Modeling of disk-type permanent magnet eddy-current ... - IOS Press

International Journal of Applied Electromagnetics and Mechanics 50 (2016) 525–535 DOI 10.3233/JAE-150096 IOS Press

525

Modeling of disk-type permanent magnet eddy-current driver based on soft measurement method and performance analysis Tongyu Shi∗ , Dazhi Wang, Zhao Li and Di Zheng School of Information Science and Engineering, Northeastern University, Shenyang, Liaoning, China Abstract. In order to solve the problem of immeasurability of eddy current density on the conductor plate and magnetic field intensity in the air gap, this paper establishes the eddy current soft measurement model. Eddy current density of the PM eddycurrent driver during run time can be calculated by measured data of slip speed and air gap in this model which is based on equivalent magnetic circuit. Then, the calculation formula of the torque can be deduced by applying this model. Transmit performance which provides an important basis for optimization calculation and preliminary design is also analyzed. All calculated soft measurement results are compared with results of finite element method using ANSYS software and experimental data. FEM results and experiment results confirm the validity of the soft measurement mechanism model. Keywords: Disk-type permanent magnet eddy-current driver, equivalent magnetic circuit method, finite-element method, soft measurement

1. Introduction According to Faraday’s law of induction and Lorentz’s formula of force, there are a series of electromagnetic connections between the permanent magnets and moving conductor. After analyzed in detail, a novel noncontact driving mode is discovered. Such mode facilitates the application of many eddy-current-based appliances, such as permanent magnet eddy-current driver, permanent magnet eddycurrent coupling (coupler) and so on [1–4]. As a typical representative among eddy-current-based appliances, permanent magnet eddy-current driver has been widely utilized in industrial transmission systems, such as draught fan and water bump. Permanent magnet eddy-current drivers have significant advantages such as soft starting of the load, vibration isolating, overload protection, installation simplicity, bigger error of centralization tolerance and non-mechanical joint. The magnetic field intensity in the air gap as well as eddy current density in the conductor are often needed when analyzing performance. But it is difficult to measure the magnitude of magnetic intensity in ∗ Corresponding author: Tongyu Shi, School of Information Science and Engineering, Northeastern University, Shenyang, Liaoning, China. Tel.: +86 13998875097; E-mail: [email protected].

c 2016 – IOS Press and the authors. All rights reserved 1383-5416/16/$35.00 

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T. Shi et al. / Modeling of disk-type permanent magnet eddy-current driver based on soft measurement method

