The purpose of this paper is to present a 2-D equivalent finite element model of a quadratic linear electromagnetic actuator that can save space and power as it does not employ an energy conversion system. A 2-D model, while being fairly accurate, is preferable to a 3-D finite element analysis for the design and analysis of a quadratic linear electromagnetic actuator as it requires significantly lower computing resources and results in faster calculations. We calculate the effective coil length for the equivalent 2-D finite element model and validate the accuracy of this model with experimental data

Byunghyun Kim

Hwa Cho Yi

Jin Ho Kim

Jung Hun Lee

Lawrence Kulinsky

Marc Madou

English

Journal of Vibroengineering

  © JVE INTERNATIONAL LTD. JOURNAL OF VIBROENGINEERING. JUNE 2014. VOLUME 16, ISSUE 4. ISSN 1392-8716 2047 
1306. 2-D Equivalent finite element model of quadratic 
linear electromagnetic actuator 
Jung Hun Lee1, Lawrence Kulinsky2, Marc Madou3, Byunghyun Kim4, Hwa Cho Yi5, 
Jin Ho Kim6 
1Compressor R&D Development Departament, LG Electronics, Changwon, Gyengnam, Korea 
2, 3Department of Mechanical and Aerospace Engineering, University of California 
Irvine, CA, 92697, USA 
4UC Center for Hydrologic Modeling, Department of Civil & Environmental Engineering 
University of California, Irvine, CA, 92617, USA 
5, 6Department of Mechanical Engineering, Yeungnam University 
Gyeongsan, Gyeongbuk, 712-749, Korea 
6Corresponding author 
E-mail: 1junghun13.lee@lge.com, 2lawrence.kulinsky@uci.edu, 3mmadou@uci.edu, 4bh.kim@uci.edu, 
5hcyi@yu.ac.kr, 6jinho@ynu.ac.kr 
(Received 5 March 2014; received in revised form 4 April 2014; accepted 13 April 2014) 
Abstract. The purpose of this paper is to present a 2-D equivalent finite element model of a 
quadratic linear electromagnetic actuator that can save space and power as it does not employ an 
energy conversion system. A 2-D model, while being fairly accurate, is preferable to a 3-D finite 
element analysis for the design and analysis of a quadratic linear electromagnetic actuator as it 
requires significantly lower computing resources and results in faster calculations. We calculate 
the effective coil length for the equivalent 2-D finite element model and validate the accuracy of 
this model with experimental data. 
Keywords: equivalent, 2D, finite element model, quadratic, electromagnetic, actuator. 
1. Introduction 
Voice coil actuator are demanding applications as precision actuators widely in IT, automobile, 
defense and medical systems and so on. It has numerous designs as the application. The flat coil 
actuaor has being used in IT systems as computer hard disk, camera module, semiconductor 
equipment and etc. [1-3] and the cylindrical actuator has being used in automobile as 
electromagnetic engine valve, vibration absorber and etc. [4, 5] Especially, the quadratic linear 
electromagnetic actuators leads to higher magnetic force due to large number of permanent 
magnets compared to other voice coil motors [6]. A 3-D Finite element (FE) model is a rather 
effective approach to model quadratic liear electromagnetic actuators [7-9], but a 3-D simulations 
utilize significant computer resourses and typically, 3-D dynamic simulations cannot be 
performed on personal computers. Therefore, it is often convinient to employ simpler 2-D 
equivalent model to simulate static and dynamic behavior of linear electromagnetic actuators; 
simulations that can be preformed using personal computers [10]. In this paper, we present and 
validate an equivalent 2-D FE model that replaces the 3-D FE model. To determine the depth of 
equivalent 2-D model, the effective coil length is calculated. The prototype of the quadratic 
electromagnetic actuator is manufactured, and the accuracy of the proposed equivalent 2-D FE 
model is verified experimentally. 
2. Structure and operation principle of electromagnetic linear actuator 
Figs. 1(a) and 1(b) show a schematic diagram and a cross-section of the quadratic linear 
electromagnetic actuator respctively. The moving part consists of an inner steel yoke with four 
small magnets and an outer steel yoke with four larger magnets [11-13]. The inner and outer 
magnets create a very dense magnetic field that a strong magnetic force. The stationary part 
consists of the electromagnetic coil and housing. Fig. 1(c) shows the operation principle of the 
quadratic linear electromagnetic actuator. The moving part is guided by mechanical plate springs 
1306. 2-D EQUIVALENT FINITE ELEMENT MODEL OF QUADRATIC LINEAR ELECTROMAGNETIC ACTUATOR.  
JUNG HUN LEE, LAWRENCE KULINSKY, MARC MADOU, BYUNGHYUN KIM, HWA CHO YI, JIN HO KIM 
2048 © JVE INTERNATIONAL LTD. JOURNAL OF VIBROENGINEERING. JUNE 2014. VOLUME 16, ISSUE 4. ISSN 1392-8716  
and moves up and down by the stored spring energy and the magnetic Lorentz force as the 
electromagnetic coil remains fixed [14, 15]. 
 
