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System level simulation of a double stator wobble electrostatic micromotor
Sensors and Actuators A 99 (2002) 312–320
System level simulation of a double stator wobble electrostatic micromotor Aitor Endeman˜o*, Marc P.Y. Desmulliez, Matthew Dunnigan Department of Computing and Electrical Engineering, MicroSystems Engineering Centre (MISEC), Heriot Watt University, Edinburgh EH14 4AS, UK Received 6 August 2001; accepted 15 November 2001
Abstract System level simulation results of an electrostatic micromotor, based on closed-form expressions of the static and dynamic torque behaviours, are presented. A double stator, double rotor, wobble motor designed and tested at Heriot Watt University is simulated using the high level language very high speed integrated circuits (VHSIC) hardware description language-analogue mixed signal (VHDL-AMS). The analytical torque model is obtained by two conformal mapping transformations of the electrostatic gap region between the stator and rotor surfaces. Torque results obtained by finite element analysis methods and the proposed simulation model show good agreement. Simulation results of the closed-loop control of the excitation of the motor, based on the dynamic model of the torque, are also presented. # 2002 Elsevier Science B.V. All rights reserved. Keywords: Micromotor; Actuator; Conformal mapping; Analytical modelling; System level simulation; VHDL-AMS; MEMS
1. Introduction The rapid development of microsystem technology into different fields of applications (microsensors, microactuators, microfluidics, micro-optics, etc.) is pending not only on the process knowledge and equipment needed to fabricate any prototype but also on the accurate modelling and simulation of these systems. Although nowadays specialised computer-aided design (CAD) software tools perform such simulation and modelling, their use requires a substantial understanding of the physical behaviour of such systems and are computationally intensive since most of them are based on finite element analysis (FEA) methods. Using system level modelling languages such as the IEEE 1076.1–1999 standard (VHDL-AMS) or Verilog-AMS can accelerate the modelling and simulation of microsystems, albeit at different levels of abstraction. All the different parts of a microsystem and the different physical domains involved (electrical, mechanical, fluidic, thermal) can be modelled within the same environment. It is also possible to integrate parts of a model constructed or simulated by different means (such as Spice, C language or FEA methods)
*
Corresponding author. Tel.: þ44-131-449-5111x4159; fax: þ44-131-451-3327. E-mail address: [email protected] (A. Endeman˜o).
into the same VHDL-AMS simulation environment, allowing the incorporation of results from previous complex simulations. The work was focused on the modelling of the static and dynamic behaviours of a double stator double rotor wobble motor designed and tested at Heriot Watt University [1]. This electrostatic device was designed as a first prototype to be integrated into a minimally invasive surgery system (catheter) for arterial plaque removal. Externally driven cutters are used to remove mechanically the harder and stiffer plaques from cardiac arteries. This novel design was intended to reduce the size of the system and not to require an external drive, thereby reducing operating time and the danger of damaging healthy tissue. This motor is briefly described in Section 2 of this article. A novel analytical model of the actuator is presented in Section 3. The model relies on conformal mapping techniques to transform the double air gap of the motor into a simple geometrical shape and then calculates the torque exerted on the rotor. The torque results obtained by previous FEA simulations are compared with the resulting analytical results and then discussed. The torque model for the whole motor is included into a dynamic simulation of the micromotor and implemented in VHDL-AMS in Section 4. The ADVanceMSTM simulation environment (from Mentor GraphicsTM) is employed and results are presented in Section 5.
0924-4247/02/$ – see front matter # 2002 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 4 - 4 2 4 7 ( 0 1 ) 0 0 8 3 3 - 0
A. Endeman˜o et al. / Sensors and Actuators A 99 (2002) 312–320
2. Description of the double stator, double rotor, micromotor The modelled micromotor has been fabricated using the LIGA process [2]. The motor relies on the wobbling of the double rotor within the plane of rotation and is based on electrostatic actuation. The inner surface of the outer rotor (radius R9) rolls on the outer surface of the bearing (radius R4). The offset between the rotor and stator axes is 5 mm, for a minimum air gap size of 5 mm in the fabricated prototype. A photograph of a mounted prototype and a sketch of the motor with the most relevant parameters are shown in Fig. 1. Dimensions of the motor are provided in Table 1. The configuration of the stator and rotor is shown in Fig. 2, along with the different electrodes (1–8 and A–H) from inner and outer stator and the contact point angle j. For a given excitation potential, the use of two rotor surfaces to generate electrostatic forces allows the approximate doubling of the output torque as compared to a single rotor version of the same wobble motor. The outer stator (radius R5) with the outer rotor (radius R10), and the inner stator (radius R2) with the inner rotor (radius R7) form the two pairs of surfaces that create the torque. This device operates in a wobbling fashion: a dc voltage (around 100 V) is set between stator electrodes and rotor in order to cause movement due to electrostatic forces created. This movement is constrained by the rolling of the rotor on the bearing (contact point), thus reducing the speed and rising the torque delivered. The heights of the stator electrodes and the rotor (H s ¼ 300 mm and H r ¼ 350 mm, respectively), allow the motor to develop torque values of the order of 107 to 106 N m, which are necessary to cut the deposit fat in cardiac arteries. The micromotor is to be included into a minimally invasive catheter system for the removal of arterial plaque [3].
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Table 1 Stator (Hs, R2, R4, R5) and rotor (Hr, R7, R9, R10) dimensions Hs (mm)
R2 (mm)
R4 (mm)
R5 (mm)
Hr (mm)
R7 (mm)
R9 (mm)
R10 (mm)
300
965
1120
1185
350
975
1125
1175
Fig. 2. Electrode labels and contact angle.
3. Torque model based on conformal mapping A simple analytical model of the torque behaviour, to be included in the VHDL-AMS description of the motor, is described in this section. The model is based on two successive conformal mappings, complex functions w ¼ f ðzÞ that transform points from a complex plane z ¼ ðx; yÞ to another complex plane w ¼ ðu; vÞ [4,5]. The calculation of the capacitance, which is a function of the gap geometry between the rotor and the stator is first simplified
Fig. 1. Photograph and parameters of the wobble motor.
