Magnetic vector potential (Electrical Machine)

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Potentials and formulations

The Maxwell equations represent the physical properties of the fields. To solve them, mainly the differential form of the equations and mathematical functions, the potentials, satisfying the Maxwell equations, are used. The proper choice of a potential depends on the type of field problem. In this section, the various scalar and vector potentials are introduced.

The electric vector potential for the displacement current density will not be introduced here, because it is only important for the calculation of fields in charge-free and current-free regions such as hollow wave-guides or in surrounding fields of antennas.

Various potential formulations are possible for the different field types. Their appropriate definition ensures the accurate transition of the field problem between continuous and discrete space.

Using these artificial field quantities reduces the number of differential equations. Considering a problem described by n differential equations, a potential is chosen in such a way that one of the differential equations is fulfilled. This potential is substituted in all other differential equations, the resulting system of differential equations reduces to n-l equations. It is distinguished between magnetic and electric vector respectively scalar potentials:

Table 4.1. Definition of the potentials.

scalar

potentials

vector

electric

tmp10-155 tmp10-156 tmp10-157

magnetic

tmp10-158 tmp10-159 tmp10-160

By using the vector identity

tmp10161_thumb

and applying eq.(iii), (3.10), introduces the magnetic vector potential A. The magnetic flux density is derived as the curl of another vector field:

tmp10162_thumb

The magnetic vector potential is suitable in regions with and without conducting currents. The vector field A is assigned right handed to the direction of the magnetic field B (Fig. 4.1).

Geometrical assignment of the vector potential A with the magnetic field vector B.

Fig. 4.1. Geometrical assignment of the vector potential A with the magnetic field vector B.

A static magnetic problem is described by

tmp10164_thumb

By using the magnetic vector potential A, the system of differential equations is reduced to

tmp10165_thumb

Applying the vector calculustmp10166_thumbto eq.(4.4) yields:

tmp10169_thumb

withtmp10170_thumband by assuming a constant permeabilitytmp10171_thumbleads to the A-formulation of a magneto-static field, a Poisson equation:tmp10176_thumb

To consider quasi-stationary fields, for example necessary for eddy current calculations, the magneto-dynamic formulations have to be employed. In addition to Ampere’s law, the Faraday law (i), (3.26) has to be considered to evaluate the contribution to the field by the eddy currents:

tmp10177_thumb

Now employing Ohm’s law to calculate the eddy currents Je yields:

tmp10178_thumb

Ampere’s law can now be rewritten, yielding the A-formulation for the quasi-stationary magnetic field in the time domain:

tmp10179_thumb

Substituting againtmp10180_thumband assuming

tmp10181_thumbresults in a similar A-formulation in the time domain for the transient magnetic field:

tmp10186_thumb

Assuming sinusoidal excitation currents with an angular frequency tmp10187_thumband thus substituting

tmp10190_thumb

yields the A-formulation in the frequency domain to solve eddy current problems.

tmp10191_thumb

This equation is the A-formulation to describe time-harmonic problems. The time dependent components of the vector potential

tmp10192_thumb

are expressed by:

tmp10193_thumb

The current is expressed in analogy in its complex representation.