Eddy current loss in double-sided cored slotless type permanent magnet linear synchronous generator using analytical method

Jang, Gang-Hyeon; Koo, Min-Mo; Seo, Sung-Won; Hong, Keyyong; Choi, Jang-Young

doi:10.1063/1.4973435

This paper describes the eddy current loss analysis of a double-sided cored slotless-type permanent magnet linear synchronous generator using the space harmonic method. To calculate the eddy current, an analytical solution was derived by applying the Maxwell equation, magnetic vector potential, and Faraday’s law in a two-dimensional (2-D) Cartesian coordinate system. Based on the armature reaction field distribution produced by the armature winding current, we obtained an analytical solution for the eddy current density distribution. Then, an analytical solution for eddy current loss induced in a permanent magnet (PM) was derived using the equivalent electrical resistance calculated from the PM’s volume and eddy current density distribution solution. Finally, the resulting current was determined based on analysis of the eddy current loss. The analytical results produced using this application of the space harmonic method were validated extensively through comparison with finite element method results.

I. INTRODUCTION

Owing to high energy densities of Nd-Fe-B permanent magnets, they can be miniaturized, which in turn enables the production of relatively lightweight, high-efficiency permanent magnet generators. One such generator type—the Nd-Fe-B permanent magnet linear synchronous generator (PMLSG)—can convert translational motion energy without the need for mechanical conversion from electrical energy.¹ Because of their outstanding performance capabilities, PMLSGs have been widely used in various industrial applications such as wave energy converter systems. Although the conductivity of rare earth permanent magnets is much higher than that of ferrite magnets, motors constructed from permanent magnets have reduced eddy current activity, as the magnet’s movement is synchronized with that of the stator armature winding.^2,3 Nevertheless, slotting effects owing to the slot structure, space harmonics caused by the magneto motive force the distribution of the armature winding, and the time harmonics of the armature current are all likely to increase the eddy currents in a permanent magnet. Sufficiently high eddy currents can cause an increase in temperature in a permanent magnet, causing it to operate at the knee point of the B–H curve, and thus tend towards demagnetization. This potential degradation of PMLSG performance makes eddy current loss analysis significantly important at the time of initial design.⁴ Eddy current analysis has been performed widely using both the finite element method (FEM) and analytical methods. However, the number of initial parameters used is often limited, and increasing the size of the initial model or changing the design parameters is not frequently done in FEM analysis and is difficult in reality. As they can be used to obtain analysis results quickly, it is often advantageous to use an analytical method in the initial design. This paper describes the use of an analytical approach using the space harmonic method to calculate the eddy current loss of a double-sided cored slotless-type PMLSG.

II. APPLICATION OF THE HALBACH MAGNET ARRAY TO A LINEAR SYNCHRONOUS GENERATOR

A. The analytical model

Fig. 1 shows the PMLSG developed for this study, which comprises a double-sided cored slotless-type generator and a permanent magnet magnetized in a Halbach magnet array. The proposed PMLSG has a two-pole, six-slot structure, with 80 poles altogether in the entire model. The armature windings follow a distributed-type pattern. The design specifications for the PMLSG are shown in Table I. Fig. 2 shows the analytical model of PMLSG that was used to predict the magnetic field of the armature windings. Several prior assumptions were used for this analysis. First, the magnetic permeability of the permanent magnet and the coil are both the same as the magnetic permeability of air; this is a valid approximation because the specific magnetic permeability of the rare-earth magnet is actually about 1.04, while the permeability of copper is approximately 0.99. Second, the relative permeability of the stator iron cores and the mover iron core are assumed to be infinite. In general, the stator and the mover iron cores are designed to be unsaturated as the relative permeability in the interval is naturally unsaturated and the B–H curve of the steel sheet is above 2,000. Third, the stator and the mover iron core are considered to have zero conductivity as the core is laminated. The characteristics of the magnetic armature current area based on these assumptions are divided into those for the coil region, including the current source, and those of the magnetic air-gap region, including the permanent magnets (PMs) and the air-gap, as shown Fig. 2.

FIG. 1.

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Manufactured double-sided cored slotless-type PMLSG.

TABLE I.

Design Specification for PMLSG.

Parameters	Values
Magnetic air-gap	27mm
Mechanical air-gap	7mm
Permanent magnet height	20mm
Coil height	4mm
Pole pitch	40mm
Pole number	80
Turn per coil	5
Velocity	1 m/s

Parameters	Values
Magnetic air-gap	27mm
Mechanical air-gap	7mm
Permanent magnet height	20mm
Coil height	4mm
Pole pitch	40mm
Pole number	80
Turn per coil	5
Velocity	1 m/s

FIG. 2.

