# Analytical Prediction Of Copper Losses Biology Essay

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The winding power loss, also called the copper loss, is caused by the current flow through the winding resistance. At dc and low frequencies, the current density Jc in the conductor is distributed uniformly through its cross section area. The current is induced in the conductor when the conductor material is subjected to an alternating magnetic field. It is known as eddy-current which oppose the penetration of the conductor by producing eddy-current magnetic field and produce ohmic losses by converting electromagnetic energy into heat. There are two kind of eddy-current effects: the skin effect and proximity effect. Both of these effects are cause non-uniform current density distribution in the alternative current carrying conductor at high frequencies.

In the alternating current carrying conductor, a magnetic field is induced in the conductor by its own current. It circulates an extra current itself. As a result, the current tends to flow near the surface of the conductor and current density tends to reduce at the middle of the conductor. At high frequencies, the entire current flow in a very narrow skin of the conductor. The eddy-current effect on the skin of the conductor is called the skin effect. With frequency increase, it increases the effective resistance as known as ac resistance ; as a result power losses will increase.

In general, the proximity effects occur when the ac current carrying adjacent conductor causes a time varying magnetic field. It induces circulating current inside the conductor. Hence, each conductor is subjected to its own field and also field produced by adjacent conductor. If the conductor carrying ac current skin and proximity effects are formed the total eddy-current. If the conductor does not carry the current, eddy-current is induced only due to the proximity effect.

To investigate these frequency dependent eddy current field, generally both fe and analytical can be used. Commonly application of fe method in the estimation of the eddy current loss requires large solver time for computation and also formation of the geometry. Especially in the eddy-current computation, it consumes larger solving time since the conductors are designed as segmented blocks filled by solid conductive material. As an alternative, analytical method can be used. It provides fastest solution and more insights.

In , the proximity losses are investigated for round and rectangular shapes conductor assuming the one directional (1d) flux flow in the slot of the electrical machine. The 1d method provides complementary results, but the results may inaccurate particularly for semi-open slot machines. In , it is confirmed that the obtained losses are inaccurate for the conductor which is placed closer to slot-opening. This inaccuracy comes from the assumption that made always the flux lines are in one directional which is not valid in that region.

For predicting the eddy-current losses accurately, 2d analytical models are developed in neglecting slotting effect, while considering in . Considering slotting effect, p.reddy and et.al proposed an analytical method to evaluate the resistance limited eddy current losses (known as proximity losses) for single and double layer winding arrangement of round conductors based on flux density estimation. However, this method does not consider the armature reaction effect and assumes the influence of the magnet is null.

Based on sub-domain field model which takes account into both armature reaction effect and coupling between the phases, the proximity losses were calculated for open slot configurations using field vector potential in . This method was improved in for semi-open slot configurations. However, these analytical methods which are only valid round and bar type of conductors, are limited only for resistance limited eddy-current losses and which cannot be used to calculated inductance limited eddy-current losses. Thus, an accurate analytical model which takes into account both resistance and inductance limited eddy-current losses is necessary, particularly vertically placed conductor where skin effects are predominant.

## Analytical field solution

Initial modelling is considered in the three-phase, 6slot 4pole concentrated wound ft-pm machine (Figure 3 .1). The geometrical parameters respect to the centre of origin (0, 0) are the inner radius of the rotor yoke r1, the radius of the surface of the permanent magnet r2, the inner radius of the stator r3, inner and outer radius of the slot r4 and r5, respectively. The slot-opening angle is β, the slot-width angle is δ and the pole-arc to pole pitch angle is α. The design includes Q number of semi-open slots and 2p number of poles. Each slot is represented with subscripts j and related slot-opening is given with subscripts i. i and j are always equivalent and they are given as follows:

i, i = 1, 2,...........Q

j, j = 1, 2,...........Q

The angular position of the ith slot-opening is defined as

For the simplification of the modelling, the following assumptions were made:

the machine's design have radial geometry as shown in Figure 3 .1;

the stator and rotor material has an infinite permeability and null conductivity;

the magnet magnetized in radial direction with µr = 1;

the current density uniformly distributed;

the end effects are neglected.

Figure 3.1 The geometric representation of considered 6 slot 4 pole ft-pm machine

In order to establish the exact analytical model, the machine's geometry is represented into four type of sub domain:

rotor pm sub-domain (AI - region I)

air gap sub-domain (AII - region II)

slot opening sub-domain (Ai - region III)

stator slot sub-domain (Aj - region IV)

where, A represents magnetic vector potential in the each sub-domain which is given by subscript. The sub-domains governing field equations are:

By using separation of variables technique solving Laplace's equation in sub-domains AII, Ai and Poisson's equation in sub-domains AI, Aj , the general solution can be obtained for each sub-domain. Finally, the magnetic vector potential can be obtain from the general solution. According to the adopted assumptions, the magnetic vector potential has only one component along the z direction and only depends on polar coordinated r and θ. For the sake of clarity in the general solution the following notations are adopted.

