ОСф) tends to 0. It can be verified that Fg then tends to the value predicted hy Eq. (13.68).

We can again express the total primary power loss in terms of the dc power loss that would he obtained using a conductor in which ф = 1. This loss is found by dividing Eq. (13.S6) by ф:

- =С,(гр) + (м=-1](о,(ф)-20г(ф))

(i3.87)

This expression is plotted in Fig. 13.35, for several values of M. Depending on the number of layers, the minimum copper loss for sinusoidal excitation is obtained forCp near to, or sotnewhat less than, unity.

13,4,6 Interleaving the Windings

One way to reduce the copper losses due to the proximity effect is to interleave the windings. Figure 13.36 illustrates the MMF diagram for a simple transformer in which the primary and secondary layers are alternated, with net layer current of magnitude i. It can be seen that each layer operates with 0 on one side, and i = j on the other. Hence, each layer operates effectively with in 1. Note that Eq. (13.74) is symmetric with respect to /(0) and (ft); hence, the copper losses of the interleaved secondary and primary layers are identical. The proximity losses of the entire winding can therefore be determined directly from Fig. 13.34 and 13.35, with M = 1. ll can be shown that the minimum copper loss for this case (with sinusoidal currents) occurs with ф =7t/2, although the copper loss is nearly constant for any ф> 1, and is approximately equal to the dc c[)pper loss obtained when ф = 1, It should be apparent that interieaving can lead to significant improvements in copper loss when the winding contains several layers.

Partial interleaving can lead to a partial improvement in proximity loss. Figure 13.37 illustrates a transformer having three primary layers and four secondary layers. If the total current carried by each primary layer is f(f), then each secondary layer should carry current 0.75f(f). The maximutn MMF again occurs in the spaces between the primary and secondary windings, but has value 1.5i(f).

To determine the value for m in a given layer, we can solve Eq. (13.7S) for in:

(13.88)

3i li

®

®

®

sec -f

®

sec -I

Fig. 13.36 MMF diagram for a simple transformer with interleaved windings. Each layer uperotes with hi = 1,

Secondary

Primary

Secondary

1.5/

0.5 1 0

-0.5 I -1 -1.5;

Fig. 13J7 A ptutially interleaved iwo-winding transformer, illustrating fractional values of m. The MMF diagram is constructed for the low-frequency limit.

The above expression is valid in general, and Eq. (13.74) i.s .symrnetrical in ./(0) and ./(A). However, when ДО) is greater in magnitude than (ft), it is convenient to interchange the roles of /(0) and (/l), .40 that the plots of Figs. 13.31 atid 13.32 can be employed.

In the leftmost secondary layer of Fig. 13.37, the layer carries current - 0.75г. The MMF changes from 0 to - 0.75l The value ofm for this layer is found by evaluation ofEq. (13.88):

nil)

-0,75i

.9{Щ-.9(Щ -0.751-0

(13.39)

The loss in this layer, relative to the dc loss of this secondary layer, can be determined using the plots of Fig.s. 13.31 and 13.32 with \n = 1. For the next .secondary layer, we obtain

l.5i

.*(/!)-.if(0) . i.5i-(-0.751)

(13. ))

Hence the loss in this layer can be determined using the plots of Figs. 13.31 and 13.32 with \n = 2.The next layer is a primary-winding layer. Its value of in can be expressed as

,/(0) -1.5,- - - .Ш - - t.i/ - (- 0,3i->

(13.91)

The loss in this layer, relative to the dc loss of this primary layer, can be determined using the plots of Fig.s. 13.31 and 1332 with m = 1.5. The center layer has an m of

0.5i

S(h)-.f(Q) O.Si-(-0,50

= 0.5

(13.92)

The loss in this layer, relative to the dc loss of this primary layer, can be determined using the plots of Figs. 13.31 and 13.32 with m = 0.5. The remaining layers are symmetrical to the corresponding layers described above, and have identical copper losses. The total loss in the winding is found by summing the losses descrii)ed above for each laver.

Interieavinji windings cm significantly reduce the proximity ioss when the primary and secondary currents are in phase. However, in some cases such as the transformers of the flybacl< and SEPIC converters, the winding currents are out of phase. Interleaving then does iittle to reduce the MMFs and magnetic fields in the vicinity of the windings, and hence the proximity loss is essentially unchanged. It should also be noted that Eqs. (13.74) to (13.82) asstirae that the winding currents are in phase. General expressions for out-of-phase currents, as well as analysis of a flyback example, are given in [10].

The above procedure can be used to determine the high-frequency copper losses of more complicated multiple-winding magnetic devices. The MMF diagrams are constructed, and then the power loss in each layer is evaluated using Eq. (13.81). These losses are summed, to find the total copper loss. The losses induced in electrostatic shields can also he determined. Several additional examples are given in [10].

It can be concluded that, for sinusoidal currents, there is an optimal conductor thickness in the vicinity off = 1 that leads to minimum copper ioss. It is highly advantageous to minimize the number of layers, and to interleave the windings. The amount of copper in the vicinity of the high-MMF portions of windings should he kept to a minimum. Core geometries that maximize the width of the layers, while minimizing the overall number of layers, lead to redticed proximity loss.

Use of Lilz wire is another means of increasing the conductor area while maintaining low proximity losses. Tens, hundreds, or more strands of small-gauge insulated copper wire are bundled together, and are externally connected in parallel. These strands are twisted, or transposed, such that each strand passes equally through each position inside and on the surface of the bundle. This prevents the circulation of high-frequency currents between strands. To be effective, the diameter of the strands should be sufficiently less than one skin depth. Also, it should be pointed out that the Litz wire bundle itself is composed of multiple layers. The disadvantages of Litz wire are its increased cost, and its reduced fill factor.

13.4.7 PWM Wavelorm Hurmunics

The pulse-width-raodulated waveforms encountered in switching converters contain significant harmonics, which can lead to increased proximity losses. The effect of harmonics on the losses in a layer can be determined via field harmonic analysis [10], in which the MMF waveforms HO,r) and .>(rf,/) of Eq. (13.74) are expressed in Fourier series. The power loss of each individual harmonic is computed as in Section 13.4,4, and the losses are summed to find the total loss in a layer. For example, the PWM waveform t)f Fig. 13.38 can be represented by the following Fourier series:

iil) = /f, + Ё V2 /,.cos ijw) (13.93)

where

sin urn

with Ш = 2я/Т. This waveform contains a dc component /[,= Df, plus harmonics of rms magnitude Ij proportional to Uj. The transformer winding current waveforms of most switching converters follow this Fourier series, or a similar series.

Effects of waveforms harmonics on proximity losses are discussed in [8-10]. The dc component of the winding currents does not lead to proximity loss, and should not be included in proximity ioss calculations. Failure to remove the dc component can lead to significantly pessimistic estimates of copper