One of the seemingly most difficult aspects of power electronics is how to design power inductors and transformers. This specialty is often reserved for magnetics experts while electronics engineers concentrate on converter circuit design. However, some understanding of magnetic component design can provide insight into optimization of both circuit and component together. With sufficient theoretical refinement, it is possible to design magnetics components optimally without solving a single differential equation from fields theory.

**Beyond Fields Math**

Magnetic components are so seemingly simple. They consist of two parts, one magnetic - the core, and one electric - the winding. There does not seem to be much going on here as far as *structure* goes, yet it is the *behavior* that is complicated.

Magnetic fields equations can induce anxiety in circuits-oriented engineers, but what is interesting is that, aside from finite-element analysis of fields in components (which a computer program does), only two fields equations are of general relevance for magnetics, and both of them were solved long ago. They are general equations that cover most of what is of concern in design. The first is the *modified Bessel equation* that solves for eddy-current effects on resistance in round wire. The second, which is more general, is the solution of the fields equation for resistance (with eddy-current effects) of parallel flat conducting plates, published by P.J. Dowell in the 1960s. (Plates are most closely approximated by foil windings, but with some geometric equivalences can be applied to layers of round and square wires and circuit-board traces.)

Dowell’s equation is general and accurate enough (within 5 %) for design in the region where feasible magnetic operating-points would be chosen, and is the only fields-equation solution needed for most design. The Dowell equation contains sinh and cosh terms, but by expanding them as a series and truncating, accurate algebraic approximations result which are suitable for use on calculators.

**The Quest for Optimal Magnetics Design**

A perusal of magnetics and power electronics literature reveals various methods for designing magnetics, and they differ. Two of the most frequent are the area-product method and the *K*_{g} method. There is a rationale for both of them, but on closer inspection, both are somewhat crude in that they make assumptions and set goals that are often suboptimal. Besides design schemes and their rationales are various flow charts in magnetics books showing procedures for inductor or transformer design. These methods are worth having an acquaintance with but can be replaced by a simpler, more comprehensive scheme.

As far as existing magnetics textbooks go, for general background I recommend two magnetics textbooks: *High-Frequency Magnetic Components*, Second Edition by Marian Kazimierczuk (Wiley, 2014) and *Transformers and Inductors for Power Electronics: Theory, Design and Applications*, by Gerard Hurley and Werner Wölfle (Wiley, 2013). They present the usual fields-oriented fundaments with differential equations set up and solved - the sometimes-dreaded aspect of magnetics - and also the established (yet usually suboptimal) design methods.

About a decade ago, I set out to design some power electronics with magnetic components, and some seemingly obvious design questions came to me for which I did not find answers in the magnetics literature. This resulted in a years-long quest and a recent book, *Power Magnetics Design Optimization* (www.innovatia.com). In it are developed the design formulas more thoroughly and explained in more detail than in this chat, which is intended to emphasize ideas. Once you grasp the ideas, the conceptual details are straightforward though sometimes tedious to work out.

One of the early questions was how to design a magnetic component as small as possible that will transfer the required power from converter input to output port. That is, how can transfer-power density in magnetics parts be maximized? The power to be maximized is not the power lost as heat but the transferred power. This can be emphasized as a bullet item:

- Power magnetics components are
*power transfer* devices

The more basic question is: what are the magnetics parameters that limit *transfer power*? The ultimate limit in any magnetics design is the breakdown of component structure - the melting of windings or insulation, or failure of magnetic properties of cores above their Curie temperature. To summarize with bullet items,

- Component power density limited by basic design parameter:
*temperature*
- Core: Curie temperature
- Windings: insulation breakdown temperature

Within those widest of bounds, a perusal of the aforementioned books shows that for the core, there are two limits: *power loss* and magnetic *saturation*. Core catalogs have graphs showing average power-loss density,*p̄*_{c}(*B̂*_{∼}, *f*) (in mW/cm^{3}), which depends on two parameters; the first is the amplitude (̂) of the ripple (∼) component of the magnetic field density, *B*, or *B̂*_{∼}. The total Δ*B* = 2x*B̂*_{∼}. A core-loss graph is shown below from a Micrometals Inc. catalog for powdered-iron 26 material, plotting *p̄*_{c}(*B̂*_{∼}, *f*) versus *B̂*_{∼} in Gauss = 10milliTesla, mT = mVxs. (Divide Gauss by 10 for mT) The other dependency is the frequency, *f* of *B* driving the core.

