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.
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:
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.
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 passes through all turns of a winding. In the transformer of Figure 9.1b, there are two windings where the current in winding 1, with bigger cross-section conductors, is in the opposite direction to that of in winding 2 with smaller cross-section conductors. In each winding, the conductor cross-section is chosen such that the peak current density is not exceeded at the maximum specified current in that winding. The core area in Figures 9.1a and 9.1b allows the flow of flux lines without exceeding the maximum flux density in the core.
The area-product method, based on preselected values of the peak flux density in the core and the peak current density in the conductors, allows an appropriate core size to be chosen, as described below.
The windows of the toroidal cores in Figures 9.1a and 9.1b accommodate the winding conductors, where the conductor cross-sectional area depends on the maximal RMS current for which the winding is designed. In the expression for the window area below, the window fill factor 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 designates a winding, where in general, there may be more than one, as in a transformer:
In Equation (9.1), the conductor cross-sectional area in winding y depends on its maximal RMS current and the maximal allowed current density that is generally chosen to be the same for all windings:
Substituting Equation (9.2) into Equation (9.1),
which shows that the window area is linearly proportional to the number of turns chosen by the designer.
The core cross-sectional area in Figures 9.1a and 9.1b depends on the peak flux and the choice of the maximal allowed flux density to limit core losses:
How the flux is produced depends on whether the device is an inductor or a transformer. In an inductor, depends on the peak current and equals the peak flux linkage . Hence,
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 , and the duty ratio , which is limited to 0.5. Therefore, we can express the peak flux in Figure 9.2 as
where the factor equals in a forward converter and typically has a maximum value of 0.5. The factor can be derived for transformers in other converter topologies based on the specified operating conditions, for example, it equals in a full-bridge converter. In general, the peak flux can be expressed in terms of any one of the windings, , for example, as
Substituting for from Equations (9.5) and (9.7) into Equation (9.4), respectively, we find:
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.
The core area-product is obtained by multiplying the core cross-sectional area with its window area :
Substituting for and from the previous equations,
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 and and not on the number of turns, which is an internal design variable.
Once we pick the appropriate material and the shape for a core, the cores by various manufacturers are cataloged based on the area-product . Having calculated the value of above, we can select the appropriate core. It should be noted that there are infinite combinations of the core cross-sectional area and the window area that yield the desired area-product . However, manufacturers take pains in producing cores such that for a given , a core has and that are individually optimized for power density. Once we select a core, it has specific and , which allow the number of turns to be calculated as follows:
In an inductor, to ensure that it has the specified inductance, an air gap of an appropriate length 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 of the air gap, such that
where
Using Equations (9.15) and (9.16), the air gap length can be calculated as:
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.
In this example, we will discuss the design of an inductor that has an inductance . The worst-case current through the inductor is shown in Figure 9.3, where the average current , and the peak-peak ripple at the switching frequency . We will assume the following maximum values for the flux density and the current density: , and (for larger cores, this is typically in a range of 3 to ). The window fill factor is assumed to be .
The peak value of the inductor current from Figure 9.3 is . The RMS value of the current for the waveform shown in Figure 9.3 can be calculated as (the derivation is left as a homework problem).
From Equation (9.11),
From the Magnetics, Inc. catalog [2], we will select a p-type material that has a saturation flux density of and is quite suitable for use at the switching frequency of . A pot core 26 × 16, which is shown in Figure 9.4 for a laboratory experiment, has the core Area and the window Area . Therefore, we will select this core, which has an Area-Product . From Equation (9.13),
Winding wire cross-sectional area . 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
The required electrical specifications for the transformer in a forward converter are as follows: and . Assume the RMS value of the current in each winding to be . We will choose the following values for this design: and . From Equation (9.12), where and ,
For the pot core 22 × 13 [2], , , and therefore . For this core, the winding wire cross-sectional area is obtained as
We will use three strands of AWG 25 wires [3], each with a cross-sectional area of , in parallel for each winding. From Equation (9.14),
Hence,
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].
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