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Strong coupling in electromechanical computation
Journal of Magnetism and Magnetic Materials 215}216 (2000) 746}748
Strong coupling in electromechanical computation JaH nos FuK zi * Electrical Engineering Department, Transilvania University, Brasov, Politehnicii 1-3, 2200 Brasov, Romania Electromagnetic Theory Department, Bolyai Ja& nos grantee, Technical University of Budapest, H 1521 Budapest, Hungary
Abstract A method is presented to carry out simultaneously electromagnetic "eld and force computation, electrical circuit analysis and mechanical computation to simulate the dynamic operation of electromagnetic actuators. The equation system is solved by a predictor}corrector scheme containing a Powell error minimization algorithm which ensures that every di!erential equation (coil current, "eld strength rate, #ux rate, speed of the keeper) is ful"lled within the same time step. 2000 Elsevier Science B.V. All rights reserved. Keywords: Dynamic hysteresis; Preisach model; Electromagnetic forces; Electromechanical coupling
Electromagnetic "eld and force computation, electrical circuit analysis and mechanical computation have to be carried out simultaneously to simulate the dynamic operation of electromagnetic actuators, due to the fact that material parameters, electromagnetic and mechanical quantities strongly in#uence each other. Simplifying assumptions are made concerning the shape of the magnetic circuit and the distribution of magnetic quantities within core sheets and in air gap to focus the attention on the process of coupling the electromagnetic, electrical circuit and mechanical computation steps. It is considered that the #ux is uniformly distributed within the core sheets and in the air gap and the #ux cross section in the air gap is taken to be equal to the magnetic circuit cross section. To illustrate the operation of the proposed method, the layout shown in Fig. 1 is considered. The parameter values are: E Core: cross-section: S"500 mm; length: l"200 mm; maximal airgap: x "4 mm.
* Correspondence address. Electrical Engineering Department, Transilvania University, Politehnicii 1-3, 2200 Brasov, Romania. Tel.: #40-68-138636, #36-1-463-2723; fax: #40-68143116, #36-1-463-3189. E-mail address: [email protected], [email protected] (J. FuK zi).
E Winding: N"600, resistance: R "0.4 ). E Additional resistor: R "200 ), freerun resistor: ? R "4000 ), supply voltage: u"100 V. B E Mass of keeper and parts attached: m"0.2 kg. E Spring: constant: k"10 N/m, pretensioning distance: a"0.5 mm. The static hysteresis characteristic of the core is modelled by the classical Preisach model [1]. The experimentally determined Everett integral surface (support of CPM) for the STABOCOR 250}50 A, 0.5 mm thick, nonoriented silicon-iron sheet is plotted in Fig. 2. The dynamic magnetic characteristic of the core is modelled by means of a fast time-dependent Preisach model (RPM) to encompass the e!ects of hysteresis as well as eddy currents and domain wall displacement due to fast #ux changes [2]. It is constructed by introducing a supplementary variable (H ) as input for the
CPM, delayed with respect to the actual "eld strength (H), governed by Eq. (1), when the average #ux density through the core cross section (B) is known or Eq. (2), when the actual "eld strength is given on the surface of the core sheets. Here M is the magnetization and dM/dH is computed by CPM as the slope of the ac tually followed hysteresis branch. Parameter values a"4000 s\; b"300 m/H; c"1 ensure good agreement of simulated dynamic loops with measured ones (Fig. 3).
0304-8853/00/$ - see front matter 2000 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 4 - 8 8 5 3 ( 0 0 ) 0 0 2 7 6 - 6
J. Fu( zi / Journal of Magnetism and Magnetic Materials 215}216 (2000) 746}748
747
Fig. 1. Modelled DC actuator.
Fig. 3. Static (st.), measured (meas.) and computed (comp.) dynamic magnetizing loops at f"100 Hz.
