# Magneto Static: solved equations (scalar model)

## Introduction

The scalar model is a general model proposed for 3D applications.

This model is used by default (automatically) by the 3D solver of Flux 3D to solve 3D applications.

## Equation solved in the scalar model (3D)

The equation allows introducing different magnetic scalar potentials ϕ, such as: . The term is the non-rotational part of the field ( ), while the term can be rotational or null.

can be chosen in different ways and a different choice of leads to a different scalar potentials. Flux 3D has different magnetic formulations that use different scalar potentials. To each region of the problem a formulation is associated and, consequently, an equation.

The general form of the equation solved in scalar model by the finite elements method in a magnetostatic application is written:

where:

• r] is the tensor of the relative permeability of the medium
• μ0 is the permeability of the vacuum; μ0 = 4 π 10-7 (in H/m)
• ϕ is a magnetic scalar potential (in A); there are two potentials written ϕtot and ϕred
• is a term corresponding to sources

(field source or electric vector potential (in A/m)

• is the remanent magnetic flux (permanent magnets) (in T)

## The proposed formulations…

The three formulations proposed in the scalar model correspond schematically to the following three situations:

• there are no current sources
• the current sources are of non-meshed type
• the current sources are of meshed type

The solved equations in these three situations are presented below.

Separately used, each of these formulations has significant limitations; used in a coupled manner, they form an efficient general model.

## The scalar total potential (3D)

The total magnetic scalar potential , ϕtot, is used when there are no current sources. The current density is null. This case corresponds to in the general equation of the scalar model.

The intensity of the magnetic field has the expression:

and the solved equation is:

The state variable is the total magnetic scalar potential: ϕtot

This variable is written V1 in Flux.

## The reduced scalar potential with respect to Hj (3D)

The reduced magnetic scalar potential with respect to Hj, ϕredHj, is used when the current sources are of the non-meshed type. This case corresponds to in the general equation of the scalar model, with Hj analytically computed by the Biot and Savart formula (see § Magneto Static: non-meshed sources (3D specific) ).

The magnetic field strength has the expression:

and the solved equation:

The state variable is the total magnetic scalar potential with respect to Hj: ϕredHj

This variable is written V1R in Flux.

## The reduced scalar potential with respect to T0 (3D)

The reduced magnetic scalar potential with respect to T0, ϕredTo, is used when the current sources are of the meshed type. This case corresponds to in the general equation of the scalar model. The quantity satisfies the relation , where is the current density in the meshed region of field sources.

The magnetic field strength has the expression:

and the solved equation is:

Thus, the state variables are:

• the reduced magnetic scalar potential with respect to T0: ϕredTo

This variable is also written V1R in Flux

• the electric vector potential , variable non directly accessible in Flux