the air gap by using teslameter because of the very small gap. Furthermore, the traditional measurement method of eddy current is to measure the electrical potential difference between two holes in which two probes are embedded. Then calculate the average eddy current density according to the conductivity of the conductor and the distance between two holes. Due to the complexity and the lack of precision, a lot of drawbacks of the method would emerge gradually. This kind of devices can be analyzed and studied with either analytical method or numerical method. Numerical methods, like finite element method (FEM) can give out a more accurate analysis. Nehl et al. [5] have done a torque characteristic modeling based on 2-D FEM for permanent magnet disc type eddy current couplers and the influences of a number of design parameters on the torque characteristics of the coupler have been examined. Salon et al. [6] and Albertz et al. [7] have done a 3-D finite element transient analysis for the braking force and braking torque of rotating conductor plate in magnetic field. Razav et al. [8] have optimized eddy-current coupling with slotted conductor disk in Lorentz force and transmission after their analysis with the help of FEM. Zhang et al. [9] developed an accurate 3-D nonlinear transient finite-element model of an eddy current brake and design an eight-pole ECB device as well. However applying FEM (like finite-element method) often requires a series of modeling, solving and post-processing. For this reason, the FEM is in some cases time consuming and unsuitable for the early design and analysis stages. On the other hand, with more flexible computation, time saving and high computational accuracy, anaylitical analysis is quite suitable for the research on preliminary design in such devices. Choi and Jang [10] have done an analytical analysis of magnetic field and braking force of eddy-current brake which has taken eddy current demagnetization effect into consideration. Amati et al. [11] have analyzed the torque dynamic performance of eddy-current dampers to determine the influence of major parameters on torque speed characteristics. Wang et al. [12] taking magnetic saturation and the shape of permanent magnet into consideration, built a 2-D model of permanent magnet eddy current couplings, from which the torque characteristics can be figured out. According to the Faraday’s law of electromagnetic induction and Ampere’s law, Mohammadi et al. [13] have modeled the central angle coordinate and analyzed drum type eddy-current couplers based on equivalent magnetic circuit method, then the sensitivity analysis was done. In paper [14], the magnetic field of eddy-current driver is analytically analyzed by the method of equivalent surface current. However, the reaction field due to induced current is neglected. The goal of this paper is to obtain the variables which are hard to measure directly by establishing the soft measurement mechanism model based on equivalent magnetic circuit method. The magnetic circuit is analyses in detail through the computation of leakage flux. In addition, this paper also analyzed and calculates the transmission performance of the PM eddy-current driver and an studied eddy-current driver test bench is established as well. Finally, all the results are numerically analyzed by the method of the finite element method. According to the comparison, FEM results and experiment results confirm the validity of the soft measurement mechanism model. The model is meaningful for the early design and analysis stages. 2. The basic structure and working principle 2.1. The basic structure The mechanical structure of the disk-type eddy-current driver is shown in Fig. 1. It is divided into three parts: an active conductor rotor which is connected to motor, a driven PM rotor which is connected to the load and a electric actuator which could adjust the length of the air gap. The torque-speed sensors

T. Shi et al. / Modeling of disk-type permanent magnet eddy-current driver based on soft measurement method

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Back iron Conductor plate y

wi

Permanent magnet Inner tube

Input shaft

Aluminum plate

Region 5

wm

b S

N

Output shaft

Region 4 Pin

tm S

Region 3 Region 2

a

N

0

g

-c

tc

x

Region 1

Fig. 1. The diagram the PM eddy-current drivers.

Fig. 2. 2D model of the studied eddy-current driver.

can detect torque and speed exactly and quickly. The actuator will adjust output speed by adjusting the air gap according to the signal from the controller. 2.2. Working principle Permanent magnet eddy-current driver works in the flexible transmission way. When the motor rotates, the conductor rotor will rotate along with the motor at the slip speed. According to the law of electromagnetic induction, the conductor plate will induce eddy current when the conductor cuts the magnetic lines. This eddy current simultaneously generates an induced magnetic field in the air gap which has the same direction with the permanent magnetic field. The induced magnetic field and permanent magnetic field interact with each other producing bonding force which can drive the permanent magnet rotor(the driven rotor) to rotate. Thus the transition of torque from motor to load is achieved by using eddy-current driver. 3. Soft measurement mechanism model of the PM eddy-current driver 2D model of the studied eddy-current driver is shown in Fig. 2, where tc and g are the thickness of conductor plate and the air gap length; tm is the thickness axially of the permanent magnet; wm and wi are the width of the permanent magnet and the space between each of them. 3.1. Soft measurement mechanism model regardless of the induced current In this model, the reaction filed due to induced currents is neglected and we suppose that the device is spread to a 2D plane in the circumferential direction with an average radius. Only one period covering adjacent magnetic poles is analyzed because the magnetic flow is in periodical distribution along circumference. In order to simplify the calculating procedures, following hypotheses are made, 1) The magnetic permeabilities of all kinds of materials are constants and not influenced by external factors such as temperature, pressure and so on.

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T. Shi et al. / Modeling of disk-type permanent magnet eddy-current driver based on soft measurement method Rs

2 Rg

Φg / 2

2 Rg

Rmm

Φm / 2

Φr / 2

Fig. 3. Magnetic flux paths of the 2D eddy-current driver.

