Section 4.2 · Worked examples · 1 of 3
Worked example 4.2.1: Toroidal inductor and its air gap
Three short calculations tied to the applications. They use only , and the Lorentz force , and no new physics. Numerical constants: , . A good way to use them: read the given quantities, produce the formula from the corresponding application or problem yourself, and only then put the numbers in.
Toroidal inductor and its air gap
#cf. Application 4.2.1
Given. A toroid, turns, mean radius (so ), core cross-section . Core: Mn–Zn ferrite, , saturation .
Find. (a) with an air core and with the ferrite, (b) the current at which the ferrite saturates and the energy stored there, (c) and after cutting a gap , (d) the energy then.
Ampère's law round the mean circumference gives for the closed core. With a short gap the same flux ( continuous, ) must cross of ferrite with and of air with , which is Eq. (4.2.23) and Eq. (4.2.24):
With , :
reaches at
, so :
.
Check (a) to (d) with the two presets and find where the energy sits.
Units: is (H/m)(m)/m H. Figure 4.2.11 shows for this inductor. The gap divides by and multiplies by the same , so the energy that can be stored before saturation grows times.
Where does the energy sit? The energy density at the same is times larger in the gap than in the ferrite: in a gapped inductor the gap is the energy store, and no longer depends on the poorly controlled of the ferrite. This is why every inductor of a switch-mode power supply has a gap.