page_title: Toroidal inductors and transformers
section_title: Toroidal inductor/transformer and magnetic vector potential
hierarchical_section_title: Toroidal inductors and transformers / Total B field confinement by toroidal inductors / Toroidal inductor/transformer and magnetic vector potential
caption_attribution_description: English: A depiction of the Magnetic Vector Potential of a Symmetric toroidal Inductor
context_page_description: Toroidal inductors and transformers are inductors and transformers which use magnetic cores with a toroidal shape. They are passive electronic components, consisting of a circular ring or donut shaped magnetic core of ferromagnetic material such as laminated iron, iron powder, or ferrite, around which wire is wound.
  Although in the past, closed-core inductors and transformers often used cores with a square shape, the use of toroidal-shaped cores has increased greatly because of their superior electrical performance. The advantage of the toroidal shape is that, due to its symmetry, the amount of magnetic flux that escapes outside the core is low, therefore it is more efficient and thus radiates less electromagnetic interference.
  Toroidal inductors and transformers are used in a wide range of electronic circuits: power supplies, inverters, and amplifiers, which in turn are used in the vast majority of electrical equipment: TVs, radios, computers, and audio systems.
context_section_description: See Feynman chapter 14 and 15 for a general discussion of magnetic vector potential.  See Feynman page 15-11  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 .  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, 
  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  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.  In fact 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.
  Since the equations ,  and    (assuming quasi-static conditions, i.e.  )  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.
caption: Showing the development of the magnetic vector potential around a symmetric toroidal inductor.
exif: {}