Fig. 13.28 MMF diagram for llie iranafomier winding exiimple ofFigs. 13.2Й ii.ic! 13.27.

Primary winding

® ® ® ® ® ®

Secondary wi?iding

® ®

It should be noted that the shape of the Щл) eurve in the vicinity of the winding conductors depends on the distribution of the current within the conductors. Since this distribution is not yet known, the . /(л)сигуе of Fig. 13.28 is arbitrarily drawn as straight line segments.

In general, the magnetic fields that surround conductors and lead to eddy currents must be deternuned using finite element analysis or other similar methods. However, in a large class of coaxial solenoidal winding geometries, the magnetic field lines are nearly parallel to the winding layers. As shown below, we can then obtain an analytical solution for the proximity losses.

13.4.3 Foil Windings und Layers

The winding symmetry described in the previous section allows simplification of the analysis. For the purposes of determining leakage inductance and winding eddy currents, a layer consisting of turns of round wire carrying current /(f) can be approximately modeled a.s an effective single turn of foil, which carries current rjfi(f). The steps in the transformation of a layer of round conductors into a foil conductor are formalized in Fig. 13.29 [6, 8-11]. The round conductors are replaced by square conductors having the same copper cross-sectional area. Fig. 13.29(b). The thickness li of the square conductors is therefore

Fig. 13.29 Approximating a layer of round О

conductois as an effective foil conductoi, Г ]

□

equal to the bare copper wire tJiaraeter, multiplied by the factor 7ft/4 :

(13.70)

These square conductors are then joined together, into a foil layer [Fig. 13.29(c)]. Finally, the width of the foil is increased, such that it spans the width of the core window [Fig, 13.29(d)], Since this stretching process increases the c[)nduct[)r cr[)ss-secti[)nal area, a compensating factor r) must be intrtxiuced such that the correct dc conductor resistance is predicted. This factor, sometimes called the conductor .ipacing factor or the winding porosity, is defined as the ratio of the actual layer copper area [Fig. 13.29(a)] to the area ofthe effective foil conductor of Fig. 13.29(d). The porosity effectively increases theresistivity p of the conductor, and thereby increases its slcin depth:

(13.71)

If a layer of width (. contains П( turns of round wire having tiiameter d, then the winding porosity r is given by

(13.72)

A typical value oft) for round conductors that span the width of the winding bobbin is 0.8. In the following analysis, the factor ф is given by /l/S for foil conductors, and by the ratio ofthe effective foil conductor thickness Il to the effective slrin depth 5 for round conductors as follows:

(13.73)

13.4.4 Power Loss in a Layer

In this section, the average power kjss P iu a uniform layer of thickness h is determined. As illustrated in Fig. 13.30, the magnetic field strengths on the left and right sides of the conductor are denoted Л(0) and H{d), respectively. It is assumed that the component of magnetic field normal tt) the conducttjr surface is zero. These magnetic fields are driven by the magnetomotive forces IH) and -i(/0, respectively. Sinusoidal waveforms are assumed, and rms magnitudes are employed. It is further assumed here that H(0) and H(l!) are in phase; the effect of a pha.se shift is treated in [10].

With these assumptions. Maxwells equations are solved to find the current density distribution in the layer. The power loss density is then computed, and is integrated over the volume of the layer to find the total copper loss in the layer [10]. The result is

.jt\h) + ,)f{0))G,(tp)-4 ,ЯА).Я0)о2(ф)

(i3.74)

я(0)

Hih)

Fig. 13,30 Tlic power loss is ileter-mined for a uniform layer. Uniform ti aential maanetic helds НЩ and (/() are applied to tlie layer surfaces.

where П( is the number of turns in the layer, and is the dc resistance of the layer. The functions G,(<p} and 0,(ф) are

sinh [Щ + sin (2ip) cosh (2ф)-со5 (2(p)

sinh (ф) cos (<p) + cosh (ф) sin (ф) - cosh (2<pb cos [2ф)

(□.75)

If the winding carries current of rms magnitude /, then we can write

Let us further express .¥{fl) in terms of the winding current /, as

.A(h) = m,l (13.77)

The quantity ni is therefore the ratio of the MMF (,h) to the layer ampere-turns rifl. Then,

(13.73)

The power dissipated in the layer, Eq. (13.74), can then be written where

0(Ф.т) = {2(иЗ-2)м+ l)C(({i)-4Hi(!n-l)Gj(C( ) tl3,aO)

We can conclude that the proximity effect increases the copper loss in the layer hy the factor

у = Фа((р, 1) (13.81)

Equation (13.81), in conjunction with the definitions (13.80), (13.77), (13.75), and (13.73), can be plotted using a computer spreadsheet or small computer program. The result is illustrated in Fig. 13.31, for several values of ni.

It is illuminating to express the layer copper loss P in terms of the dc power loss P.l [ that would be obtained in a foil conductor having a thickness ф = l.This loss is found hy dividing Eq. (13.81) by the effective thickness ratio (p:

- = е(Ф. ) (13.32)

Equation (13.82) is plotted in Fig. 13.32. Large copper loss is obtained for small tp simply because the layer is thia and hence the dc resistance of the layer is large. For large m and large rp, the proximity effect leads to large power loss; Eq. (13.66) predicts that Q{<p, m) is asymptotic to + (m - 1) for large (p. Between these extremes, there is a value of ф which minimizes the layer copper loss.