2 Rm

Rl

Rl

2 Rm

Fig. 4. Equivalent magnetic circuit for Fig. 3.

2) The influence of end effect and magnetic saturation on magnetic field is neglected in the analysis of this paper. 3) The influence of magnetic field generated by induced current on conductors on the magnetic field distribution of eddy current drivers is neglected when analyzing the no-load magnetic field of eddy current drivers in this section. Figure 3 is a topological graph of closed flux loop of eddy-current drivers. In the figure the equivalent magnetic circuits is obtained through magnetic circuits analysis. And the equivalent magnetic circuit is then drawn based on Fig. 3 as shown in Fig. 4. Equivalent magnetic circuit method contains not only main flux, but also leakage flux, with which magnetic flux density of air gap and copper plates under the no-load condition(regardless of the induced magnetic flux) can be figured out. The variables are defined as follows: Rs is the reluctance of the stator iron; Rg is the total reluctance of the air-gap and the CS; Rm is the reluctance of one magnet; Rl is the reluctance corresponding to magnet-to-iron leakage flux; Rr is the reluctance of the rotor iron; Rmm is the reluctance corresponding to magnet-to-magnet leakage flux; Φr is the inner flux source of one magnet pole; Φg is the air-gap flux for one magnet pole; Φm is the outer flux source of one magnet pole. Considering the symmetry of the equivalent magnetic circuit, the equivalent magnetic circuit shown in Fig. 4 can be simplified as shown in Fig. 5. The reluctance of the magnet and the total reluctance of the air-gap and the CS are calculated as follows. Rm = tm /μ0 μr wm L

(1)

Rg = ge /μ0 wm L

(2)

ge = g + tc

(3)

μr = −Br /μ0 Hc

(4)

where

where μr is the relative recoil permeability of PMs. The expression of Rl can be obtained by the corresponding permeability. The circular-arc straightline permeance model offers a number of advantages for modeling flux flowing in the air gap [15,16]. In order to simplify the calculation, exit angle of leakage magnetic flux is set as π/2. we can obtain

T. Shi et al. / Modeling of disk-type permanent magnet eddy-current driver based on soft measurement method

529

Φg / 2

Rs

Φm / 2

Φr / 2

4 Rm

2 Rl

Rmm

Rs1 Rs 2

Rs1

4 Rg

Fig. 5. A reduced form of Fig. 4.

Fig. 6. Iron reluctances.

the length of leakage magnetic flux tm + 2xπ/2. Thus, the expression of the magnet-to-iron leakage permeance is:  L1 Pl = μ0 [L/(πx + tm )]dx (5) 0

where L1 is the minimum of the half of the space between each permanent magnet and the air gap. Its equation is shown as follows: L1 = min{g, wi /2}

(6)

Based on Eqs (5) and (6), the formula of permeance of magnet-to-iron leakage flux Pl is: Pl =

μ0 L ln(1 + π min{g, wi /2}/tm ) π

(7)

Similarly, the formula permeance of magnet-to-magnet leakage flux Pmm : Pmm =

μ0 L ln(1 + πg/wi ) π

(8)

The reluctance corresponding to magnet-to-iron leakage flux Rl and the reluctance corresponding to magnet-to-magnet leakage flux Rmm are the reciprocal of Pl and Pmm . In order to increase the accuracy of calculation, the reluctance of back iron (active rotor side) is divided into two parts. The expressions of reluctance are shown as below. Rs = 2Rs1 + Rs2

(9)

where Rs1 = 0.5wm /μ0 μs (LLs + 0.5wm L)

(10)

Rs2 = wi /μ0 μs LLs

(11)

According to the principle of circuit theory, expression of the air-gap flux for one magnet pole Φg is as below: Φg =

Φr [4Rm 2Rl  Rmm ] (4Rg + Rs ) + [4Rm 2Rl  Rmm ]

(12)

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T. Shi et al. / Modeling of disk-type permanent magnet eddy-current driver based on soft measurement method

