Physical Process Modeling

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Magnetic field



    From the university physics course it is well known that magnetic field is just one side of the electromagnetic field. The theory of the electromagnetic field is based on Maxwell's equations. They can be significantly simplified by assuming that the magnetic field quantities are solely determined by the instantaneous values of source currents and that the time variations of the magnetic fields follow directly from the variations of the sources. It can be said simply that when a current flows it creates magnetic field. An opposite statement is also valid - if a magnetic field is observed, there is a current responsible for that field creation.
    Magnetic field is basically characterized by two vector quantities:

    B - magnetic flux density.
    H - magnetic field intensity.

    Both of these quantities are specified for a given space point. Magnetic flux density B is associated with the mechanical force which a current carrying element exerts in that same point. B is measured in teslas [T], 1T = 1 V.s/m2 and strongly depends on the magnetic properties of the material penetrated by an external magnetic field. The magnetic field intensity H is measured in A/m m and depends only on the currents creating the field and geometry factors.
    The relationship between B and H is of main interest. In free vacuum it is expressed introducing magnetic permeability o as the ratio o = B/H having in mind that both quantities are vectors in one and the same direction. The unit is o = 4..10-7 (henry per meter).
    To define this relationship in a medium different than vacuum, the atomic structure of matter has to be involved. Electrons cycling around the atom nucleus are considered as micro currents loops developing their own (internal) magnetic field, which results in the so called Magnetization M of the material. This quantity is defined per unit volume, its measuring unit being equal to that of H i.e. A/m. Thus:

magnetic field - equations

As M for isotropic materials is usually proportional to H (M = xH):  magnetic field - equations
where r = (l+x) is called relative magnetic permeability of the material concerned.
    An additional and very important quantity associated with the magnetic field is the magnetic flux . It is defined as = B.S, where S is introduced as a certain surface in space, as for example depicted in figure:

magnetic flux - theory


    It should be noted that in case the distribution of B over the surface S is not uniform, than actually a surface integral shall be created to find the flux :
magnetic field - equations

    Here ds is the surface element vector. It is clear now from the dot product B.ds that the flux is actually scalar quantity. Nevertheless in engineering it is convenient that the flux has the same as B direction (it is said to flow in that same direction).
    In practical engineering when the conductor, creating the loop to form the surface S as depicted in figure, is not a single one but has many turns, an additional important quantity is introduced - the flux linkage magnetic field - equations
    Here w is the number of the turns. Such configuration is named in engineering a coil (winding).
    The flux linkage is used mainly, when we apply the Faraday's law of electromagnetic induction (when a voltage is induced in a coil by a time-varying magnetic field). The flux linkage is also important when the magnetic energy stored in a winding is calculated. It helps to introduce the winding inductance L.



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modeling magnetic field FEM

modeling magnetic field FEM

modeling magnetic field FEM

modeling magnetic field FEM

modeling magnetic field FEM

modeling magnetic field FEM

modeling magnetic field FEM

modeling magnetic field FEM




ELECTROMAGNETIC FIELD COMPUTATION AND MODELLING :
UNIT I INTRODUCTION
Review of basic field theory - electric and magnetic fields - Maxwell's equations - Laplace, Poisson and Helmoltz equations - principle of energy conversion - force/torque calculation - Electro thermal formulation.
UNIT II SOLUTION OF FIELD EQUATIONS
Limitations of the conventional design procedure, need for the field analysis based design, problem definition , solution by analytical methods-direct integration method - variable separable method - method of images, solution by numerical methods- Finite Difference Method.
UNIT III SOLUTION OF FIELD EQUATIONS II
Finite element method (FEM) - Differential/integral functions - Variational method - Energy minimization - Discretisation - Shape functions -Stiffness matrix -1D and 2D planar and axial symmetry problem.
UNIT IV FIELD COMPUTATION FOR BASIC CONFIGURATIONS
Computation of electric and magnetic field intensities- Capacitance and Inductance - Force, Torque, Energy for basic configurations.
UNIT V DESIGN APPLICATIONS
Insulators- Bushings - Cylindrical magnetic actuators - Transformers - Rotating machines.

REFERENCES :
1. K.J.Binns, P.J.Lawrenson, C.W Trowbridge, "The analytical and numerical solution of Electric and magnetic fields", John Wiley & Sons, 1993.
2. Nathan Ida, Joao P.A.Bastos , "Electromagnetics and calculation of fields", Springer-Verlage, 1992.
3. Nicola Biyanchi , "Electrical Machine analysis using Finite Elements", Taylor and Francis Group, CRC Publishers, 2005.
4. S.J Salon, "Finite Element Analysis of Electrical Machines." Kluwer Academic Publishers, London, 1995, distributed by TBH Publishers & Distributors, Chennai, India.
5. User manuals of MAGNET, MAXWELL & ANSYS software.
6. Silvester and Ferrari, "Finite Elements for Electrical Engineers" Cambridge University press, 1983.



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