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using suitable conformal mapping functions. Torque is then calculated, as the derivative of the capacitance with respect to the angle of rotation; this analysis is carried out for each of the torque creating pairs of surfaces. Only the calculations for the inner pair of surfaces are presented in detail in this section. The dynamic simulation of the total torque on the rotor requires of course the contribution of both pairs of surfaces. 3.1. One electrode torque model for the inner pair of surfaces (radii R2 and R7) The stators are composed of segmented electrodes, which are selectively activated in order to generate Coulombic forces. Considering one active electrode (A in Fig. 3) of the
inner stator, the torque expression depends on the angle from the contact point to the centre of the active electrode. Neglecting the effects of the fringing fields in the region above the stator, a 2D representation is projected on a z-plane with (x, y) coordinates as shown in Fig. 3. The origin is taken from the centre of the rotor. The contact point between the bearing and the inner surface of the outer rotor (not shown in Fig. 3a) matches the minimum air gap all the time. D is the electrode width; points J, H, K indicate the beginning, centre and end of the active electrode. W is the angle taken from the contact point at the centre of the active electrode. The bearing sets the offset between the rotor and stator axes: offset ¼ R9 R4
(1)
From Fig. 3, the points of intersection of the x-axis and the stator circumference, x1 and x2 are x1 ¼ R2 þ offset ¼ R2 þ R9 R4
(2)
x2 ¼ R2 þ offset ¼ R2 þ R9 R4
(3)
The first conformal mapping w ¼ u þ jv ¼
z R7 A Az R7
(4)
transforms the stator and rotor circles (of radius R2 and R7, respectively) into concentric circles of radius R0 > 1 and unity as shown in Fig. 3b. The resulting values for A and R0, which are determined by the conditions: z ¼ R7 ) w ¼ 1; z ¼ R7 ) w ¼ 1; z ¼ x1 ) w ¼ R0 ; z ¼ x2 ) w ¼ R0 , are qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R27 þ x1 x2 þ ðR27 þ x1 x2 Þ2 R27 ðx1 þ x2 Þ2 (5a) A¼ R7 ðx1 þ x2 Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðR27 þ x1 x2 Þ2 R27 ðx1 þ x2 Þ2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R0 ¼ x1 x22 R27 x1 þ x2 ðR27 þ x1 x2 Þ2 R27 ðx1 þ x2 Þ2 R7 x22 R37 R7
(5b) An arbitrary z point in the rotor circle can be expressed in its exponential form: z ¼ R7 ejW . The corresponding point in the w-plane is then expressed as w ¼ ejfs ðWÞ , with fs: ðA2 1Þsin W fs ðWÞ ¼ tan1 (6) 2A ð1 þ A2 Þcos W
Fig. 3. Three planes of transformation for the inner pair of surfaces: (a) z-plane projection; (b) W-plane; (c) w-plane.
When the angle W in the z-plane increases in a counterclockwise fashion from 0 to 3608, the fs angle in the w-plane increases in a clockwise way from 180 to 1808 with a single corresponding point in the w-plane for each point in the z-plane. The derivative with respect to W of the stored electrical energy, U, is calculated in order to calculate the torque exerted on the rotor by the active electrode. As the expression of U depends on the angle-dependent capacitance
A. Endeman˜ o et al. / Sensors and Actuators A 99 (2002) 312–320
C(W) and the latter depends on fs(W), the expression of the derivative of fs(W) with respect to W is needed @fs 1 A2 ¼ gðWÞ ¼ 2 @W A þ 1 2A cos W
(7)
The logarithmic transformation w ¼ log w
(8)
maps each point in the w-plane to another point in the w-plane so that the modelled air gap between the active electrode and the rotor in the w-plane has a square shape allowing therefore the simple calculation of the capacitance as shown in Fig. 3(c). The stator electrode is itself converted into a flat electrode in plane w. Using the parallel plate approximation, the capacitance is e0 Area e0 LjqðK 00 Þ qðJ 00 Þj CðWÞ ¼ ¼ d log R0 e0 L½fs ðW þ D=2Þ fs ðW D=2Þ
¼ log R0
(9)
Points J00 and K00 are the images of points J and K in the wplane. The maximum capacitance is reached when the contact point is at the centre of the active electrode and keeps decreasing until W ¼ 1808 (Fig. 4a). The torque on the rotor can then be calculated from an energy balance. Taking into account the ideal gear ratio of this motor: R9 W n¼ (10) ¼ offset j where j is the rotating angle of the rotor (same as the contact point angle), the generated torque is equal to the change in the electrical stored energy between the pair of surfaces (U ¼ CV 2 /2) with respect to j dU dU dW dU ¼ ¼ n (11) G¼ dj dW dj dW V2 @ fs ðW þ D=2Þ fs ðW D=2Þ GðWÞ ¼ ne0 L @W log R0 2 2 V e0 L ntðWÞ (12) ¼ 2
315
where t(W) is the normalised torque. The analytic expression for t(W) can then be developed 1 D D g W (13) g Wþ tðWÞ ¼ log R0 2 2 where g(W) is the expression obtained in Eq. (7). The torque behaviour as a function of the contact angle for one active electrode is shown in Fig. 4(b). As expected, the value of the torque is zero at 1808, when the electrode is aligned with the contact point and at a maximum of around 0.18 mN m for one active electrode. 3.2. Gross torque calculation and validation for the inner pair of surfaces The calculation of the overall torque for the inner pair of surfaces depends on the number of electrodes and the excitation scheme applied to the electrodes. Four adjacent electrodes, excited at any given time, have been demonstrated to provide the best torque performance. The one electrode torque expression obtained in Eq. (12), gives the torque values as a function of the angle W. A superposition of the torque values of the four active electrodes must take into account the four different angles from the contact point to each of the active electrodes. First the gross torque is calculated for the four active electrodes (A–D) as the rotor contact point angle j varies from 22.5 to 22.58 (Fig. 5). The fixed angles for the centres of the four electrodes are aE ¼ 22:58, aF ¼ 67:58, aG ¼ 112:58 and aH ¼ 157:58. For every simulation step (i.e. for every increment of the contact point angle j), the angles from the contact point to the centre of each active electrode are WE ¼ aE j, WF ¼ aF j, WG ¼ aG j and WH ¼ aH j. The gross torque expression is therefore Ggross ¼ GE ðWE Þ þ GF ðWF Þ þ GG ðWG Þ þ GH ðWH Þ (14) where j varies from 22.5 to 22.58. The torque behaviour, shown by a solid line in Fig. 6, increases from 22.5 to 08 with the effective gap for the active electrodes and decreases down from 0 to 22.58 due to the opposite effect. The maximum value for the torque created by the inner surfaces of the motor is 0.3806 mN m.
