Lumped Parameter Thermal Model of Heat-Generating General Geometry in Cylindrical Coordinate System

In this study, an equivalent thermal network for a general geometry with internal heating in cylindrical coordinates is derived. The equivalent thermal circuit consists of three thermal resistances: two resistances to the external heat flow and one resistance to compensate for the internal heating. The inclusive expressions can analytically derive the well-known equivalent thermal network of a hollow cylinder and one of the conventional thermal resistances to the external heat flow, as a special case. Simulations are performed for an insulator geometry bounded by equilateral triangles and a circle sector. The developed network outperforms the cylindrical approximation with a large radius ratio in terms of accurate estimation of average temperature and heat flow. For example, when an external heat flow and internal heat generation coexist and the radius ratio is 0.85, the error rate of the temperature rise in the developed network is 1.58%, whereas it is 10.8% in the cylindrical approximation network. The computation time of the developed network is less than 1 s, which is much shorter than that of the finite element method. The developed model is adaptable for any application, with or without heat generation, which is expected to be applied to the homogenization process of the winding and insulator.


I. INTRODUCTION
Thermal problems limit the miniaturization and output enhancement of electrical devices such as motors and transformers.In particular, the windings in the slots of electromagnetic devices are the hottest components because losses in the windings account for most of the total loss.A rise in temperature exponentially reduces the life of the resin that insulates the windings.Therefore, the accurate prediction of the temperature in a slot is required during the design process.
The temperature prediction methods can be classified into two main categories: numerical methods, such as the finite element method (FEM), and spatial analytical methods, such as the lumped parameter thermal network (LPTN) [1], [2], [3], [4].The FEM is an accurate but computationally expensive tool.It performs calculations on fine cells that The associate editor coordinating the review of this manuscript and approving it for publication was Agustin Leobardo Herrera-May .
divide the geometry into smaller elements.Simulating the inside of a slot is particularly time-consuming because the windings and insulators that it contains are smaller than the entire machine and the number of cells must be large.The layered winding model (LWM) provides a concise and accurate representation of the inside of a slot [5], [6].It models numerous conductors in several conductor layers.However, an LWM is unsuitable for structures with asymmetric temperature distributions, such as the heat sinks and coolant channels in a slot, because it distributes the conductor layers along isotherms equidistant from the slot wall.
Another analytical approach is to represent the inside of a slot as a homogeneous equivalent material [7], [8], [9].Modeling a slot using a single material helps construct an LPTN and estimate the motor temperature with low computational cost [3].Important thermal properties of an equivalent material are heat capacity and thermal conductivity.Heat capacity is typically calculated as the sum of those of the conductors and insulators in the slot.
Equivalent thermal conductivity has been estimated theoretically and experimentally.Hashin and Shtrickman estimated its upper and lower limits for macroscopically homogeneous and isotropic multiphase materials [10].Milton added microstructural information and estimated the structural parameters of various geometries [11], [12], [13].A simple model of equivalent thermal conductivity considers conductors and insulators in series [14], [15], [16].However, this method underestimates the equivalent thermal conductivity in the cross-section of a slot because it neglects the parallel thermal resistance of the insulators [7].Furthermore, adequately approximating the conductor geometry using a series model is difficult.
The structure within a slot exhibits anisotropic equivalent thermal conductivity.The cross-sectional thermal conductivity is sensitive to the parameters of the insulators, whereas the axial thermal conductivity is dominated by those of the conductors.The random arrangement of the conductors and insulators causes thermal anisotropy in the cross-section.This anisotropy is almost negligible, particularly when the winding fill factor is high; this is because the winding arrangement is similar to a hexagonal circle filling [17], [18], [19].Recent studies have analyzed equivalent properties by considering a single wire and its surrounding insulation as the minimum unit [17], [20], [21].
Thermal circuits with the internal heat generation in basic shapes such as hollow cylinders and rectangles have been established [22], [23].They consist of three thermal resistances: two to the external heat flow and one to compensate for the heating.However, many studies on the LPTNs of slots ignore internal heating.Our final goal is to model the insulator and heat-generating winding as a homogeneous material.To construct a homogenized LPTN for a winding and insulator unit, a thermal circuit that accounts for heat generation in complex geometries is needed.
This paper presents the derivation of an LPTN in the radial direction for a general geometry with internal heating in cylindrical coordinates.The developed model is adaptable for any application, with or without heat generation.Moreover, the simulation of the LPTN of a specific insulator shape and comparison of its results with those of the FEM are discussed.

