Inductor selection for very-low-power or very-high-power energy conversion is not as simple a process as it is for the majority of power-supply designs. Off-the-shelf inductors from major suppliers are targeted at the mainstream designs and are generally not suitable for specialized designs. When output inductors are needed for low-power buck regulator applications that require high efficiency, inductor selection criteria change and the data may not be readily available.

The typical application is a small battery-powered device that must operate for an extended period of time. The engineering tradeoffs involved are battery capacity (cost and size) versus the efficiency and size of the buck regulator.

To reduce power-supply size, the highest practical frequency is used. Switching losses in the power stage and losses in the output inductor increase as the frequency goes up and may become the dominant losses affecting efficiency. This creates a challenging engineering problem. For designers of low-power, high-frequency buck regulators, it's important to understand how high-frequency operation impacts inductor losses and dictates core and wire selection, and how to take simple measurements that will determine whether a given inductor is suitable for a particular application.

### Inductors for Buck Regulators

Ferromagnetic devices are probably the most nonlinear circuit elements an engineer is likely to encounter. The information provided in inductor catalogs is of no use when trying to predict a given inductor's losses in critical high-frequency applications. In manufacturers' catalogs, power inductors are generally specified for open-circuit inductance, operating current, saturation current, dc resistance and self-resonant frequency.

Most power-conversion design can be accomplished using these few basic parameters, and selecting inductors is ordinarily quite simple. However, very-low-current, high-frequency applications are particularly sensitive to the nonlinear properties of the core materials in the inductor and, to a lesser degree, the frequency dependent losses in the windings.

When inductor properties other than the basic catalog information are required, off-the-shelf parts may not suffice. Usually the only frequency-dependent specification provided for inductors will be self-resonant frequency. Core loss information is almost never provided, because most applications are only sensitive to the dc I^{2}R and current-dependent inductance effects, while time-varying losses are considered negligible. However, in very-low-power, high-frequency systems operating from batteries, these losses often dominate over the dc losses.

There are two major sources of power dissipation in an inductor: winding losses and core losses. Winding losses include the dc I^{2}R losses and the ac losses due to skin effect and proximity effect. Skin-effect losses result from the inability of moving charges to penetrate conductors (wire) at high frequency, thus reducing the effective copper cross section and increasing the resistance to ac current.

For instance, at 2 MHz, the skin depth of copper is approximately 0.00464 cm. This is the depth into the copper that the current density falls to 1/e (about 0.37), where e is Euler's constant. Proximity-effect losses are caused by magnetic fields from each turn of wire in a winding interacting with adjacent turns. These magnetic fields influence the distribution of current in each winding and cause current crowding, which again raises the effective ac resistance of the winding.

Some of the skin-depth effects can be reduced by using multiple strands of thinner wire wrapped into a bundle (Litz wire). But if the ac ripple current in the inductor is low compared to the dc current, Litz wire provides little reduction in total loss.

Core losses are caused by magnetic hysteresis and the magnetic induction interacting with conductive and other nonlinear properties of the core. In a buck converter, the first quadrant of the B-H loop is the region of greatest interest. The minor loop that the inductor experiences is from its initial induction state to the peak induction and back to the initial state. If the converter is operating steady state in discontinuous conduction mode, the minor loop is from the remanence induction (B_{R}) to the peak induction (**Fig. 1**).

If the converter is in continuous conduction mode, the excursion will be from the dc-bias point on the upper part of the curve to the peak value and back. The exact shape of the minor hysteresis loop can be determined experimentally and is shown arbitrarily as an oval in the diagram.

Most cores consist of powdered magnetic material held together with a ceramic or other type of binding material. A new core can be thought of as a collection of randomly oriented grains of magnetized material separated by thin layers of the binding material. There is no universally accepted model that fully explains core loss. Research is still being done in this field. The simple model explained here is generally accepted as a practical first-order explanation of core losses. The references at the end of this article give a more in-depth look at magnetic modeling.^{[1-3]}

Adjacent grains that are similarly magnetized can influence each other and come into alignment. They are physically separated by the binder, but magnetically aligned, forming what is called a domain. The boundary of the domain, outside of which the magnetic material is aligned differently from the inside, is called a domain wall. This model can be used to explain many of the basic properties of magnetic cores.

In the presence of an applied magnetic field, the randomly oriented domains are forced into alignment with each other. The winding and the current flowing in it are the source of the magnetic field. When a sufficiently high current is applied to a randomly magnetized magnetic core, the magnetic domains closest to the wire are brought into alignment first because the field is strongest there (**Fig. 2**).

Farther away from the wire, deeper inside the core, the magnetic domains are initially unaffected. The boundary between the aligned domains and the randomly magnetized interior of the core forms a moving domain wall that propagates into the core. If the current in the wire is not withdrawn or reversed, the domain wall will finally propagate to the center of the core and all of the domains in the core will be aligned. When this condition occurs, the core is said to be saturated.

Manufacturers' published B-H loops show this initial magnetization of the core as a curved line from the origin to the saturation point. If the current is then decreased, the domains that have been forced into alignment will relax toward their initial random state. However, some of the domains that were forced into alignment by the applied field will remain in alignment. This incomplete return to a random state gives rise to the remanence induction (B_{R}) seen on the hysteresis curve. The magnetic realignment of the domains and the moving domain walls create mechanical stresses that contribute to the core loss.

The amount of energy per cycle lost to hysteresis is:

W_{H} = ∫H × dI,

where the integration is performed from the initial induction, to the peak induction, and back to the initial induction and is the area bound by the minor hysteresis loop. The power lost is the energy per cycle times the switching frequency (F_{SW}):

P_{H} = F_{SW} × ∫H×dI.