Fig. 2. Everett surface for the core sheets.
dB dH dH
"a (H!H )!b #c ,
dt dt dt
(1)
a (H!H )#(c!k b)dH/dt dH
" . (2) dt 1#k b(dM/dH )
The electromagnetic force is computed according to the principle of virtual work with constant current condition d= "d=#Fdx, (3) where d= is the energy transferred to the coil to keep its current constant, d= is the variation of the energy in the iron core and the air gap, F the force acting on the keeper and dx is its displacement. The received energy is d= "u i dt"NiS dB. The variation of the stored energy
(4)
d="H dB < #H dB < # k H d< . (5) $ The "eld strength in the region of the iron and the air gap are, respectively, Ni 2x B B H" ! ; H " k l l k
(6)
Fig. 4. Electromagnetic quantities versus time.
and the force acting on the keeper (as obtained by Maxwell's stress tensor [3]) d= !d= BS F" "! , dx k In the equation of the switched on supply circuit u"Ri#N
du , dt
(7)
(8)
at the beginning R"R , and after the air gap is halved, R"R #R . When the supply is switched o!, the
J. Fu( zi / Journal of Magnetism and Magnetic Materials 215}216 (2000) 746}748
748
Fig. 6. Mechanical quantities versus time. Fig. 5. Track of core operation point.
circuit is closed through the freerun diode: du 0"(R #R )i#N , dt
(9)
The #ux variation rate is du dB dH k N(di/dt)!2B(dx/dt) "S "S . dt dH dt lk (dH/dB)#2x The movement of the keeper follows: m
dx "F#k(a#x !x), dt
Eqs. (2), (6)}(11) in time domain by means of a predictor}corrector scheme containing a Powell error minimization algorithm. The latter minimizes the error square of each di!erential equation (coil current, "eld strength rate, #ux rate, speed of the keeper) with respect to the current and airgap increments within each time step. This method provides both accuracy and stability for the solver.
(10)
Acknowledgements
(11)
This work has been performed in the frame of the Bolyai Ja& nos Grant BO/00920/98 awarded by the Hungarian Academy of Science.
with the conditions !F'ka; x'0 in the switching on process and !F(k(a#x ); x(x through the switching o! process. The results of simulation can be followed in Figs. 4}6. The supply circuit is switched on at t"0 and switched o! at t"12 ms. Strong coupling * required due to the tight interaction between the operation of the supply circuit, the magnetic state of the core, position and speed of the keeper * is realized by solving the system containing
References [1] I.D. Mayergoyz, Mathematical Models of Hysteresis, Springer, New York, 1991. [2] FuK zi J., Computationally E$cient Rate Dependent Hysteresis Model, Proceedings of the Eighth IGTE Symposium on Numerical Field Calculation in Electrical Engineering, Graz, 1998, Part II, pp. 397}402. [3] De Medeiros, G. Reyne, G. Meunier, IEEE Trans. Magn. 34 (1998) 3560.
Strong coupling in electromechanical computation JaH nos FuK zi * Electrical Engineering Department, Transilvania University, Brasov, Politehnicii 1-3, 2200 Brasov, Romania Electromagnetic Theory Department, Bolyai Ja& nos grantee, Technical University of Budapest, H 1521 Budapest, Hungary
Abstract A method is presented to carry out simultaneously electromagnetic "eld and force computation, electrical circuit analysis and mechanical computation to simulate the dynamic operation of electromagnetic actuators. The equation system is solved by a predictor}corrector scheme containing a Powell error minimization algorithm which ensures that every di!erential equation (coil current, "eld strength rate, #ux rate, speed of the keeper) is ful"lled within the same time step. 2000 Elsevier Science B.V. All rights reserved. Keywords: Dynamic hysteresis; Preisach model; Electromagnetic forces; Electromechanical coupling
Electromagnetic "eld and force computation, electrical circuit analysis and mechanical computation have to be carried out simultaneously to simulate the dynamic operation of electromagnetic actuators, due to the fact that material parameters, electromagnetic and mechanical quantities strongly in#uence each other. Simplifying assumptions are made concerning the shape of the magnetic circuit and the distribution of magnetic quantities within core sheets and in air gap to focus the attention on the process of coupling the electromagnetic, electrical circuit and mechanical computation steps. It is considered that the #ux is uniformly distributed within the core sheets and in the air gap and the #ux cross section in the air gap is taken to be equal to the magnetic circuit cross section. To illustrate the operation of the proposed method, the layout shown in Fig. 1 is considered. The parameter values are: E Core: cross-section: S"500 mm; length: l"200 mm; maximal airgap: x "4 mm.
* Correspondence address. Electrical Engineering Department, Transilvania University, Politehnicii 1-3, 2200 Brasov, Romania. Tel.: #40-68-138636, #36-1-463-2723; fax: #40-68143116, #36-1-463-3189. E-mail address: [email protected], [email protected] (J. FuK zi).
