What every designer should know about magnetics in switch-mode power supplies
Power is often an afterthought in system design, but the choice and design of the magnetic elements at the heart of an SMPS are crucial. Acquaint or reacquaint yourself with the fundamentals of this frequently overlooked area.
By Sameer Kelkar, Power Integrations -- EDN, 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.
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 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.
Transformers, inductors, and SMPSs
You normally construct transformers using an iron core because iron is highly permeable, enabling it to be efficient at transferring energy from the primary to the secondary winding. The purpose of the transformer core is not to store energy but to act as an instantaneous conduit. Practically speaking, a transformer does store energy in its magnetizing and leakage inductances. These inductances degrade performance, and the goal of transformer design is normally to minimize them. Mutual inductance is a measure of the coupling between the primary and the secondary winding. Leakage inductance occurs when the magnetic flux does not fully couple to the secondary winding.
An inductor stores and releases energy to smooth the current through it. A flyback transformer is an inductor with multiple windings; it stores energy it takes from the input during one portion of the switching period—that is, the on-time—and then delivers energy to the output during a subsequent interval—that is, the off-time. For a core to act as an efficient conduit for magnetic flux, it must contain highly permeable material. Such materials are inherently incapable of storing significant energy, however. When an application requires energy storage, you accomplish the task by creating one or more nonmagnetic gaps in series with the core and store the energy across the gap. The following equation defines stored energy in a magnetic circuit per unit volume:
where W is the energy stored and µ is the permeability of the material. Because magnetic materials are highly permeable, they can store little energy. The addition of an air gap reduces the effective permeability and allows for energy storage.
Figure 2b shows the effect of the introduction of an air gap on the BH characteristic of the magnetic circuit. The air gap tilts the BH curve to the right. The ungapped core would saturate at a field intensity of H1, whereas a gapped core can be useful up to a field intensity of H2, where H2 is greater than H1. Because current is the prime driver of magnetic-field intensity, you can effectively push more current through the core without saturating it.
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This concept is analogous to storing energy across the insulation layer between the two plates of a capacitor. You create the air gaps by constructing the core with either physical discrete air gaps or from a granular composite material with distributed air gaps. The magnetic particles are isolated from one another in a solid, nonmagnetic binder. In this way, the gap is effectively distributed throughout the core. The core’s function in a flyback transformer remains the same whether you construct it with distributed air gaps or composite material: It provides the flux-linkage path between the primary winding and the gap and between the gap and the secondary windings.
Magnetic-core materials
Because of its low cost and high-saturation flux density, laminated silicon steel is the most popular material for low-frequency applications; however, it exhibits high core losses at higher frequencies. At higher frequencies, cores require more exotic materials, such as low-loss amorphous metal alloys and composite powdered-metal cores, including powdered iron, Kool Mu, and Permalloy powder and ferrites.
Ferrites are the most popular core materials for SMPS applications because they exhibit low losses and are inexpensive. Ferrites are ceramic materials manufacturers develop by sintering a mixture of iron oxide with oxides or carbonates of either manganese and zinc or nickel and zinc. The main disadvantage of ferrite is that, being a ceramic, the core is mechanically less robust than other materials and may be unacceptable in high-shock environments.
Ideal magnetic materials have a square-loop characteristic with high permeability and insignificant energy storage until you finally drive them into saturation—that is, when they develop a sharp saturation characteristic (Figure 3). With metal-alloy cores, the characteristic approaches the square-loop characteristic of that in Figure 3. Composite metal-powder and ferrite cores exhibit a rounded, or soft, characteristic due to the particulate structure of the core material. In composite metal-powder cores, nonmagnetic gaps exist between the discrete magnetic particles. Similar nonmagnetic occlusions occur among the sintered particles in ferrite cores. These tiny gaps cause the distribution of flux and flux changes across the entire core, rather than at a discrete flux-change boundary.
At low flux densities, flux tends to concentrate in the paths with the lowest resistance, in which the particles are in close proximity (Figure 4). As the flux density increases, these paths are the first to saturate. Any incremental flux must then move to adjacent paths in which the magnetic material has not saturated but the gap is somewhat wider. This process continues, effectively widening the incremental distributed gap as the flux increases. The incremental permeability and inductance thus progressively decrease, creating the rounded shape of the BH characteristic.
When you add a discrete gap to a ferrite core for energy storage in a filter or flyback application, the rounding of the ferrite characteristic disappears because the high reluctance of the gap swamps it, and the inductance characteristic becomes linear until it reaches saturation—another advantage of using ferrite cores in SMPS applications.