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Simplified model for the analytical method.

B. Current density modeling

Fig. 3 shows a Fourier series expansion model of the current density, distributed between the stator core and the mover cores, of a three-phase armature assuming the equilibrium condition in the analytical model of Fig. 2. The armature winding is of the distributed type. Equation (1) gives a Fourier series expression of Fig. 3:

J = \sum_{m = - \infty}^{\infty} Re [I_{m} {i_{a} e^{- j k_{m} x} + i_{b} e^{- j k_{m} (x - 2 / 3 τ_{p})} + i_{c} e^{- j k_{m} (x - 4 / 3 τ_{p})}}],

(1)

where I_m is the m-th order Fourier coefficient of the current density distribution, x is a stator based on the displacement, and $τ_{p}$ is the pole pitch. The Fourier coefficients I_m are given by

I_{m} = \frac{J_{0}}{2 j k_{m} τ_{p}} {(e^{j k_{m} c l} - 1) + (e^{j k_{m} (3 c l + 3 d)} - e^{j k_{m} (4 c l + 3 d)})},

(2)

where k_m is the space harmonic coefficient, J₀ is the magnitude of the current density, cl is the width of the coil, and d is the distance between the two coils.

FIG. 3.

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Modeling of the armature current density.

Three phase currents can then be expressed as follows:

\begin{matrix} i_{a} = \sum_{n = 1}^{\infty} I_{n} \cos (2 n π f t) \\ i_{b} = \sum_{n = 1}^{\infty} I_{n} \cos (2 n π f t - \frac{2}{3} n π), \\ i_{c} = \sum_{n = 1}^{\infty} I_{n} \cos (2 n π f t - \frac{4}{3} n π) \end{matrix}

(3)

where n is the mean time harmonic order of the phase current, I_n is the magnitude of the phase current, and f is the frequency of the phase current, which is expressed as

f = \frac{V_{s}}{2 τ_{p}},

(4)

where V_s is the mechanical speed of the magnet mover.

Eddy current loss is caused by asynchronous space harmonics of the magneto motive force and time harmonics of the armature current. Using the relation X = V_st + x, the stator displacement x can be expressed by amending the mover displacement equation (1) in order to simplify the analysis. After amending equation (1) and substituting it into the phase current equation (3), the final armature current density distribution equation can be obtained as follows:

J = \sum_{n = 1}^{\infty} \sum_{m = - \infty}^{\infty} R e (I_{m} I_{n} K) .

(5)

C. Armature reaction field distribution

The magnetic flux density B and the magnetic intensity H satisfy Maxwell’s equations, B = $μ_{0}$ (H + M); taking the curl of this gives $\nabla \times$ B = $μ_{0}$ (⁠ $\nabla \times$ H) + $μ_{0}$ (⁠ $\nabla \times$ M). Using the magnetic vector potential A defined from $\nabla \times$ A = B, Laplace’s and Poisson’s equations in terms of the Coulomb gauge $\nabla \cdot$ A = 0 are given by⁵

\nabla^{2} 𝐀 = - {μ_{0} 𝐉 + μ_{0} (\nabla \times 𝐌)} .

(6)

The magnetic air-gap is region I that is assumed to have no magnetization component and no current sources. Therefore, it is possible to obtain a Laplace equation in I as

\nabla^{2} 𝐀^{I} = 0.

(7)

In region II (the coil region), which has only current sources and no magnetization component, it is possible to obtain a Poisson equation as follows:

\nabla^{2} 𝐀^{I I} = - μ_{0} 𝐉 .

(8)

And then, the magnetic vector potential is determined by the direction of the current and the current density distribution can be expressed in the form of expression.

𝐀 = A_{z n, m} (y) K 𝐢_{𝐳} .

(9)

Summarizing the above equations by substituting in the governing equation produces the following differential equations:

\begin{matrix} \frac{\partial^{2}}{\partial y^{2}} A_{z n m}^{I} (y) - k_{m}^{2} A_{z n m}^{I} (y) = 0 \\ \frac{\partial^{2}}{\partial y^{2}} A_{z n m}^{I I} (y) - k_{m}^{2} A_{z n m}^{I I} (y) = - μ_{0} J \end{matrix} .

