., the steady-state temperatures, are drawn up as a basis for validating the transient calculation because they can be checked relatively easily with test results. The investigations are based on a PM machine with 300 / 600 kW continuous / peak power at speeds up to 3500 rpm. Rotor losses are low with the electric machine design that is considered here and are not taken into account.

The stator losses are conducted to the cooling jacket via the electromagnetically active part of the motor, the laminated core. To be able to calculate with a flat mod-el, an axially homogeneous temperature distribution is first assumed. The uneven distribution due to the introduction of heat at the end from the winding heads is superimposed in a later calculation step. In the area of its active length, the motor resembles a multi-layer tube with a rotationally symmetrical cross-section (see fig. 3).

Fig. 3

*Layered model: simplified cross section through a cylindrical PM machine.*

*Shaft, material steel**Laminated core rotor, material electrical steel (magnetic material neglected)**air gap**Spool area with teeth, material properties averaged from copper, potting material and electrical steel according to their mass fractions**Winding-free area of the stator, material electrical steel**Massive area of the cooling jacket, material aluminum**Area with cooling channels, material properties averaged out of aluminum and cooling medium**Welded cooling channel cover, material aluminum*

The losses occurring in the copper wires of the winding are divided according to the proportionate wire lengths or copper masses on the winding heads and the winding portion in the grooves of the laminated core. Each winding head has a certain power loss, which is evenly distributed over the unwound wire length of the winding head (here 160 mm, see fig. 4) and is assumed to be dissipated symmetrically on both sides into the grooves exclusively by heat conduction in the copper wire.

*Fig. 4: Temperature profile in a wire of the winding head from groove to groove*

To derive the equation, it is therefore sufficient to consider half of the wire loop with the positive length coordinate from l = 0 to lanb = 80 mm and the proportional power loss PV. The heat flux Q ̇ to be conducted thus increases linearly from the middle of the wire loop to both sides with the amount of the length coordinate l:

(eq. 1)

Transformation and integration of the equation yield the temperature curve in the winding head. Based on the temperature Tanb at the connection points at the transi-tion to the groove, this corresponds to a square parabola whose vertex at l = 0 repre-sents the maximum temperature in the middle of a wire in the winding head (fig. 4).

Within the groove area, the final introduction of heat from the winding heads leads to an increase in temperature compared to the temperature already calculated from the total power dissipation in the layer model. For the course of this temperature increase over the longitudinal coordinate x of the machine, an analytical approach similar to that of a cooling fin can be selected when heat is introduced from one winding head: Heat is transported over a certain cross-section with a defined thermal conductivity in the longitudinal direction and simultaneously dissipated laterally over the surface with a certain heat transfer coefficient. The lateral heat dissipation reduces the heat flow remaining in the longitudinal direction and thus also the temperature gradient between the copper wire and the cooling medium that drives the lateral heat dissipation. Transformation and integration of the approach produce a differential equation with the following solution for the temperature curve along the wire in the groove:

with: TU: ambient temperature [° C]

h: heat transfer coefficient wire to cooling [W / (m^{2} · K)]

L: half of the groove length [m]

U: circumference of the wire [m]

λ: thermal conductivity of the wire material (copper) [W / (K · m)]

A: cross-sectional area of the wire [m²]

ΔT_{1} is the temperature difference between the wire and the cooling medium at the beginning of the groove where the winding head is connected (x = 0). The heat flow at this point is:

(eq. 3)

Since this heat flow from the winding head has already been calculated, the temperature difference between the wire and the cooling medium at the inlet to the laminated core can be calculated by changing the equation:

(eq. 4)

The end of a cooling fin is characterized by the fact that heat conduction in the longitudinal direction is no longer possible there. In our case with a copper conductor in the groove, this must be true if the machine is symmetrically constructed in the middle of the laminated core, i.e., at half the active length, because a heat flow of the same magnitude would have to occur there from the opposite side, so that the sum is zero. We call this place x = L and get:

Fig. 5 shows the graph for the length coordinates x = 0 to x = L = 180 mm (center of the active length) for the heat introduced from one half of the winding head (blue line). For symmetry reasons, a mirror image is to be expected for the other half of the active length.