Fig. 4. (a) Capacitance as a function of contact angle; (b) one electrode torque function as a function of contact angle
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Fig. 5. Scheme of the active electrodes(in black) for the torque calculation at j ¼ 08.
For comparison, the FEA simulations for the gross torque are plotted on the same figure as well a different analytical (but more computationally intensive) approach based on numeric integration of a torque formula including an analytical model for the electric field [1]. Our results match fairly well with these previous FEA simulations. In the worst case that corresponds to the maximum torque case, a discrepancy of less than 5% has been calculated between the FEA model and the analytical model. This discrepancy can be explained by neglecting the closed-form model fringing field effects at the top edges of stator and rotor (3D) and at the edges of the electrodes (2D). As expected, this discrepancy decreases as the electrode to gap spacing ratio increases. A further refinement of the model will be done in the future by importing FEA results for the electric field into the VHDL-AMS model, thus taking into account these second order effects. Both FEA simulations and the analytic model for integration developed for the micromotor (torque results in Fig. 6) are not suitable for VHDL-AMS implementation. System level simulation can be done with the results of FEA through parameter extraction, however this method assumes that
Fig. 6. Gross torque function (solid line) compared to FEA results (circles) and another analytical model (crosses).
FEA has been carried out previously, at the expense of time-consuming iterative computations. The developed conformal mapping model for the static torque is accurate, simple and easy to implement in VHDL-AMS as closedform equations.
4. VHDL-AMS implementation Once the torque model has been validated, the next step is to implement the whole model in VHDL-AMS. The VHDLAMS model of the system is divided into four blocks: dc power supply, excitation control unit, micromotor torque calculation and micromotor dynamics (these last two blocks define the behaviour of the micromotor itself). Each of the blocks is represented in VHDL-AMS code as an ‘‘Entity’’, as seen in Fig. 7. Each of the entities has one or more different ‘‘Architectures’’.
Fig. 7. Basic structure of the VHDL-AMS model of the system.
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For example in our model, the excitation entity has two different architectures depending on the excitation strategy implemented. For all those architectures, 16 outputs drive the 16 segmented electrodes of the inner and outer stators (eight segments each) of the micromotor. When the motor is operated in a closed-loop configuration, the switching of the drive signals depends on the rotor position feedback; if operated in an open-loop configuration, the switching has a fixed frequency. The micromotor torque entity implements the conformal mapping model for the gross torque. The micromotor dynamics entity has the gross torque as an entry and implements the dynamic model of the motor. VHDLAMS as a language offers the possibility to implement differential and algebraic equations that define the physical behaviour of a system [6]. The dynamic equation proposed for the micromotor model is ¼ Ggross Bj_ ðC1 þ C2 V 2 Þsgn j_ Jj
(15)
317
where J is the moment of inertia of the rotor mathematically calculated as 1:48 1011 kg m2, Ggross is the instantaneous motor torque calculated by the model, B is the coefficient of viscous friction and C1, C2 are the constant and voltage dependant kinetic (Coulomb) friction terms. These constants have been estimated analytically from micromotor geometric and material properties compared to previous works [7–9] and are B ¼ 5 1011 N m s, C1 ¼ 1 1012 N m, C2 ¼ 3 1012 N m V2.
5. Simulation and results 5.1. Fixed step responses Using the VHDL-AMS model, at j ¼ 08 position, different electrodes are activated in order to study the micromotor response from rest to the center of the active
Fig. 8. Active electrode configuration for step responses.
Fig. 9. Motor step responses for different number of electrodes active in both stators (1–4): (a) continuous; (b) dashed; (c) dashed/dotted; (d) dotted lines). Angular position j (rad) and rotor speed o (rpm).
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electrode. The active electrodes in the inner stator are the opposite of the ones active in the outer stator, so that the maximum torque is developed as the contribution of both pairs of surfaces is summed. Results for 1–4 active electrodes (shown in Fig. 8) in both stators are presented in Fig. 9. The most performant configuration is the four electrodes case, for which the rotor reaches the highest values of acceleration (50,000 rad s2). Damped or nearly damped responses are observed in all the cases.
6. Continuous operation In order to simulate a steady and sustained operation of the micromotor, the excitation control block can activate the different electrodes of each stator with different excitation sequences. The switching moments for those
sequences can be a periodic function of time (open-loop) or function of the rotor position (closed-loop). One of the aims of developing this model of the micromotor is to verify the influence of a closed-loop control system for real applications. Fig. 10 shows the closed-loop operation of the motor for a four-electrode-scheme which results in a excitation frequency of 20.25 Hz. In this case, the obtained average torque is the maximum achievable (7:8 107 N m), because the electrodes are switched on and off at the exact moment required in order not to lose energy. The optimum open-loop excitation frequency that corresponds to the maximum achievable torque in those conditions is 19.25 Hz, quite near to the closed-loop one. Any higher frequency applied to the stator electrodes results in motor operation failure, since the rotor cannot follow the electrical frequency of the stator. Different open-loop four
Fig. 10. Closed-loop results: gross torque (N m), angular speed (Rpm).
Fig. 11. Open loop results: gross torque (N m); angular speed (rpm); switching frequency: f ¼ 19:25 Hz (continuous line), f ¼ 14:25 Hz (dotted line).
