Field equations in partial differential form (Electrical Machine)

By

Every electromagnetic phenomenon can be attributed to the seven basic equations, the four Maxwell equations of the electro-dynamic and those equations of the materials. The latter can be

• isotropic or an-isotropic

• linear or non-linear

• homogenous or non-homogenous.

The Maxwell equations are linked by interface conditions. Together with the material equations they form the complete set of equations describing the fields completely.

In this section the Maxwell equations, necessary for the calculation of electromagnetic fields, are discussed in their differential form. The seven equations describe the behaviour of the electromagnetic field in every point of a field domain. All electric and magnetic field vectors E, D, B, H, and J and the space charge density p are in general functions of time and space. The conducting current density can be distinguished by a material/field dependent part Jc and by an impressed and given value J0 . It is assumed that the physical properties of the material permitivity e, permeability fx and conductivity a are independent of the time. Furthermore it is assumed that those quantities are piecewise homogenous.

Three groups of equations can be distinguished:

group 1:

group 1:

group 2:

group 2:

group 3 for isotropic material:

for isotropic material:

In the literature published in different languages different operators are used. Here for the identity of the operatorstmp1074_thumb

tmp1077_thumb

is chosen. Iftmp1078_thumbare constant in a domain their position in the equations can be exchanged for the geometrical vector operators tmp1079_thumbIf they are constant in time, their position can be exchanged for the time derivativestmp1080_thumb

The three groups of equations are called the main equations, the laws of conservation and the material equations. The first equation (i), eq.(3.8) is known as the law of Faraday-Lenz with E the electric field strength and B the flux density. Eq. (ii), (3.9) is known as Ampere’s law with magnetic field strength H, conducting current density J and D the electric flux density. The termtmp1081_thumbthe displacement current density,is neglected from now on, as already argued. Equations (iii), (3.10) and (iv), (3.11) describe the constitutive properties of the magnetic flux density and the displacement current with the space charge density p. All six field quantities E, D, H, B, J and p are dependent on each other.

Motion

Electric and magnetic fields form a unit with phenomena depending on the point of view of the observer. An observer at rest looking at a moving charge perceives an electric field caused by the charge and an additional magnetic field. The observer moving with the same speed as the charge does notices only the electric field. The field quantities can be represented in different co-ordinate systems by different electric and magnetic field quantities. The Lorentz-transformation can be used to link both resting and moving systems. The field strength in a resting system x,y,z and of a uniformly in x-direction moving systemtmp1086_thumbcan be given by:

 

tmp1088_thumb

A distinction is made between the parallel and perpendicular direction of motion. The vectors for the field quantities are calculated by:

tmp1090_thumb

Using the Lorentz transformation for the charge density p’ and the vector of the conducting current density J’ for example for a particle beam moving in the x-direction with

tmp1091_thumb

it can be written

tmp1092_thumb

In a similartmp1093_thumbthe scalar potentialtmp1094_thumband vector potentialtmp1095_thumbof a moving charge in the x-direction is transformed by:

tmp10102_thumbtmp10103_thumb

In practically all electrical engineering problems of technical importance, observed phenomena concerning motion are slow in an electromagnetic sense, when the speed v is compared to the speed of light c. A possible exception is a high energy particle acceleratorin pulsed-power-technology. Observing an uniformly moving systemtmp10104_thumb from a resting co-ordinate system x,y,z and assumingtmp10105_thumbthe transformation is simplified (Schwab l01) to:

tmp10108_thumb

Applying the quantities within the appropriate co-ordinate system, the Maxwell equations remain valid. The field equations from groups 1 and 2 are Lorenz invariant. The same can be stated for the forces on charges caused by electric and magnetic fields. Forces depend on the frame of reference and can be of electric or magnetic origin. The Lorentz force is:

tmp10109_thumb

Tn contrast to the Lorentz invariance of the field equations, the materia! relation (vii), (3.14) changes for the moving system and the resting observer to:

tmp10110_thumb

and respectively for the resting system with the moving observer using

tmp10111_thumb

Interface conditions

Technical devices are constructed using piecewise homogenous materials. The boundary of such materials can be identified as a surface inhomogeneity. To consider this boundary, the associated interface conditions for the electric and magnetic field are discussed in the following section.