II. THERMAL MODEL OF A HOLLOW CYLINDER
This section presents the derivation of the radial LPTN of a hollow cylinder with internal heating.The results discussed in this section will help obtain the radial LPTN of a general geometry in cylindrical coordinates.
The thermal circuit including internal heat generation is obtained by solving the heat diffusion equation.Under steady-state conditions and neglecting the circumferential and axial temperature variations, the heat diffusion equation is represented by (1).
In (1), T denotes the temperature, r is the radius variable, H is the internal heat generation, k is the thermal conductivity, r 1 is the inner radius, r 2 is the outer radius, T 1 is the temperature of the inner surface, and T 2 is the temperature of the outer surface.Equation ( 1) is a non-homogeneous partial differential equation under non-homogeneous Dirichlet boundary conditions.The solution of ( 1) is expressed as the sum of solutions T A of the homogeneous partial differential equation in (2) under non-homogeneous boundary conditions and T B of the non-homogeneous partial differential equation in (3) under homogeneous boundary conditions.
In (4), Q is the heat flow in the radial direction, φ is the central angle, and h is the height of the cylinder.The average temperature T A,ave for T A is calculated using (5).
Fig. 1 shows the relationship between the average temperature and the thermal resistances from the boundaries.The thermal resistance between the average temperature node and the inner surface node R 1 is expressed in (6).
Similarly, the thermal resistance between the average temperature node and inner surface node R 2 is expressed in (7).
139250 VOLUME 11, 2023 Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.Solving (3) yields (8).
Fig. 2 shows the relationship between the thermal resistance from the boundary and the average temperature with internal heat generation Q in .The thermal resistance R 3 from the boundary to the average temperature node is calculated using (9).
The objective of this study was to integrate the networks in Fig. 1 and Fig. 2 to obtain the general LPTN shown in Fig. 3.
The LPTN in Fig. 3 is equivalent to the circuit in Fig. 4 under homogeneous boundary conditions.Therefore, the thermal resistance R m satisfies (10).
Equations ( 6), ( 7), (10), and Fig. 3 represent the commonly known LPTN for a hollow cylinder.R 1 and R 2 represent the thermal resistances to the external heat flow, and R m is the correction term for the average temperature to the internal heating.R m can be omitted if there is no internal heating.High transient response accuracy can be achieved by inserting the heat capacity of the geometry as a capacitor element at the mean temperature node [7].The LPTN can be obtained not only in the cylindrical coordinate system but also in the Cartesian coordinate system by the same procedure.For the latter, the LPTN is implied to be connected in three dimensions via the average temperature node [24].However, LPTNs with internal heating as shown in Fig. 3 have only been obtained for simple geometries such as hollow cylinders and rectangles.This is because of the difficulty in solving the heat diffusion equation for complex boundary geometries.

