9
DESIGN OF HIGH-FREQUENCY INDUCTORS AND TRANSFORMERS

9.1 INTRODUCTION

As discussed in Chapter 8, inductors and transformers are needed in switch-mode DC power supplies, where switching frequencies are in excess of 100 kHz. High-frequency inductors and transformers are generally not available off-the-shelf and must be designed based on the application specifications. A detailed design discussion is presented in [1]. In this chapter, a simple and commonly used approach called the area-product method is presented, where the thermal considerations are ignored. This implies that the magnetic component built on the design basis presented here should be evaluated for its temperature rise and efficiency, and the core and the conductor sizes should be adjusted accordingly.

9.2 BASICS OF MAGNETIC DESIGN

In designing high-frequency inductors and transformers, a designer is faced with countless choices. These include the choice of core materials, core shapes (some offer better thermal conduction whereas others offer better shielding to stray flux), cooling methods (natural convection versus forced cooling), and losses (lower losses offer higher efficiency at the expense of larger size and higher weight), to name a few. However, all magnetic design-optimization programs calculate two basic quantities from given electrical specifications:

  1. The peak flux density upper B Subscript max in the magnetic core to limit core losses, and
  2. The peak current density upper J Subscript max in the winding conductors to limit conduction losses.

The design procedure presented in this chapter assumes values for these two quantities based on the intended applications of inductors and transformers. However, they may be far from optimal in certain situations.

9.3 INDUCTOR AND TRANSFORMER CONSTRUCTION

Figures 9.1a and 9.1b represent the cross-section of an inductor and a transformer wound on toroidal cores. In Figure 9.1a, for an inductor, the same current i passes through all upper N turns of a winding. In the transformer of Figure 9.1b, there are two windings where the current i 1 in winding 1, with upper N 1 bigger cross-section conductors, is in the opposite direction to that of i 2 in winding 2 with upper N 2 smaller cross-section conductors. In each winding, the conductor cross-section is chosen such that the peak current density upper J Subscript max is not exceeded at the maximum specified current in that winding. The core area upper A Subscript c o r e in Figures 9.1a and 9.1b allows the flow of flux lines without exceeding the maximum flux density upper B Subscript max in the core.

FIGURE 9.1 Cross-sections.

9.4 AREA-PRODUCT METHOD

The area-product method, based on preselected values of the peak flux density upper B Subscript max in the core and the peak current density upper J Subscript max in the conductors, allows an appropriate core size to be chosen, as described below.

9.4.1 Core Window Area Awindow

The windows of the toroidal cores in Figures 9.1a and 9.1b accommodate the winding conductors, where the conductor cross-sectional area upper A Subscript c o n d depends on the maximal RMS current for which the winding is designed. In the expression for the window area below, the window fill factor k Subscript w in a range from 0.3 to 0.6 accounts for the fact that the entire area of the window cannot be filled, and the subscript y designates a winding, where in general, there may be more than one, as in a transformer:

upper A Subscript w i n d o w Baseline equals StartFraction 1 Over k Subscript w Baseline EndFraction sigma-summation Overscript zero width space Endscripts Underscript y Endscripts left-parenthesis upper N Subscript y Baseline upper A Subscript c o n d comma zero width space y Baseline right-parenthesis period (9.1)

In Equation (9.1), the conductor cross-sectional area in winding y depends on its maximal RMS current and the maximal allowed current density upper J Subscript max that is generally chosen to be the same for all windings:

upper A Subscript c o n d comma zero width space y Baseline equals StartFraction upper I Subscript rms comma zero width space y Baseline Over upper J Subscript max Baseline EndFraction period (9.2)

Substituting Equation (9.2) into Equation (9.1),

upper A Subscript w i n d o w Baseline equals StartFraction sigma-summation Overscript zero width space Endscripts Underscript y Endscripts left-parenthesis upper N Subscript y Baseline upper I Subscript rms comma y Baseline right-parenthesis Over k Subscript w Baseline upper J Subscript max Baseline EndFraction comma (9.3)

which shows that the window area is linearly proportional to the number of turns chosen by the designer.

