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 | 10−1 | 10−2 |
| Encoding | log | log | log | log | log | log |
| Chromosome | 1 | 1 | 1 | 1 | 1 | 1 |
Figure 1.23 Single‐objective optimization study.
The next step in the design process is to determine the parameter space Ω. This is tabulated in Table 1.7. Some level of engineering estimation is required to select a reasonable range. However, situations where a range is incorrectly set are usually easy to detect by looking at the population distribution. We will return to this point.
We have now set forth a fitness function and a domain for the parameter vector, and so we can proceed to conduct an optimization. We will begin with a single‐objective case. To conduct this study, a MATLAB‐based genetic optimization toolbox known as GOSET was used. This open‐source code and the code for this particular example are available