*Fig. 5: Temperature component from the initiation of the heat loss from the windings, plotted for the slot area (active length)*

In order to obtain correct temperature values in the layer model, the heat flow from the winding heads had to be taken into account. It is now a matter of determining the temperature increase resulting from the lateral introduction of this heat from the winding heads at their connection point. This is read off as the difference between the temperature at the beginning of the groove (blue line, length x = 0) and the arithmetic mean value (red line): 4.3 K. Due to the linearity of the heat flow in relation to the temperature difference valid for heat conduction, the derived heat flow is proportional to the area under the lines and thus the same for both lines. The arithmetic mean value thus corresponds to the temperature from the layer model for the warmest copper conductor in the groove, 42 °C in the example. A temperature T_{anb} = 46.3 °C can be expected at the end of the groove.

In order to obtain the maximum temperature in the winding head, the previously calculated temperature range of 3.3 K within the winding head must now be added, and T_{max} = 49.6 °C is obtained (fig. 6: blue dot at the actual start of the winding at radius r = 180 mm). Towards the outer radius r = 211 mm of the coil, a decrease of the winding head temperature corresponding to the temperature in the groove is expected.

*Fig. 6: Radial temperature profile from the air gap (r = 180 mm) to the cooling jacket outside (r = 273 mm)*

#### 2.4.2 Transient temperature behavior

The calculation of the stationary temperature behavior can only be regarded as a preliminary stage of the actual task: the calculation of the time-dependent, i.e., transient, temperature behavior as a function of the power losses that occur in the electric machine. The transient calculation requires the consideration of heat capacities in addition to the quantities for heat transfer by conduction, convection, and radiation (as far as this is not negligible).

##### 2.4.2.1 Analogies to electrical components

Under certain conditions, a system of bodies with a heat capacity that touch each other so that heat conduction occurs behaves similarly to an electrical system of ca-pacitors and resistors. One of the prerequisites is that the thermal conductivity within the heat-storing bodies is very good compared to the transition resistances to the adjacent bodies.

When the thermal behavior of the electric machine is calculated, heat current sources result from the electrical power losses. Since heat also flows through the material during a warm-up process, the heat flow that enters a thermal mass is the sum of the heat flowing off and the heat required for a temperature change at a given heat capacity of the mass.

An electrical model that reproduces this process consists, for each thermal mass, of a current source, a capacitor as a storage device, and an ohmic resistor connected in parallel through which current flows to the next component or directly to "earth".

##### 2.4.2.2 Creation of a thermal model with three masses

Applied to the type of PM machines to be calculated, a model is created which has three thermal masses (fig. 7): 1. rotor, 2. winding heads, 3. rest of the stator without the winding heads: groove areas, laminated core, and cooling jacket.

*Fig. 7: Thermal model of a PM machine with 3 heat capacities*

The combination of the copper found in the grooves, the insulating and impregnating varnish between them, the laminated core, and the aluminum housing with its cooling channels is regarded as one thermal mass, or "stator mass" for short. This simplification is permissible because the temperature gradient within these components is relatively flat compared with the temperature difference from the winding heads (see fig. 6). The heat capacity of the stator C_{St} is the sum of the heat capacities of the contained materials, each multiplied by the proportional masses. Their temperature T_{St} refers to the arithmetic mean over the active length for the warmest copper conductor in the groove as calculated in the layer model. The thermal resistance R_{St} is determined from the temperature difference T_{St}-T_{K} to the cooling water and the heat flow Q ̇ from the total power loss of the winding heads and stator:

(eq. 5)

The power loss in the stator P_{V_St} contains proportions of iron and copper losses. The winding heads with the power loss P_{V_Wi} (only proportional copper losses) and the heat capacity C_{Wi} are connected via the thermal resistance R_{Wi-St}, which takes into account the temperature increase at the beginning or end of the groove where the winding heads are connected as well as the temperature difference from there to the warmest point in the middle of the individual wire loops of the winding heads.