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electrodes excitation frequencies have been used. Applying the maximum 19.25 Hz frequency results in quite a uniform motion of the rotor, but with bigger losses than in the closedloop case, due to the torque shape (as shown in Fig. 11). Applying an excitation frequency of 14.25 Hz (lower than the optimum) results in irregular operation. The average torque produced for this case (4:4 107 N m) is much lower than the closed-loop one and produces a much bigger ripple (Fig. 11 in dotted line). The ripple leads to undesirable accelerations and decelerations that cause an unnecessary loss of energy. The torque supplied to the load, in a future real application would also be affected by these fluctuations and make an open-loop operation of the motor unsuitable for most applications. Measurement results of existing micromotors operating in open loop are currently underway to validate these system level simulation results. As the 3D FEA has been known to provide a good agreement with experimental results, we are confident that our model, based on closed form expressions detailed in Section 3, gives a sufficiently good agreement with experiments.
7. Conclusions A new analytical model for a double stator double rotor electrostatic micromotor has been developed based on conformal mapping techniques. The purpose of developing this mathematical model is to integrate it into a system level simulation of the microdevice together with its excitation control electronics. It was necessary to develop a simple closed-form model in order to implement it in VHDL-AMS, but with accuracy comparable to that of a FEA simulation or an integration based model. The theoretical model obtained has been essential in order to further develop a complete dynamic model of the micromotor and implement it using the VHDL-AMS high level modelling language. The model was fast and simple to simulate and it has been the first step to assess the important influence of closed-loop excitation control for the micromotor performance. Open-loop operation of the motor was shown to be irregular and easily affected by small changes in excitation frequency and therefore unsuitable for real micromotor applications. Closedloop operation has been shown to provide the maximum torque, but there is still much work to do in changing patterns of excitation voltages in order to minimize ripple and maximize torque. Four VHDL-AMS blocks have been developed and tested for the micromotor system: power supply, excitation, micromotor torque creation and micromotor dynamics.
Acknowledgements The author would like to acknowledge useful discussions with Prof. Alan Sangster (Heriot-Watt University) concerning
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the fabrication and torque behaviour of the micromotor. This work would not have been possible without the financial help from Ikerlan Research Centre in Mondragon—Basque Country (Spain). References [1] V.D. Samper, A.J. Sangster, R.L. Reuben, U. Wallrabe, Multistator LIGA-fabricated electrostatic wobble motors with integrated synchronous control, J. Microelectromech. Syst. 7 (2) (1998) 214– 223. [2] W. Bacher, W. Menz, J. Mohr, The LIGA technique and its potential for microsystems—a survey, IEEE Transact. Indust. Electron. 42 (5) (1995) 431–441. [3] K.S. Shea, V.D. Samper, A.J. Sangster, R.L. Reuben, S.J. Yang, An electrostatic harmonic microactuator for arterial plaque removal, J. Micromech. Microeng. 5 (4) (1995) 297–304. [4] S.-J. Yun, E.-W. Lee, D.-J. Lee, Torque Characteristics Analysis of Harmonic Side Drive Motor by Conformal Mapping Transactions of the Korean Institute of Electrical Engineers, Vol. 48, No. 3, 1999, pp. 104–109. [5] R.V. Churchill, J.W. Brown, Complex Variables and Applications, McGraw-Hill, New York, 1984. [6] IEEE Standard VHDL Analog and Mixed-Signal Extensions, IEEE Manual, March 1999. [7] V.R. Dhuler, M. Mehregany, S.M. Phillips, An experimental technique and a model for studying the operation of harmonic side-drive micromotors, IEEE Trans. Electron. Devices 40 (11) (1993) 1977– 1984. [8] S.F. Bart, M. Mehregany, L.S. Tavrow, J.H. Lang, S.D. Senturia, Measurements of electric micromotor dynamics, microstructures, Sens. Actuators DSC 19 (1990) 19–29. [9] S.F. Bart, M. Mehregany, L.S. Tavrow, J.H. Lang, S.D. Senturia, Electric micromotor dynamics, IEEE Trans. Electr. Devices 39 (3) (1992) 566–575.
Biographies Aitor Endeman˜ o Isasi received his degree in Robotics and Industrial Electronics Engineer from Mondragon University (Spain) in 1999. Mr. Endeman˜ o is currently completing his PhD degree in the Department of Computing and Electrical Engineering at Heriot Watt University (Edinburgh, Scotland). His work is mainly focused on the analytical modelling and system level simulation of microsystems. Dr. Desmulliez born in Lille (France) in 1963, graduated from the Ecole Supe´ rieure d’Electricite´ of Paris (Supe´ lec) in 1987. In the same year, he received a college diploma in microwave and modern optics from University College London. Dr. Desmulliez has also a college diploma in theoretical physics (Tripos III) from the University of Cambridge and a PhD in optoelectronics from the Department of Physics at Heriot Watt University. He became lecturer in 1996 in the Department of Computing and Electrical Engineering and senior lecturer in 1999. Dr. Desmulliez became in 2000 the director of the Microsystems Engineering Centre (MISEC), organisation that regroups the expertise in MEMS of 12 members of staff across four departments. His current research interests are in MEMS, optoelectronics and advanced packaging. Dr. Desmulliez has published over 65 papers in these fields of research. Dr. M.W. Dunnigan’s (http://www.cee.hw.ac.uk/power/front.htm) background is in the area of modelling and control. He is currently the leader of the ‘‘High Performance Control of Electric Drives’’ Research Group, a sub-group of electrical power. His research has concentrated on the
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development and implementation of adaptive, state-space and non-linear control methods applied to different application domains (electric motors, electrodynamic actuators and robotic manipulators/underwater vehicles). Control techniques researched include self-tuning adaptive, non-linear sliding-mode, fuzzy/neural, state-space estimators, and adaptive inverse control. This work has resulted in more than 60 publications (25 of these in
refereed journals) that have originated from PhD, industrial and EPSRC/ EU funded research. He is currently principal investigator on the EPSRC project (GR/R28447, A New Sensorless High-Performance Induction Motor Drive). He has been a co-investigator on two EPSRC collaborative grants and an EU grant, which involved adaptive self-tuning control for a robotic manipulator and an electrodynamic shaker.