III. EXTENDED THERMAL MODEL
This section presents the derivation of the radial LPTN for a general boundary shape with internal heating in cylindrical coordinates.Moreover, the following part of the study describes the LPTN of a shape bounded by an equilateral triangle and a circle sector, specifically designed to simulate the insulator.
A. THERMAL MODEL OF A GENERAL SHAPE Fig. 5 shows a generalized boundary geometry C. C has internal heating, and the temperatures of the inner and outer boundaries are maintained at T 1 and T 2 , respectively.The circumferential and axial heat flows are neglected.In principle, as in the previous section, the LPTN of geometry C is obtained by adding the solutions for the cases with no internal heating under non-homogeneous boundary conditions and those with internal heating under homogeneous boundary conditions.The mean temperature T C,ave is defined using (11) if the axial temperature distribution is uniform.
In (11), T C is the temperature distribution in geometry C, and The temperature distribution T C can be derived when the heat flows only in the radial direction.In the case where the inner and outer boundary temperatures are T 1 and T 2 , respectively, and the internal heat generation H is zero, T C is expressed as (12), with reference to (4).
R 1 is the thermal resistance from the inner boundary to the average temperature node and is defined in (13).
In ( 13), Q C denotes the heat flow in the radial direction for geometry C. Substituting ( 11) and ( 12) into ( 13), thermal resistance R 1 is calculated using (14).
Following the same process, thermal resistance R 2 from the mean temperature node to the outer boundary is obtained using (15).
In the second case where the inner and outer boundary temperatures are the same and the internal heat generation is H , T C is expressed as (16), with reference to (8).
Thermal resistance R 3 from the boundary to the mean temperature node is defined in (17).
In (17), Q C,in denotes the total heat generation inside geometry C. To construct an equivalent thermal circuit for the form shown in Fig. 3, the thermal resistance R m needs to be defined as in (10).Substituting ( 16) and ( 17) into (10) yields the solution expressed in (18) for thermal resistance R m .
Equations ( 14), ( 15), ( 18), and Fig. 3 represent the radial LPTN for geometry C. R 1 and R 2 represent the thermal resistances to the external heat flow, and R m is the correction term of the average temperature to the internal heating.R m can be omitted if there is no internal heating.

B. THERMAL MODEL OF AN INSULATOR
This section presents the calculation of the LPTN for a shape bounded by a regular triangle and a circle sector, as shown in Fig. 6.This shape represents a portion of the insulator when the motor windings are densely arranged in a hexagonal lattice.In the actual motor, the outer cylindrical surface is cooled, and heat flows in the transverse plane.Moreover, the insulator shape in the actual coil arrangement is very thin.Therefore, the axial and circumferential heat flows are assumed to be negligible [17].In this study, internal heating is considered because one of our next objectives is to develop an LPTN for a homogeneous material with heat generation.
The integrals in ( 14), (15), and ( 18) are computed based on symmetry.The shapes of the inner and outer boundaries are expressed in (19) and (20), respectively.
The results are expressed using the radius ratio N , defined in (21).
139252 VOLUME 11, 2023 Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.The calculation results are expressed in ( 22)- (24).
φ = π/3 when the triangle is equilateral.Numerical integration is performed to calculate the integrals over θ after the radius ratio N and the central angle φ are provided because they cannot be solved analytically.The total internal heating Q in is as expressed in (25).
Fig. 7 shows the LPTN for the insulator.The composite thermal resistance R 1 + R 2 to the external heat flow can be further solved analytically, as expressed in (26).
2 )−ln(N cos θ) dθ (26) Note that (26) is identical to the equation verified in [17], which partially confirms the validity of ( 14) and (15).In general, the computational complexity of the LPTN is sufficiently lower than that of the FEM.When practically solving the whole circuit, the numerical integrations for the developed equations are pre-computed and they are treated as constant values, resulting in a simple circuit calculation.Furthermore, the numerical integrals can be computed with few evaluation points of the function as the Gaussian quadrature.
For a large-scale system, circuit equations based on Kirchhoff's law are formulated with the resistances.The equations are nonlinear differential algebraic equations and generally solved numerically, whereas solved analytically if only static elements such as resistances and DC power supplies are included.The developed LPTN only contains the static elements, therefore, the complexity of the whole system depends on the other circuits connected to the T-shaped LPTN.
A notable feature of the developed LPTN, compared to conventional methods for insulator geometry, is its ability to represent internal heating and calculate the average temperature.In the process of modeling the insulator and conductors as a homogeneous material, for example, it is necessary to model the homogeneous part which replaces the insulator shape and includes homogeneous internal heating.The developed method can represent internal heating as a current source, which is suitable for that homogeneous part.Furthermore, the average temperature of the homogenized material is necessary to express the temperature dependence of the heating rate.The proposed method is valuable because the volume-averaged temperature can be easily calculated as the voltage at a node.
In addition, once the thermal resistances are calculated for a desired radius ratio and central angle, it can be adjusted and reused for any thermal conductivity and height.As shown in ( 22)- (24), the parameters on which the thermal resistances of the developed LPTN depends are the radius ratio, the central angle, the thermal conductivity, and the height.The inner and outer radii are theoretically independent of the resistances.The height, which is related to the cross-sectional area, and the thermal conductivity are inversely proportional to the thermal resistance.This is the general trend in thermal resistance.The authors verified and confirmed that these relationships are true by changing the thermal conductivity, the height, and the radius in the FEM.