Figure 3.2 Assigned boundary conditions in a slot-opening sub-domain

## Solution in the slot-opening sub- domain

Figure 3 .2 shows the ith slot domain with its associated boundaries. In order to obtain the solution, well known Laplace's equation is employed in the slot sub-domain which enclosed by the angles θi and θi+β in-between the radius r3 and r4 as explained in .

As shown in Figure 3 .2, the tangential component of the magnetic field at the wall of the slot-opening is null since the assumption is made that the stator material has infinite permeability. Hence, the associated boundary conditions are:

The interface conditions between the slot sub-domain (which related to the ith slot-opening) and airgap sub-domain provide additional continuity boundaries:

Now, the Laplace's equation can be solved by using separation variable techniques, considering the assigned boundary conditions above. The separation of variables technique is a method to solve the partial differential equation by separating variables involved in. The variable can be one or more than one. Thus, this method of separation of variables can be used to analyse not only 2d problems and also 3d problems too. However, in this study only 2d problem is considered due to the complexity of solving 3d problems.

As an initial step of solving process of the Laplace's problem , assuming that the solution will be a product of two functions the solution can be written as follows:

Plug the product solution into the partial differential equation , a separation constant (λ) is introduced to separate the variables. This will produce two following ordinary differential equations.

Using the boundary conditions in , Eigen values and Eigen functions of can be evaluated. The general method of evaluating the Eigen values and Eigen functions are explained in X. Hence, the Eigen values of problem is given by

where, K represents the number of harmonics which is a positive integer. The Eigen functions corresponding to λo, and λk are given as follows:

For λo, and λk, the solutions of differential equation are given by

Hence, the general solution of the magnetic vector potential in the slot-opening can be obtained by multiplying the solutions and .

Taking into account the boundary conditions and the interface conditions , the general solution obtained in can be given in the following form for ith slot-opening sub-domain.

where, Ekπ/β(r,R4) is defined in and Aoi, Boi, Aki and Bki are constants. The constants Aoi, Boi, Aki and Bki are evaluated using Fourier series expansions of slot magnetic vector potential Aj(r,θ) and airgap magnetic vector potential AII(r,θ) over the slot-opening interval. Thus, the Fourier series expansions of these coefficients are:

The solution for the magnetic vector potential in the slot and airgap domains are described in the next section and the expansions of the coefficients Aoi, Boi, Aki and Bki are given in the Appendix E.

## Solution in the air gap sub- domain

Solution for the airgap sub-domain can be obtained by solving the Laplace's equation in the similar way as methods of separation of variables explained before. The associated boundary conditions of the airgap sub-domain are given in Figure 3 .3. The solution can be attained from the following Laplace's problem.

Using interface between the PM sub-domain and the airgap at r = R2, the following interface condition is yield form the continuity of the tangential component of the magnetic flux.

Figure 3.3 Assigned boundary conditions in the airgap sub-domain

In similar way as , the interface condition can be assigned at r = R3. However it becomes more complex since the inner surface of the stator tooth give a boundary condition and also existence of the slot-opening sub-domain provides a interface condition at r = R3 within the interval of θi and θi+β. Hence, the condition which takes into accounts both boundary and interface conditions can be written as follows:

Considering the boundary conditions and , the general solution of the magnetic vector potential in the air gap sub domain can be obtained as

Where, n is the harmonics term which is a positive integer and Pn(r,R3), En(R2,R3) are expressed in and , respectively. The constants AnII, BnII, CnII and DnII are evaluated using Fourier series expansions as follows:

The expansions of the coefficients AnII, BnII, CnII and DnII are given in the Appendix E.

Figure 3.4 Assigned boundary conditions in a slot sub-domain

## Solution in the slot sub domain

According to super position principle, the general solution of slot sub domain is sum of the general solution of corresponding Laplace's equation and a particular solution which satisfies the Xequation. Hence, general solution of the magnetic vector potential in the jth slot is given by,

where, only DC field is accounted and resulting eddy current field is neglected.

## Solution in the permanent magnet sub domain

Applying Poisson's equation with associated boundary conditions the general solution of the magnetic vector potential in the PM sub domain can be written as,

(A-1)

where,

(A-2)

(A-3)

Back-emf, flux density and torque predication

## Eddy current loss prediction

Fe verification

problem description

Analytical field solution considering reaction effect

In order to solve the eddy current field, from the quasi static Maxwell equations, we have to solve the Helmholtz equation in the slot sub domain. Using obtained solution in (1) the problem simplified defining the interface boundary Dj as shown in fig.2. Boundary Dj is representing source current as an equivalent current sheet taking into account coupling between the slot opening, air gap, magnet and other slots. Hence, the solution can be obtained solving Helmholtz equation only in the slot by separating the problem from other sub domain. The effective fields in the rest of the sub domains are assumed to be same.