Eddy-current loss
Eddy current is an induced current that flows around the core. Eddy-current loss can occur in both transformer cores and windings at high frequencies. The parasitic eddy-current characteristic is a function of the volts you apply per turn and duty cycle. At high SMPS frequencies, eddy currents can cause serious problems.
You can think of the core itself as a single-turn secondary winding that links to all the windings. It induces a voltage, equal to the volts per turn you apply to the windings, around the core’s periphery. This voltage induces a current flow around the core, resulting in energy losses that manifest themselves as heat due to the internal resistance of the battery. If the core comprises high-resistivity material, such as ferrite, the eddy current is low, and eddy-current loss is insignificant in SMPS applications. For metal-alloy cores, resistivity is low; in solid-metal cores, eddy current would cause a shorted turn. One approach to this problem is to break the core into electrically insulated laminations (Figure 5). The flux divides between the laminations, and they present a higher resistance than that of the bulk core. Thus, the laminated core presents an effective eddy-current resistance many times greater than that of an equivalent solid core. For this reason, all iron-alloy transformer cores have laminations.
Windings and high frequencies
The behavior of electric currents in high-frequency windings can differ greatly from those in low-frequency applications, resulting in high-frequency skin effects and proximity effects. To understand the skin effect, consider that electricity is like water: Both always take the easiest path available—that is, they follow paths requiring the lowest expenditure of energy. At low frequencies, electricity accomplishes this goal by minimizing heat-caused losses. At high frequency, current flows in the paths that minimize inductive energy. Conservation of energy causes high-frequency current to flow near the surface of a thick conductor, even though this flow may result in higher losses (Figure 6).
In the figure, you can see the surface and the center of the wire. L is the inductance of the wire. LI is the inductance within the wire from the surface to the center, and RI is the distributed resistance. At dc or low frequency, the effect of LI is trivial, and the current distributes itself evenly from the surface to the center, minimizing loss. However, at high frequency, the inductive reaction of LI becomes larger, so LI effectively blocks the current from flowing in the center of the wire, and it concentrates at the surface. The penetration, or “skin,” depth is the distance from the conductor surface to a point at which the current density is one electric-field current times the surface current density. At 100 kHz in copper, penetration depth is 0.24 mm, meaning that, at a 100-kHz frequency, the largest diameter of wire you can use is 0.48 mm—that is, AWG #25. If your design requires thicker wire, you should instead consider paralleling two thinner wires.
Another important high-frequency effect that has an impact on transformer windings is the proximity effect. When two conductors thicker than the penetration density are in proximity and carry current in the same direction, the magnetic-flux lines are denser near the wire junction (Figure 7). Consequently, current tends to flow along the halves of the wire that are not in close proximity to each other. You must consider this effect when deciding the strategy for placement of transformer windings.
Proximity losses increase exponentially as the number of layers increases. To combat proximity losses, the cores for high-frequency SMPS applications often have a window shape with the winding much wider than it is high, thus minimizing the number of layers. This approach is not a panacea, however, because stretching the windings increases capacitance between them. An interleaving winding strategy, on the other hand, alternately places the primary and secondary windings on top of each other, avoiding the need for an elongated window. Contrary to popular belief, in high-frequency SMPSs, it is often better to leave the window area unused rather than pack it with copper to reduce dc resistance. The increased layers can increase ac losses by as much as 10 times as compared with dc losses.
Other considerations
Minimizing losses is vital to SMPS design, but minimizing EMI (electromagnetic interference) is also critical. One cause of EMI is interwinding capacitance that couples common-mode noise from the primary to the secondary windings. You can reduce this capacitance by using a Faraday shield—a thin foil or metallized film in the intervening space between the primary and secondary windings. The shield should be thinner than the penetration density to avoid eddy-current losses and should connect to the low-impedance—that is, nonswitching—node of the transformer’s primary. Special winding configurations, such as those that Power Integrations developed, can also reduce EMI (Reference 1). The unterminated windings on the transformer act as electrostatic shields within the transformer. The transformer shields cancel some of the switching signals, significantly attenuating the composite signal across the parasitic capacitance and significantly reducing the transformer’s parasitic (capacitive) coupling paths (Figure 8).
Core hysteresis, eddy-current, and winding losses, which generate temperature rise, occur in all transformers. In buck-derived applications, under fixed-frequency operation, volt seconds and flux swing are constant. Hysteresis loss is therefore constant, regardless of changes in input voltage or load current. Core eddy-current loss is proportional to the input voltage; thus, worst-case loss occurs at high input voltages. Winding losses are proportional to the duty cycle, which is greatest at low input voltage.