For a given design, *f* is usually determined by the converter switching frequency, *f*_{s}, which can be chosen to optimize the magnetics design for maximum transfer power. The magnetic design (apart from circuit limitations) determines the optimal frequency for maximum transfer power. Given frequency, then Δ*B* is limited by acceptable power loss, and that is a thermal design consideration, one of the most obscure aspects of magnetics design. I have found a simple rationale for handling it that gives numbers in agreement with manufacturer measurement data. (More on that in a future series of chats or see *PMDO*.) The size of the core affects the allowable core power loss density, *p̄*_{c}(*B̂*_{∼}, *f*). The larger the core, the harder it is to “get the heat out” and the power-loss density must be lower. Thermal considerations determine, *p̄*_{c}(*B̂*_{∼}, *f*) and along with *f* determines Δ*B*. To bulletize this simple point,

- Core temperature is a function of
- Magnetic switching frequency,
*f*_{s}
- Magnetic-field ripple, Δ
*B* = twice amplitude, *B̂*_{∼}
- Core power-loss density,
*p̄*_{c}(*B̂*_{∼}, *f*)

The other core limitation is magnetic saturation. It is quantified by the fraction of saturation, *k*_{sat}, shown on another catalog graph below, for the same 26 material. (See www.micrometals.com for newer graphs with SI units.)

Click here for larger image

The left vertical axis of this graph is *k*_{sat}. The graph shows the core relative permeability, *μ*_{r}, decreasing with increasing magnetic field intensity, *H*. (Multiply Oe by 80 for A/m - or 79.56 to be more accurate.) Saturation reduces *μ* and both circuit inductance, *L*, and field-referred inductance, L = *L*/*N*^{ 2}, where *N* is turns. Turns decrease with increasing current, which by Ampere’s Law is *N*x*i* = *H*x*l*, where *l* is the core magnetic path length. The main point about saturation is that:

*Saturation *

- refers to the sublinearity of
*B*(H) or *L*(*i*) curves
- is quantified by the
*saturation factor*, the ratio of μ or L or L at a given current to the zero-current value:

Saturation as *k*_{sat}(*H*) can be modeled as a mathematical function. Manufacturers use polynomial functions with multiple parameters to numerically fit the saturation curves. A simpler, yet useful saturation model, is a

- Linearized approximation at
*k*_{sat} = 0.5 of the saturation region, drawn on the catalog curve for 26 material:

Click here for larger image

Saturation-region curves can be approximated more simply using line segments on semi-log plots:

- log(
*H*_{T} /*H*_{0}) parameter = decades of range of transition of curve, a measure of how gradually the core saturates
- The larger log(
*H*_{T} /*H*_{0}) is, the more range over which saturation occurs
- Iron-powder (Fe-pwd) saturates over a 1.17-decade range, the most gradual transition

The following table corresponds to the above graph, with parameters for some common core materials.

Saturation is associated with the magnetic operating-point (op-pt) of the component, at the average *H* = *H̄*. Then on a *B*(*H*) graph, the op-pt is set at (*H̄*, *B̄)*.

**Maximum Transfer-Power Density**

Now we return to the main goal of maximizing transfer-power density;

maximum transfer-power density

Linear transfer energy each cycle =

where *V* is the magnetic core volume. Δ*W*_{L}/*V* is the linear energy density to be maximized for a given *f*_{s}.

Power transfer is proportional to *f*_{s}:

Therefore, to maximize transfer power, we must maximize the energy product, Δ*B* x *̄H* and *f*_{s}. Δ*B* is related to core power loss and *̄H* to saturation. When the maximum core loss and saturation (within design margins) are achieved, and with optimal *f*_{s}, Δ*P* is maximized. To close with a summary of this point,