In order to be able to record the losses in the rotor and to calculate its temperature T_{Ro}, the model contains the associated power loss source P_{V_Ro} and a mass with the heat capacity C_{Ro}. The air gap is modelled as the heat conduction resistance R_{L}.

##### 2.4.2.3 Differential equations for three thermal masses

The thermal power P absorbed by a mass (with a negative sign: emitted) is equal to the product of thermal capacity C and the temporal temperature change dT/dt:

(eq. 6)

The product of the heat conduction resistance R and the flowing heat output P is the temperature difference ΔT_{R}:

(eq. 7)

The desired temperature vector y ⃗ contains the temperatures of the winding head, stator, and rotor. The input variables of the calculation in the vector u ⃗ are the power losses of the winding heads P_{V_Wi}, in the stator P_{V_St}, in the rotor P_{V_Ro}, and the cooling water temperature T_{K}.

The model requires the calculation of the following differential equation system:

(eq. 8)

The influence coefficients a_{ij} and b_{ik} contain terms from the heat capacities and heat resistances specific to the machine, which are derived from the static calculation model. Some are zero. The method can also be extended to a larger number of thermal masses. A model with four masses is currently being developed.

### 2.4 Efficiencies and temperature calculations

Losses in the inverter occur at diodes and IGBTs. Simulation tools from the manu-facturers are used to calculate these losses. The heat sink temperature is calculated from the coolant flow rate and the coolant input temperature according to the design; the heat sink temperature and the power loss produce the temperature of the chips. This temperature limits the current that the inverter can supply.

The temperatures that result from the losses limit the torque and power for the elec-tric machine as well. In principle, electric machines are very well suited for the oper-ating requirements of vehicles, as these machines can be strongly overloaded for a short time beyond their continuous output limit. The enamel of the copper conductors in the coils, especially in the so-called winding heads (fig. 2), i.e., the loops of the winding wire that extend beyond the laminated core and are cooled the worst in typical PM machines, is critical with regard to overheating. Another limitation is caused by the properties of the permanent magnets in the rotor.

*Fig. 2: Winding head of a permanent magnet-excited electric machine for commercial vehicles with 300/600 kW continuous / peak power*

It is therefore important to limit the amount and duration of the current load to ensure that the temperature limit is not exceeded at any point on the machine at any time. However, this should not result in a hard switch-off of the current. The power should instead be reduced so that the driver can recognize it and adapt his or her driving style accordingly.

Constant temperature measurement leads to high construction costs and is not pos-sible at all during the project planning phase. This is why simulation is so important here. The converted power of vehicle drives can vary greatly over time. The simula-tion task therefore requires the calculation of the transient, i.e., time-dependent, temperature behavior.

Because it is to be used in project planning, the method described here must do without CAD data from the electric machine. In the area of thermodynamics, it there-fore works with analytical equations derived from basic equations of heat conduction. Therefore, suitable simplifications are necessary to reduce the complexity and still consider the essential effects.

- The rotor and rotor shaft are considered as one body with a homogeneous tem-perature distribution.
- The heat input to be assigned to the stator at an imaginary cylinder surface is as-sumed at a stator diameter close to the air gap. Temperature differences between the copper wires of the coils and the teeth between them made of electrical steel, the coil cores, are neglected. Averaged values for temperature, heat conduction, and heat capacity are assumed for both of them together. These assumptions are supported by an FEM temperature model of a single tooth of the coil that was studied as an example.
- Heat radiation and convection on the external surface of the housing are negligible in comparison to the heat transfer via the cooling medium.