E Winding: N"600, resistance: R "0.4 ). E Additional resistor: R "200 ), freerun resistor: ? R "4000 ), supply voltage: u"100 V. B E Mass of keeper and parts attached: m"0.2 kg. E Spring: constant: k"10 N/m, pretensioning distance: a"0.5 mm. The static hysteresis characteristic of the core is modelled by the classical Preisach model [1]. The experimentally determined Everett integral surface (support of CPM) for the STABOCOR 250}50 A, 0.5 mm thick, nonoriented silicon-iron sheet is plotted in Fig. 2. The dynamic magnetic characteristic of the core is modelled by means of a fast time-dependent Preisach model (RPM) to encompass the e!ects of hysteresis as well as eddy currents and domain wall displacement due to fast #ux changes [2]. It is constructed by introducing a supplementary variable (H ) as input for the
CPM, delayed with respect to the actual "eld strength (H), governed by Eq. (1), when the average #ux density through the core cross section (B) is known or Eq. (2), when the actual "eld strength is given on the surface of the core sheets. Here M is the magnetization and dM/dH is computed by CPM as the slope of the ac tually followed hysteresis branch. Parameter values a"4000 s\; b"300 m/H; c"1 ensure good agreement of simulated dynamic loops with measured ones (Fig. 3).
0304-8853/00/$ - see front matter 2000 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 4 - 8 8 5 3 ( 0 0 ) 0 0 2 7 6 - 6
J. Fu( zi / Journal of Magnetism and Magnetic Materials 215}216 (2000) 746}748
747
Fig. 1. Modelled DC actuator.
Fig. 3. Static (st.), measured (meas.) and computed (comp.) dynamic magnetizing loops at f"100 Hz.
Fig. 2. Everett surface for the core sheets.
dB dH dH
"a (H!H )!b #c ,
dt dt dt
(1)
a (H!H )#(c!k b)dH/dt dH
" . (2) dt 1#k b(dM/dH )
The electromagnetic force is computed according to the principle of virtual work with constant current condition d= "d=#Fdx, (3) where d= is the energy transferred to the coil to keep its current constant, d= is the variation of the energy in the iron core and the air gap, F the force acting on the keeper and dx is its displacement. The received energy is d= "u i dt"NiS dB. The variation of the stored energy
(4)
d="H dB < #H dB < # k H d< . (5) $ The "eld strength in the region of the iron and the air gap are, respectively, Ni 2x B B H" ! ; H " k l l k
(6)
Fig. 4. Electromagnetic quantities versus time.
and the force acting on the keeper (as obtained by Maxwell's stress tensor [3]) d= !d= BS F" "! , dx k In the equation of the switched on supply circuit u"Ri#N
du , dt
(7)
(8)
at the beginning R"R , and after the air gap is halved, R"R #R . When the supply is switched o!, the
J. Fu( zi / Journal of Magnetism and Magnetic Materials 215}216 (2000) 746}748
748
Fig. 6. Mechanical quantities versus time. Fig. 5. Track of core operation point.
circuit is closed through the freerun diode: du 0"(R #R )i#N , dt
(9)
The #ux variation rate is du dB dH k N(di/dt)!2B(dx/dt) "S "S . dt dH dt lk (dH/dB)#2x The movement of the keeper follows: m
dx "F#k(a#x !x), dt
Eqs. (2), (6)}(11) in time domain by means of a predictor}corrector scheme containing a Powell error minimization algorithm. The latter minimizes the error square of each di!erential equation (coil current, "eld strength rate, #ux rate, speed of the keeper) with respect to the current and airgap increments within each time step. This method provides both accuracy and stability for the solver.
(10)
Acknowledgements
(11)
This work has been performed in the frame of the Bolyai Ja& nos Grant BO/00920/98 awarded by the Hungarian Academy of Science.
with the conditions !F'ka; x'0 in the switching on process and !F(k(a#x ); x(x through the switching o! process. The results of simulation can be followed in Figs. 4}6. The supply circuit is switched on at t"0 and switched o! at t"12 ms. Strong coupling * required due to the tight interaction between the operation of the supply circuit, the magnetic state of the core, position and speed of the keeper * is realized by solving the system containing
References [1] I.D. Mayergoyz, Mathematical Models of Hysteresis, Springer, New York, 1991. [2] FuK zi J., Computationally E$cient Rate Dependent Hysteresis Model, Proceedings of the Eighth IGTE Symposium on Numerical Field Calculation in Electrical Engineering, Graz, 1998, Part II, pp. 397}402. [3] De Medeiros, G. Reyne, G. Meunier, IEEE Trans. Magn. 34 (1998) 3560.