System level simulation of a double stator wobble electrostatic micromotor Aitor Endeman˜o*, Marc P.Y. Desmulliez, Matthew Dunnigan Department of Computing and Electrical Engineering, MicroSystems Engineering Centre (MISEC), Heriot Watt University, Edinburgh EH14 4AS, UK Received 6 August 2001; accepted 15 November 2001
Abstract System level simulation results of an electrostatic micromotor, based on closed-form expressions of the static and dynamic torque behaviours, are presented. A double stator, double rotor, wobble motor designed and tested at Heriot Watt University is simulated using the high level language very high speed integrated circuits (VHSIC) hardware description language-analogue mixed signal (VHDL-AMS). The analytical torque model is obtained by two conformal mapping transformations of the electrostatic gap region between the stator and rotor surfaces. Torque results obtained by finite element analysis methods and the proposed simulation model show good agreement. Simulation results of the closed-loop control of the excitation of the motor, based on the dynamic model of the torque, are also presented. # 2002 Elsevier Science B.V. All rights reserved. Keywords: Micromotor; Actuator; Conformal mapping; Analytical modelling; System level simulation; VHDL-AMS; MEMS
1. Introduction The rapid development of microsystem technology into different fields of applications (microsensors, microactuators, microfluidics, micro-optics, etc.) is pending not only on the process knowledge and equipment needed to fabricate any prototype but also on the accurate modelling and simulation of these systems. Although nowadays specialised computer-aided design (CAD) software tools perform such simulation and modelling, their use requires a substantial understanding of the physical behaviour of such systems and are computationally intensive since most of them are based on finite element analysis (FEA) methods. Using system level modelling languages such as the IEEE 1076.1–1999 standard (VHDL-AMS) or Verilog-AMS can accelerate the modelling and simulation of microsystems, albeit at different levels of abstraction. All the different parts of a microsystem and the different physical domains involved (electrical, mechanical, fluidic, thermal) can be modelled within the same environment. It is also possible to integrate parts of a model constructed or simulated by different means (such as Spice, C language or FEA methods)
*
Corresponding author. Tel.: þ44-131-449-5111x4159; fax: þ44-131-451-3327. E-mail address: [email protected] (A. Endeman˜o).
into the same VHDL-AMS simulation environment, allowing the incorporation of results from previous complex simulations. The work was focused on the modelling of the static and dynamic behaviours of a double stator double rotor wobble motor designed and tested at Heriot Watt University [1]. This electrostatic device was designed as a first prototype to be integrated into a minimally invasive surgery system (catheter) for arterial plaque removal. Externally driven cutters are used to remove mechanically the harder and stiffer plaques from cardiac arteries. This novel design was intended to reduce the size of the system and not to require an external drive, thereby reducing operating time and the danger of damaging healthy tissue. This motor is briefly described in Section 2 of this article. A novel analytical model of the actuator is presented in Section 3. The model relies on conformal mapping techniques to transform the double air gap of the motor into a simple geometrical shape and then calculates the torque exerted on the rotor. The torque results obtained by previous FEA simulations are compared with the resulting analytical results and then discussed. The torque model for the whole motor is included into a dynamic simulation of the micromotor and implemented in VHDL-AMS in Section 4. The ADVanceMSTM simulation environment (from Mentor GraphicsTM) is employed and results are presented in Section 5.
0924-4247/02/$ – see front matter # 2002 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 4 - 4 2 4 7 ( 0 1 ) 0 0 8 3 3 - 0
A. Endeman˜o et al. / Sensors and Actuators A 99 (2002) 312–320
2. Description of the double stator, double rotor, micromotor The modelled micromotor has been fabricated using the LIGA process [2]. The motor relies on the wobbling of the double rotor within the plane of rotation and is based on electrostatic actuation. The inner surface of the outer rotor (radius R9) rolls on the outer surface of the bearing (radius R4). The offset between the rotor and stator axes is 5 mm, for a minimum air gap size of 5 mm in the fabricated prototype. A photograph of a mounted prototype and a sketch of the motor with the most relevant parameters are shown in Fig. 1. Dimensions of the motor are provided in Table 1. The configuration of the stator and rotor is shown in Fig. 2, along with the different electrodes (1–8 and A–H) from inner and outer stator and the contact point angle j. For a given excitation potential, the use of two rotor surfaces to generate electrostatic forces allows the approximate doubling of the output torque as compared to a single rotor version of the same wobble motor. The outer stator (radius R5) with the outer rotor (radius R10), and the inner stator (radius R2) with the inner rotor (radius R7) form the two pairs of surfaces that create the torque. This device operates in a wobbling fashion: a dc voltage (around 100 V) is set between stator electrodes and rotor in order to cause movement due to electrostatic forces created. This movement is constrained by the rolling of the rotor on the bearing (contact point), thus reducing the speed and rising the torque delivered. The heights of the stator electrodes and the rotor (H s ¼ 300 mm and H r ¼ 350 mm, respectively), allow the motor to develop torque values of the order of 107 to 106 N m, which are necessary to cut the deposit fat in cardiac arteries. The micromotor is to be included into a minimally invasive catheter system for the removal of arterial plaque [3].
313
Table 1 Stator (Hs, R2, R4, R5) and rotor (Hr, R7, R9, R10) dimensions Hs (mm)
R2 (mm)
R4 (mm)
R5 (mm)
Hr (mm)
R7 (mm)
R9 (mm)
R10 (mm)
300
965
1120
1185
350
975
1125
1175
Fig. 2. Electrode labels and contact angle.
3. Torque model based on conformal mapping A simple analytical model of the torque behaviour, to be included in the VHDL-AMS description of the motor, is described in this section. The model is based on two successive conformal mappings, complex functions w ¼ f ðzÞ that transform points from a complex plane z ¼ ðx; yÞ to another complex plane w ¼ ðu; vÞ [4,5]. The calculation of the capacitance, which is a function of the gap geometry between the rotor and the stator is first simplified
Fig. 1. Photograph and parameters of the wobble motor.