C. SIMULATIONS FOR THE SHAPE OF THE INSULATOR
Table 1 lists the properties of the insulator.The radius ratio N is in the range 0.05-0.85 in increments of 0.05.The desired radius ratio was obtained by adjusting the outer radius while keeping the inner radius constant.The specific length parameters were arbitrarily set because they were unrelated to the thermal resistances of the LPTNs.The central angle was π/3 rad because the insulator and windings were arranged in a regular hexagon pattern, and the simulated unit corresponded to a portion of that hexagon.
The thermal parameters were set to simulate realistic conditions.For example, the temperatures of the boundary surfaces were set to 300 K or 400 K to represent room temperature and the elevated temperature when the motor heated up [14].For simplicity, the inner and outer boundary surfaces were kept at constant temperatures, whereas the other surfaces were set as adiabatic.The thermal conductivity was set to 0.30 W/m • K, close to the value of epoxy resin.The magnitude of uniform heating was modified depending on the volume of the geometry, as summarized in Table 2.This is because, as the volume decreases, the total heating decreases, resulting in minimal temperature increases.For example, when the magnitude of uniform heating is 1.0 × 10 6 W/m 3 and the radius ratio is 0.85, the average temperature rise becomes lower than 0.001 K, which is beyond the sensitivity threshold determined by the simulation's convergence conditions.Therefore, the values of heat generation were carefully selected to ensure a sufficient impact on the average temperatures.
The average temperature of the insulators and the total heat flow through the inner cylinder surface were simulated using three different methods: the developed LPTN, a cylinderapproximated LPTN, and FEM.The developed LPTN is shown in Fig. 7, and its respective thermal resistances are expressed in ( 22)-( 24).The cylinder-approximated LPTN is the approximation of a conventional hollow-cylinder LPTN.The equivalent thermal circuit is shown in Fig. 3, and the thermal resistances are expressed in ( 6), (7), and (10).The equal-area condition in (27) was used to approximate the shape of the insulator as a hollow cylinder.