9.4.2 Core Cross-Sectional Area Acore

The core cross-sectional area in Figures 9.1a and 9.1b depends on the peak flux ModifyingAbove phi With ˆ and the choice of the maximal allowed flux density upper B Subscript max to limit core losses:

upper A Subscript c o r e Baseline equals StartFraction ModifyingAbove phi With ˆ Over upper B Subscript max Baseline EndFraction period (9.4)

How the flux is produced depends on whether the device is an inductor or a transformer. In an inductor, ModifyingAbove phi With ˆ depends on the peak current and equals the peak flux linkage upper N ModifyingAbove phi With ˆ . Hence,

StartLayout 1st Row 1st Column ModifyingAbove phi With ˆ equals StartFraction upper L ModifyingAbove upper I With ˆ Over upper N EndFraction 2nd Column left-parenthesis inductor right-parenthesis EndLayout period (9.5)

In a transformer, based on Faraday’s law, the flux depends linearly on the applied volt-seconds and inversely on the number of turns. This is shown in Figure 9.2 for a forward converter transformer with upper N 1 zero width space equals upper N 3 , and the duty ratio upper D , which is limited to 0.5. Therefore, we can express the peak flux in Figure 9.2 as

StartLayout 1st Row 1st Column ModifyingAbove phi With caret equals StartFraction k Subscript c o n v Baseline zero width space upper V Subscript i n Baseline Over upper N 1 f Subscript s Baseline EndFraction 2nd Column left-parenthesis transformer right-parenthesis comma EndLayout (9.6)

FIGURE 9.2 Waveforms in a transformer for a forward converter..

where the factor k Subscript c o n v equals upper D in a forward converter and typically has a maximum value of 0.5. The factor k Subscript c o n v can be derived for transformers in other converter topologies based on the specified operating conditions, for example, it equals upper D slash 2 in a full-bridge converter. In general, the peak flux can be expressed in terms of any one of the windings, y , for example, as

StartLayout 1st Row 1st Column ModifyingAbove phi With caret equals StartFraction k Subscript c o n v Baseline upper V Subscript y Baseline Over upper N Subscript y Baseline f Subscript s Baseline EndFraction 2nd Column left-parenthesis transformer right-parenthesis EndLayout period (9.7)

Substituting for ModifyingAbove phi With ˆ from Equations (9.5) and (9.7) into Equation (9.4), respectively, we find:

StartLayout 1st Row 1st Column upper A Subscript c o r e Baseline equals StartFraction upper L ModifyingAbove upper I With caret Over upper N upper B Subscript max Baseline EndFraction 2nd Column left-parenthesis inductor right-parenthesis EndLayout (9.8)
StartLayout 1st Row 1st Column upper A Subscript c o r e Baseline equals StartFraction k Subscript c o n v Baseline upper V Subscript y Baseline Over upper N Subscript y Baseline f Subscript s Baseline upper B Subscript max Baseline EndFraction 2nd Column left-parenthesis transformer right-parenthesis EndLayout (9.9)

Equations (9.8) and (9.9) show that in both cases, the core cross-sectional area is inversely proportional to the number of turns chosen by the designer.

9.4.3 Core Area-Product upper A Subscript upper P Baseline left-parenthesis equals upper A Subscript c o r e Baseline upper A Subscript w i n d o w Baseline right-parenthesis

The core area-product is obtained by multiplying the core cross-sectional area upper A Subscript c o r e with its window area upper A Subscript w i n d o w :

upper A Subscript p Baseline equals upper A Subscript c o r e Baseline upper A Subscript w i n d o w Baseline period (9.10)

Substituting for upper A Subscript w i n d o w and upper A Subscript c o r e from the previous equations,