#### 2.4.1 Stationary temperature behavior

The equations for calculating the stationary temperature behavior, i.e., the steady-state temperatures, are drawn up as a basis for validating the transient calculation because they can be checked relatively easily with test results. The investigations are based on a PM machine with 300 / 600 kW continuous / peak power at speeds up to 3500 rpm. Rotor losses are low with the electric machine design that is considered here and are not taken into account.

The stator losses are conducted to the cooling jacket via the electromagnetically active part of the motor, the laminated core. To be able to calculate with a flat mod-el, an axially homogeneous temperature distribution is first assumed. The uneven distribution due to the introduction of heat at the end from the winding heads is superimposed in a later calculation step. In the area of its active length, the motor resembles a multi-layer tube with a rotationally symmetrical cross-section (see fig. 3).

Fig. 3

*Layered model: simplified cross section through a cylindrical PM machine.*

*Shaft, material steel**Laminated core rotor, material electrical steel (magnetic material neglected)**air gap**Spool area with teeth, material properties averaged from copper, potting material and electrical steel according to their mass fractions**Winding-free area of the stator, material electrical steel**Massive area of the cooling jacket, material aluminum**Area with cooling channels, material properties averaged out of aluminum and cooling medium**Welded cooling channel cover, material aluminum*

The losses occurring in the copper wires of the winding are divided according to the proportionate wire lengths or copper masses on the winding heads and the winding portion in the grooves of the laminated core. Each winding head has a certain power loss, which is evenly distributed over the unwound wire length of the winding head (here 160 mm, see fig. 4) and is assumed to be dissipated symmetrically on both sides into the grooves exclusively by heat conduction in the copper wire.

*Fig. 4: Temperature profile in a wire of the winding head from groove to groove*

To derive the equation, it is therefore sufficient to consider half of the wire loop with the positive length coordinate from l = 0 to lanb = 80 mm and the proportional power loss PV. The heat flux Q ̇ to be conducted thus increases linearly from the middle of the wire loop to both sides with the amount of the length coordinate l:

(eq. 1)

Transformation and integration of the equation yield the temperature curve in the winding head. Based on the temperature Tanb at the connection points at the transi-tion to the groove, this corresponds to a square parabola whose vertex at l = 0 repre-sents the maximum temperature in the middle of a wire in the winding head (fig. 4).

Within the groove area, the final introduction of heat from the winding heads leads to an increase in temperature compared to the temperature already calculated from the total power dissipation in the layer model. For the course of this temperature increase over the longitudinal coordinate x of the machine, an analytical approach similar to that of a cooling fin can be selected when heat is introduced from one winding head: Heat is transported over a certain cross-section with a defined thermal conductivity in the longitudinal direction and simultaneously dissipated laterally over the surface with a certain heat transfer coefficient. The lateral heat dissipation reduces the heat flow remaining in the longitudinal direction and thus also the temperature gradient between the copper wire and the cooling medium that drives the lateral heat dissipation. Transformation and integration of the approach produce a differential equation with the following solution for the temperature curve along the wire in the groove:

with: TU: ambient temperature [° C]

h: heat transfer coefficient wire to cooling [W / (m^{2} · K)]

L: half of the groove length [m]

U: circumference of the wire [m]

λ: thermal conductivity of the wire material (copper) [W / (K · m)]

A: cross-sectional area of the wire [m²]

ΔT_{1} is the temperature difference between the wire and the cooling medium at the beginning of the groove where the winding head is connected (x = 0). The heat flow at this point is:

(eq. 3)

Since this heat flow from the winding head has already been calculated, the temperature difference between the wire and the cooling medium at the inlet to the laminated core can be calculated by changing the equation:

(eq. 4)

The end of a cooling fin is characterized by the fact that heat conduction in the longitudinal direction is no longer possible there. In our case with a copper conductor in the groove, this must be true if the machine is symmetrically constructed in the middle of the laminated core, i.e., at half the active length, because a heat flow of the same magnitude would have to occur there from the opposite side, so that the sum is zero. We call this place x = L and get:

Fig. 5 shows the graph for the length coordinates x = 0 to x = L = 180 mm (center of the active length) for the heat introduced from one half of the winding head (blue line). For symmetry reasons, a mirror image is to be expected for the other half of the active length.