S insulator = S cylinder (27)
In (27), S insulator and S cylinder denote the areas of the base of the insulator and cylinder, respectively.From (27), the approximation of the inner radius r C,1 and the outer radius r C,2 are derived using (28) and (29), respectively.The thermal resistances of the cylindrical approximation LPTN are obtained by substituting ( 28) and ( 29) into ( 6), (7), and (10).Fig. 8 shows R m for the developed network and the cylindrical approximation network.R m is the correction term for the average temperature with internal heat generation; it is negative in conventional thermal circuits.R m is calculated with R 1 , R 2 , and R 3 as defined in (10).The positive resistance R 3 has the physical meaning of thermal resistance between the homogeneous boundaries and the node of the average temperature with internal heat generation.Conversely, R 1 and R 2 represent the thermal resistances for the external heat flow, and the total resistance of the two in parallel is larger than R 3 .Therefore, the correction term R m becomes negative.The R m of the developed network generally agrees with that of the cylindrical approximation network.However, the difference increases as the radius ratio increases, and the R m of the developed network is positive when the radius ratio exceeds approximately 0.84.
LTspice was used to calculate the average temperature and heat flow of the LPTN.Python was used to implement a series of simulations of LTspice with a for loop.To calculate the thermal resistance, numerical integrals were computed with the quad() function in the scipy.integratemodule.This function uses a technique from the Fortran library QUADPACK, and integrates over finite intervals by the global adaptive quadrature based on 21-point Gauss-Kronrod quadrature within each subinterval.Although the accuracy can be increased by adding more evaluation points, the number of points is sufficient to obtain the temperature accuracy of 0.001 K in the simulation.
The other compared method was the FEM.FEM is an effective tool for simulating ideal thermal conditions.The insulator shape was simple for the radial thermal analysis, and the heat generation was set to be spatially uniform and constant.Therefore, the model did not exhibit extreme temperature distributions that degrade the accuracy.From the above, the evaluation by the FEM was reliable.
It was performed using ANSYS Fluent, with four solver processors in parallel.Fig. 9 shows the simulated geometries 139254 VOLUME 11, 2023 Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.and Fig. 10 shows an overlaid representation of the FEM mesh and temperature contour plot for the case when the radius ratio is 0.50.The mesh was generated by the sweep method and setting the number of divisions for each edge such that the shapes of the radial, circumferential, and axial elements are not excessively distorted.The number of meshes was set as 50000-200000 for all geometries.In the FEM simulation, the convergence condition was a temperature variation smaller than 0.001 K.Both the FEM and LPTN simulations were performed on an Intel(R) Core i7-9700K CPU @3.60 GHz with 32 GB RAM.
The discretization and simulation run times in FEM and LPTN were separately investigated in detail.The FEM meshing time wes taken from ANSYS Meshing's Performance Diagnostics.The simulation wall-clock time in ANSYS FLUENT was used as a reference for simulation run time of FEM.The numerical calculation time of the resistances was measured using the timeit() function in the timeit module to obtain an average time of 10,000 cycles.Finally, the computation time in LTspice was measured by the perf_counter() function in the time module in Python, and the average of 10 computation times, excluding outliers, was obtained.Note that the simulation runtimes are not pure CPU computation times, but the total simulation time including overhead such as disk I/O.Three different cases were simulated: no internal heating and an external temperature difference, with internal heating and no external temperature difference, and the combined condition.The first simulation assumed no internal heating, Q in = 0, and R m = 0.The temperature at the outer boundary T 2 was fixed at 400 K, and the heat flow was from the outer boundary to the inner boundary.This simulation verified the accuracy of thermal resistances R 1 and R 2 to the external heat flow.
In the second simulation, the outer boundary temperature T 2 was fixed at 300 K, equal to the inner boundary temperature T 1 .The magnitude of the heating was changed depending on the radius ratio, as listed in Table 2.This simulation verified the accuracy of the equivalent thermal circuit for internal heating.
For the third simulation, the temperature at the outer boundary T 2 was fixed at 400 K, and the magnitude of the internal heating was changed depending on the radius ratio, as listed in Table 2.This condition integrates the first and second simulations.This simulation verified the overall accuracy of the developed LPTN.

IV. RESULTS
Table 3 summarizes the average calculation times for each process to obtain the results of the third simulation.The table shows that the FEM took a total of 22.5 s on average to generate the mesh and calculate the temperature distribution.In addition, the FEM required considerably longer to change the geometry in computer-aided design.In contrast, the resistances calculation time was 0.26 ms, and the LPTN simulation time was 1.03 s on average.When LTspice was controlled by Python, equations including numerical integration of each thermal resistance were easily calculated, and the geometry parameters were rapidly changed.Thus, it was confirmed that the total simulation time of LPTN is shorter than that of FEM.

