[102] Custom Inductors – ONE Design Equation

Dr. Ray Ridley discusses his introduction into Power Electronics. An early foray into design optimization software revealed a curious fact about magnetics design - there is only ONE design equation that must be obeyed. 

My Introduction to Power Electronics


Over 35 years ago, I started my career in power electronics. I completed a practical power project in my senior year in college, building a 6 KV, 20 kHz AC output power supply. It was a crash-course introduction into the world of power and set the course for my career. Pushing power through a transformer where nothing on the primary touched anything on the secondary seemed like magic, and I was hooked.

After university, my first interviews were with computer companies in the Boston area, and I really thought that my future was in logic design. However, as interviewers learned that I knew something – anything! – about magnetics and power supply design, I was shunted off to the power department.

In those days, switching power supplies were just beginning to take over the computer industry, and there was a desperate need for design expertise. My first officemate became my mentor – a wonderful man named Spriadon (Dan) Balulescu. On my first day on the job, he put Dr. Cuk’s dissertation on my desk and asked me to explain it to him in a couple of days, I think to prove that I was worthy of his time.

Soon after, we embarked on a 1 kW full-bridge power supply for a minicomputer. There was little guidance. FETs were just becoming practical, and there were no current-mode control chips.

I took on the job of simulation, electronics and control design, and Dan did the magnetics design, thermal, and layout. He would work for days at a time on the magnetics, huddled over a notebook, and a working specification would emerge to send off to our vendor. A few weeks later, samples would return. We placed them in the circuit to determine what worked and what didn’t. Two iterations were usually enough to get the job done properly.

Throughout the process, over a period of two years, the magnetics design remained a mystery to me. It was the domain of the older, experienced engineers and the magnetics companies. Dan promised me that he would give me all his notes when I left the company to return to graduate school. He was good to his word – but I was no more enlightened after reading them than I was before.

Graduate School and Magnetics

After working for a few years on products at my first job, I went back to graduate school to study under Dr. Fred Lee at Virginia Tech. Dr. Lee had already built a reputation for hiring the best in the world, and I was honored to be included in his group. All graduate students were assigned projects for industry. I was given two – analysis of parallel power converters for IBM, and power converter design optimization for NASA. The NASA project involved writing nonlinear optimization code with Lagrangian multipliers to get to an optimum solution for selection of power parameters

This is where I was first exposed to the inner workings of magnetics design. The nonlinear optimization code worked to move parameters around (switching frequency, core sizes, etc.) to try to find a minimum objective function. In this case—the weight of a converter. The power loss in components was penalized by the weight of a required heatsink. There were many calculations to estimate the performance of the design, but I found something remarkable at the center of the code. There was only ONE equation that had to be obeyed to make magnetics work.

This was a revelation to me – ONE equation? How could that be? Why did magnetics books make it so complicated? They all had this equation, but quickly moved on to discuss hundreds more. I posed the question to everyone in the research group at Virginia Tech. Did they know that there is only one equation for magnetics? Everyone was puzzled. They each had a methodology, but could not describe the process.

ONE Equation

Figure 1 shows the basic structure of a generic core with a winding. This shows everything you need to build a custom inductor. The core can be any shape, with any cross section – square, rectangular, round, or some other shape. The core material sets the permeability, and the ability to open a gap allows adjustment in the permeability of the structure. The gap can also be embedded in the material, but this is a more specialized case for future discussion. There are negative consequences that come with a pre-gapped core material, along with some advantages. For peak performance the inductor will be a gapped ferrite in most cases.


Figure 1: The Basic Constituents of a Custom-Designed Inductor.

There are a lot of design freedoms offered by this structure – core size, number of turns, wire size, permeability, and gap. But where to start with a design? We can look at the basic magnetics equations to decide. The inductance of the structure is given by:


The area and the path length are shown in Figure 1. The effective permeability of the core, µ, is given by:


We don’t have any idea how to deal with the permeability at the outset. It is very variable in the core with a wide tolerance in the specification. We will control this later with the gap in the core

The maximum excitation, H, is given in terms of the peak current in the inductor by:


The maximum flux of the core is then


But we do not know how to select any of these variables. So, we do some math tricks:


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