Practical Demonstration of an Electromagnetic Levitation for a Cylindrical Rod

Subrata Banerjee 1, Rupam Bhaduri 2 and Pabitra Biswas 3

1 Department of Electrical Engineering, National Institute of Technology, Durgapur-713209, W.B., India, E-mail: bansub2004@rediffmail.com.

2 Department of Electrical & Electronics Engineering, N.F.E.T., Durgapur- 713212, WB, India, E-mail: rupam.bhd@gmail.com.

3 Department of Electrical Engineering, Asansol Engg. College, Asansol-713305, WB, India, E-mail: pabitra2001@rediffmail.com.

Abstract- In this paper an attempt has made for stable suspension of a cylindrical rod under two electromagnets controlling different degrees of freedom movement. The two actuators are controlled independently through two identical controllers and the stable levitation of the rod is achieved through single input and single output (SISO) control of each air-gap corner. The focus is on to design and development of controller unit for two actuators. The proposed single switch based power circuit simplifies the overall hardware and it can be extended to any number of magnet-coil. A cascade lead-lag compensation control scheme utilizing inner current loop and outer position loop has been designed and implemented for stabilization of such highly unstable and strongly non-linear system. The prototype has been successfully tested and stable levitation was demonstrated at the desired operating gap.

Index Terms— Electromagnetic levitation, actuator, multi-magnet based levitation, SISO control, switched mode power amplifier, lead-lag compensation.

I. INTRODUCTION

Electromagnetic levitation system (EMLS) is inherently unstable and strongly nonlinear in nature [1-8]. To determine the overall close loop stability for such unstable EMLS, a cascade lead compensation technique [1,2,5,6,8] is mostly reported. In this work, two actuators based EMLS for the suspension of a cylindrical rod in air utilizing two independent controllers have been successfully designed, fabricated and tested. Though the single magnet, single axis levitation scheme may be useful for some industrial applications, majority of the applications require multi-axis levitation control [11,12] where one may need to use a multiple-magnet based levitation system. This part of the work reports control of an electro-magnetically levitated cylindrical rod under two independent controlled attraction type magnets. Any cylindrical rod has only two-degrees of freedom (2-DoF) unlikely of any sphere which has 6-DoF. Therefore it is convenient to levitate a ferromagnetic sphere under the force of an electromagnet having any kind of core structure. The suspension of a long cylindrical rod without tilting is an interesting and challenging task. Initially an attempt has made to make stable suspension of a cylindrical rod under single electromagnet. One position sensor has placed centrally under the cylindrical rod. It has been

observed that the rod gets tilted one side that exerts more levitating force due to non-uniformity of the distributed field flux. The schematic block diagram of individual unit for the proposed EMLS is shown in Fig.1. In each case the current of the electromagnet is controlled through the DC to DC switch mode chopper circuit [7-10] utilizing an outer position control loop and an inner current feedback control loop. The parameters of the maglev systems are given in Table-1. The photograph of the experimental setup is shown in Fig.2. The magnet current is controlled by a single switch based DC to DC chopper circuit (Fig.9) Since the electromagnetic levitation system is inherently unstable, the selection and design of the controller is important so that the overall closed loop system becomes stable and gives satisfactory performance. A lead-lag compensator [15,16] is used in cascade with the position control loop for maintaining overall closed loop stability. A linear inductive type position sensor [5,6] is used to measure the actual gap between the magnet pole-face and the object. Output of the position loop (current reference signal) is compared with the actual coil-current signal sensed by an LEM make (LA-55P) Hall-effect current sensor. For better dynamic response and steady state accuracy the current error is processed through a PI controller and its output is used to control the chopper output voltage through PWM control logic [7-10]. The duty ratio of the MOSFET switches varies as the object moves up and down within the electromagnetic field. The stable suspension of the prototype is achieved through the control of air-gap between the magnet and the cylindrical rod by controlling the current flowing through the magnet-coil. The two magnets are controlled independently through two identical controllers and stable levitation of the rod is achieved through SISO control of each air-gap.

Fig. 1. Schematic block diagram of individual unit for the proposed EMLS

978-1-4244-9312-8/11/$26.00 ©2011 IEEE 685

Fig. 2. Photograph of the experimental setup

II. SYSTEM MODELING AND CONTROLLER DESIGN

The flux pattern of the system at 10mm operating air-gap using finite element method (FEM) software ANSYS is shown in Fig.3. The two magnet-coils used in this work are identical each having almost same mechanical and electrical parameters. Assuming uniform distribution of the load, the upward lift force to be generated by each magnet is equivalent to 0.061 kg and this leads to the magnet parameters being derived by the technique used in the single magnet levitation.