So, the air gap axial magnetic flux density which is directly faces the PMs can be calculated. And axial magnetic flux density in the air gap which is between the PMs is approximately zero. Then the expression of air gap magnetic flux density can be written uniformly as Eq. (13). ⎧ ⎨ Φg = Φg Lwm Bg = Ag (13) ⎩ 0 Where Ag is the area of the air gap magnetic flux. 3.2. Soft measurement mechanism model considering the induced current As a result of the action of alternating magnetic field, the radial eddy current will be induced in the area where is just the opposite part of the permanent magnet in the conductor plate. The eddy current also excites magnetic field at the same time. So, it is necessary to take the reaction filed produced by eddy currents into account when analyzing the magnetic field. And the total magnetic field equals to the sum of the magnetic field resulting from the two magnetic source. The total magnetic flux density is shown as below, B = Bg + Bcs

(14)

Where Bcs is magnetic flux density produced from induced current on the conductor plate. Radial induced current density on the conductor plate of the eddy-current driver can be expressed by the Eq. (15). J = σBv

(15)

where v is the relative velocity and B is the total magnetic flux density inducted by PMs and the eddy current. According to the Ampere circuital theorem, we can get Eq. (16),  x2  tc  Hdl = Jdxdy (16) c

x1

0

Suppose that the corresponding MMF drop of the reaction flux in the iron regions is neglected. Substitute Eqs (14)–(15) and B = μ0 H into Eq. (16). Then this formula could be equated to the formula as below.  2(tc + tm + g)Bcs /μ0 = tc σv(Bg + Bcs )dx (17) So, differential equation of Bcs could be obtained according to Eq. (17):     μ0 σvtc μ0 σvtc dBcs /dx − Bcs = Bg 2tc + 2tm + 2g 2tc + 2tm + 2g To facilitate the calculation, we divide the copper region into three parts.

(18)

T. Shi et al. / Modeling of disk-type permanent magnet eddy-current driver based on soft measurement method Table 1 Structure parameters of the model of eddy-current driver Parameter Thickness of conductor plate, tc Thickness axially of the PM, tm Space between magnest, wi Width of the PM, wm Length radially of the PM, L Relative permeability of PM µr The air gap length, g Resistivity of copper Overhang lenth, H Number of PMs, N PM corecivity, Hc PM remanence, Br

Bcs1

Values 10 mm 20 mm 10.14 mm 30 mm 30 mm 1.05 3 mm 1.7 × 10−8 Ω/m2 10 mm 18 875000 A/m 123000 mT

Bcs 2

531

Bcs 3

S

N

S

N

S

N

Fig. 7. The magnetic field distribution diagram in the conductor plate.

Magnetic induction intensity produced from induced current on the conductor plate Bcs could be solved by substituting boundary conditions of the magnetic flux density Eqs (19)–(21) into differential Eq. (18). Bcs2 (x0 ) = 0

(19)

Bcs1 (x = −wm /2) = Bcs2 (x = −wm /2)

(20)

Bcs2 (x = wm /2) = Bcs3 (x = wm /2) ⎧ μ0 σvtc x ⎪ 2 t +2 t +2 g ; ⎪ ⎨Bcs1 = k1 e c μ σvtmc 0 Bcs = Bcs2 = k2 e 2 tc +2 tm +2 g x − Bg ; ⎪ μ0 σvtc ⎪ x ⎩B = k e 2 tc +2 tm +2 g ;

(21)

3

cs3

−(wi + wm )/2  x  −wm /2 −wm /2  x  wm /2

(22)

wm /2  x  (wi + wm )/2

x0 is the symmetric points of magnetic field distribution [17], x0 is determined through the following,  x0  tc  (wi +wm )/2  tc Jdxdy = Jdxdy (23) −(wi +wm )/2

0

x0

0

The total magnetic flux density inducted by PMs and the eddy current could be solved by substituting into Eq. (14). ⎧ Bcs1 ; −(wi + wm )/2  x  −wm /2 ⎪ ⎪ ⎪ ⎨ Φg B= (24) + Bcs2 ; −wm /2  x  wm /2 ⎪ Lw ⎪ m ⎪ ⎩ wm /2  x  (wi + wm )/2 Bcs3 ; 4. The soft measurement and verification results 4.1. The structure and material parameters In order to perform validation and analysis, an experimental model is established. The detailed parameters are shown in the Table 1.