To derive those interface conditions, the integral form of the Maxwell equations is used.

Table 3.1. Maxwell equations for quasi-stationary fields.

differential form

integral form

(i) (ii)

tmp10-112 tmp10-113

(3.26)

(3.27)

(iii)

tmp10-114 tmp10-115

(3.28)

(iv)

tmp10-116 tmp10-117

(3.29)

Here,tmp10119_thumbis the magnetic flux, I the conducted current, Q the charge, C indicates the contour integral and S the surface integral.

Normal component Maxwell equation (iii), (3.28) in integral form is used to derive the interface conditions at the boundary of different materials for the normal component (Fig. 3.9). Using tmp10120_thumbwith n the unit vector in normal direction yields:

tmp10123_thumb Interface between material O and ©with different properties.

Fig. 3,9. Interface between material O and ©with different properties.

The surface S can be of arbitrary shape and the integral only vanishes if the integrand is zero. This yields the interface conditions for the electromagnetic field. The components of the magnetic flux density B are continuous at material boundaries even if they have different ferromagnetic properties.

tmp10125_thumb

With respect to the finite element method, this means that the normal component of the flux density must be constant at the boundary between finite elements (Fig. 3.10).

Normal component of the flux density at the interface of two triangular finite elements.

Fig. 3.10. Normal component of the flux density at the interface of two triangular finite elements.

Similar to the normal component of the magnetic flux density, the normal component of the displacement current density can be derived. Here, Maxwell equation (iv) in integral form eq.(3.29) is evaluated in the same way yielding:

tmp10127_thumb

Wheretmp10128_thumbis the space charge density, V indicates the volume integral andtmp10129_thumbthe surface charge density,

tmp10134_thumb

With the presence of a surface charge density, the normal component of the displacement current density is discontinuous at the interface of a material boundary. Without surface charge density, the normal component of the displacement current density is continuous at boundaries.

With the presence of a surface charge density, the normal component of the displacement current density is discontinuous at the interface of a material boundary. Without surface charge density, the normal component of the displacement current density is continuous at boundaries.

Normal component of the displacement current density at the interface of two triangular finite elements.

Fig. 3.11. Normal component of the displacement current density at the interface of two triangular finite elements.

To discuss the interface conditions for the conducting current density /, the Maxwell equation (ii), eq.(3.27) is considered. With (vi), (3.13), the same conditions found for the magnetic field can be applied for the current density as well:

tmp10136_thumbNormal component of the current density.

Fig. 3.12. Normal component of the current density.

Tangential component The interface conditions valid for the tangential component of the electrical field E can be derived from the integral form of the first Maxwell equation (i), (3.26).

Interface and path of integration between material O and © with different material properties.

Fig. 3.13. Interface and path of integration between material O and © with different material properties.

Applying the path of integration on the contour drawn in Fig. 3.13 and counting positive as indicated, yields:

tmp10139_thumb

It is assumed thattmp10140_thumbwith t the unit vector in tangential direction. With a finite B and w->0 the surface integral over B vanishes and this yields:

tmp10142_thumbContinuous tangential component of the electric field strength.

Fig. 3.14. Continuous tangential component of the electric field strength.

The tangential component of the electric field strength is continuous at interfaces of a boundary with different material properties.

Analogous to the electric field strength and employing the integral form of Maxwell equation (ii) eq.(3.27) yields:

tmp10145_thumb

Heretmp10146_thumbis a possible surface current density.

tmp10149_thumb

Herewith, the interface conditions for the tangential component of the magnetic field strength H can be written by:

tmp10150_thumbContinuous tangential component of the magnetic field strength

Fig. 3.15. Continuous tangential component of the magnetic field strengthtmp10152_thumb

With vanishing surface current density the tangential component of the magnetic field strength is continuous at interfaces.