hit)

Fig, J4.iO TransfoE [псг wiivelbmis, i(t) t PWM iLill-brJilgtf tonvci-te) схщпрЬ.

0.51

0.5/

27, t

be allocated to each half of the center-tapped secondary. The total copper loss at this optimal design point is found from evaluation ofEq. (14.34):

р(л-/;-Т)

(14.421

1 +2rt-l-2vD(.l + D)

14.3.2 Coupled Inductor Design Cunstraints

Let us now consider how to design a A-winding coupled inductor, as discussed in Section 13.5.4 and illustrated in Fig. 14.11. It is desired that the magnetizing inductance be a specified value L. referred to winding 1. h is also desired that the numbers of turns for the other windings be chosen according to desired turns ratios. When the magnetizing current i/{t) reaches its maximum value /f, , . the coupled inductor should operate with a given maximum flnx density With rms currents /j,.,., applied to the respective windings, the total copper loss should he a desired value given by Eq. (14.34). Hence, the design procedure involves selecting the core size and number of primary turns so that the desired magnetizing inductance, the desired flux density, and the desired total copper loss are achieved. Other quantities, such as air gap length, secondary turns, and wire sizes, can then be selected. The derivation follows the derivation for the single winding case (Section 14.1), and incorporates the window area optimization of Section 14.3.1.

The magnetizing current /(/) can be expressed in terms of the winding cun-ents /(0> ijW,--. /ДО by solution of Fig. 14.11 (a) (or by u.se of Amperes Law), as follows:

iM(o = (i(o + -;,(() + + -/ДО

1 Hi

(14.43)

By solution of the magnetic circuit model of Fig. 14.1 1(b), wecan write

14.3 Miiltiple-Windm!; Magnelicji Design via the KMeilwd 551

3 b<.t)

ЛЛ/-

; tt

Fig, 14.11 A t-winding magnetit device, with specified turns rati ind waveforms: (a) electrical circuit model, (h) a magnetic circuit model.

(14.44)

This equatitm is aiialogt>us tt) Eq. (14.4), and assumes that the reluctance ofthe air gap is much larger than the rcluctatiec r:Jf.of the core. As usual, the total flux Ф(0 is equal to lf(f)A. Leakage inductances cire ignored.

To avoid saturation of the core, the instantaneous flux density B(t) must be less than the saturation flux density ofthe core tnaterial, 1/,,. Let us define l/f, as the niaxiniitm value ofthe magnetizing current According toEq. (14.44), this will lead to a maximum fiuxdensity й, givenby

(14.45)

For a value of lja given by the circuit application, we should use Bq. (14.45) to choose the turns tii and gap length such that the maximum fiuxdensity is less than the saturation density Й, ,. Equation (14.45) is similar to Eq. (14.6), but acct>unts for the magnetizations produced by multiple winding currents.

Tlie magnetizing inductance £,д, referred to winding 1, is equal to

(14.46)

This equation is analogous to Eq. (14.7).

As shown in Section 14.3.1, the total copper loss is minitnized when the core window area is allocated to the various windings according to Eq. (14.35) or (14.36). The total copper loss is then giveti by Eq. (14.34). Equation (14.34) can be expressed in the form

(14.47)

S52 Inductor Design where

(14,43)

is tlie sura ot the rms winding currents, referred to winding 1.

We can now eiiminate the unlcnownquantities and n, from Eqs. (14.45), (14.46), and (14.47). Equation (14.47) then becomes

We can now reiurange this equation, by grouping terras that involve the core georaetry on the left-hand side, and specitlcations on the right-hand side:

AlWPiA (14.50)

The left-hand side of the equation can be recognized as the same A , term defined in Eq. (14.15), Tliere-fore, to design a coupled inductor that meets the requirements of operating with a given maxiraura tlux density C i,(j, given primary inagnetizing inductanceL, and with a given total copper lossPj, we must .select a core that satisfies

f. PiSiii (14.51)

* a } p

Once such a core is found, then the winding 1 turns and gap length can be selected to satisfy Eqs. (14.45) and (14.46). The turns of windings 2 through it are selected according to the desired turns ranos. The window area is allocated araoiig the windings according to Eq. (14.35), and the wire gauges are cho.sen using Eq. (14.27).

The procedtue above is applicable to design of coupled inductors. The resuits are applicable to design of flyback and SEPIC transforraers as well, although it should be noted that the procedtue does not account for the effects of core or proximity loss. It also can he extended to design of other devices, such as conventional transformers-doing so is left as a homework problem.

14.33 Design Procedure

The following quantities are specified, using the units noted:

Wire effective resistivity p (Q-cm)

Total rms winding currents, referred lo winding I

Peak magnetizing currenl, referred lo winding 1 M.miu (A)

Desired turns ratios г !- h\>