T. Shi et al. / Modeling of disk-type permanent magnet eddy-current driver based on soft measurement method

Table 2 Comparison and error analysis between the soft measurement model and the FEM Air-gap Slip speed Soft measurement Simulation Error (mm) (rpm) value (A/m2 ) value (A/m2 ) (%) 8 3.0 300 0.510 × 10 0.526 × 108 3.08 8 3.0 500 0.665 × 10 0.681 × 108 2.35 3.0 700 0.824 × 108 0.849 × 108 2.94 8 3.0 1000 1.043 × 10 1.012 × 108 3.06 8 5.0 300 0.454 × 10 0.445 × 108 2.02 5.0 500 0.596 × 108 0.610 × 108 2.30 8 5.0 700 0.739 × 10 0749 × 108 1.34 8 5.0 1000 0.936 × 10 0.920 × 108 1.74

1

0.8

0.6

400rpm 400rpm(FEM) 200rpm 800rpm

0.4

0.2

B(T)

532

0

-0.2

-0.4

-0.6

-0.8 -0.025

-0.02

-0.015

-0.01

-0.005

0 x(m)

0.005

0.01

0.015

0.02

0.025

Fig. 8. Magnetic flux density of permanent magnet eddycurrent driver at speed of 200 rpm, 400 rpm and 800 rpm.

4.2. Magnetic flux density and eddy current Figure 8 shows comparison of predictions with FEM results at different slip speeds, namely, 200 rpm, 400 rpm and 800 rpm. As shown in Fig. 8, excellent agreement is observed. The deviations between soft measurement mechanism results and FEM results are very small. According to the curve in Fig. 8, the larger slip speed of the eddy-current driver is, the greater the impact on the total magnetic field will be. Comparison results between FEM data and soft measurement data under different speeds and air gap length without other design parameter changed are shown in Table 1 (mechanical angle is 0◦ ). As the table shows that the maximum error of eddy current density between model prediction and experimental testing is about 4%, and average error is about 2%. So, the soft measurement is comparatively accurate, which provides important basis for engineering calculation and heat dissipation analysis. Figure 9 shows changing trend of eddy current density at the speed of 100 r/min, 300 r/min and 500 r/min in 3 mm air gap. According to the curve, the eddy current density changes as an alternating curve with the change of the slip velocity. And it can be seen that the amplitude of the eddy current density increases with an increase at the slip speed. The reason is that the influence of magnetic field generated by eddy current on resultant field becomes larger with the increase of slip speed. 4.3. Calculation and verification of electromagnetic force of eddy current driver In engineering calculation, we suppose slip loss and the eddy current loss on copper plate P are approximately equal. Therefore, the magnetic force generated by each magnet can be calculated in the following Eq. (25). F =

P v

(25)

According to the literature [5], eddy-current loss corresponding to each pole can be obtained by multiplying J 2 /σ by the volume of conductor. Suppose there are N magnets in the eddy-current driver. Then

T. Shi et al. / Modeling of disk-type permanent magnet eddy-current driver based on soft measurement method

533

8

1.5

x 10

80 n=300r/min n=100r/min n=500r/min

electrMagnetic Torque(N.m)

Current Density [A/m2]

1

0.5

0

-0.5

-1

-1.5 -0.8

T(Soft Measurement) T(FEM) T(Experimental)

70 60 50 40 30 20 10 0

-0.6

-0.4

-0.2 0 0.2 0.4 Mechanical Angular Position [Rad]

0.6

0.8

Fig. 9. Curve of current density against slip speed using soft measurement model.