SMPS designers must select cores that will remain within a safe working temperature under any of these worst-case conditions. They should also consider start-up and transients when a sudden increase in load current occurs. The control loop calls for full current, pushing the duty cycle to its absolute-maximum limit. If the input voltage is at the maximum, the volt seconds you apply to the transformer windings could be several times larger than normal, driving the core into saturation. Fortunately, modern control ICs include soft-start and sophisticated current- or volt-second-limiting circuitry to protect the system from both operational and fault conditions.
To design a flyback transformer for an SMPS, first define the circuit parameters and select the core material and geometry. Next, determine the peak current in the circuit and the maximum flux density and flux swing. After making those determinations, tentatively select the core’s shape and size and determine the loss limit for flux density or core losses. You then calculate the number of turns, the gap length, the conductor size, and the winding resistance. Last, calculate flux density, winding loss, total loss, and temperature rise and then adjust these values to the size of the core. Available software, such as PI Expert from Power Integrations, can help with this process (Reference 2). The available packages can automate power-supply design, including component selection and transformer design, enabling even less-experienced design engineers to successfully complete an SMPS design that complies with international standards for efficiency, safety, and EMI. Detailed knowledge of magnetics is not necessary, but it helps to have a basic understanding of what the software is doing.
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Corrections, continued. e = 2.71828... not 2.71728...
Sorry, my mistake.
Dallas Raty - 2010-16-3 13:20:00 PDT -
The article is good and filled with good information, thank you, but unless I'm just misinterpreting, the following points concern me, and perhaps deserve clarification from the author:
1. The article says "Ideal magnetic materials have a square-loop characteristic", and that is illustrated in figure 3. Yet the article also says "The area the S encloses represents the hysteresis curve, or the total lost energy in the cycle", and figure 2 indicates that the "Width of hysteresis determines losses". A wide loop is indicative of memory effects, or (by analogy to a superball) "inelastic" behavior. A square-loop is ideal for a magnetic core computer memory. Hysteresis is generally unwelcome in switched-mode power supplies, and the ideal characteristic is usually a straight line (with no loop area, with perfectly elastic energy storage, and with no saturation non-linearities). If there is saturation, then ideal saturation also exhibits no inelasticity (no area in the saturation portion of the loop).
2. The article says "The penetration, or “skin,” depth is the distance from the conductor surface to a point at which the current density is one electric-field current times the surface current density." I believe the "e" in the (1/e) factor is the base of natural logarithms, 2.71728..., and not "electric-field current". That is, the skin-effect depth is defined as that depth at which the current density reduces by a factor of (1/e) from the density at the surface.
3. To deal with skin effect, a useful rule of thumb often may be that "the largest diameter of wire you can use" is twice the skin effect depth; but that is not "always" the only effective strategy. If you use larger diameter wire you can sometimes still end up with a low resistance and an economical design, it's just that a lot of your copper volume won't be carrying much current. Sometimes, for example, when you have only a few secondary turns, large-diameter wire can economically give you the low resistance you need (by virture of its large surface area), despite low skin depth. In some cases, hollow copper pipe may be suitable. I'm just pointing out that engineers new to transformer and inductor design shouldn't feel that they absolutely cannot use larger diameter wire. There are trade-offs that might dictate otherwise.
4. Litz wire as mentioned in another reader comment is yet another strategy for dealing with skin effect. It differs from regular stranded wire in that each strand is insulated from every other (except at the two endpoints where we connect to it). The wire bundle then becomes so many thin wires, wound in parallel, and connected in parallel. The strands are all thin compared to the penetration depth, so that essentially all of the copper in each strand carries current. The result is high copper utilization and therefore low resistance (compared to a normal stranded or solid wire of the same overall diameter). I believe Litz wire is also effective in dealing with the proximity effect.
Despite the concerns, I still found the article contained useful information for me, and I appreciate the author's contributions and expertise.
Dallas Raty - 2010-16-3 11:03:00 PDT -
Thanks for a well-written article on a subject that is little written about. I'll clip and save. In the 80's, EE's in the USA were taught that small inductors would go away, while power inductors were just 60 Hz. When HV transistors made switchmode possible, it seems only Asian engineers were ready; perhaps from TV horizontal circuit experience. I note that you did not mention Litz wire, beloved of the audiophile, for reducing skin effect. And your Fig. 7 (Proximity effect) shows why Litz doesn't work for that.
TedC - 2009-9-12 08:40:00 PST -
Great presentation of some much-needed information.
Noor Khalsa - 2009-30-11 10:48:00 PST