(10)

By solving the above equations, a general solution of the differential equation for the armature current can be obtained as

\begin{matrix} A_{z n m}^{I} (y) = \sum_{n = 1}^{\infty} \sum_{m = - \infty}^{\infty} Re ⟨ C_{n}^{I} e^{k_{m} y} + D_{n}^{I} e^{- k_{m} y} ⟩ \\ A_{z n m}^{I I} (y) = \sum_{n = 1}^{\infty} \sum_{m = - \infty}^{\infty} Re ⟨ C_{n}^{I I} e^{k_{m} y} + D_{n}^{I I} e^{- k_{m} y} + \frac{μ_{0} 𝐉}{k_{m}^{2}} ⟩ \end{matrix} .

(11)

Based on the definition of the magnetic vector potential $\nabla \times$ A = B, equation (11) can be used to derive a magnetic flux density equation for each region as follows:

𝐁 = \frac{\partial A_{z}}{\partial y} 𝐢_{x} - \frac{\partial A_{z}}{\partial x} 𝐢_{y} .

(12)

Using this flux density equation and the general solution of the differential equation, we then obtain an armature reaction magnetic flux density in each region as follows:

\begin{matrix} 𝐁_{𝐱}^{𝐈} & = & \sum_{n = 1}^{\infty} \sum_{m = - \infty}^{\infty} Re ⟨ k_{m} [C_{n}^{I} e^{k_{m} y} - D_{n}^{I} e^{- k_{m} y}] K ⟩ 𝐢_{𝐱} \\ 𝐁_{𝐲}^{𝐈} & = & \sum_{n = 1}^{\infty} \sum_{m = - \infty}^{\infty} Re ⟨ j k_{m} [C_{n}^{I} e^{k_{m} y} + D_{n}^{I} e^{- k_{m} y}] K ⟩ 𝐢_{𝐲} \\ 𝐁_{𝐱}^{𝐈𝐈} & = & \sum_{n = 1}^{\infty} \sum_{m = - \infty}^{\infty} Re ⟨ k_{m} [C_{n}^{I I} e^{k_{m} y} - D_{n}^{I I} e^{- k_{m} y}] K ⟩ 𝐢_{𝐱} \\ 𝐁_{𝐲}^{𝐈𝐈} & = & \sum_{n = 1}^{\infty} \sum_{m = - \infty}^{\infty} Re ⟨ j k_{m} [C_{n}^{I I} e^{k_{m} y} + D_{n}^{I I} e^{- k_{m} y} + \frac{μ_{0} 𝐉}{k_{m}^{2}}] K ⟩ 𝐢_{𝐲} \end{matrix} .

(13)

The coefficients of the above equations can be derived using the boundary conditions in Table II.

TABLE II.

Boundary Condition of each boundary.

Boundary	Tangential component	Normal component
y = h_a	${𝐁_{𝐱}}^{𝐈} = 0$	$\cdot$
y = 0	${𝐁_{𝐱}}^{𝐈} = {𝐁_{𝐱}}^{𝐈 𝐈}$	${𝐁_{𝐲}}^{𝐈} = {𝐁_{𝐲}}^{𝐈 𝐈}$
y = −h_c	${𝐁_{𝐱}}^{𝐈 𝐈} = 0$	$\cdot$

Boundary	Tangential component	Normal component
y = h_a	${𝐁_{𝐱}}^{𝐈} = 0$	$\cdot$
y = 0	${𝐁_{𝐱}}^{𝐈} = {𝐁_{𝐱}}^{𝐈 𝐈}$	${𝐁_{𝐲}}^{𝐈} = {𝐁_{𝐲}}^{𝐈 𝐈}$
y = −h_c	${𝐁_{𝐱}}^{𝐈 𝐈} = 0$	$\cdot$

D. Eddy current density analysis

The eddy current density equation can be obtained from the magnetic vector potential using the definition of Faraday’s Law as follows:⁶

\begin{matrix} \nabla \times 𝐄 & = & - \frac{\partial 𝐁}{\partial t} = - \frac{\partial (\nabla \times 𝐀)}{\partial t} = \nabla \times (- \frac{\partial 𝐀}{\partial t}) \\ ∵ 𝐄 & = & - \frac{\partial 𝐀}{\partial t} \end{matrix} .

(14)

The eddy current induced in a permanent magnet with electric conductivity σ is given by $𝐉_{𝐞} = σ 𝐄$ .

Combining this with equation (14), the eddy current density can be expressed using the following equation:

𝐉_{𝐞} = - σ \frac{\partial 𝐀}{\partial t} .

(15)

The eddy current density induced in the permanent magnet by the three-phase armature current can be expressed using equation (15) as

𝐉_{𝐞} = - σ \frac{\partial 𝐀_{z n m} K}{\partial t} .