*Fig. 5: Temperature component from the initiation of the heat loss from the windings, plotted for the slot area (active length)*

In order to obtain correct temperature values in the layer model, the heat flow from the winding heads had to be taken into account. It is now a matter of determining the temperature increase resulting from the lateral introduction of this heat from the winding heads at their connection point. This is read off as the difference between the temperature at the beginning of the groove (blue line, length x = 0) and the arithmetic mean value (red line): 4.3 K. Due to the linearity of the heat flow in relation to the temperature difference valid for heat conduction, the derived heat flow is proportional to the area under the lines and thus the same for both lines. The arithmetic mean value thus corresponds to the temperature from the layer model for the warmest copper conductor in the groove, 42 °C in the example. A temperature T_{anb} = 46.3 °C can be expected at the end of the groove.

In order to obtain the maximum temperature in the winding head, the previously calculated temperature range of 3.3 K within the winding head must now be added, and T_{max} = 49.6 °C is obtained (fig. 6: blue dot at the actual start of the winding at radius r = 180 mm). Towards the outer radius r = 211 mm of the coil, a decrease of the winding head temperature corresponding to the temperature in the groove is expected.

*Fig. 6: Radial temperature profile from the air gap (r = 180 mm) to the cooling jacket outside (r = 273 mm)*

#### 2.4.2 Transient temperature behavior

The calculation of the stationary temperature behavior can only be regarded as a preliminary stage of the actual task: the calculation of the time-dependent, i.e., transient, temperature behavior as a function of the power losses that occur in the electric machine. The transient calculation requires the consideration of heat capacities in addition to the quantities for heat transfer by conduction, convection, and radiation (as far as this is not negligible).

##### 2.4.2.1 Analogies to electrical components

Under certain conditions, a system of bodies with a heat capacity that touch each other so that heat conduction occurs behaves similarly to an electrical system of ca-pacitors and resistors. One of the prerequisites is that the thermal conductivity within the heat-storing bodies is very good compared to the transition resistances to the adjacent bodies.

When the thermal behavior of the electric machine is calculated, heat current sources result from the electrical power losses. Since heat also flows through the material during a warm-up process, the heat flow that enters a thermal mass is the sum of the heat flowing off and the heat required for a temperature change at a given heat capacity of the mass.

An electrical model that reproduces this process consists, for each thermal mass, of a current source, a capacitor as a storage device, and an ohmic resistor connected in parallel through which current flows to the next component or directly to "earth".

##### 2.4.2.2 Creation of a thermal model with three masses

Applied to the type of PM machines to be calculated, a model is created which has three thermal masses (fig. 7): 1. rotor, 2. winding heads, 3. rest of the stator without the winding heads: groove areas, laminated core, and cooling jacket.

*Fig. 7: Thermal model of a PM machine with 3 heat capacities*

The combination of the copper found in the grooves, the insulating and impregnating varnish between them, the laminated core, and the aluminum housing with its cooling channels is regarded as one thermal mass, or "stator mass" for short. This simplification is permissible because the temperature gradient within these components is relatively flat compared with the temperature difference from the winding heads (see fig. 6). The heat capacity of the stator C_{St} is the sum of the heat capacities of the contained materials, each multiplied by the proportional masses. Their temperature T_{St} refers to the arithmetic mean over the active length for the warmest copper conductor in the groove as calculated in the layer model. The thermal resistance R_{St} is determined from the temperature difference T_{St}-T_{K} to the cooling water and the heat flow Q ̇ from the total power loss of the winding heads and stator:

(eq. 5)

The power loss in the stator P_{V_St} contains proportions of iron and copper losses. The winding heads with the power loss P_{V_Wi} (only proportional copper losses) and the heat capacity C_{Wi} are connected via the thermal resistance R_{Wi-St}, which takes into account the temperature increase at the beginning or end of the groove where the winding heads are connected as well as the temperature difference from there to the warmest point in the middle of the individual wire loops of the winding heads.

