Analytical Calculation and Experimental Validation of Litz Wires Axial Thermal Conductivity

Litz wires are widely used in electrical machines with reduced ac losses, compared to traditional random winding. The reduced ac losses can improve both electrical machines’ electromagnetic performance, such as efficiency and thermal performance. Thermal aspect is identified as a key enabler for next-generation high power density electrical machines, where thermal modeling plays a critical role. Equivalent slot thermal conductivity is one of the most challenging parameters to be determined in the process of thermal model development, due to various components, such as copper, insulation, and impregnation resin. There are extensive literature where the litz wire thermal conductivity in the radial direction is reported. However, the thermal conductivity in the axial direction is not well studied, which is critical to determine the heat transfer, such as for electrical machines with end-winding spray-cooling system. In this article, the axial thermal conductivity is investigated with analytical calculations and experimental validation. First, nine litz wire samples are selected with various types, such as rectangular and circular shapes, with varnish process and without. Two different analytical methods are then introduced and presented to calculate the litz wire thermal conductivity in axial direction, based on equivalent length theory and equivalent medium theory, respectively. Finally, experimental tests are conducted, and results are compared to those obtained from the proposed analytical methods. Guidelines are also provided to predict the litz wire axial thermal conductivity in this field.

toward higher power density.Meanwhile, the high-frequency technology has become more important in electric motors, transformers, and other equipment [1], [2], [3].However, the higher ac losses due to the skin and proximity effects are problematic for high-frequency electrical machines, which results in lower efficiency for the electromagnetic aspect and higher winding temperature for the thermal aspect.The application of litz wires with transposition features can effectively reduce this effect, which are widely used in high-performance motors [7], [8], [9], [10].Fig. 1 shows a litz wire electrical machine [7].
For electrical machines, in particular high-performance electrical machines, thermal management is critical, where accurate thermal model is an essential tool [5].The thermal conductivity including the equivalent slot thermal conductivity is critical in the thermal model building up process.However, most existing literature focused on the thermal conductivity calculation in the radial direction, i.e., xy direction in Fig. 2 [5], [6], [7], [8], which is important for cooling technique such as water jacket cooling.However, there is much less research focused on the axial thermal conductivity, i.e., z direction in Fig. 2. The axial thermal conductivity is important to build up the 3-D model in the cooling system, where most of the heat is removed axially in winding, such as the end-winding oil sprayed stator [11] and axial flux motor [13].The methodology used to calculate the radial thermal conductivity [5] cannot be applied to the axial thermal conductivity calculation, in particular for litz wires.This is due to the fact that litz wire has a unique twisted structure with wrapped insulating material.The anisotropy heat transfer characteristic is more pronounced compared to that of traditional windings.This article will focus on litz wire thermal conductivity determination in the axial direction, with two proposed analytical methods.
Generally, there are two main methods to estimate the coils' thermal behavior for thermal conductivity calculation: finite element (FE) and analytical calculations.The FE analysis is the most widely used method at present [8], [14], [15], [16], [17], [18], [19].Compared to analytical calculations, it can provide more accurate results if the coils can be properly modeled.The effects of transposition and parallel arrangement on the litz wire thermal conductivity with FE model are discussed in [14].FE is also used to simulate the equivalent model of winding and impregnation resin materials [16].However, for much more complicated coil made of multilayer litz, FE method may not be suitable due to the numerous time to build and calculate.
Analytical methods provide quicker solutions compared to FE.An equivalent model of a 3-D porous cube was proposed in [7] to calculate litz wire thermal conductivity in the radial direction, which was validated with experimental results.Similarly, four 2-D rectangular equivalent models on litz wire in the radial direction are proposed and compared in [9].Analytical methods are quicker compared to FE model.In this article, two analytical methods to consider litz wire complex features are proposed to calculate the equivalent axial thermal conductivity of litz wires.The novelties within the articles are. 1) The complex structure of the litz wire, such as bundle distribution is considered within the two analytical calculations.
2) The two analytical methods are proposed and compared for litz wires with various dimensions, validated with experimental results.3) Empirical formulas are proposed, and guidelines are presented to use the analytical methods.The article structure is as follows.Section II presents the structure of various litz wire samples used for this study.Analytical calculation methodologies are introduced in Section III, including the equivalent length model (ELM) and equivalent medium model (EMM).Axial thermal conductivities of nine litz wire samples are experimentally measured and compared with the analytical results proposed in Section IV.Finally, Section V concludes this article.