Fig. 3. Flux pattern for the system at 10mm air-gap

Assuming all flux generated by the electromagnet passes through the ferromagnetic guide-way, the instantaneous coil inductance may be expressed as:

TT R

Nti

NzL2

)()( =Φ= (1)

where, N = No of turns of the coil,

)(ti = Instantaneous current through the coil,

TΦ = Total flux in the magnetic circuit,

TR = Reluctance of the entire magnetic circuit. If the reluctance of the magnetic core is assumed to be

negligible when compared to the two air gaps [total length = ] )(2 tz

)(2)(

20

tzAN

zLμ

= (2)

At any instant of time, the force of attraction between the electromagnet and the ferromagnetic rod is given by

⎥⎦⎤

⎢⎣⎡−= 2)()(

21),( tizL

dzdziF (3)

Now putting the inductance value from Equation 2 into the force Equation 3, one can write:

220

)()(

4),( ⎥

⎦

⎤⎢⎣

⎡=

tztiAN

ziFμ (4)

So, at the equilibrium position )( 0,0 zi the normalized force

equation is

mgziAN

ziF =⎥⎦

⎤⎢⎣

⎡=

2

0

02

0000 4),(

μ (5)

The dynamics of the cylindrical rod is given by the following equations

;),()( mgziFtzm +−=

;)()(

4

220 mg

tztiAN

+⎥⎦

⎤⎢⎣

⎡−=

μ (6)

If the mass of the rod is displaced by an amount )( tzΔ from the stable point, then let the corresponding change in current be )( tiΔ . The small perturbation linear equations (discounting second order effects) of the system are:

mgtzztiiANtzm +⎥⎦

⎤⎢⎣

⎡Δ+Δ+=Δ

2

0

02

0

)()(

4)( μ

)()( tzKtiK zi Δ+Δ−= ; (7)

where, ( )di

zidFKi,= and ( )

dzzidFK z

,= (8)

So, these two force constants iK and zK are respectively the slope at the operating point of the force-current and force-distance characteristics of the electromagnetic suspension system. Taking Laplace transform on both sides of Equation (7) and after rearranging, the transfer function of the plant is written as:

⎟⎠

⎞⎜⎝

⎛ −

⎟⎠

⎞⎜⎝

⎛

−=ΔΔ=

mK

s

mK

sIsZsG

z

i

p2)(

)()( (9)

The negative sign in the expression indicates the decrease of object position with the incremental change of coil-current or force. The inductance profile with distance for the levitated system (Coil-1) as found experimentally is shown in Fig. 4. The inductance value around an operating point (in the medium gap range) mostly varies inversely with respect to rod position and has been approximated as:

zzL

LzL c00)( += (10)

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Fig. 4. Inductance profile of the levitated system

where LC is the inductance of the coil in the absence of the

guide-way and L0 is the additional inductance contributed by the ferromagnetic guide-way.

Now the basic force equation becomes ( ) 22

2)( ⎟

⎠⎞

⎜⎝⎛=−=

ziC

dzzdLizF (11)

where 2

00 zLC = is a constant, which can be determined

experimentally. The partial derivatives of the force expression in Equation (11) w.r.t. ‘i’ and ‘z’ (at the nominal current iO and air-gap zO ) will respectively be equal to Ki and KZ .

Thus:

⎟⎟⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛= 3

0

20

20

0 2 and 2ziCK

ziCK zi (12)

The pick-up current versus air-gap for the EMLS is shown in Fig.5.

So the transfer function of the plant (at operating air-gap 10 mm) can be found from equation 5

)97.26)(97.26(98.13

)()()(

2 −+=

⎟⎠⎞

⎜⎝⎛ −

⎟⎠⎞

⎜⎝⎛

=ΔΔ=

ssmKs

mK

sIsZsG

z

i

p

(13)

Fig. 5. Current profile of the levitated system

TABLE I LEVITATED SYSTEM PARAMETERS

Parameters Values Current sensor gain Mass of the object No. of magnets Effective mass of object per magnet Length of the rod Diameter of the rod Resistance of the coil-1 Resistance of the coil-2 Self-Inductance of the coil-1 Self-Inductance of the coil-2 Position sensor gain

