Sufficient conditions for total internal confinement of the B field

Fig. 1. Coordinate system. The Z-axis is the nominal axis of symmetry. The X-axis chosen arbitrarily to line up with the starting point of the winding. ρ is called the radial direction. θ is called the circumferential direction.

Fig. 2. An axially symmetric toroidal inductor with no circumferential current.

The absence of circumferential current ^[4] (the path of circumferential current is indicated by the red arrow in figure 3 of this section) and the axially symmetric layout of the conductors and magnetic materials ^[4][5][6] are sufficient conditions for total internal confinement of the B field. (Some authors prefer to use the H field). Because of the symmetry, the lines of B flux must form circles of constant intensity centered on the axis of symmetry. The only lines of B flux that encircle any current are those that are inside the toroidal winding. Therefore, from Ampere's circuital law, the intensity of the B field must be zero outside the windings.^[6]

Fig. 3. Toroidal inductor with circumferential current

Figure 3 of this section shows the most common toroidal winding. It fails both requirements for total B field confinement. Looking out from the axis, sometimes the winding is on the inside of the core and sometimes on the outside of the core. It is not axially symmetric in the near region. However, at points a distance of several times the winding spacing, the toroid does look symmetric.^[7] There is still the problem of the circumferential current. No matter how many times the winding encircles the core and no matter how thin the wire, this toroidal inductor will still include a one coil loop in the plane of the toroid. This winding will also produce and be susceptible to an E field in the plane of the inductor.

Figures 4-6 show different ways to neutralize the circumferential current. Figure 4 is the simplest and has the advantage that the return wire can be added after the inductor is bought or built.

Fig. 4. Circumferential current countered with a return wire. The wire is white and runs between the outer rim of the inductor and the outer portion of the winding.

Fig. 5. Circumferential current countered with a return winding.

Fig. 6. Circumferential current countered with a split return winding.

E field in the plane of the toroid

Fig. 7. Simple toroid and the E-field produced. ±100 Volt excitation assumed.

Fig. 8. Voltage distribution with return winding. ±100 Volt excitation assumed.

There will be a distribution of potential along the winding. This can lead to an E-Field in the plane of the toroid and also a susceptibility to an E field in the plane of the toroid, as shown in figure 7. This can be mitigated by using a return winding, as shown in Figure 8. With this winding, each place the winding crosses itself; the two parts will be at equal and opposite polarity, which substantially reduces the E field generated in the plane.

Toroidal inductor/transformer and magnetic vector potential

Showing the development of the magnetic vector potential around a symmetric toroidal inductor.

See Feynman chapter 14^[8] and 15^[9] for a general discussion of magnetic vector potential. See Feynman page 15-11 ^[10] for a diagram of the magnetic vector potential around a long thin solenoid which also exhibits total internal confinement of the B field, at least in the infinite limit.

The A field is accurate when using the assumption ${\displaystyle bf{A}=0}$. This would be true under the following assumptions:

• 1. the Coulomb gauge is used
• 2. the Lorenz gauge is used and there is no distribution of charge, ${\displaystyle \rho =0\,}$
• 3. the Lorenz gauge is used and zero frequency is assumed
• 4. the Lorenz gauge is used and a non-zero frequency that is low enough to neglect ${\displaystyle {\frac {1}{c^{2}}}{\frac {\partial \phi }{\partial t}}}$ is assumed.

Number 4 will be presumed for the rest of this section and may be referred to the "quasi-static condition".

Although the axially symmetric toroidal inductor with no circumferential current totally confines the B field within the windings, the A field (magnetic vector potential) is not confined. Arrow #1 in the picture depicts the vector potential on the axis of symmetry. Radial current sections a and b are equal distances from the axis but pointed in opposite directions, so they will cancel. Likewise, segments c and d cancel. All the radial current segments cancel. The situation for axial currents is different. The axial current on the outside of the toroid is pointed down and the axial current on the inside of the toroid is pointed up. Each axial current segment on the outside of the toroid can be matched with an equal but oppositely directed segment on the inside of the toroid. The segments on the inside are closer than the segments on the outside to the axis, therefore there is a net upward component of the A field along the axis of symmetry.

Representing the magnetic vector potential (A), magnetic flux (B), and current density (j) fields around a toroidal inductor of circular cross-section. Thicker lines indicate field lines of higher average intensity. Circles in the cross-section of the core represent B flux coming out of the picture. Plus signs on the other cross-section of the core represent B flux going into the picture. Div A = 0 has been assumed.

Since the equations ${\displaystyle \nabla \times \mathbf {A} =\mathbf {B} \ }$, and ${\displaystyle \nabla \times \mathbf {B} =\mu _{0}\mathbf {j} \ }$ (assuming quasi-static conditions, i.e. ${\displaystyle {\frac {\partial E}{\partial t}}\rightarrow 0}$ ) have the same form, then the lines and contours of A relate to B like the lines and contours of B relate to j. Thus, a depiction of the A field around a loop of B flux (as would be produced in a toroidal inductor) is qualitatively the same as the B field around a loop of current. The figure to the left is an artist's depiction of the A field around a toroidal inductor. The thicker lines indicate paths of higher average intensity (shorter paths have higher intensity so that the path integral is the same). The lines are just drawn to look good and impart general look of the A field.

Toroidal transformer action in the presence of total B field confinement

The E and B fields can be computed from the A and ${\displaystyle \phi \,}$ (scalar electric potential) fields

${\displaystyle \mathbf {B} =\nabla \times \mathbf {A} .}$^[11] and  :${\displaystyle \mathbf {E} =-\nabla \phi -{\frac {\partial \mathbf {A} }{\partial t}}}$ ^[11] and so even if the region outside the windings is devoid of B field, it is filled with non-zero E field.

The quantity ${\displaystyle {\frac {\partial \mathbf {A} }{\partial t}}}$ is responsible for the desirable magnetic field coupling between primary and secondary while the quantity ${\displaystyle \nabla \phi \,}$ is responsible for the undesirable electric field coupling between primary and secondary. Transformer designers attempt to minimize the electric field coupling. For the rest of this section, ${\displaystyle \nabla \phi \,}$ will assumed to be zero unless otherwise specified.

Stokes theorem applies,^[12] so that the path integral of A is equal to the enclosed B flux, just as the path integral B is equal to a constant times the enclosed current

The path integral of E along the secondary winding gives the secondary's induced EMF (Electro-Motive Force).

${\displaystyle \mathbf {EMF} =\oint _{path}\mathbf {E} \cdot {\rm {d}}l=-\oint _{path}{\frac {\partial \mathbf {A} }{\partial t}}\cdot {\rm {d}}l=-{\frac {\partial }{\partial t}}\oint _{path}\mathbf {A} \cdot {\rm {d}}l=-{\frac {\partial }{\partial t}}\int _{surface}\mathbf {B} \cdot {\rm {d}}s}$

which says the EMF is equal to the time rate of change of the B flux enclosed by the winding, which is the usual result.

Toroidal transformer Poynting vector coupling from primary to secondary in the presence of total B field confinement

In this figure, blue dots indicate where B flux from the primary current comes out of the picture and plus signs indicate where it goes into the picture.