II. STUDIED OBJECTS
This section presents up to nine litz wire samples used in this article, as shown in Fig. 3, ordered from " # a" to " # i".The physical structure of litz wire sample is briefly described first, and the main parameters of these samples are listed in Table I.More structure analysis discussions are followed in order to build up the theoretical models to estimate the equivalent axial thermal conductivities for various litz wires.

A. Parameters of the Samples
In Table I, Shape is litz wires appearance, such as rectangular or circular.Cross section, A e , and α are the litz wires cross-sectional profile, i.e., single wire cross-sectional area, and bundle twist angle, respectively.It should be noted that the bundle twist angle is the same as the wire twist angle inside each single bundle at the opposite direction if the litz wire is made of multilayer bundles.Therefore, for litz wire you can see at the out layer of enameled wires is aligned with the cable direction, as shown in Fig. 4.
Structure consists of the litz wire major parameters of construction, including the number of litz wire layers N l .There are three types of layer numbers in the studied objects, singlelayer litz wire, double-layer litz wire, and triple-layer litz wire.Single-layer litz wire is directly twisted with u enameled wires, double-layer litz wire is twisted again using v singlelayer litz wires as its basic bundles, and triple-layer litz wire is using w double-layer litz wires as its subbundles to build, as shown in Fig. 5.
N w is the total number of enameled wires in a litz wire.D w is the bare copper diameter of each enameled wire.Material includes Enamel Wires, Wrap, and Varnish, which represents the insulation material in the litz wire, and their thermal properties are listed in Table II using standard NFC 31010.The twisting process of two-layer litz wires is detailed in Fig. 5.As we can see, single wires are first twisted to form a basic bundle, which are further twisted as subbundles and then can be twisted again to form a litz wire finally.The twisting times are depending on the layers of litz wire.Further compression or extrusion can be done if a rectangular cross section litz wire is required.The litz wire can be wrapped to get better insulation or varnished to keep the coil in shape.Among the nine samples, samples # b, # c, # h, and # i are wrapped without varnish, and samples # a, # d, # e, and # g are investigated for both conditions with varnish and without.

B. Bundles Distribution
The internal structure of litz wires is further clarified in this section.There are 5, 6, 7, 10, and 12 bundles for the litz wires in Fig. 3.The litz wires are twisted based on the internal middle bundle shown in Fig. 6.
Therefore, there are two kinds of bundles in litz wires, middle bundles, and outer bundles, respectively.To specify the twisting times and thus the required wire length for axial thermal conductivity calculation in Section III, the middle and outer layers bundle are introduced in Fig. 7.
Depending on the conductors' locations, two categories of "middle bundles" and "outer bundles" are defined in Fig. 8, with the ratio of 1:6 referring to samples # a, # e, and # h.
Fig. 8 also shows an example of the total twisting times for samples # a, # e, and # h with one middle bundle and six outer bundles.0 refers to the untwisted conductors, 1 means   that the wires have been twisted once and 2 refers to the case that enameled wires have been twisted twice.For samples # b, and # d, the structure with less than six bundles, they do not have a middle bundle, all the bundles are twisted at least once, as shown in the gray circle in Fig. 8.
For the nine samples in this article, the middle bundle number is 0, 1, or 2. The bundles length can be further calculated, respectively, through the thread pitch.The total number of bundles in a litz wire, and the number of middle and outer bundles in each sample investigated in this article are listed in Table III.In general, middle bundle number in litz wires is determined by the total number of bundles.For example, there are no middle bundles for litz wires with less than six bundles, and there is one middle bundle for litz wires with six-seven bundles.There are two middle bundles for litz wires with more than ten bundles.