Calculating these ac losses would seem to be a straightforward procedure, but at high frequency and moderate flux density, it can be quite tricky. Each application circuit has specific characteristics that affect core loss, including stray capacitance, board layout, drive voltage, pulse width, loading conditions, input and output voltages, and other factors. Core loss is highly dependent on the properties of the magnetic material.

Every core material has an electrical conductivity that is nonlinear and contributes to the core loss. Because of this conductivity, eddy currents in the core are induced by the applied magnetic field of the windings. These currents contribute to the total core loss.

For a given operating frequency, increasing the ac flux in the core will increase the core losses (**Fig. 3**). Likewise, for a constant flux level, core losses increase roughly with the square of the frequency. The exponents associated with frequency and flux density vary with core materials and manufacturing processes. Most core manufacturers supply empirical formulas for the core loss along with graphs of core loss.

### Inductor Characteristics

The magnetic induction, or B field, of a core operating as an inductor in a forward converter is given by the following formula:

B_{PK} = E_{AVG}/(4 × A × N × F),

where B_{PK} is the peak ac flux density in Teslas, E_{AVG} is the average ac voltage per half cycle, A is the cross-sectional area of the core in square meters, N is the number of turns on the core and F is the frequency in Hertz.

Magnetic material manufacturers usually evaluate the inductance ratings of the cores and use the term A_{L} to give the user a convenient way to calculate inductance:

L = N^{2}A_{L}.

The A_{L} value is proportional to the permeability of the magnetic material and the cross-sectional area of the toroid divided by its magnetic path length.

Total core loss is related to the volume of the core multiplied by factors involving B_{PK} and frequency, and is given in watts per cubic meter or similar units, depending on the manufacturer.

### Measuring Core Losses

The best way to screen inductors for power supplies would be to put them in the final power circuit and take efficiency measurements. This is a tedious process. Instead, a simple test fixture can be used to evaluate the core loss at its design frequency.

The core is placed in series with a very-low-loss capacitor (such as a silver mica) and driven in series-resonant mode. Capacitors are chosen so that the resonant frequency is the same as the switching frequency of the power supply. The simplest way to run these tests is with a network analyzer, but a signal generator and an RF voltmeter or power meter will work as well. The mechanical design of one such fixture is shown in **Fig. 4**, and the complete test setup with network analyzer is shown in **Fig. 5**.

At resonance, a very-low-loss core will form a series L-C circuit with the capacitor, and the approximate loss can be represented as a resistance (this will include the wiring and the core loss, which is what we are interested in). In the **Fig. 5** test fixture, the A and R ports are terminated in 50 Ω. The open-circuit equivalent (no inductor installed) is the oscillator driving a 150-Ω resistive divider. In decibels, the analyzer will display:

20 × Log(A/R) = 20 × Log(50 Ω/150 Ω) = -9.54 dB.

In this test circuit, the resonance capacitor is 2000 pF. The inductors tested were 2.5 µH to 2.8 µH, measured at 1 kHz. Permeability is not linear with frequency, and the inductors may have a different value at high frequency.

### Experimental Data

A single 125-µ Molypermalloy Thinz core was wound using 16 turns of 10/44 Litz and a second double-stacked 250-µ permeability MPP Thinz core with eight turns of 10/44 Litz. These inductors measured 2.75 µH and 2.78 µH, respectively. The 16-turn core had twice the turns but half the cross section as the eight-turn core. The inductors were driven with the same amplitude signal. Both had high loss, with equivalent resistances of 360 Ω and 300 Ω, respectively.

Another inductor (2.5 µH) was wound using a very-low-permeability material (T25-6 carbonyl SF with µ = 8.5) from Micrometals, and 34 turns of 10/44 Litz. At the same drive level, its equivalent resistance was about 22,000 Ω. **Fig. 6** shows the experimental results of the low-loss inductor.

A high-efficiency, low-power buck power stage was built to test the inductors. At 3.6-V input and 1.8-V output at 20 mA, the efficiency with the 16-turn single-stacked MPP Thinz inductor was 66%, the 8-turn double-stacked core was 55% and the T25-6 core with 34 turns was 92%. **Fig. 7** shows a high-efficiency, fully integrated buck regulator using the MAX8460Y.

### Tips for Inductor Selection

Inductors for low-power electronics have unique requirements. The conventional catalog or data sheet information that is available is generally insufficient to evaluate inductors for low power and high efficiency. Most commercial power-supply inductors are gapped ferrites, which are typically not low-loss material and will generally not perform well in low-power, high-frequency applications.

A relatively simple test fixture for evaluating inductor losses can be used to compare the relative performance expected at the power supply's operating frequency.

When designing inductors for low losses, use low-permeability materials to get the B field down, choose low-loss core materials and consider using Litz wire. It is often best to use the IC manufacturer's recommended magnetic components that provide the specified efficiency curves or to consult professional magnetics specialists who have experience designing and manufacturing magnetics for demanding applications.

### References

- Globus, A., and Duplex, P., “Separation of Susceptibility Mechanisms for Ferrites of Low Anisotropy,”
*IEEE Transactions on Magnetics*, Vol. 2, September 1966, pp. 441-445, and other papers by the same authors. - Jiles, D.C., and Atherton, D.L., “Theory of Ferromagnetic Hysteresis,”
*Journal of Magnetism and Magnetic Materials*, Vol. 61, 1986, pp. 48-60. - Roshen, Waseem A., “Magnetic Loss in Soft Ferrites,”
*Journal of Applied Physics*, Vol. 101, May 2007, 09M522.