StartLayout 1st Row 1st Column upper A Subscript p Baseline equals StartFraction upper L ModifyingAbove upper I With ˆ upper I Subscript rms Baseline Over k Subscript w Baseline upper J Subscript max Baseline upper B Subscript max Baseline EndFraction 2nd Column left-parenthesis inductor right-parenthesis EndLayout (9.11)
StartLayout 1st Row 1st Column upper A Subscript p Baseline equals StartFraction k Subscript c o n v Baseline sigma-summation Overscript zero width space Endscripts Underscript y Endscripts left-parenthesis upper V Subscript y Baseline upper I Subscript y comma rms Baseline right-parenthesis Over k Subscript w Baseline upper B Subscript max Baseline upper J Subscript max Baseline f Subscript s Baseline EndFraction 2nd Column left-parenthesis transformer right-parenthesis EndLayout period (9.12)

Equations (9.11) and (9.12) show that the area-product that represents the overall size of the device is independent (as it ought to be) of the number of turns. After all, the core and the overall component size should depend on the electrical specifications and the assumed values of upper B Subscript max andupper J Subscript max and not on the number of turns, which is an internal design variable.

9.4.4 Design Procedure Based on Area-Product Ap

Once we pick the appropriate material and the shape for a core, the cores by various manufacturers are cataloged based on the area-product upper A Subscript p . Having calculated the value of upper A Subscript p above, we can select the appropriate core. It should be noted that there are infinite combinations of the core cross-sectional area upper A Subscript c o r e and the window area upper A Subscript w i n d o w that yield the desired area-product upper A Subscript p . However, manufacturers take pains in producing cores such that for a given upper A Subscript p , a core has upper A Subscript c o r e and upper A Subscript w i n d o w that are individually optimized for power density. Once we select a core, it has specific upper A Subscript c o r e and upper A Subscript w i n d o w , which allow the number of turns to be calculated as follows:

StartLayout 1st Row 1st Column upper N equals StartFraction upper L ModifyingAbove upper I With ˆ Over upper B Subscript max Baseline upper A Subscript c o r e Baseline EndFraction 2nd Column left-parenthesis inductor semicolon from Equation 9 .8 right-parenthesis EndLayout (9.13)
StartLayout 1st Row 1st Column upper N Subscript y Baseline equals StartFraction k Subscript c o n v Baseline upper V Subscript y Baseline Over upper A Subscript c o r e Baseline f Subscript s Baseline upper B Subscript max Baseline EndFraction 2nd Column left-parenthesis inductor semicolon from Equation 9 .9 right-parenthesis EndLayout period (9.14)

In an inductor, to ensure that it has the specified inductance, an air gap of an appropriate length script l Subscript g is introduced in the path of flux lines. Assuming the chosen core material to have very high permeability, the core inductance is primarily dictated by the reluctance German upper R Subscript g of the air gap, such that

upper L asymptotically-equals StartFraction upper N squared Over German upper R Subscript g Baseline EndFraction period (9.15)

where

German upper R Subscript g Baseline asymptotically-equals StartFraction script l Subscript g Baseline Over normal mu Subscript o Baseline upper A Subscript c o r e Baseline EndFraction period (9.16)

Using Equations (9.15) and (9.16), the air gap length script l Subscript g can be calculated as:

script l Subscript g Baseline equals StartFraction upper N squared normal mu Subscript o Baseline upper A Subscript c o r e Baseline Over upper L EndFraction (9.17)

The above equations are approximate because they ignore the effects of finite core permeability and the fringing flux, which can be substantial. Core manufacturers generally specify measured inductance as a function of the number of turns for various values of the air gap length. In this section, we used a toroidal core for descriptive purposes in which it will be difficult to introduce an air gap. If a toroidal core must be used, it can be picked with a distributed air gap such that it has the effective air gap length as calculated above. The above procedure explained for toroidal cores is equally valid for other types of cores. The actual design described in the next section illustrates the introduction of an air gap in a pot core.

