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Electromagnetic Actuator Modelling with the Extended Modelica Magnetic Library

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Electromagnetic Actuator Modelling with the Extended Modelica Magnetic Library Electromagnetic Actuator Modelling with the Extended Modelica Magnetic Library Electromagnetic Actuator Modelling with the Extended Modelica Magnetic Library Thomas Bödrich Dresden University of Technology, Institute of Electromechanical and Electronic Design 01062 Dresden, Germany [email protected] Abstract fort needed for creation of rough magnetic network models and to the low computational effort for dy- Recently, an improved and extended version of a namic simulation compared to finite element tech- Modelica library for lumped network modelling of niques. electromagnetic devices has been released [1]. This A first realisation of a Modelica library for lumped library is intended for both rough design of the mag- network modelling of electromagnetic devices was netic circuit of those devices as well as for their sys- presented in [2]. This library was improved and ex- tem simulation. Improvements compared to a first tended and is now available [1]. Some of the impor- realisation of this library [2] will be discussed and tant improvements of the extended library compared utilization of the extended library for modelling of to the prior version are: electromagnetic actuators will be illustrated. To sup- • a more general approach for calculation of mag- port the work with this library, focus will be on netic reluctance forces (section 2), newly-implemented features and on peculiarities of • redesigned and extended components for represen- lumped network modelling of electromagnetic actua- tation of typical flux tube shapes (section 2), tors rather than on an introductory explanation of the • additional soft magnetic and permanent magnetic underlying concept of magnetic flux tubes. materials (section 3), Keywords: lumped magnetic network; electromag- • more accurate modelling of magnetic leakage netic/electrodynamic actuator; system design fields for dynamic simulation by splitting of the magnetomotive force imposed by a coil into sepa- rate sources (section 4) and 1 Introduction • additional examples of different modelling depths. The well-established concept of magnetic flux tubes enables the modelling of magnetic fields with 2 Reluctance Force Calculation lumped networks. In the decades prior to the broad availability of software packages based on finite Generally, the thrust F developed by a translatory element techniques, this was the only efficient means electro-magneto-mechanical actuator (similar for the for model-based design of the magnetic circuit of rotational case with torque and angular position) is electromagnetic devices such as transformers, motors equal to the change of magnetic co-energy W * with and electromagnetic or electrodynamic actuators. m armature position x according to The method of magnetic flux tubes, its utilization for ∂ ∂ the design of electromagnetic devices as well as F = Ψ di = W * ∫ m (1) derivation of the permeance of many flux tube ∂x (i) ∂x shapes are explained in-depth for example in [3], [4] and [5] and in condensed form in the documentation (Ψ flux linkage, i actuator current) [4]. In lumped of [1]. magnetic network models, the above equation sim- plifies to Even though finite element techniques allow for n linear more accurate calculation of magnetic fields, lumped 1 2 dGmi F = ∑ Vmag i (2) magnetic network models are still an efficient means 2 i=1 dx for initial rough design of electromagnetic devices and for simulation of their dynamic behaviour during where nlinear is the number of flux tube elements with system design. This is due to the relatively little ef- constant relative permeability that change their per- The Modelica Association 221 Modelica 2008, March 3rd 4th, 2008 − T. B¨odrich meance Gm i with armature position (index i), Vmag i pends on the design of the actuators magnetic circuit the magnetic voltage across each respective flux tube and on the definition of the armature coordinate x. and dGm i/dx the derivative of the respective per- To cover all possible conditions in a uniform way, meances with respect to armature position. Transi- the above derivative is calculated as follows for all tion from the general formula based on magnetic flux tubes with force generation of sub-package co-energy (Eq. (1)) to Eq. (2) is outlined in [4] for FluxTube.Force: the reciprocal of the permeance, i.e. for the magnetic dG dG dl reluctance. Compared to the reluctance force calcula- m = m . (4) tion with Maxwell’s formula used in [2], the dx dl dx newly-implemented approach according to Eq. (2) For the flux tubes defined in this package with their simplifies force calculation for air gaps different rather simple shapes, the derivative dGm /dl is given from a simple cylindrical or prismatic shape with analytically (Table 1). l is the flux tube dimension axial magnetic flux (see below). that changes with armature motion. dl /dx is an inte- The usability of Eq. (2) is not restricted solely to net- ger parameter that must be set to +1 or -1 according to the magnetic circuit and the definition of the ar- work models with constant relative permeabilities µr of the flux tubes. However, it is required that flux mature coordinate x of an actuator. For more com- tubes with a dependency of the permeability on the plex shapes and variations of dimensions with arma- flux density B such as ferromagnetic components ture motion, the derivative dGm /dl must be provided analytically during model development, prefer- with non-linear characteristics µr(B) do not change their shape with armature motion (e.g. portion of a ably by extending the partial model Flux- solenoid plunger where the magnetic flux passes Tube.Force.PartialForce. through in axial direction). This limitation is not a strong one, since the permeance of non-linear, high Table 1: Selected flux tube elements with reluctance permeable ferromagnetic flux tube elements and its force generation (subset of FluxTube.Force), change with armature position compared to that of permeance Gm and analytic derivative dGm /dl air gap flux tubes can be neglected in most cases. Because of this constraint, the dimensions of possi- (Hollow) cylinder with axial magnetic flux bly non-linear flux tube elements in sub-package µ µ A FluxTube.FixedShape are fixed, whereas the G = 0 r dimension l in direction of motion of the linear flux m l tube elements in sub-package FluxTube.Force dG µ µ A can vary during simulation. Elements with fixed m 0 r = − 2 shape are intended e.g. for modeling of transformer dl l cores or ferromagnetic sections of actuators. Force with A = π (r 2 − r 2 ) elements are to be used for modelling of working air o i gaps or for moving permanent magnets of actuators. Hollow cylinder with radial magnetic flux In order to fulfill Eq. (2) in a magnetic network model of a translatory actuator, the reluctance force µ µ 2π l F of each flux tube with force generation is calcu- G = 0 r m m ln r r lated in the respective element accordingly: ()o i 1 dG dG µ µ 2π F = V 2 m . (3) m = 0 r m mag 2 dx dl ln()ro ri Vmag denotes the magnetic voltage across the flux tube and dGm /dx is the derivative of flux tubes per- Simple leakage flux tube around cylindrical or pris- meance Gm with respect to armature position x. matic poles Summation of all particular reluctance forces to the actuators net force F according to Eq. (2) is induced 2 µ0 t ⎛ π rleak ⎞ Gm = ln ⎜1+ ⎟ by connecting the translatory flange connectors of all π ⎝ 2 l ⎠ flux tube elements with force generation in an actua- dG µ t r tor model (see example in Figure 4b). m = − 0 leak dl ⎛ π r ⎞ In a particular actuator model, the derivative dGm /dx l 2 ⎜1+ leak ⎟ of the models flux tubes with force generation and ⎝ 2 l ⎠ hence the sign of the generated reluctance force de- The Modelica Association 222 Modelica 2008, March 3rd 4th, 2008 − Electromagnetic Actuator Modelling with the Extended Modelica Magnetic Library The leakage flux tube shown in Table 1 provides a of a device. It must be noted that a measured charac- simple but efficient means to account for leakage teristics µr(B) strongly depends on the shape, ma- around prismatic or cylindrical poles. In the latter chining state and heat treatment of a sample, and on case, depth t is equal to the circumference of a circle the measurement conditions. Hence, different char- given by the average radius of the flux tube. Due to acteristics are possible for similar materials as Figure the constant radius rleak of the leakage field, the 1 indicates. model is rather simple. In reality, rleak is approxi- mately constant for air gap lengths l greater than this M330-50A (machined, packeted) M350-50A (sheet sample) radius, but decreases with air gap lengths less than 8000 r M530-50A (sheet sample) rleak. This decrease for small air gaps is neglected in µ 7000 M700-100A (sheet sample) M940-50A (sheet sample) the model since the influence of the leakage flux 6000 tube compared to that of the enclosed main air gap 5000 (connected in parallel, see example in Figure 4b) 4000 decreases for decreasing air gap length l. 3000 The sub-package FluxTube.Leakage contains tive permeability 2000 flux tube shapes typical for leakage flux around Rela 1000 prismatic or cylindrical poles that do not change their 0 shape with armature motion. Hence, the permeance 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 of these flux tubes does not depend on armature po- Flux density B in T sition and these elements do not contribute to the Figure 1: Approximated magnetization characteris- thrust of a reluctance actuator.
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