Power Magnetic Devices. Scott D. Sudhoff

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СКАЧАТЬ in our design are the number of turns N, the slot depth d_s, the slot width w_s, the core thickness w_c, the core depth l_c, and the air gap g. Thus, our parameter vector may be expressed as

(1.10-1)

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

(1.10-2)

      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

(1.10-3)

In (1.10-3), ρ_mc and ρ_wc denote the mass density of the magnetic core and wire conductor, respectively, and k_pf 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

(1.10-4)

In (1.10-4), σ_wc 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

(1.10-5)

(1.10-6)

(1.10-7)

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

(1.10-14)

      For the multi‐objective case, the fitness function will be taken as

(1.10-15)

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 B_mx = 0.617 T and ρ_mc = 4680 kg/m³, and consider copper for the wire with ρ_wc = 8890 kg/m³ and J_mx = 7.5 A/mm². We will take rated current to be 10 A and take the minimum inductance L_mn to be 1 mH. Finally, let us take the maximum allowed mass as M_mx = 1kg, and the maximum allowed loss to be P_mx = 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 СКАЧАТЬ В начало < 26 27 28 29 30 31 32 33 34 > В конец