III. CALCULATION TECHNIQUES
Based on the litz wire structure in Section II, this section presents the following two methods to calculate the equivalent axial thermal conductivity of litz wire: 1) ELM and 2) EMM.It is assumed with ELM that dominated heat transfer is along the enamel wires, i.e., path#1 in Fig. 9(a).For ELM, the enamel wires are assumed separated from each other.Whilst it is assumed with EMM that dominated heat transfer is along the litz wire axial direction, i.e., path#2 in Fig. 9(b).For EMM, litz wire is assumed a uniform mixture of copper and insulation materials.The heat is transferred from one end to another end of a mixture material cylinder with the same cross section area of litz wire.

A. Equivalent Length Model
It is assumed with ELM that there is no heat transfer between different enameled wires in the xy direction.Thus, Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.the litz wire length is a critical factor for the axial thermal conductivity calculation, which can be calculated based on the bundles' descriptions in Section II.Enameled wires of the same lengths are classified as one type, which are determined by the twisting times in Fig. 8.The enamel wires' length distribution in Fig. 10, corresponds to the structure in Fig. 8, i.e., one middle bundle and six outer bundles.
The lengths l 0 , l 1 , and l 2 in Fig. 9 are calculated with (1)-(3).l 0 , l 1 , and l 2 corresponds to the wire length that is untwisted, twisted once, and twisted twice, respectively, where l e is the length of litz wires.The three aforementioned types of enamel wires in Fig. 10 and the thermal transfer through the varnish are in parallel, as shown in Fig. 11.In Fig. 11, λ ELM is the litz wire thermal conductivity calculated with ELM, λ 0 , λ 1 , and λ 2 are the thermal conductivity of each kind of enamel wires corresponding to wires with length l 0 , l 1 , and l 2 , respectively, λ V is the varnish thermal conductivity.Since the varnish process is conducted after litz wires are twisted, its length is equivalent to litz wire length l e .λ i in (4) is the transformed thermal conductivity based on copper and wires that corresponds to the various lengths in Fig. 10, i = 0, 1, 2 where λ Cu and λ En represent the thermal conductivity of the bare copper and enamel, respectively, and σ is the ratio of the enamel thickness to copper diameter, which is 0.05 in this article.The calculation method of each kind of single enameled wire length and varnish is shown in the following: where A i is the sum cross section area of the enamel wires in length l i , A V is the cross section area of varnish, and A e is the cross section area of the litz wire.A i and A V can be obtained with the following: where D w is the diameter of bare copper wires, n i is the sum number of the enamel wires in length l i , and as for the structure of one middle bundle and six outer bundles in Fig. 10, n i can be obtained by the number of enamel wires N M as follows: A e is the cross section area of the litz wire.The λ ELM is obtained as follows:

B. Equivalent Medium Model
With EMM, it is assumed there is heat transfer in the xy direction between different components, including copper, enamel, and varnish.In this case, the cross area in each bundle is different in Fig. 12, which illustrates the samples # a, # e, and # h, with one middle bundle and six outer bundles.
The cross area for each component is calculated with twisting angles seen in Fig. 13.Based on the twisting times in Section II-B, twisting angles θ 0 , θ 1 , θ 2 , θ 3 , and θ 4 are calculated.θ 0 is close to 0 • for the untwisted wires in (12).θ 1 is the enameled wires twisted once, which is close to the thread pitch α in (13).θ 2 corresponds to the wires that are  twisted twice in opposite directions, which thus is close to 0 • in (14).θ 3 in (15) means the angle of the wires that is twisted twice in the same direction and is close to 2 α in (15).θ 4 in ( 16) is twisted twice by two vectors at an angle of 60 • , as shown in Fig. 14.In Fig. 14, vector 1 shows the direction that the enamel wires are twisted within bundles and vector 2 indicates that the bundles are twisted within the litz wire With EMM, the length is the same for all the components (copper, enamel, and varnish), with different cross areas, as seen in Fig. 13 that can be calculated with the aforementioned twisting angles.Based on EMM idea, the thermal conductivity of the mixed material is similar to the volume ratio of the material in differential length, as shown in Fig. 15.
The cross section of enameled wires is similar to elliptical columns in a small section of litz wire in the following: where A i is the cross-sectional area for each ellipse in the enamel wires at angle θ i , i = 0, 1, 2, 3, 4. a and b represent the long and short radii of the ellipse.Set A ′ Cu , A ′ En , and A ′ V as the sum of cross section area at θ i , for bare copper, enamel, and varnish, respectively.The cross section area of copper A ′ Cu is calculated with the following: where n i is the total number of the wires at the same angle, i = 0, 1, 2, 3, 4. As for the structure of one middle bundle and six outer bundles in Fig. 13, n i can be obtained as follows: where N M is the total number of enamel wires.According to the properties of the enameled wires, the cross section area of the enamel material A ′ En is shown in the following: where σ is the ratio of the enamel thickness in each enamel wires to copper diameter, which is 0.05 in this case.The cross area of epoxy resin A ′ V is where A e is the total litz wire cross section area.The litz wire equivalent thermal conductivity λ EMM obtained with EMM is