Fig. 2. jth slot sub domain with boundary conditions FIG. 2 HERE

## Solution of the jth slot's boundary Dj from quasi static field solution

Taking into account the boundary conditions in the slot-opening sub domain, the general solution in the slot can be written as follows ,

(2)

As can be seen in fig.2, Dj corresponds to the continuity of tangential component of the magnetic field between jth slot sub domain and ith slot-opening sub domain and thus, the boundary condition at r = R4 is given by,

(3)

Hence,

(4)

## Solution of Helmholtz's equation in the jth slot

The jth slot domain and the associated boundary conditions are shown in fig.2. In order to obtain the AC field distribution following complex Helmholtz equation is solved in the slot domain.

(5)

Since the stator iron has infinite permeability, the tangential and radial component of the magnetic field at the slot wall is null. Hence, the boundary conditions are given by,

(6)

(7)

In addition the interface boundary Dj (R4,θ) takes into account the armature reaction field from the DC field solution. Hence, taking into account the boundary condition (6) the general solution of (5) can be written as,

(8)

Considering the boundary conditions (3) and (7) the general solution can be simplified with two coefficients Coj and Cmj as follows:

(9)

where,

(10)

(11)

(12)

(13)

The coefficients Coj, Cmj are determined using a Fourier expansion of Dj (R4,θ) over the slot-opening interval (θi, θi+β).

(14)

(15)

The detailed expansions of coefficients Coj, Cmj are given in appendix.

## Eddy current loss prediction

The total current density in the conductor can be expressed from complex magnetic vector potential by,

(16)

Thus, the eddy current density in the conductor can be separated from the actual current density by,

(17)

where, C(t) is introduced to ensure the total current flowing through each conductor is equal to the source current. Hence, the copper losses over a fundamental electrical period in a conductor can be calculated from complex current density by,

(18)

where, the first term represents complex current density in a considered conductor and the second one is its conjugate. By replacing Je in (18) instead of Jactual the induced eddy current losses can be obtained.

TABLE I

Parameters of the PM machine

Inner radius of the rotor yoke (R1)

27.5 mm

Stator inner radius (R3)

31.5 mm

Stator outer radius (R5)

50.0 mm

Magnet depth (R2-R1)

3 mm

Tooth-tip height (R4-R3)

2.5 mm

Depth of stator back iron

4 mm

Axial length (lstk)

100 mm

Slot width angle (δ)

20 mech.degrees

Slot opening angle (β)

3 mech.degrees

Magnet span (α)

0.85

Number of pole pairs (p)

7

Number of stator slots (Q)

12

Number of turns per phase (Nph)

90

Current density (Jrms)

1.89 A/mm2

Remanence flux density of the PM (Brem)

1.08 T

## TABLE 1 HERE

Fig. 3. Representation of VSW in the jth slot sub domain FIG. 3 HERE

Fe validation

In order to validate the proposed analytical model 12 slot 14 pole PM machine (fig.1) is considered. The specification of the PM machine is given in Table I. In the FE analysis, non oriented silicon steel (M250-35A) is used, where non linear BH characteristic and magnetic saturation are accounted. The VSW is placed in the each slot domain by creating separated conductor with solid (copper) conductive material as shown in fig.3. The mesh refinement has been done each sub domain until convergent results are obtained. The 2D FE transient with motion simulation is carried out for the whole cross section of the machine by exciting the current into each phase winding with its phase shift.

In the analytical solution of (1), the PM, air gap, slot-opening and slot regions are computed with 30 harmonic terms n, k and m, respectively. In the solution of eddy current field 15 harmonic terms are considered. Each conductors are divided equally into 50 section radially and 3 section tangentially and then the information of magnetic vector potential in each part to compute the power loss. The accuracy of the solution of course, depends on the finite number of harmonic terms and also the amount of information is accumulated from the each conductor. However, a good precision is obtained with given harmonic terms and observed that there are no much changes with increasing number of harmonic terms. Results for solution (1) neglecting eddy current field

Fig.4 shows the radial and tangential components of flux density in the bottom of the slot (r = 34.5mm), along the slot angle (δ). The results are obtained for different radius in the slot, but results were obtained closer to slot-opening is included since the region is quite sensitive. The obtained results has good agreement with FE ones everywhere in the slot. However, there is a negligible discrepancy between the FE and analytical obtained results closer to the slot opening region as shown in fig.4. It may come from numerical error of the analytical tool or mesh refinement of FE. These issues are discussed in section V in detail.