9.5 DESIGN EXAMPLE OF AN INDUCTOR

In this example, we will discuss the design of an inductor that has an inductance upper L equals 100 normal mu normal upper H . The worst-case current through the inductor is shown in Figure 9.3, where the average current upper I equals 5.0 normal upper A , and the peak-peak ripple upper Delta upper I equals 0.75 normal upper A at the switching frequency f Subscript s Baseline equals 100 kHz . We will assume the following maximum values for the flux density and the current density: upper B Subscript max Baseline equals 0.25 normal upper T , and upper J Subscript max Baseline equals 6.0 normal upper A slash mm squared (for larger cores, this is typically in a range of 3 to 4 normal upper A slash mm squared ). The window fill factor is assumed to be k Subscript w Baseline equals 0.5 .

FIGURE 9.3 Inductor current waveforms.

The peak value of the inductor current from Figure 9.3 is ModifyingAbove upper I With caret equals upper I plus StartFraction upper Delta upper I Over 2 EndFraction equals 5.375 normal upper A . The RMS value of the current for the waveform shown in Figure 9.3 can be calculated as upper I Subscript rms Baseline equals StartRoot upper I squared plus one-twelfth upper Delta upper I squared EndRoot asymptotically-equals 5.0 normal upper A (the derivation is left as a homework problem).

From Equation (9.11),

StartLayout 1st Row 1st Column Area hyphen Product 2nd Column upper A Subscript p Baseline equals StartFraction 100 times 10 Superscript negative 6 Baseline times 5.375 times 5 Over 0.5 times 0.25 times 6 times 10 Superscript 6 Baseline EndFraction times 10 Superscript 12 Baseline equals 3587 mm Superscript 4 EndLayout

From the Magnetics, Inc. catalog [2], we will select a p-type material that has a saturation flux density of 0.5 normal upper T and is quite suitable for use at the switching frequency of 100 kHz . A pot core 26 × 16, which is shown in Figure 9.4 for a laboratory experiment, has the core Area upper A Subscript c o r e Baseline equals 93.1 mm Superscript 2 and the window Area upper A Subscript w i n d o w Baseline equals 39 mm Superscript 2 . Therefore, we will select this core, which has an Area-Product upper A Subscript p Baseline equals 93.1 times 39 equals 3631 mm Superscript 4 . From Equation (9.13),

upper N equals StartFraction 100 normal mu times 5.375 Over 0.25 times 93.1 times 10 Superscript negative 6 Baseline EndFraction asymptotically-equals 23 Turns period

FIGURE 9.4 Pot core mounted on a plug-in board.

Winding wire cross-sectional area upper A Subscript c o n d Baseline equals upper I Subscript rms Baseline slash upper J Subscript max Baseline equals 5.0 slash 6.0 equals 0.83 mm Superscript 2 . We will use five strands of American Wire Gauge AWG 25 wires [3], each with a cross-sectional area of 0.16 mm2, in parallel. From Equation (9.17), the air gap length can be calculated as

script l Subscript g Baseline equals StartFraction 23 squared times 4 pi times 10 Superscript negative 7 Baseline times 93.1 times 10 Superscript negative 6 Baseline Over 100 normal mu EndFraction asymptotically-equals 0.62 mm period

9.6 DESIGN EXAMPLE OF A TRANSFORMER FOR A FORWARD CONVERTER

The required electrical specifications for the transformer in a forward converter are as follows: f Subscript s Baseline equals 100 kHz and upper V 1 equals upper V 2 equals upper V 3 equals 30 normal upper V . Assume the RMS value of the current in each winding to be 2.5 normal upper A . We will choose the following values for this design: upper B Subscript max Baseline equals 0.25 normal upper T and upper J Subscript max Baseline equals 5 upper A slash mm Superscript 2 . From Equation (9.12), where k Subscript w Baseline equals 0.5 andk Subscript c o n v Baseline equals 0.5 ,

upper A Subscript p Baseline equals StartFraction k Subscript c o n v Baseline Over k Subscript w Baseline f Subscript s Baseline upper B Subscript max Baseline upper J Subscript max Baseline EndFraction sigma-summation Underscript y Endscripts ModifyingAbove upper V With caret Subscript y Baseline upper I Subscript rms comma y Baseline equals 1800 mm Superscript 4 Baseline period