IV. EXPERIMENTAL VALIDATION
A specially made test rig is used to test the nine litz samples in Section II to validate the proposed analytical methods in Section III.The results calculated from the analytical methods and experimental testing are compared and analyzed.

A. Test Principle
A test rig in Fig. 16 was designed to measure the axial thermal conductivity of a litz wire.In Fig. 16, A is a copper plate with heaters embedded, which temperature is controlled with a constant heat source.Fixtures B and D are used to hold the litz wire samples C. E is made of copper with Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.cooling channels inside, which temperature can be controlled by the coolant flowing through the cooling channels.With the measured temperature difference across the sample and temperature differences at two ends of heat and cold plates, one can derive the axial thermal conductivity of measured sample.Insulation bracket is applied to the samples and the test rig to stop the heat dissipation from sample to environment.
In Fig. 16, points 1-13 are temperature measuring points.Temperature difference between points 1, 2, and 3 can be used to calculate the heat flux q B flowing through B, while points 8, 9, and 10 can be used to calculate the heat flux q D flowing through D. Temperature at point 4 can be used to check the contact between the litz wire sample and B. Similarly, point 7 is used to check the contact between the D and litz wire samples.Temperature difference from points 5 and 6 is used to calculate axial thermal conductivity, COMPARISON OF TEST RESULTS OF THE TEN SAMPLES using the calculated heat flux across litz wire samples C. Points 11 and 12 correspond to the coolant inlet and outlet temperature, respectively.Point 13 is used to measure the temperature of hot plate for the control of power supply.
Under a thermally steady condition, the heat flux q B flowing through B should be equal to q D through D, which is the same with that q C through C (litz wire sample).Equations ( 27) and (28) can then be used to calculate the axial flux and the axial thermal conductivity of measured litz wire sample where A B is the cross section area of fixture B and A C is the cross section area of each test sample C. T i, j is the temperature difference between point i and point j, x i, j is the physical distance between point i and point j, and λ Sample is the thermal conductivity to be tested in each sample.The test rig is shown in Fig. 17, which is covered with thermal insulating asbestos (calcium-magnesium silicate ther-

B. Calibration
As shown in Fig. 17, the test rig has been thermally insulated with calcium-magnesium silicate thermal insulating sheet and foam box to minimize the losses that are dissipated to the environment.Furthermore, the following procedures are conducted to ensure the testing accuracy with samples tested.
Three samples in Fig. 18 whose cross section areas and lengths are similar to litz wire samples in Fig. 3 are made from pure copper [HC101] with known thermal conductivity (391 W/m • K), and three other samples are made from aluminum [HE30TF] (180 W/m • K).With the same dimensions, the losses dissipated to the ambient will be the same for the test and litz wire samples.These two types of material are chosen as their thermal conductivities falling in the range of litz wires axial thermal conductivities based on prediction of thermal models developed.
All the test samples in Fig. 18 and litz wire samples are tested under the same conditions: high temperature end is approximately 50 • C, 70 • C, and 90 • C, respectively, while the low temperature end is approximately 5 • C. Utilizing ( 27) and (28), the thermal conductivity of the litz wire samples can be obtained.
During the validation process, multiple measurements in different length of strips are conducted to ensure the reliabilities and accuracies of measurements, one set of the test data is shown in Fig. 19(a) and (b), and all of the results are listed in Table IV.From the data shown in Table IV, one can see that the measured axial thermal conductivities are close to the samples and the measurement errors are within 1.15%.The same measurement techniques are then applied to the litz wire samples measurement to ensure the accuracy of the measurement of litz wires.