A. THERMAL RESISTANCE TO EXTERNAL HEAT FLOW
Fig. 11 shows the simulation results of the average temperature without internal heat generation.The developed and conventional methods lead to errors within 3 K when the temperature difference of 100 K is applied to the boundary surfaces.The developed network outperforms the cylindrical approximation network regardless of the radius ratio.The mean absolute percentage error (MAPE) values for the developed and cylindrical approximation networks are 1.45% and 1.60% for the average temperature rise, respectively, assuming the FEM results to be true values.A large radius ratio implies a large error compared to the FEM.Fig. 12 shows the total heat flow through the inner cylinder surface obtained using the different methods under the same condition.The developed network outperforms the cylindrical approximation network regardless of the radius ratio.The MAPE for the heat flow is 1.03% for the developed method and 4.85% for the cylindrical approximation.Particularly for  the cylindrical approximation, the error rate for large radius ratios is 38.8%.Fig. 11 shows that the ratios of R 1 and R 2 are almost the same using the conventional and developed LPTNs.A large radius ratio implies a large error in the average temperature.This is because the circumferential heat flow is nonnegligible with a large radius ratio, and the assumptions in the derivation process of the developed LPTN are violated.Fig. 12 shows that the amount of heat passing through the inner cylinder surface is significantly improved using the developed method as the radius ratio increases.A high accuracy of the sum of R 1 and R 2 implies closeness to the true heat flow.These results indicate that both R 1 and R 2 are accurately obtained using the developed LPTN, and the cylindrical approximation of LPTN becomes less accurate as the radius ratio increases.

B. THERMAL RESISTANCE TO INTERNAL HEAT GENERATION
Fig. 13 shows the simulation results of the average temperature with internal heating when the boundary temperatures are equal.Fig. 14 shows the error rate of the temperature rise for each network relative to the FEM temperature rise.The error rate of the temperature rise for the cylindrical approximation network is lower than for the developed network when the radius ratio is smaller than 0.75.However, the error rate of the cylindrical approximation network increases rapidly when the radius ratio is larger than 0.75.Fig. 15 shows the ratio of the heat flow through the inner surface to the total internal heat generation.The developed network outperforms the cylindrical approximation network regardless of the radius ratio.As the radius ratio increases, the developed network becomes more accurate than the cylindrical approximation; however, the difference is smaller than 0.5%.The MAPE for the heat flow through outer surface is 1.46% for the developed method and 1.61% for the cylindrical approximation.Note that the heat that passes through the outer boundary surface is the portion of the total internal heat generation that does not flow to the inner boundary surface.The simulation confirms that the sum of the heat flow passing through the outer and inner boundary surfaces is equal to the total internal heat generation.Fig. 13 shows the accuracy of the entire network, including R m .Fig. 14 shows that the developed network can calculate the average temperature with good accuracy for internal heating.For the developed network, the error in the average temperature increases as the radius ratio increases.Conversely, the cylindrical approximation network has a good accuracy when the radius ratio is approximately 0.65.As shown in Fig. 13, the approximation network output average temperature is higher than that of FEM when the radius ratio is low, whereas it outputs a lower value when the radius ratio is high.Therefore, the conventional network gives exactly the same average temperature as FEM for a certain radius ratio.This is why the cylindrical approximation network yields good results when the radius ratio is approximately 0.65.Fig. 14 shows the absolute value of the error rate, which would exhibit a monotonically decreasing trend for the approximation network if the graph was presented without taking absolute values.
Fig. 15 shows that as the radius ratio increases, the error in the LPTNs increases, with the developed network being slightly more accurate.This trend is similar to the results based on Fig. 11, which shows the ratio of R 1 to the sum of R 1 and R 2 .This is because the currents representing the internal heating are divided depending on the ratio of R 1 and R 2 in the LPTNs.Fig. 15 shows the ratio of R 2 to the sum of R 1 and R 2 , and the upside-down graph of Fig. 15 shows almost the same curves as Fig. 11.
However, the MAPEs for the developed network are slightly different for each result; 1.45% for the average temperature rise with external heat flow, and 1.46% for the heat flow through the outer surface with internal heat generation.The former represents the error between the average temperature node of the developed LPTN and that of the FEM in relation to external heat flow.The latter represents the error between the node in which the internally generated-heat flows in the developed LPTN with respect to that of FEM.Therefore, the difference of the MAPEs means that these two nodes do not exactly match.The difference becomes larger as the radius ratio increases; however, the difference is 0.01% on average for the insulator shape.This result indicates the validity of the developed LPTN, as it accurately captures the flow of internally generated heat into the node located between R 1 and R 2 .
The heat flow and the temperature rise at any node are expected to be proportional to the heating rate because the heating rate is equivalent to the current source in the developed T-shaped LPTN.Therefore, to eliminate the influence of the discrete magnitude of the heating rate, Fig. 14 and Fig. 15 show the percentage or error rate relative to the reference value.These graphs imply that the ratio is on a smooth line even when the heating rate is discrete; the thermal resistances in the model are independent of the heating rate and adaptable for any heating rate.