(a)

(b)

Fig. 4. (a) Radial (b) tangential flux density component in bottom of the slot (r = 34.5mm, âˆ† = π/4 elec.degree) FIG. 4 HERE

Using the obtain field from solution (1) neglecting eddy current reaction field eddy current and losses are calculated in (17), (18) respectively. From fig.5 as can be seen that the analytically calculated losses have good agreement with low frequency where the reaction field can be neglected and then start to diverge at a point (i.e. ~150Hz) which depends on machine's geometry and the conductor's height. These results clearly demonstrate that the eddy current reaction fields are acting a vital role at high frequencies, importantly VSW type of conductor. It reduces the total copper losses by opposing proximity phenomena.

Fig. 5. VSW copper losses vs frequency neglecting eddy current reaction field FIG. 5 HERE

## Results accounting eddy current field

Fig. 6 and 7 shows both real and imaginary part of radial and tangential components of flux density in the middle of the slot (r = 40mm) at frequency f = 200Hz and bottom of the slot (r = 34.5mm) at frequency f = 500Hz, along the slot angle (δ). The obtained results have good agreement with results were obtained in FE ones.

In the analytical solution the interaction of the slot opening and slot due to the reaction field is not accounted and thus, it can be influence the results. It is evident in the analytically computed imaginary tangential field in fig 7d that shows noticeable discrepancy at slot opening region. However, the influence can be neglected since the magnitude of magnetic vector potential used to calculate the eddy current losses.

(a)

(b)

(c)

(d)

Fig. 6. (a) Radial (b) tangential flux density component in bottom of the slot (r = 34.5mm, âˆ† = π/4 elec.degree) at frequency f = 200Hz

## FIG. 6 HERE

From fig. 8, it can be seen that the magnitude flux density dramatically decreases with frequency increase and there are no much changes in the reaction effect after at certain frequency (i.e. ~500Hz) which may vary with geometry of the machine. These results clearly explain that the reaction field is highly influence the VSW type of winding and limiting the additional AC losses by opposing the slot leakage flux.

(a)

(b)

(c)

(d)

Fig. 7. (a) Radial (b) tangential flux density component in bottom of the slot (r = 34.5mm, âˆ† = π/4 elec.degree) at frequency f = 500Hz FIG.7 HERE

(a)

(b)

Fig. 8. Flux density component in the (a) bottom (r = 34.5mm ) (b) middle of the slot (r = 40) .vs frequency FIG. 8 HERE

Fig. 9 shows the comparison between analytically calculated and FE computed losses with frequency. Both results have good agreement. The error between the two methods is less than one percent (~0.79%) at frequency over 400Hz. From these obtained results it can be concluded that proposed analytical method predicting the eddy current losses effectively and quickly accounting reaction field.

Fig. 9. VSW copper losses vs frequency accounting eddy current reaction field

## Limitation of analytical method and comparison with FE method

The analytical tools are powerful and becoming popular since the FE methods are taking significantly larger solving time. For example of considered PM machine , to calculate the losses at a frequency, a solution of electrical cycle in analytical tool takes only 46.66s while FE method requires around 2 - 3 hours. However, this time is not certain and it may vary for different level of frequency level. Thus, analytical tool can be solution in terms of cuts on time and cost.

Even though analytical tools are powerful and fastest, they have some limitations itself. In section IV, it already mentioned that the accuracy of analytical tools depends on choice of number of harmonic order as equivalent to mesh refinement in the FE. However the choice of number of harmonic order is limited by the software (i.e. Matlab and Maple) used for the calculations. For example, solution for the geometry of the machine designed to have narrow slot opening forced to choose a lower order of slot opening harmonic, result in inaccuracy in the solution.

Generally the analytical tools are modeled importantly, neglecting saturation. It is a major assumption which has high influence on the results. Thus, the analytical method is not always satisfies the actual condition. Hence, there are always a tradeoff between the accuracy, time and cost.

## Conclusion

An analytical model to evaluate the eddy current losses in the VSW wound, surface mounded radial flux PM machines accounting reaction effect has been proposed. The effectiveness of proposed method is validated through the FE analysis. The limitations of tool are discussed comparing FE method. This model can be used as optimization tool to calculate the eddy current losses in the VSW type of winding. In addition the analytical tool provides degree of freedom to investigate the parametric study in foil wound machines.

Appendix - E

Computation of coefficients:

To simplify the computation, firstly integral term is calculated as follows:

(B-1)

Expansion of (A-7) gives following functions that will be replaced in the computation of the Fourier coefficient Cmj.

for

(B-2)

for

(B-3)

Expansion of (15) given by,

(B-4)

Development of (16) can be expressed as follows:

(B-5)