handbook of modern sensors

3.3 Magnetism 55
3.3.3 Toroid
Another useful device that can produce a magnetic field is a toroid (Fig. 3.14B), which
we can describe as a solenoid bent into the shape of a doughnut. A calculation of the
magnetic field inside the toroid gives the following relationship:
B = µ 0 i 0 N
2π r , (3.32)
where N is the total number of turns and r is the radius of the inner circular line where
the magnetic field is calculated. In contrast to a solenoid, B is not constant over the
cross section of a toroid. Also, for an ideal case, the magnetic field is equal to zero
outside a toroid.
The density of a magnetic field, or the number of magnetic lines passing through
a given surface, is defined as the magnetic flux B for that surface:
∫
B = B dS. (3.33)
The integral is taken over the surface, and if the magnetic field is constant and is
everywhere at a right angle to the surface, the solution of the integral is very simple:
B = BA, where A is the surface area. Flux, or flow of the magnetic field, is analogous
to the flux of electric field. The SI unit for magnetic flux, as follows from the above,
is tesla meter 2 , to which is named weber. It is abbreviated as Wb:
1Wb = 1T m 2 . (3.34)
3.3.4 Permanent Magnets
Permanent magnets are useful components for fabricating magnetic sensors for the
detection of motion, displacement, position, and so forth. To select the magnet for
any particular application, the following characteristics should be considered:
• Residual inductance (B) in gauss—how strong the magnet is
• Coercive force (H ) in oersteds—how well will the magnet resist external demagnetization
forces
• Maximum energy product, MEP, (BH ) is gauss oersteds times 10 6 . A strong
magnet that is also very resistant to demagnetization forces has a high MEP.
Magnets with a higher MEP are better, stronger, and more expensive.
• The temperature coefficient in %/ ◦ C shows how much B changes with temperature.
Magnets are produced from special alloys (see Table A.6). Examples are rare
earth (e.g., samarium)–cobalt alloys. These are the best magnets; however, they are
too hard for machining and must be ground if shaping is required. Their maximum
MEP is about 16 × 10 6 . Another popular alloy is Alnico, which contains aluminum,
nickel, cobalt, iron, and some additives. These magnets can be cast or sintered by