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using suitable conformal mapping functions. Torque is then calculated, as the derivative of the capacitance with respect to the angle of rotation; this analysis is carried out for each of the torque creating pairs of surfaces. Only the calculations for the inner pair of surfaces are presented in detail in this section. The dynamic simulation of the total torque on the rotor requires of course the contribution of both pairs of surfaces. 3.1. One electrode torque model for the inner pair of surfaces (radii R2 and R7) The stators are composed of segmented electrodes, which are selectively activated in order to generate Coulombic forces. Considering one active electrode (A in Fig. 3) of the
inner stator, the torque expression depends on the angle from the contact point to the centre of the active electrode. Neglecting the effects of the fringing fields in the region above the stator, a 2D representation is projected on a z-plane with (x, y) coordinates as shown in Fig. 3. The origin is taken from the centre of the rotor. The contact point between the bearing and the inner surface of the outer rotor (not shown in Fig. 3a) matches the minimum air gap all the time. D is the electrode width; points J, H, K indicate the beginning, centre and end of the active electrode. W is the angle taken from the contact point at the centre of the active electrode. The bearing sets the offset between the rotor and stator axes: offset ¼ R9 R4
(1)
From Fig. 3, the points of intersection of the x-axis and the stator circumference, x1 and x2 are x1 ¼ R2 þ offset ¼ R2 þ R9 R4
(2)
x2 ¼ R2 þ offset ¼ R2 þ R9 R4
(3)
The first conformal mapping w ¼ u þ jv ¼
z R7 A Az R7
(4)
transforms the stator and rotor circles (of radius R2 and R7, respectively) into concentric circles of radius R0 > 1 and unity as shown in Fig. 3b. The resulting values for A and R0, which are determined by the conditions: z ¼ R7 ) w ¼ 1; z ¼ R7 ) w ¼ 1; z ¼ x1 ) w ¼ R0 ; z ¼ x2 ) w ¼ R0 , are qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R27 þ x1 x2 þ ðR27 þ x1 x2 Þ2 R27 ðx1 þ x2 Þ2 (5a) A¼ R7 ðx1 þ x2 Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðR27 þ x1 x2 Þ2 R27 ðx1 þ x2 Þ2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R0 ¼ x1 x22 R27 x1 þ x2 ðR27 þ x1 x2 Þ2 R27 ðx1 þ x2 Þ2 R7 x22 R37 R7
(5b) An arbitrary z point in the rotor circle can be expressed in its exponential form: z ¼ R7 ejW . The corresponding point in the w-plane is then expressed as w ¼ ejfs ðWÞ , with fs: ðA2 1Þsin W fs ðWÞ ¼ tan1 (6) 2A ð1 þ A2 Þcos W
Fig. 3. Three planes of transformation for the inner pair of surfaces: (a) z-plane projection; (b) W-plane; (c) w-plane.
When the angle W in the z-plane increases in a counterclockwise fashion from 0 to 3608, the fs angle in the w-plane increases in a clockwise way from 180 to 1808 with a single corresponding point in the w-plane for each point in the z-plane. The derivative with respect to W of the stored electrical energy, U, is calculated in order to calculate the torque exerted on the rotor by the active electrode. As the expression of U depends on the angle-dependent capacitance
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C(W) and the latter depends on fs(W), the expression of the derivative of fs(W) with respect to W is needed @fs 1 A2 ¼ gðWÞ ¼ 2 @W A þ 1 2A cos W
(7)
The logarithmic transformation w ¼ log w
(8)
maps each point in the w-plane to another point in the w-plane so that the modelled air gap between the active electrode and the rotor in the w-plane has a square shape allowing therefore the simple calculation of the capacitance as shown in Fig. 3(c). The stator electrode is itself converted into a flat electrode in plane w. Using the parallel plate approximation, the capacitance is e0 Area e0 LjqðK 00 Þ qðJ 00 Þj CðWÞ ¼ ¼ d log R0 e0 L½fs ðW þ D=2Þ fs ðW D=2Þ
¼ log R0
(9)
Points J00 and K00 are the images of points J and K in the wplane. The maximum capacitance is reached when the contact point is at the centre of the active electrode and keeps decreasing until W ¼ 1808 (Fig. 4a). The torque on the rotor can then be calculated from an energy balance. Taking into account the ideal gear ratio of this motor: R9 W n¼ (10) ¼ offset j where j is the rotating angle of the rotor (same as the contact point angle), the generated torque is equal to the change in the electrical stored energy between the pair of surfaces (U ¼ CV 2 /2) with respect to j dU dU dW dU ¼ ¼ n (11) G¼ dj dW dj dW V2 @ fs ðW þ D=2Þ fs ðW D=2Þ GðWÞ ¼ ne0 L @W log R0 2 2 V e0 L ntðWÞ (12) ¼ 2
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where t(W) is the normalised torque. The analytic expression for t(W) can then be developed 1 D D g W (13) g Wþ tðWÞ ¼ log R0 2 2 where g(W) is the expression obtained in Eq. (7). The torque behaviour as a function of the contact angle for one active electrode is shown in Fig. 4(b). As expected, the value of the torque is zero at 1808, when the electrode is aligned with the contact point and at a maximum of around 0.18 mN m for one active electrode. 3.2. Gross torque calculation and validation for the inner pair of surfaces The calculation of the overall torque for the inner pair of surfaces depends on the number of electrodes and the excitation scheme applied to the electrodes. Four adjacent electrodes, excited at any given time, have been demonstrated to provide the best torque performance. The one electrode torque expression obtained in Eq. (12), gives the torque values as a function of the angle W. A superposition of the torque values of the four active electrodes must take into account the four different angles from the contact point to each of the active electrodes. First the gross torque is calculated for the four active electrodes (A–D) as the rotor contact point angle j varies from 22.5 to 22.58 (Fig. 5). The fixed angles for the centres of the four electrodes are aE ¼ 22:58, aF ¼ 67:58, aG ¼ 112:58 and aH ¼ 157:58. For every simulation step (i.e. for every increment of the contact point angle j), the angles from the contact point to the centre of each active electrode are WE ¼ aE j, WF ¼ aF j, WG ¼ aG j and WH ¼ aH j. The gross torque expression is therefore Ggross ¼ GE ðWE Þ þ GF ðWF Þ þ GG ðWG Þ þ GH ðWH Þ (14) where j varies from 22.5 to 22.58. The torque behaviour, shown by a solid line in Fig. 6, increases from 22.5 to 08 with the effective gap for the active electrodes and decreases down from 0 to 22.58 due to the opposite effect. The maximum value for the torque created by the inner surfaces of the motor is 0.3806 mN m.