a) 
 
b) 
 
 
c) 
Fig. 1. Quadratic electromagnetic linear actuator: a) schematic diagram,  
b) cross section of magnetic actuator, c) operating principle 
3. Equivalent 2D FE Model 
Fig. 2(a) is the 3-D FE model of the quadratic linear electromagnetic actuator. Each of the four 
sides of the square- shape coil interacts with the magnetic field of the outer and inner magnets. 
We assume that the magnetic force of the four sides of the coil in 3-D model is twice the magnetic 
force of FE model. Therefore, we set the depth of the equivalent 2-D model to be twice the length 
of the inner magnet and outer magnet or the mean length of the inner and outer magnets. The 
length of the permanent inner magnet is 30 mm, that of the permanent outer magnet is 50 mm and 
the average length is 40 mm. Therefore, the corresponding depths of the equivalent model are 
60 mm, 100 mm, and 80 mm, respectively. We have created a 2-D static FE equivalent model and 
a 3-D static FE full model utilizing the commercial “MAXWELL” FEA software. The resulting 
magnetic force calculated by these two models are then compared. The nonlinear B-H 
characteristic of steel 1010 is applied to the steel yoke and the magnetic properties of the 
permanent magnet are shown in Table 1. The modeling results are presented in Table 2. 
 
a) 
 
b) 
Fig. 2. Simulation model: a) 3-D model, b) equivalent 2-D model 
1306. 2-D EQUIVALENT FINITE ELEMENT MODEL OF QUADRATIC LINEAR ELECTROMAGNETIC ACTUATOR.  
JUNG HUN LEE, LAWRENCE KULINSKY, MARC MADOU, BYUNGHYUN KIM, HWA CHO YI, JIN HO KIM 
 © JVE INTERNATIONAL LTD. JOURNAL OF VIBROENGINEERING. JUNE 2014. VOLUME 16, ISSUE 4. ISSN 1392-8716 2049 
Table 1. Specification of the Neodymium permanent magnet 
Neodymium permanent magnet 
Symbol Parameter Value  Residual induction 1.23 T  Coercive –890 kA/m  Relative permeability 1.1 
Table 2. Result of the FE analysis with 2-D and 3-D FE models 
Case 
3D model 2D equivalent model 
Magnetic force (N) Magnetic force (N) Ratio 
Inner PM (60 mm) 
11.7 
9.6 0.82 
Average (80 mm) 12 1.1 
Outer PM (100 mm) 16 1.367 
The ratios of magnetic forces calculated for the 2-D equivalent models to that of the full 3-D 
model are 0.82 if the depth is of the system is assumbed to be double the length of the inner PM 
and that ratio is 1.367, if the depth of the system is double the legth of the outer PM. In the case 
when the depth of the modelled system is assumed to be twice the average length of PM, the ratio 
is 1.1. There is at least 10 % error using this method. Accordingly, we need to find a new way to 
calculate the depth of the equivalent 2-D model. The depth of the equivalent 2-D FE model to 
replace the 3-D model is proposed. The Lorentz force to drive the actuator is expressed by a cross 
vector between the flux density and coil length in the Eq. (1) [16]: 
	
    , (1)
where  is the number of coil turns,  is the residual flux density of the PM in the air gap,  is 
the input current and l is the effective length of coil in the air gap. 
 