1 122 gm 2 61 gm 14.5 cm 10 mm 4.1 Ohm 3.9 Ohm 0.16648 H 0.15778 H 1000 V/m

The selection of the operating point for the levitated object

plays a significant role in design and development of a DC electromagnetic levitation system. There are many reasons why the operating air-gap between the pole-face of magnet and the object cannot be made arbitrarily too small or too large. In the higher operating gap zone the required current for levitation is very high and the high value of coil-current becomes a major constraint in the design of actuator, power amplifier and controller. The large magnitude of coil-inductance in the lower gap zone leads to a high (L/R) ratio, which consequently makes it more difficult to achieve faster control of the coil-current and levitation force. Considering all these points the operating air-gap for the present system has been selected as 10 mm. The transfer-function of the levitated system (considering coil-1) at operating gap of 10 mm is given by

( )( )97.2697.2698.13

)()()(

−+=

ΔΔ=

sssIsZsG p

(14)

From Equation 17, the plant GP(s) has one stable pole at s = -26.97 and an unstable pole at s = 26.97. The closed loop system has been stabilized by a cascade lead-lag compensation technique utilizing inner current loop and outer position control loop. The design of control loop starts from the innermost (fastest) current loop and proceeds to the slowest position loop [13]. The parameters of the current controller (PI controller) and position controller (lead-lag compensator) have been designed based on the linear control theory utilizing the root-locus and frequency domain techniques. The pole and zero are to be placed on the negative real axis and there are many possible pole-zero locations for stabilizing the system. Simulation studies are carried out to find a pole-zero and gain combination so that the system is stable as well as acceptable performance (16% overshoot, 1% steady-state error and settling time 0.25 sec) is obtained. It must be mentioned that since the system is inherently unstable, in general, both stability and good performance cannot be achieved by using a single controller. The transfer function of the designed lead-lag controller for operating air-gap of 10 mm is given by

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠⎞

⎜⎝⎛

++

⎟⎠⎞

⎜⎝⎛

++=

5.05

665252.21)(

ss

sssGc (15)

The root-locus plot of the overall closed loop system

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utilizing lead-lag compensator is shown in Fig.6. Fig.7 shows the time response plot of the closed loop stable system.

Fig. 6. Root-locus plot of the overall closed loop system utilizing lead-lag

compensator

Fig. 7. Time response plot of the overall closed loop system utilizing

lead-lag compensator

III. POWER AMPLIFIER

In this work two electro-magnets with their individual air-gap controllers have been used for levitating the cylindrical. These magnets have their separate power amplifiers and they are separately controlled. The asymmetric H-bridge converter [3], [7], [9], [10], [14] (Fig. 8) is ideal for high power levitation systems, not only for its energy efficiency but also for better dynamic performance (in the sense that the amplifier may be made to behave like a simple gain block). For two-magnet case one need to have two bridge circuits, each having two controlled switches and two diodes. Now instead of asymmetrical bridge circuit if the simplified circuit with only one switch (and a diode with energy-dump circuit) (Fig.9) per magnet is used [5]-[7], [13], the overall saving in the power amplifier components may be significant. Two switches of the simplified power circuit may be connected such that their gate drive signals may not need ohmic isolation. Moreover two amplifiers may share a common energy dump capacitor-resistor circuit. The exact connection diagram is shown in Fig.9. This circuit, uses half the number of switches and diodes used in the asymmetrical bridge

circuit and as a result the total EMI generation [14] is also less. The circuit is less energy-efficient compared to asymmetrical bridge circuit but more efficient compared to the linear amplifier.

Fig. 8. Schematic diagram of the asymmetrical bridge (H-bridge) converter

circuit

Fig. 9. Proposed simplified chopper circuit for four-coil levitation system

IV. EXPERIMENTAL RESULTS

The main objective of this work is the stable suspension of the cylindrical rod at the desired operating air-gap position. The complete hardware set-up (Fig.10) has been fabricated in the laboratory and stable levitation of the cylindrical rod around 10 mm has been demonstrated in Fig.11. The advantage of the proposed system is the decoupled control of air-gap. Either through reference gap control or change in controller parameters, any side of the rod may be controlled independently. Fig.12 shows demonstration of ‘tilting levitation’ when the reference gap of the left system (from front side) is less compare to right system.