C. Test Data
The average measured data is used to calculate the samples' axial thermal conductivity, using (29) and (30).One set of three times test data for samples, such as # a, with varnish and without are shown in Fig. 20(a) and (b).Similarly, testing results are plotted for all other samples in 100 mm (from # b to # i) and put in Appendix where δ i is the correction coefficient of the measured temperature by thermocouples under i condition, which is obtained by placing the thermocouples together in a specific temperature hot bath.The testing results for all the samples, as well as the calculated results using the analytical methods based on ELM and EMM, are listed in Table V, and plotted in Fig. 21 for a clearer comparison.
In Table V, Error 1 is the tolerances between the prediction result from ELM and the test result of the samples.Error 2 is  the tolerances between the prediction result from EMM and the test result.
From Table V and Fig. 21, it can be noticed that ELM provides more accurate results compared to EMM for samples without varnish, while EMM is more suitable for samples with varnish.However, it should be noted that the thermal conductivity in both of the varnished and without varnished samples # d is closer to the results obtained with ELM.The reason is the diameter of enameled wires is large and translate to a clear heat transfer path via copper.
It can be concluded that ELM can be used for any type litz wires without varnish process, transposition litz wire and round varnished litz wires, with less than 8.5% error.EMM is suitable for rectangular litz wires with varnish process, with less than 2.5% error.
Considering the litz wire shape and dimensions, quick (31) and (32) are provided in this article for litz wires based on ELM model and EMM, respectively.Wrapped round litz wire Varnished rectangular litz wire where n M is the number of middle bundles, n O is the number of outer bundles in Table III and l b is the bundles equivalent Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.length.Litz wire axial thermal conductivity is mainly affected by the pitch angle, the distribution of bundles, the compress process, as well as the varnish process.
V. CONCLUSION Thermal model is critical to evaluate electrical machines' thermal performance, which is important for next-generation high power density electrical machines.Slot thermal conductivity is a complex but critical value in the process to build up 3-D thermal models.The equivalent slot thermal in the axial direction is always ignored compared to that in the radial direction.However, the axial thermal conductivity is important to predict the thermal performance for electrical machines with enhanced cooling methods, where the heat is mostly dissipated to the ambient environment via the axial direction, such as end-winding sprayed cooling.This article proposes two models (ELM and EMM) based on the twist angle and number of middle bundles to calculate litz wires axial thermal conductivity, which are more difficult compared to traditionally wound wires, due to the twisting characteristics.Nine litz samples are used as case studies in this article covering various litz wire types, such as being varnished and without, rectangular or round, and different thread pitches.A specially designed and manufactured test rig is used to validate the models.It is concluded that ELM is more suitable for litz wires without varnish, round litz wires, and transposition litz wires, while EMM is more accurate to predict the thermal conductivity of varnished rectangular litz wires.Analytical formulas are also presented in this article to provide guidelines for researchers in this field.APPENDIX See Figs.22-29.Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.

Fig. 8 .
Fig. 8. Number of twisting times for each kind of conductors.

Fig. 9 .
Fig. 9. Two heat transfer paths.Dominated heat transfer direction is along the (a) enamel wires and (b) litz wire.

Fig. 10 .
Fig. 10.Length distribution of the conductors in each litz wire.

Fig. 11 .
Fig. 11.Thermal conductivity in each kind of conductor length.

Fig. 13 .
Fig. 13.Angle distribution of the enamel wires in each bundle.

Fig. 20 .
Fig. 20.Testing results of sample a.(a) Sample # a with varnished in 100 mm.(b) Sample # a without varnished in 100 mm.

Fig. 21 .
Fig. 21.Comparison of the results collected by the test rig, and the results predicted by ELM and EMM.

TABLE I SAMPLE
PARAMETERS Fig. 2. Defining the direction of litz wires.

TABLE II THERMAL
CONDUCTIVITY OF MATERIALS

TABLE III MIDDLE
BUNDLES QUANTITY RELATIONSHIP

TABLE IV TOLERANCE
OF TEST BENCH