Fig. 4. (a) Capacitance as a function of contact angle; (b) one electrode torque function as a function of contact angle
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Fig. 5. Scheme of the active electrodes(in black) for the torque calculation at j ¼ 08.
For comparison, the FEA simulations for the gross torque are plotted on the same figure as well a different analytical (but more computationally intensive) approach based on numeric integration of a torque formula including an analytical model for the electric field [1]. Our results match fairly well with these previous FEA simulations. In the worst case that corresponds to the maximum torque case, a discrepancy of less than 5% has been calculated between the FEA model and the analytical model. This discrepancy can be explained by neglecting the closed-form model fringing field effects at the top edges of stator and rotor (3D) and at the edges of the electrodes (2D). As expected, this discrepancy decreases as the electrode to gap spacing ratio increases. A further refinement of the model will be done in the future by importing FEA results for the electric field into the VHDL-AMS model, thus taking into account these second order effects. Both FEA simulations and the analytic model for integration developed for the micromotor (torque results in Fig. 6) are not suitable for VHDL-AMS implementation. System level simulation can be done with the results of FEA through parameter extraction, however this method assumes that
Fig. 6. Gross torque function (solid line) compared to FEA results (circles) and another analytical model (crosses).
FEA has been carried out previously, at the expense of time-consuming iterative computations. The developed conformal mapping model for the static torque is accurate, simple and easy to implement in VHDL-AMS as closedform equations.
4. VHDL-AMS implementation Once the torque model has been validated, the next step is to implement the whole model in VHDL-AMS. The VHDLAMS model of the system is divided into four blocks: dc power supply, excitation control unit, micromotor torque calculation and micromotor dynamics (these last two blocks define the behaviour of the micromotor itself). Each of the blocks is represented in VHDL-AMS code as an ‘‘Entity’’, as seen in Fig. 7. Each of the entities has one or more different ‘‘Architectures’’.
Fig. 7. Basic structure of the VHDL-AMS model of the system.
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For example in our model, the excitation entity has two different architectures depending on the excitation strategy implemented. For all those architectures, 16 outputs drive the 16 segmented electrodes of the inner and outer stators (eight segments each) of the micromotor. When the motor is operated in a closed-loop configuration, the switching of the drive signals depends on the rotor position feedback; if operated in an open-loop configuration, the switching has a fixed frequency. The micromotor torque entity implements the conformal mapping model for the gross torque. The micromotor dynamics entity has the gross torque as an entry and implements the dynamic model of the motor. VHDLAMS as a language offers the possibility to implement differential and algebraic equations that define the physical behaviour of a system [6]. The dynamic equation proposed for the micromotor model is ¼ Ggross Bj_ ðC1 þ C2 V 2 Þsgn j_ Jj
(15)
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where J is the moment of inertia of the rotor mathematically calculated as 1:48 1011 kg m2, Ggross is the instantaneous motor torque calculated by the model, B is the coefficient of viscous friction and C1, C2 are the constant and voltage dependant kinetic (Coulomb) friction terms. These constants have been estimated analytically from micromotor geometric and material properties compared to previous works [7–9] and are B ¼ 5 1011 N m s, C1 ¼ 1 1012 N m, C2 ¼ 3 1012 N m V2.
5. Simulation and results 5.1. Fixed step responses Using the VHDL-AMS model, at j ¼ 08 position, different electrodes are activated in order to study the micromotor response from rest to the center of the active
Fig. 8. Active electrode configuration for step responses.
Fig. 9. Motor step responses for different number of electrodes active in both stators (1–4): (a) continuous; (b) dashed; (c) dashed/dotted; (d) dotted lines). Angular position j (rad) and rotor speed o (rpm).
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electrode. The active electrodes in the inner stator are the opposite of the ones active in the outer stator, so that the maximum torque is developed as the contribution of both pairs of surfaces is summed. Results for 1–4 active electrodes (shown in Fig. 8) in both stators are presented in Fig. 9. The most performant configuration is the four electrodes case, for which the rotor reaches the highest values of acceleration (50,000 rad s2). Damped or nearly damped responses are observed in all the cases.
6. Continuous operation In order to simulate a steady and sustained operation of the micromotor, the excitation control block can activate the different electrodes of each stator with different excitation sequences. The switching moments for those
sequences can be a periodic function of time (open-loop) or function of the rotor position (closed-loop). One of the aims of developing this model of the micromotor is to verify the influence of a closed-loop control system for real applications. Fig. 10 shows the closed-loop operation of the motor for a four-electrode-scheme which results in a excitation frequency of 20.25 Hz. In this case, the obtained average torque is the maximum achievable (7:8 107 N m), because the electrodes are switched on and off at the exact moment required in order not to lose energy. The optimum open-loop excitation frequency that corresponds to the maximum achievable torque in those conditions is 19.25 Hz, quite near to the closed-loop one. Any higher frequency applied to the stator electrodes results in motor operation failure, since the rotor cannot follow the electrical frequency of the stator. Different open-loop four
Fig. 10. Closed-loop results: gross torque (N m), angular speed (Rpm).
Fig. 11. Open loop results: gross torque (N m); angular speed (rpm); switching frequency: f ¼ 19:25 Hz (continuous line), f ¼ 14:25 Hz (dotted line).