Fig. 3. Direction of flux at part1  
and part2 
Fig. 4. The direction of the flux in five divided edge parts 
The 2-D FE model considers the geometry of the coils presented in Fig. 3 where the magnetic 
field is perpendicular to the long axis of coil wire, a situation where Lorentz force is maximized. 
We subdivide each side of the coil into part ① where the magnetic field is perpendicular to the 
direction of coil length and part ② where it is not. 
Accordingly, the total coil length of part ①, which is equal to the length of the inner magnet, 
is fully converted to the depth of the 2-D model while the coil length of part ② cannot be 
accurately converted to the depth of the 2-D model. In order to arrive at the accurate conversion 
of the coil length of part ②, it first needs to be modified to the effective coil length, which can be 
considered to be perpendicular to the magnetic field and then it will be directly converted to the 
depth of 2-D model. In order to calculate the effective coil length of part ②, we divide the coil at 
1306. 2-D EQUIVALENT FINITE ELEMENT MODEL OF QUADRATIC LINEAR ELECTROMAGNETIC ACTUATOR.  
JUNG HUN LEE, LAWRENCE KULINSKY, MARC MADOU, BYUNGHYUN KIM, HWA CHO YI, JIN HO KIM 
2050 © JVE INTERNATIONAL LTD. JOURNAL OF VIBROENGINEERING. JUNE 2014. VOLUME 16, ISSUE 4. ISSN 1392-8716  
part ② into five segments. Fig. 4 shows the direction of the magnetic field in the five segments of 
the coil. 
 through  are defined as the length of each coil and  is defined as the angle of the 
magnetic field at each segment with respect to the direction of the long axis of the wire of the coil. 
Using finite element analysis, the magnetic flux distribution at part ② was evaluated. Fig. 5 
demonstrates the calculated flux density in the airgap in the edge parts. The magnitude of flux 
density from  to  varies. We will use the weight () defined as the ratio of  to , to 
calculate the effective coil length, from which result we will deduce the depth of the 2-D model. 
The weight () is: 
  . (2)
 
Fig. 5. The flux density at the airgap of edge parts ② (Fig. 4) 
4. Verification of equivalent 2-D model by experiment 
In this section, we validate the 2-D equivalent model. In section 4.1, FE dynamic simulation 
is performed based on the 2-D equivalent model. In section 4.2, the simulation results are validated 
against the empirical data collected using the fabricated quadratic linear electromagnetic actuator. 
4.1. 2-D Dynamic finite element analysis 
The dynamic FE analysis calculated the coupled system composed of magnetic, electric and 
mechanical subsystems. Table 4 and Table 5 summarize the data for the electric and mechanical 
subsystems, respectively. The current through the coils was of a sine waveform described by 
Eq. (5) below and the resultant magnetic force oscillated as a sine waveform [17]: 
    sin#2%&'(. (3)
 
Table 4. Specification of the electric subsystem 
Electric subsystem 
Symbol Parameter Value  Input current 3 A & Input current frequency 50 Hz 
Turns Number of coil turns turns 
 
Table 5. Specification of the mechanical subsystem 
Mechanical subsystem 
Symbol Parameter Value ) Yoke and PM mass 2.15 kg * Spring coefficient 110 kN/m + Natural frequency 22 Hz 
 