Fig.13 shows gate-pulses and coil-currents of both the systems during stable levitation of the ferromagnetic cylindrical rod at the desired operating air-gap. When the switches are turned on, full supply voltage appears across the coils and the currents in the coil increases. Thus energy flows from source to the magnet-coil. Duration of current flow through the coils in each switching cycle is equal to the ON duration of the gate-driving pulses. During ON time the energy dump capacitor (10 µF, 400V) that holds some charge due to previous operation, discharges partially through the rheostat (185 Ω ). During OFF period of the switch, the coil currents find a path through fast-recovery power diodes to the resistor capacitor combination. The capacitor voltage effectively appears across the coils with a negative polarity (reverse of the polarity during ON duration). With reversal of coil voltage the coil current starts decreasing at a slope

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decided by the coil inductance and the capacitor voltage. The two coil-voltages and two coil currents during stable

suspension of the rod is shown in Fig.14. During ON time of the switches full supply voltage appears across the magnet-coils and current through the coils increases. It is seen from the Fig.14 that both the actuators are excited by same amount of voltage (100 volt) and current (1 Amp) during stable levitation of the object at desired operating air-gap (10mm). Fig.15 shows a typical oscillogram of the two coil-voltages, capacitor voltage, coil current and gate-pulse (during stable levitation of rod) when the DC input supply is 100 volts and the 185 ohm rheostat is set at 175 ohms. The coil voltage is made equal positive and negative supply voltage (100 volts) with the predefined circuit parameters (resistor and capacitor).

Fig.16 shows two positions (CH1 & CH2) and two current signals (CH3 & CH4) of the levitated system during stable levitation of the cylindrical rod at the desired operating air-gap (10mm). It is clear from the Fig.16 that both side of cylindrical rod has been lifted by same distance (around 10mm shown in Fig.11) from the platform. The two actuators are drawing almost same amount of current from the controlled current source. The symmetrical pattern of position and current signals indicates that the two systems are working in synchronism. Fig.17 shows the dynamic position responses of two corners of rod. When the main switch of DC link power supply is suddenly ON both end of rod gets vertical lift from the platform and the dynamic position responses for the two ends have been recorded. Similarly when the DC link power supply is switched OFF, the levitated rod falls due to gravity and this response is much faster then the response during lifting. This demonstration shows the stable movement control of the rod that lifts from a platform. The closed loop position responses are stable and have satisfactory transient response.

Fig. 10. Complete hardware set-up for the levitation system

Fig. 11. Photograph of the levitated rod at 10 mm air-gap

Fig. 12. Demonstration of tilting levitation of the rod

Fig. 13. Gate pulses (CH-1 & CH-2) and coil currents (CH-3 & CH-4)

during stable levitation of cylindrical rod

Fig. 14. Coil voltages (CH-1 & CH-2) and coil currents (CH-3 & CH-4)

during stable levitation of cylindrical rod

Fig. 15. Coil voltage (CH-1), capacitor voltage (CH-2), coil current (CH-3),

Gate pulse (CH-4) during stable levitation of cylindrical rod

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Fig. 16. Two position signals (CH1 & CH2) and two current signals (CH3

& CH4) during stable levitation of cylindrical rod

Fig. 17. Dynamic position responses of two corners of rod

V. CONCLUSION

Two identical controllers have been utilised for the two electromagnets independently and the cylindrical rod has been levitated by employing independent SISO control of each gap. The whole unit was considered to be composed of two decoupled identical subsystems. The use of decoupling technique simplified the overall analysis and design of the controller units. The cascade compensators for the two units have been designed by using classical synthesis method. The prototype has been successfully tested and its stable levitation has been demonstrated at the desired gap position. The two independent SISO controllers designed for controlling and maintaining a fixed air-gap at the two side of the rod are found to work successfully in unison. The SISO controllers are able to rectify and compensate errors introduced due to non-identical coils, and mechanical imperfections caused by in-house laboratory production. Due to the use of single switch based switched mode power amplifier the overall hardware circuit becomes simpler and compact. The overall cost of the power amplifier as well as DCALS has also been reduced considerably by the use of this power circuit. The use of two actuator based controller demonstrates better pitching movement than single controller based system. By applying suitable vertical gap commands to the two electromagnets, to some extent the roll, pitch and heave is also being controlled.

ACKNOWLEDGMENT

The author wishes to acknowledge DST, Govt. of India for sponsoring the Project No.SR/S3/EECE/0008/2010 entitled “Development of DC Electromagnetic Levitation Systems –Suitable for Specific Industrial Applications”.

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