(16)

By applying equation (11), the general solution of the magnetic vector potential, into equation (16), an expression for the eddy current density in the region containing the permanent magnet can be obtained as follows:

𝐉_{𝐞} = \sum_{n = 1}^{\infty} \sum_{m = - \infty}^{\infty} Re ⟨ - σ_{m} [C_{n}^{I} e^{k_{m} y} + D_{n}^{I} e^{- k_{m} y}] Π ⟩ .

(17)

The eddy current losses induced in one pole of the combined PM can then be calculated using the following equation:⁷

P_{𝐞} = \frac{1}{t} \int_{o}^{t} \int_{y 1}^{h a} \int_{- τ_{p}}^{τ_{p}} ρ J_{e}^{2} d x d y d t,

(18)

where ρ is the resistivity of the permanent magnet. The calculation of eddy current loss using the above equation involves three multiplexed integrations of the square of the eddy current density, which can involve very complex mathematical operations. Therefore, in this study, we can obtain the eddy current loss using the volume of the PM which is substituted with the resistance based on the conductivity of the PM:

P_{𝐞} = I_{e}^{2} R_{P M} = {(J_{e}^{S} S)}^{2} l_{z} / (σ_{m} S) = {(J_{e}^{S})}^{2} S l_{z} / σ_{m},

(19)

where I_e is the eddy current flowing to the permanent magnet, R_PM is the resistance value for each pole pair and S, and l_z represent each area and length, respectively, of the flowing eddy current, i.e., $S = (2 τ_{p} p) (y_{2 k + 1} - y_{1})$ ⁠, and l_z is the length of the permanent magnet in the z-axis direction. $J_{e}^{S}$ is the effective value of the eddy current density flowing per pair of poles; $J_{e}^{S}$ changes according to height. Therefore, the eddy current loss can be predicted by dividing the PM into 2k equal parts, as shown in Fig. 4. The eddy current loss can be predicted using the following equation:

P_{𝐞} = \frac{l_{z}}{σ_{m}} \sum_{k = 1} {(J_{e}^{y_{2 k}})}^{2} (2 τ_{p}) (y_{2 k + 1} - y_{2 k - 1}),

(20)

where $y_{2 k + 1} = h_{a}$ and $J_{e}^{y_{2 k}}$ is the effective value of the eddy current density at magnet height $y_{2 k}$ ⁠.

FIG. 4.

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Schematic for the calculation of eddy current losses.

III. ANALYSIS AND MEASUREMENT RESULTS

A. The phase current waveform and fast Fourier transform analysis according to ac and dc loads

Fig. 5 shows the phase current waveform obtained under ac and dc loads. Fig. 5(a) shows the phase current waveforms measured for an ac and a dc load, i.e., a rectifier load at an output condition of 800 W. Fig. 5(b) shows the results of fast Fourier transform analysis of the waveforms in (a). In the case of the PMLSG with the rated ac load, it is seen that the harmonic components of the phase current are composed of first, third, and seventh orders, with the first being the dominant harmonic order. In the PMLSG with the rated dc load, the harmonic components of the phase current consist of first, third, fifth, seventh, and eleventh orders, which are dominated by the first and fifth orders.

FIG. 5.

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Measured current based on load condition: (a) phase currents for AC and DC loads, and (b) comparison of the FFT results.

B. Comparison of armature reaction field and eddy current density

Fig. 6 shows a comparison between the armature reaction fields induced by the ac and dc load phase currents. These results are also compared with the FEM results. Figs. 6(a) and (b) show the armature reaction field analysis results under the ac and dc load conditions, respectively. Fig. 7 shows, in accordance with the load state, the eddy current density results of analysis for the first and fifth harmonic currents. All of these analytical results are also compared with the FEM results. It is seen from Fig. 7 that the eddy current is generated primarily by the fifth harmonic in the dc load state.

FIG. 6.

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Armature reaction field analysis results based on load condition: (a) for AC load, and (b) for DC load.

FIG. 7.

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Eddy current density analysis results based on time harmonic components of armature winding current: (a) AC load condition, and (b) DC load condition.

C. Comparison of eddy current distribution and eddy current losses

Fig. 8(a) shows the distribution of the generated eddy currents of the permanent magnet in accordance with the load state. It is seen from the Fig. 8(a) that, under a given output condition, the harmonic-produced eddy currents are greater under the dc load. Fig. 8(b) shows a comparison of the analytical and FEM results obtained for loss occurring using the ac and dc loads. The analytical and FEM results are relatively well matched and confirm that the eddy current loss under a given output condition is further increased when using the dc load.