In order to be able to record the losses in the rotor and to calculate its temperature T_{Ro}, the model contains the associated power loss source P_{V_Ro} and a mass with the heat capacity C_{Ro}. The air gap is modelled as the heat conduction resistance R_{L}.

##### 2.4.2.3 Differential equations for three thermal masses

The thermal power P absorbed by a mass (with a negative sign: emitted) is equal to the product of thermal capacity C and the temporal temperature change dT/dt:

(eq. 6)

The product of the heat conduction resistance R and the flowing heat output P is the temperature difference ΔT_{R}:

(eq. 7)

The desired temperature vector y ⃗ contains the temperatures of the winding head, stator, and rotor. The input variables of the calculation in the vector u ⃗ are the power losses of the winding heads P_{V_Wi}, in the stator P_{V_St}, in the rotor P_{V_Ro}, and the cooling water temperature T_{K}.

The model requires the calculation of the following differential equation system:

(eq. 8)

The influence coefficients a_{ij} and b_{ik} contain terms from the heat capacities and heat resistances specific to the machine, which are derived from the static calculation model. Some are zero. The method can also be extended to a larger number of thermal masses. A model with four masses is currently being developed.

### 2.5 Calculation results

Fig. 8 shows a simulation of the heating process of the example machine with the following initial temperatures: rotor: 27.0 °C, stator and winding head: 30.0 °C. The temperature of the cooling water is kept constant at 30.0 °C.

*Fig. 8: Simulation of the heating process of a PM machine *

The power loss is zero for 100 seconds and then jumps to the following values: copper losses 2038 W, iron losses 182 W, thus 2220 W in total. This simple example shows how the temperature of the winding head rises faster and reaches higher values than that of the stator. Both temperatures approach the steady-state temperatures determined in the stationary calculation (see fig. 6).

The rotor temperature only rises very slowly, because the rotor has a power loss of only 22.4 W (1% of the total losses) and because the air gap has a very high thermal resistance, so that hardly any heat is dissipated from the stator to the rotor.

### 2.5 Calculation results

Fig. 8 shows a simulation of the heating process of the example machine with the following initial temperatures: rotor: 27.0 °C, stator and winding head: 30.0 °C. The temperature of the cooling water is kept constant at 30.0 °C.

*Fig. 8: Simulation of the heating process of a PM machine *

The power loss is zero for 100 seconds and then jumps to the following values: copper losses 2038 W, iron losses 182 W, thus 2220 W in total. This simple example shows how the temperature of the winding head rises faster and reaches higher values than that of the stator. Both temperatures approach the steady-state temperatures determined in the stationary calculation (see fig. 6).

The rotor temperature only rises very slowly, because the rotor has a power loss of only 22.4 W (1% of the total losses) and because the air gap has a very high thermal resistance, so that hardly any heat is dissipated from the stator to the rotor.

## 3. Summary and outlook

The simulation tool has been developed to such an extent that temperature calcula-tions can be carried out in the time domain. The time-varying power loss, divided into iron and copper losses, is determined from the route simulation.

In the near future, several motors will be driven on a load test bench with load collectives derived from the route simulation to make realistic operating conditions available. The measurement of the relevant winding temperatures during these tests will provide data for the calibration of the simulation tool.

It will also be investigated whether a model with four thermal masses can deliver more accurate results than the existing model with three masses.

## 3. Summary and outlook

The simulation tool has been developed to such an extent that temperature calcula-tions can be carried out in the time domain. The time-varying power loss, divided into iron and copper losses, is determined from the route simulation.

In the near future, several motors will be driven on a load test bench with load collectives derived from the route simulation to make realistic operating conditions available. The measurement of the relevant winding temperatures during these tests will provide data for the calibration of the simulation tool.

It will also be investigated whether a model with four thermal masses can deliver more accurate results than the existing model with three masses.