C. VERIFICATION OF INTEGRATED THERMAL CONDITIONS
Fig. 16 and Fig. 17 show the simulation results of the average temperature of the insulator and the total heat flow through the inner surface, respectively, under the integrated conditions where the outer boundary temperature T 2 is fixed at 400 K and the internal heating is set as listed in Table 2. Noticeably, the developed network has better accuracy than the cylindrical approximation network.For example, when the radius ratio is 0.85, the error rate of the temperature rise in the developed network is 1.58%, whereas it is 10.8% in the cylindrical approximation network.
The developed LPTN exhibits lower average temperature output in response to external heat flow compared to FEM as depicted in Fig. 11, wheares it yields a higher temperature output in response to internal heat generation, as shown in Fig. 13.Under integrated conditions, these errors are cancelled out and the developed LPTN yields more accurate results.Conversely, the cylindrical approximation network outputs lower average temperatures to both external heat flow and internal heating than FEM when the radius ratio is large, resulting in a larger error under simulation with integrated conditions.
Note that the cylindrical approximation network is more accurate for internal heat generation for small radius ratios, as shown in Fig. 14.If the internal heating dominates the temperature difference of boundaries, the cylindrical approximation network yields more accurate results than the developed network for small radius ratios.

V. DISCUSSION
A notable feature of the developed LPTN is its ability to represent internal heating and calculate the average temperature.The volume-averaged temperature can be easily obtained as the voltage at a node.Therefore, the model is applicable for the geometry with the temperature dependent heating source.In addition, once the thermal resistances are calculated for a desired radius ratio and central angle, it can be adjusted and reused for any thermal conductivity and height.
The LPTNs are improved by adjusting the thermal resistances according to the radius ratios.Fig. 12 shows the validity of the total thermal resistance of R 1 and R 2 , with a MAPE of 1.03% for the developed LPTN.The total resistance can be adjusted accordingly based on this validation.Fig. 11 and Fig. 15 show the ratios of R 1 and R 2 , and it is implied that R 1 is underestimated.The corrected total resistance should be divided into larger R 1 and smaller R 2 than ( 22) and ( 23) in order to obtain the proper ratios.Fig. 13 and Fig. 14 show that the thermal resistance R 3 in the developed network exhibits an increasing error as the radius ratio increases.The exact LPTN is obtained by calculating R m in (10) with the corrected R 1 , R 2 , and R 3 .However, this correction method is an experimental and laborious process because it involves conducting FEM simulations to calculate thermal resistances in Fig. 7.The developed LPTN with ( 14), (15), and ( 18) is a reasonable solution that does not require any previous experiments or numerical simulations.
One of the limitations of the developed LPTN is that it ignores the circumferential and axial heat flows.The simulation results show that the error of the developed LPTN becomes larger as the radius ratio increases, which means that, in this particular shape, the circumferential heat flow can not be ignored.In actual applications, heat flows in other directions should be taken into account.For a cuboid shape, the general LPTN implies that one-directional LPTNs are connected to other directional LPTNs via the average temperature node [24].Further verifications are required to examine the scenario where circumferential and axial LPTNs are connected to the developed radial LPTN.
This study assumes that the straight radius of the circle sector in Fig. 5 is applicable.However, for more constricted shapes, the thermal resistances are expected to vary based on the narrow area.In such cases, the area should be divided into parts and the LPTNs of each part should be then connected.
It is important to note that the developed LPTN does not account for the effect of temperature changes on thermal parameters.For example, the electrical resistance and the heat generation rate usually depend on temperature, and this dependency is not reflected in the developed LPTN.To ensure accuracy, the LPTN simulations for the average temperature and updates of the heating rate should be repeated until convergence.
Further experiments are needed to apply the developed LPTN to real-world scenarios.Whereas FEM and LPTN are used to simulate ideal thermal conditions, exact simulations for actual applications present challenges, such as the inability of FEM to represent impurities in the insulator and non-uniform coil arrangements [19], [24].To address such practical issues, experiments are required to correct the thermal properties.Furthermore, the validation of experiments for achieving high reliability is a topic for future investigation.

