in our design are the number of turns N, the slot depth ds, the slot width ws, the core thickness wc, the core depth lc, and the air gap g. Thus, our parameter vector may be expressed as
In (1.10-1), N* is the desired number of turns rather than the actual number of turns N because, as a design parameter, we will let the number of turns be represented as a real number rather than an integer. This is because for large N this variable acts in a more continuous rather than discrete fashion. The actual number of turns is calculated from the desired number as
In order to perform the optimization, we will need to analyze the device. It is assumed that windings occupy the entire slot, that the core is infinitely permeable, and that the fringing and leakage flux components are negligible. Again, for the reader unfamiliar with these terms, the analysis can be taken as a set of arbitrary mathematical equations; we will spend the rest of the book defining and developing more accurate expressions for these quantities.
With these assumptions, the mass of the design may be expressed as
In (1.10-3), ρmc and ρwc denote the mass density of the magnetic core and wire conductor, respectively, and kpf is the fraction of the U‐core window occupied by conductor. Ideally, it would be 1, but 0.7 is a very high number in practice.
The next step is the computation of loss. The power dissipation of the winding at rated current may be expressed as
In (1.10-4), σwc denotes the conductivity of the wire conductor.
There are constraints both on the inductance, flux density at rated current, and current density at rated current. These quantities may be expressed as
In (10.1‐5) and (1.10-6), μ0 is the magnetic permeability of free space, a constant equal to 4π10−7 H/m.
In order to formulate a fitness function, expressions (1.10-1)–(1.10-7) can be sequentially evaluated. Then constraint functions can be evaluated as
(1.10-8)
(1.10-9)
(1.10-10)
(1.10-11)
(1.10-12)
Keeping with (1.9-4), we find the aggregate constraint
(1.10-13)
We will consider both single‐ and multi‐objective optimization. For the single‐objective case, we will minimize mass and our fitness is given by
For the multi‐objective case, the fitness function will be taken as
In (1.10-14) and (1.10-15), we will take ε = 10−10.
For our design, let us consider a ferrite material for the core with Bmx = 0.617 T and ρmc = 4680 kg/m3, and consider copper for the wire with ρwc = 8890 kg/m3 and Jmx = 7.5 A/mm2. We will take rated current to be 10 A and take the minimum inductance Lmn to be 1 mH. Finally, let us take the maximum allowed mass as Mmx = 1kg, and the maximum allowed loss to be Pmx = 1W.
Table 1.7 Domain of Design Parameters
| Parameter | N | ds (m) | ws (m) | wc (m) | lc (m) | g (m) |
|---|---|---|---|---|---|---|
| Min. value | 1 | 10−3 | 10−3 | 10−3 | 10−3 | 10−5 |
| Max. value | 103 | 10−1 | 10−1 |
10−1
|
