1 Introduction - Finite Element Method Magnetics

purposes, like determining the RMS amplitude of a periodic flux distribution.
5.10 Block Integrals
Once a closed contour has been specified in Block mode and the block appears highlighted in
green, Block Integrals over the specified area. These integrals are performed by analytically integrating
the specified kernel over each element in the defined region, and summing the results for
all elements.
To perform an integration, press the “integral” icon on the toolbar (as shown in Figure 22). A
small dialog will appear with a drop list. Choose the desired integral from the drop list and press
OK. Generally, volume integrals take several seconds to evaluate, especially on dense meshes. Be
patient. When the evaluation of the integral is completed, the answer appears on the screen in a
pop-up box.
The block integrals currently supported are:
• A.J This integral is performed to evaluate inductance for linear problems. Generally, the
self-inductance of a coil is:
A · JdV
Lsel f =
(23)
where i is the current flowing through the coil.
• A This integral can be used to evaluate mutual inductances between coils. Similar to the
formula for self inductance, mutual inductance is:
A1 · J2dV2
(24)
Lmutual =
where A1 is the component of A produced by the first coil, J2 is the current in the second
coil, and i1 and i2 are the current in the first and second coils, respectively. dV2 is meant to
denote that the integral is taken over the volume of the second coil. We can rearrange (24)
into a somewhat simpler form by noting that n2 ∗ i2 = J2 ∗ a2. That is, the total amps times
turns for the second coil equals the current density in the second coil times the second coil’s
cross-section area. Substituting for J2 in (24) yields:
Lmutual = n2
i1a2
i 2
i1i2
A1dV2 −
J2+
A1dV2
J2−
where the first bracketed term in (25) is the contribution from the turns of coil 2 that are
pointed out of the page and the second term is the contribution from the turns of coil 2 that
are pointed into the page. To evaluate mutual inductance with FEMM, one substitutes values
into (25). First, run the model with only “coil 1” turned on. Then, integrate A over the
volume in which the second coil lies (although the second coil is not turned on). For planar
problems, you will typically have to make two separate integrations–one over the region
where the turns in “coil 2” are pointed out of the page (i.e. that part of the coil in which
a positive current results in current flowing in the out-of-the-page direction), and one over
the region in which the turns in “coil 2” are pointed into the page. Add these two results
together to get the total A1dV2 integral. Lastly, multiply the integral result times n2/(i1a2) to
get mutual inductance.
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(25)