FIG. 8.

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Eddy current loss analysis results: (a) FEM results for eddy current density in permanent magnet region produced by armature winding current under AC, DC load conditions, and (b) comparison of FEM results with analytical results for eddy current loss produced by armature winding current according to load condition.

IV. CONCLUSION

This paper described an analytically driven interpretation of eddy current losses in a double-sided cored slotless-type permanent magnet linear synchronous generator with a Halbach array. After modeling the armature current density in terms of space and time harmonics, the magnetic field distribution owing to the armature current based on the magnetic vector potential in a 2-D Cartesian coordinate system was applied to obtain the governing equation coefficients. Refined governing equations were then obtained by substituting the boundary condition at each boundary surface. The eddy current density equation was derived on the basis of the induced magnetic field distribution equation; using the eddy current density and volume of the permanent magnet in terms of equivalent resistance, eddy current loss in the permanent magnet could be calculated. Finally, the eddy loss was calculated based on the eddy currents in the load state of the PMLSG. Analytical results demonstrated the validity of the proposed method, which produced results in close agreement with those produced by finite element analysis. The method, equations, and properties derived in the paper should be very useful in modeling eddy current loss during the PMLSG initial design analysis stage.

ACKNOWLEDGMENTS

This work is a result of the project “Development of Wave Energy Converters Applicable to Breakwater and Connected to Micro-Grid with Energy Storage System” (20160254).

APPENDIX

The undefined coefficients are given as follows:

K = e^{- j k_{m} (X - V_{s} t)} \cdot \cos (2 n π f t) + e^{- j k_{m} (X - V_{s} t - \frac{2}{3} τ_{p})} \cdot \cos (2 n π f t - \frac{2}{3} n π) + e^{- j k_{m} (X - V_{s} t - \frac{4}{3} τ_{p})} \cdot \cos (2 n π f t - \frac{4}{3} n π)

Π = j k_{m} V_{s} {e^{- j k_{m} (X - V_{s} t)} \cdot \cos (2 n π f t) - 2 n π f \cdot \sin (2 n π f t) \cdot e^{- j k_{m} (X - V_{s} t)} + e^{- j k_{m} (X - V_{s} t - \frac{2}{3} τ_{p})} \cdot \cos (2 n π f t - \frac{2}{3} n π) - 2 n π f \cdot \sin (2 n π f t - \frac{2}{3} n π) \cdot e^{- j k_{m} (X - V_{s} t - \frac{2}{3} τ_{p})} + e^{- j k_{m} (X - V_{s} t - \frac{4}{3} τ_{p})} \cdot \cos (2 n π f t - \frac{4}{3} n π) - 2 n π f \cdot \sin (2 n π f t - \frac{4}{3} n π) \cdot e^{- j k_{m} (X - V_{s} t - \frac{4}{3} τ_{p})}}

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Author(s)

Eddy current loss in double-sided cored slotless type permanent magnet linear synchronous generator using analytical method

I. INTRODUCTION

II. APPLICATION OF THE HALBACH MAGNET ARRAY TO A LINEAR SYNCHRONOUS GENERATOR

A. The analytical model

B. Current density modeling

C. Armature reaction field distribution

D. Eddy current density analysis

III. ANALYSIS AND MEASUREMENT RESULTS

A. The phase current waveform and fast Fourier transform analysis according to ac and dc loads

B. Comparison of armature reaction field and eddy current density

C. Comparison of eddy current distribution and eddy current losses

IV. CONCLUSION

ACKNOWLEDGMENTS

APPENDIX

REFERENCES

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Eddy current loss in double-sided cored slotless type permanent magnet linear synchronous generator using analytical method Open Access

I. INTRODUCTION

II. APPLICATION OF THE HALBACH MAGNET ARRAY TO A LINEAR SYNCHRONOUS GENERATOR

A. The analytical model

B. Current density modeling

C. Armature reaction field distribution

D. Eddy current density analysis

III. ANALYSIS AND MEASUREMENT RESULTS

A. The phase current waveform and fast Fourier transform analysis according to ac and dc loads

B. Comparison of armature reaction field and eddy current density

C. Comparison of eddy current distribution and eddy current losses

IV. CONCLUSION

ACKNOWLEDGMENTS

APPENDIX

REFERENCES

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Eddy current loss in double-sided cored slotless type permanent magnet linear synchronous generator using analytical method