For the pot core 22 × 13 [2], upper A Subscript c o r e Baseline equals 63.9 mm Superscript 2 , upper A Subscript w i n d o w Baseline equals 29.2 mm Superscript 2 , and therefore upper A Subscript p Baseline equals 1866 mm Superscript 4 . For this core, the winding wire cross-sectional area is obtained as

upper A Subscript c o n d comma 1 Baseline equals StartFraction upper I Subscript 1 comma rms Baseline Over upper J Subscript max Baseline EndFraction equals StartFraction 2.5 Over 5 EndFraction equals 0.5 mm squared period

We will use three strands of AWG 25 wires [3], each with a cross-sectional area of 0.16 mm Superscript 2 , in parallel for each winding. From Equation (9.14),

upper N 1 zero width space equals StartFraction 0.5 times 30 Over left-parenthesis 63.9 times 10 Superscript negative 6 Baseline right-parenthesis times left-parenthesis 100 times 10 cubed right-parenthesis times 0.25 EndFraction asymptotically-equals 10 period

Hence,

upper N 1 equals upper N 2 equals upper N 3 equals 10

9.7 THERMAL CONSIDERATIONS

Designs presented here do not include eddy current losses in the windings, which can be very substantial due to proximity effects. These proximity losses in a conductor are due to the high-frequency magnetic field generated by other conductors in close proximity. These proximity losses can be minimized by designing inductors with a single-layer construction. In transformers, windings can be interleaved to minimize these losses, as described in detail in [1]. Therefore, the area-product method discussed in this chapter is a good starting point, but the designs must be evaluated for the temperature rise due to additional losses. A more detailed analysis is presented in [4].

REFERENCES

  1. 1. N. Mohan, T.M. Undeland, and W.P. Robbins, Power Electronics: Converters, Applications and Design, 3rd Edition (New York: John Wiley & Sons, 2003).
  2. 2. Magnetics, Inc. Ferrite Cores: www.mag-inc.com.
  3. 3. Wire Gauge Comparison Chart: http://www.engineeringtoolbox.com/wire-gauges-d_419.html.
  4. 4. N. Mohan, W. Robins, T. Undeland, and S. Raju, Power Electronics for Grid-Integration of Renewables: Analysis, Simulations and Hardware Lab (New York: John Wiley & Sons, 2023).

PROBLEMS

Inductor Design

  • 9.1 Derive the expression for the RMS current for the current waveform in Figure 9.3.
  • 9.2 In the design example of the inductor in Section 9.5 of this chapter, the core has an area upper A Subscript c o r e Baseline equals 93.1 mm Superscript 2 , the mean magnetic path length script l equals 37.6 mm , and the relative permeability of the core material is normal mu Subscript r Baseline equals normal mu Subscript m Baseline slash normal mu Subscript o Baseline equals 5000 . Calculate the inductance with 23 turns if the air gap is not introduced in this core in the flux path.
  • 9.3 In the inductor design presented in Section 9.5 of this chapter, what is the reluctance offered by the magnetic core as compared to that offered by the air gap if the mean magnetic path length of the core is script l equals 37.6 mm ?
  • 9.4 In Problem 9.2, what is the maximum current that will cause the peak flux density to reach 0.25 normal upper T ?
  • 9.5 In the inductor designed in Section 9.5 of this chapter, what will be the inductance and the maximum current that can be passed without exceeding the upper B Subscript max specified, if the air gap introduced by mistake is only one-half of the required value?

Transformer Design

  • 9.6 In the design example of the transformer in Section 9.6 of this chapter, the core has an area upper A Subscript c o r e Baseline equals 63.9 mm Superscript 2 , the magnetic path script l equals 31.2 mm , and the relative permeability of the core material is normal mu Subscript r Baseline equals normal mu Subscript m Baseline slash normal mu Subscript o Baseline equals 5000 . Calculate the peak magnetizing current at duty ratio of 0.5 in this example.
  • 9.7 What is the tertiary winding conductor diameter needed for the magnetizing current calculated in Problem 9.6?
  • 9.8 Derive k Subscript c o n v for transformers in two-switch forward, half-bridge, and push-pull converters.
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