Eq. (6) below describes the motion of the mechanical subsystem: 
),-   .  /,0   .  *,    	
#(, (4)
where ) is a total mass of the moving part, / is the damping coefficient, * is the spring coefficient 
1306. 2-D EQUIVALENT FINITE ELEMENT MODEL OF QUADRATIC LINEAR ELECTROMAGNETIC ACTUATOR.  
JUNG HUN LEE, LAWRENCE KULINSKY, MARC MADOU, BYUNGHYUN KIM, HWA CHO YI, JIN HO KIM 
 © JVE INTERNATIONAL LTD. JOURNAL OF VIBROENGINEERING. JUNE 2014. VOLUME 16, ISSUE 4. ISSN 1392-8716 2051 
and + is the natural frequency of the actuator. In this FEA, the damping force is neglected. 
The initial conditions are: ,#0(    0, ,0#0(  0. 
To solve the coupled system, the dynamic FEA is performed with a time step of 0.1 millisecond 
for a time period of 0.1 seconds. 
  
Fig. 6. Plot of the magnetic force at the input 
frequency of 50 Hz 
Fig. 7. Plot of the spring force at the input  
frequency of 50 Hz 
 
Fig. 8. Plot of the excitation force at the input frequency of 50 Hz 
Figs. 6 and 7 show the resultant magnetic force and spring force, respectively. The total 
excitation force of the actuator is calculated by subtracting the spring force from the magnet force. 
Excitation force is shown in Fig. 8. 
 
Fig. 9. Prototype of the quadratic linear actuator 
1306. 2-D EQUIVALENT FINITE ELEMENT MODEL OF QUADRATIC LINEAR ELECTROMAGNETIC ACTUATOR.  
JUNG HUN LEE, LAWRENCE KULINSKY, MARC MADOU, BYUNGHYUN KIM, HWA CHO YI, JIN HO KIM 
2052 © JVE INTERNATIONAL LTD. JOURNAL OF VIBROENGINEERING. JUNE 2014. VOLUME 16, ISSUE 4. ISSN 1392-8716  
4.2. Experiment 
In order to verify the result of dynamic 2-D finite element analysis, a prototype of the quadratic 
linear actuator (see Fig. 9) was manufactured. 
Fig. 10 demonstrates the experiment setup consisting of power supply, load cell, NI-board and 
a personal computer [18]. With the power supply, a sinusoidal wave of 3 A 50 Hz is applied to 
the coil and the force of moving part versus time is measured by the load cell every 0.1 millisecond 
[19, 20]. 
 
a) 
 
b) 
Fig. 10. a) Schematic diagram of experimental set up, b) Experimental set up 
Fig. 11 shows the measured excitation force and the simulated excitation force with the 
equivalent 2-D FE model. The graph of experiment is not smooth due to the error of experiment, 
but the difference between the model results and the experimental data is less than 5.6 % which 
shows the validity of 2-D FE model. 
 
