# What every designer should know about magnetics in switch-mode power supplies

By Sameer Kelkar, Power Integrations -November 26, 2009

The application of electromagnetics has been in practice for more than a century: In 1831, English chemist and physicist Michael Faraday invented the transformer, although he called it an induction coil. Unfortunately, engineering schools rarely provide instruction in practical magnetics relevant to SMPS (switch-mode-power-supply) applications. Part of the problem is that the classic design equations for magnetics target sinusoidal waveforms, but SMPSs operate with rectangular waveforms.

The starting point for understanding magnetics is to look at the relationships between current flow and electric and magnetic fields. Figure 1 shows a simple air-cored winding. A current-carrying conductor creates its own magnetic field (B), which produces flux lines around the conductor. In this example, 10 turns of wire carry a dc current, and each turn creates its own magnetic field. The fields combine to create a concentrated and fairly linear field within the winding; the field diverges and weakens outside the winding. The magnetic field inside the winding is the primary storage area for energy, but the external field can also store a significant amount.

Figure 1 In a simple air-cored winding, a current-carrying conductor creates its own magnetic field, which produces flux lines around the conductor (a); 10 turns of wire carry a dc current, and each turn creates its own magnetic field (b).

If you place an object comprising a magnetic material, such as iron, within the winding, the magnetic field exerts an EMF (electromotive force) on the object. If you then place a second winding within the field and the primary winding is carrying ac current so the field is changing with time, the magnetic field will induce a current to flow within the second winding. Lenz’s Law, which Russian chemist and physicist Heinrich Lenz postulated in 1834, states that an induced current always flows in a direction opposing the motion or change causing it.

Thus, you can describe the properties of a magnetic field in terms of its intensity or its density. The magnetic-field intensity defines the field’s ability—in ampere turns per meter—to exert forces on magnetic poles. The magnetic-flux density (B) is the ability of the magnetic field, in teslas, to induce an electric field when it changes. This property introduces the dimension of time.

Two laws—Ampere’s and Faraday’s—jointly govern the relationship between magnetic components and their characteristics you see from the terminals. Ampere’s Law, which French physicist and mathematician André-Marie Ampère postulated in 1826, relates the integrated magnetic field around a closed loop to the electric current passing through the loop. Faraday’s Law, which Faraday postulated in 1834, states that the induced EMF or EMF in any closed circuit equals the time rate of change of the magnetic flux through the circuit.

You may wonder why magnetic circuits require cores. Answering this question requires consideration of another characteristic, permeability—a measure of the amount of flux a magnetic field can push through a unit volume of material. You would not expect the winding in Figure 1 to perform well as an electromagnet because it has no core. However, if you insert an iron core in the center of the windings, it can make a powerful electromagnet because the permeability of iron is about 10,000 times that of free space, enabling the concentration of a relatively large amount of magnetic flux between the windings. Permeability is roughly analogous to conductivity in the electrical realm. Table 1 shows the equivalence between the magnetic and the electrical domains. Just as a conductor is a conduit for energy to flow in the form of an electrical current, a high-permeability magnetic material acts as a conduit for energy to flow as magnetic flux.

It is important to account for leakage in magnetic circuits. Many parallels exist between the electrical and the magnetic realms. However, compared with free space, the conductivity of common conductors, such as copper, at approximately 1020, is much higher than the permeability of magnetic materials, at approximately 104. Thus, you can easily ignore leakage currents, but not leakage flux, in low-frequency systems. Although permeability is analogous to conductivity, it is not a linear characteristic for many materials and takes a different value depending on what previously occurred (Figure 2).

Figure 2 A BH curve shows the relationship between magnetic-field intensity, H, and magnetic-flux density, B, in a ferromagnetic material (a). The slope of this curve at any moment is the instantaneous permeability of the material. For low values of H, permeability is constant and relatively high. The introduction of an air gap tilts the BH curve to the right (b). The ungapped core would saturate at a field intensity of H1, whereas a gapped core can be useful at a field intensity as high as H2, where H2 is greater than H1. Because current, I, is the prime driver of magnetic-field intensity, you can push more current through the core without saturating it.

Figure 2a shows the relationship between magnetic-field intensity, H, and magnetic-flux density, B, in a ferromagnetic material. The slope of this curve at any given time is the instantaneous permeability (µR) of the material. For low values of intensity, permeability is constant and relatively high. However, for larger values of intensity, permeability starts decreasing to the point that the material starts resembling free space (µR=1). Thus, you need a larger and larger magnetic-field intensity to produce a small increase in the density field. At this point, the material reaches saturation.

The area between the rising BH curve and the vertical axis represents energy stored in the material. If you then reduce the field intensity from the saturation point, you can recover energy from the material; however, the energy you recover is less than that stored, so the BH curve follows a different path. The result for a complete cycle is that the BH curve forms a closed S shape. The area the S encloses represents the hysteresis curve, or the total lost energy in the cycle. The area of the curve is a function of the frequency; thus, at higher frequencies, the area of the hysteresis curve increases, and so do the losses.

The total flux, or flux linkage, relates to the electrical current through the inductance constant. Thus,

where LM is the inductance constant, Nφ is the flux linkage, and I is the electrical current. Further,

where AE is the cross-sectional area of the core, B is the magnetic-flux density, µ0 is the permeability of free space, µR is the relative permeability of the core material, H is the magnetic-field intensity, N is the number of turns, I is the current, and lM is the magnetic path length. Therefore,

The parameter

is called reluctance, R, and is purely material- and geometry-dependent. It is analogous to resistance in the electrical domain.

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