VI. CONCLUSION
This study constructed an equivalent thermal network in cylindrical coordinates for general geometry with internal heat generation.The developed model is adaptable, with or without heat generation.The network consisted of three thermal resistances: two for the external heat flow and one for compensating the internal heat generation.The equations could analytically derive the well-known equivalent thermal network for a hollow cylinder as a special case.Simulations were conducted for the shape of an insulator surrounding a coil.The results showed that the developed LPTN was more accurate than the conventional LPTN and was lesser computationally expensive than the FEM.
These results provide new insight into the problem of heat generation in winding structures of electrical equipment.The final goal of this study is to model the insulator and heatgenerating winding as a homogeneous material.To achieve this, a thermal circuit that accounts for heat generation is needed.The study also presents a highly rigorous solution for applications that were previously approximated using rectangular or hollow cylinders, or that involved heat generation.Additionally, the general LPTN in the Cartesian coordinate system can be obtained via a similar process.
Some studies have shown the potential for connecting oneaxial LPTNs to another axial LPTN via the mean temperature node in simple shapes [24].Moreover, simulation of transient responses can be achieved by inserting capacitor elements 139258 VOLUME 11, 2023 Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
that represent the heat capacitors at the mean temperature nodes.The feasibility of these schemes should be further investigated for a general shape in the cylindrical coordinate system.

FIGURE 1 .
FIGURE 1. Equivalent thermal network for a hollow cylinder without internal heat generation.

FIGURE 2 .
FIGURE 2. Thermal network for internal heat generation.

FIGURE 3 .
FIGURE 3. Equivalent thermal network for a general hollow cylinder with internal heat generation.

FIGURE 4 .
FIGURE 4. Equivalent thermal network for a hollow cylinder under homogeneous boundary conditions.

FIGURE 5 .
FIGURE 5. Area C with arbitrary boundary shape.

FIGURE 6 .
FIGURE 6. Unit of insulator when windings are hexagonally arranged.

FIGURE 7 .
FIGURE 7. Equivalent thermal network for the insulator unit.

FIGURE 8 .
FIGURE 8. Comparison of thermal resistance R m .

FIGURE 9 .
FIGURE 9. Outline of the model of the insulator for each radius ratio.

FIGURE 10 .
FIGURE 10.The mesh example and temperature distribution.

FIGURE 11 .
FIGURE 11.Average temperature without internal heat generation.

FIGURE 12 .
FIGURE 12. Heat flow through the inner cylinder surface without internal heat generation.

FIGURE 13 .
FIGURE 13.Average temperature with internal heat generation and homogeneous boundary temperature.

FIGURE 14 .
FIGURE 14. Error rate of temperature rise with internal heat generation and homogeneous boundary temperature.

FIGURE 15 .
FIGURE 15.Ratio of heat flow through inner surface to total internal heat generation.

FIGURE 16 .
FIGURE 16.Average temperature under integrated conditions.

FIGURE 17 .
FIGURE 17.Heat flow through inner cylinder surface under integrated conditions.

TABLE 1 .
Properties of the insulator.

TABLE 2 .
Magnitude of uniform heat generation.

TABLE 3 .
Computational cost of FEM and LPTN.