0

20

40 60 80 Relative Speed n(r/min)

100

120

Fig. 10. The comparison curves of results of torque of eddy-current driver with FEM, soft measurement method and experimental data.

the electromagnetic torque will be N (Φg )2 rav (tc + tm + g) 2 μ Lwm 0 (26)   μ0 σωrav tc (wi + wm ) μ0 σωrav tc wi 2 μ0 σωrav tc wi tanh − sinh 2 cosh 4(tc + tm + g) 4(tc + tm + g) 2(tc + tm + g) where rav is the average radius. Reference [12] presents a correction factor which could be adopted to amendment on the real eddy current path and the actual coupling geometry. According to the structure of eddy-current driver in this paper, we can express ks in the following formula: tanh λm ks = 1 − (27) λm (1 + tanh λm tanh λcm ) where T2d =

λm = πL/ [2(wi + wm )] λcm = πH/ [(wi + wm )]

(28) (29)

where L is length radially of overlapping area between the permanent magnets and the conductor and H is length of overhang region of the conductor from both sides. The 3D electromagnetic torque after correction can be calculated. The formula is shown as below. T3d = ks T2d

(30)

It is remarkable that the eddy currents in the back iron (active rotor) also supply torque of eddy-current driver. In other words, the role of the eddy currents in the back iron cannot be neglected. According to calculation, the iron region accounts for 8.4% of the total torque. Figure 10 shows torque-speed characteristic curves of eddy-current driver after correction obtained by using soft measurement method. As can be observed in Fig. 10, driving torque is almost in direct proportion to the relative speed between the driving shaft and driven shaft. Increasing extent of torque reduces because the influence of eddy current becomes larger as the relative speed increased.

534

T. Shi et al. / Modeling of disk-type permanent magnet eddy-current driver based on soft measurement method

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$&PRWRU

(GG\FXUUHQW 'ULYHU WRUTXHPHW HU

$&PRWRU

'&PRWRU

,QVWUXPHQWEDVH

Fig. 11. The overall structure of the experimental test bench.

Fig. 12. The structure of the eddy-current driver test bench.

4.4. Experimental testing results According to structure parameters in this paper, prototype of the disk-type eddy-current driver has been manufactured and a experiment test bench has also been established. The overall structure of the experimental test bench is shown in Fig. 11. The apparatus consists of (on the test bench from left to right) variable-frequency drive, AC motor, eddy-current driver, torque/speed meter, AC motor and DC motor. Figure 12 shows the actual test bench. The active rotor of the permanent magnet eddy-current driver is driven by an asynchronous motor at the rated speed. A torque meter is installed between the permanent magnet eddy-current driver and the load in order to obtain the real-time torque. Ac motor is the prime mover. Its rated power is 15 kw and rated speed is 1455 r/min. Torque experiment results of PM eddy-current driver are also shown in Fig. 10. As can be observed, soft measurement method has a good agreement with the experimental ones and did not cost much computation resource. 5. Conclusion Soft measurement method based on equivalent magnetic circuit provides many benefits for predicting and analyzing eddy current driver. Eddy current density and magnetic flux density of the PM eddycurrent driver during run time can be calculated by measured data of slip speed and air gap in this model. We also use a nonlinear transient finite-element model for analyzing to verify that the method is accurate. After theoretical deducing and simulating, transmit performance has been tested on the test bench. The results from finite element method and experiments are in good agreement with those from the soft measurement method indicating that the soft measurement method is reasonable, correct and can provide a quite accurate reference for the design and development of the permanent magnet eddy-current driver. Acknowledgments The work described in this paper was supported by the Chinese Technical Innovation Major Projects of Liaoning Province (201309001) and National Science Foundation of China (61433004) financially.

T. Shi et al. / Modeling of disk-type permanent magnet eddy-current driver based on soft measurement method

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