Fig. 11. Plot of experimental vs. simulated force 
5. Conclusion 
In this work, we propose an equivalent 2-D FE model of the quadratic actuator to replace the 
3-D FE model. The accuracy of the equivalent 2-D FE model was verified with experimental data 
and the deviation of the experimental results from the 2-D FEA model was below 5.6 %. The 
equivalent 2-D FE model can be used to design and analyze the performance of quadratic linear 
electromagnetic actuators by employing low computational power resources such as personal. 
1306. 2-D EQUIVALENT FINITE ELEMENT MODEL OF QUADRATIC LINEAR ELECTROMAGNETIC ACTUATOR.  
JUNG HUN LEE, LAWRENCE KULINSKY, MARC MADOU, BYUNGHYUN KIM, HWA CHO YI, JIN HO KIM 
 © JVE INTERNATIONAL LTD. JOURNAL OF VIBROENGINEERING. JUNE 2014. VOLUME 16, ISSUE 4. ISSN 1392-8716 2053 
Acknowledgment 
This research was supported by Yeungnam University research grant in 2014. 
References 
[1] Nakazato Hiroshi Rotary voice coil motor with a flat coil. U.S. Patent No. 5, 122, 702, 1992. 
[2] Oveyssi Kamran, Alireza Rahimi Dual coil voice coil motor. U.S. Patent No. 5, 621, 591, 1997. 
[3] Yeom D. H., Park N. J., Jung S. Y. Digital controller of novel voice coil motor actuator for optical 
image stabilizer. International Conference on Control, Automation and Systems, 2007. 
[4] Boldea Ion, Syed A. Nasar Linear electric actuators and generators. Electric Machines and Drives 
Conference Record, 1997. 
[5] Chen Yi-De, Chyun-Chau Fuh, Pi-Cheng Tung Application of voice coil motors in active dynamic 
vibration absorbers. Magnetics, Vol. 41, Issue 3, 2005, p. 1149-1154. 
[6] Stratton Julius Adams Electromagnetic Theory. John Wiley & Sons, Vol. 33, 2007. 
[7] Kuk J. Y., Lim J. H. A study on vibration characteristic of engine mount system of a medium duty 
truck at the key on/off. Trans. of KASE, Vol. 16, Issue 4, 2008, p. 97-102. 
[8] Choi J. Y., Lee W. H., Shon H. C., Kim S. S., Kim J. H. A study on decoupler of active engine 
mount for vehicle vibration reduction. Conference KASE, 2009, p. 1142-1426. 
[9] Citipitioglu A. M., Haj-Ali R. M., White D. W. Refined 3D finite element modeling of 
partially-restrained connections including slip. Journal of Constructional Steel Research, Vol. 58, 
Issue 5, 2002, p. 995-1013. 
[10] Bastos João Pedro A., Nelson Sadowski Electromagnetic Modeling by Finite Element Methods. CRC 
press, 2003. 
[11] Liang Tian-Ye, Shi Wen-Ku Study on electromagnetic actuator active engine mount with fuzzy 
control. International Conference on Mecahtronics and Automation, 2009, p. 3010-3014. 
[12] Sumali Hartono, Harley Cudney An active engine mount with a piezoelectric stacked actuator. 
Proceedings of the 35th SDM Conference, 1994. 
[13] Bohn C., et al. Active control of engine-induced vibrations in automotive vehicles using disturbance 
observer gain scheduling. Control Engineering Practice, Vol. 12, Issue 8, 2004, p. 1029-1039. 
[14] Sohn J. W., Paeng Y. S., Choi S. B. An active mount using an electromagnetic actuator for vibration 
control: experimental investigation. Proceedings of the Institution of Mechanical Engineers Part C, 
Journal of Mechanical Engineering Science, Vol. 224, 2009, p. 1-9. 
[15] Chai Hi-Dong Electromechanical Motion Devices. Prentice Hall, 1998. 
[16] Hwang K. I., Kim J. H. Single-axis flat electro-magnetic actuator using shorted turn for fast initial 
response. Journal of Korean Medical Science, Vol. 19, Issue 6, 2009, p. 222-226. 
[17] Yoon S. M., Kim J. H., Kim J. H. Mathematical modeling and analysis of vibration characteristics 
of smart-phone. International Journal of Precision Engineering and Manufacturing, Seoul, South 
Korea, p. 135-010. 
[18] Gene F. Franklin, Powell J. David, Abbas Emami-Naeini Feedback Control of Dynamic System. 
4th ed., Prentice Hall, 2004, p. 368-500. 
[19] Fuller C. R., Elliott S. J., Nelson P. A. Active Control of Vibration. Academic Press Limited, London, 
1996. 
[20] Rao S. S. Mechanical vibration. Pearson Education International, p. 409-422, 2004. 


2-D Equivalent finite element model of quadratic linear electromagnetic actuator

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2-D Equivalent finite element model of quadratic linear electromagnetic actuator

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