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electrodes excitation frequencies have been used. Applying the maximum 19.25 Hz frequency results in quite a uniform motion of the rotor, but with bigger losses than in the closedloop case, due to the torque shape (as shown in Fig. 11). Applying an excitation frequency of 14.25 Hz (lower than the optimum) results in irregular operation. The average torque produced for this case (4:4 107 N m) is much lower than the closed-loop one and produces a much bigger ripple (Fig. 11 in dotted line). The ripple leads to undesirable accelerations and decelerations that cause an unnecessary loss of energy. The torque supplied to the load, in a future real application would also be affected by these fluctuations and make an open-loop operation of the motor unsuitable for most applications. Measurement results of existing micromotors operating in open loop are currently underway to validate these system level simulation results. As the 3D FEA has been known to provide a good agreement with experimental results, we are confident that our model, based on closed form expressions detailed in Section 3, gives a sufficiently good agreement with experiments.
7. Conclusions A new analytical model for a double stator double rotor electrostatic micromotor has been developed based on conformal mapping techniques. The purpose of developing this mathematical model is to integrate it into a system level simulation of the microdevice together with its excitation control electronics. It was necessary to develop a simple closed-form model in order to implement it in VHDL-AMS, but with accuracy comparable to that of a FEA simulation or an integration based model. The theoretical model obtained has been essential in order to further develop a complete dynamic model of the micromotor and implement it using the VHDL-AMS high level modelling language. The model was fast and simple to simulate and it has been the first step to assess the important influence of closed-loop excitation control for the micromotor performance. Open-loop operation of the motor was shown to be irregular and easily affected by small changes in excitation frequency and therefore unsuitable for real micromotor applications. Closedloop operation has been shown to provide the maximum torque, but there is still much work to do in changing patterns of excitation voltages in order to minimize ripple and maximize torque. Four VHDL-AMS blocks have been developed and tested for the micromotor system: power supply, excitation, micromotor torque creation and micromotor dynamics.
Acknowledgements The author would like to acknowledge useful discussions with Prof. Alan Sangster (Heriot-Watt University) concerning
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the fabrication and torque behaviour of the micromotor. This work would not have been possible without the financial help from Ikerlan Research Centre in Mondragon—Basque Country (Spain). References [1] V.D. Samper, A.J. Sangster, R.L. Reuben, U. Wallrabe, Multistator LIGA-fabricated electrostatic wobble motors with integrated synchronous control, J. Microelectromech. Syst. 7 (2) (1998) 214– 223. [2] W. Bacher, W. Menz, J. Mohr, The LIGA technique and its potential for microsystems—a survey, IEEE Transact. Indust. Electron. 42 (5) (1995) 431–441. [3] K.S. Shea, V.D. Samper, A.J. Sangster, R.L. Reuben, S.J. Yang, An electrostatic harmonic microactuator for arterial plaque removal, J. Micromech. Microeng. 5 (4) (1995) 297–304. [4] S.-J. Yun, E.-W. Lee, D.-J. Lee, Torque Characteristics Analysis of Harmonic Side Drive Motor by Conformal Mapping Transactions of the Korean Institute of Electrical Engineers, Vol. 48, No. 3, 1999, pp. 104–109. [5] R.V. Churchill, J.W. Brown, Complex Variables and Applications, McGraw-Hill, New York, 1984. [6] IEEE Standard VHDL Analog and Mixed-Signal Extensions, IEEE Manual, March 1999. [7] V.R. Dhuler, M. Mehregany, S.M. Phillips, An experimental technique and a model for studying the operation of harmonic side-drive micromotors, IEEE Trans. Electron. Devices 40 (11) (1993) 1977– 1984. [8] S.F. Bart, M. Mehregany, L.S. Tavrow, J.H. Lang, S.D. Senturia, Measurements of electric micromotor dynamics, microstructures, Sens. Actuators DSC 19 (1990) 19–29. [9] S.F. Bart, M. Mehregany, L.S. Tavrow, J.H. Lang, S.D. Senturia, Electric micromotor dynamics, IEEE Trans. Electr. Devices 39 (3) (1992) 566–575.
Biographies Aitor Endeman˜ o Isasi received his degree in Robotics and Industrial Electronics Engineer from Mondragon University (Spain) in 1999. Mr. Endeman˜ o is currently completing his PhD degree in the Department of Computing and Electrical Engineering at Heriot Watt University (Edinburgh, Scotland). His work is mainly focused on the analytical modelling and system level simulation of microsystems. Dr. Desmulliez born in Lille (France) in 1963, graduated from the Ecole Supe´ rieure d’Electricite´ of Paris (Supe´ lec) in 1987. In the same year, he received a college diploma in microwave and modern optics from University College London. Dr. Desmulliez has also a college diploma in theoretical physics (Tripos III) from the University of Cambridge and a PhD in optoelectronics from the Department of Physics at Heriot Watt University. He became lecturer in 1996 in the Department of Computing and Electrical Engineering and senior lecturer in 1999. Dr. Desmulliez became in 2000 the director of the Microsystems Engineering Centre (MISEC), organisation that regroups the expertise in MEMS of 12 members of staff across four departments. His current research interests are in MEMS, optoelectronics and advanced packaging. Dr. Desmulliez has published over 65 papers in these fields of research. Dr. M.W. Dunnigan’s (http://www.cee.hw.ac.uk/power/front.htm) background is in the area of modelling and control. He is currently the leader of the ‘‘High Performance Control of Electric Drives’’ Research Group, a sub-group of electrical power. His research has concentrated on the
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development and implementation of adaptive, state-space and non-linear control methods applied to different application domains (electric motors, electrodynamic actuators and robotic manipulators/underwater vehicles). Control techniques researched include self-tuning adaptive, non-linear sliding-mode, fuzzy/neural, state-space estimators, and adaptive inverse control. This work has resulted in more than 60 publications (25 of these in
refereed journals) that have originated from PhD, industrial and EPSRC/ EU funded research. He is currently principal investigator on the EPSRC project (GR/R28447, A New Sensorless High-Performance Induction Motor Drive). He has been a co-investigator on two EPSRC collaborative grants and an EU grant, which involved adaptive self-tuning control for a robotic manipulator and an electrodynamic shaker.