# Inductor Saturation

While writing a column for Circuit Cellar, I managed to utterly botch the choice of a boost converter inductor. The inductor (the rightmost one in the lineup) was a common-mode choke from a power supply, so I figured it had enough meat for 300 mA of boost current.

Bzzzzt! Wrong choice!

So it’s time to document how to measure this stuff. The intent is to plot a BH curve: core flux density B versus magnetizing force H.

Quick inductor tutorial using (ptooie) Imperial units and avoiding Greek letters for simplicity:

Magnetomotive force mmf = 0.4 * pi * (N = turns) * (I = current)

induced voltage e = – N * (d phi / d t) * (10^-8) where phi = flux linking the N turns

Magnetizing force H = mmf / (MPL = mean path length around core)

Flux density B = phi / (Ac = effective core area)

also B = (mu = permeability) * H

You might think that B & H are lashed linearly together, but mu is most savagely nonlinear. It depends on H, temperature, core material, and a bunch of other stuff you don’t want to know about. So, by and large, plotting B against H shows you how mu varies: it’s the slope of the curve at any point. Vertical slope = high mu (good), horizontal slope = zero mu (bad).

Stirring all that together, you get the Fundamental Transformer Equation:

E in RMS volts = 4 * F * f * N * Ac * B * (10^-8)

F = form factor = rms / average

• sine = 1.11
• bipolar square = 1.00
• unipolar square = 1.41

So…

• H is proportional to mmf and thus to primary current
• B is proportional to phi and thus the integral of secondary voltage

Sounds scary, but primary current & secondary voltage are easy to measure.

The basic lashup goes a little something like this: stick a sampling resistor in the primary, run the secondary through an RC integrator, feed both into an oscilloscope that can do XY plotting, drive the primary from a Variac, turn the knob, and watch at the results. You might want a bit more finesse than this, but, eh, it works.

The primary voltage will be relatively low, so plug a 12 VAC wall wart into the Variac. This gives you galvanic isolation from the power line, finer control over the primary voltage (more of a full turn on the Variac), and will prevent you from killing yourself or burning out your basement laboratory. You want a fairly husky wart if you’re measuring husky inductors. Remember: this is an AC measurement, so you want an AC wall wart, not one with a nice filtered DC output.

The resistance of the sampling resistor should be much less than the inductive reactance of the coil. We’ll measure it at 60 Hz because that’s easy, so for a coil of inductance L, the reactance at 60 Hz is 2 * pi * 60 * L. That simplifies to 377 * L, so a 1 mH coil has about 400 m-ohm reactance. I have a nearly infinite stash of 100 m-ohm sandbox power resistors, so that’s what we’ll use; you’d want less, but this is quick & easy.

If the inductor doesn’t have a secondary, poke some magnet wire through the core. Use some simple number of turns and remember that number, just in case you want to calibrate the results.

The time constant of the RC integrator must be a lot bigger than the frequency you’re integrating. For a 60 Hz signal, you want maybe a 6 Hz integrator: 60 Hz -> 17 ms, so pick 200 ms. Having a nice 1 uF film cap in my heap, the resistor works out to (200×10^-3) / (1×10^-6) = 200 k. Anything in that range will work fine. A larger cap gives you a smaller resistor and more signal to the scope.

Wired it up about like that and here’s what happens for the inductor I picked

The calibration of H is 100 mV / amp (from the 100 m-ohm resistor) and the scope display is thus 2 A/div. The calibration for B is whatever the secondary winding provides, scaled by 1/RC from the integrator.

For this coil, the value of mu is essentially infinite (B rises vertically for changes in H), but the core saturates at very small values of H (and thus primary current). It was a common-mode choke for a power supply, so that’s exactly what you want: high inductance for very small currents and no need for much in the way of core flux because the opposing line currents in the windings cancel.

I harvested some inductors from a bunch of dead power supplies in my heap and measured them similarly, using an existing winding as the secondary wherever possible. The results look like this:

The area inside the loop is proportional to the core power loss, so it’s obvious that some inductors are better than others.

If you’re going to do this for real, you’ll want actual calibrations for both axes. You can work the numbers out from the equations and read real values for both B and H directly from the ‘scope graticule. Pretty slick, mmm?

The primary current can get surprisingly high, so turn the Variac knob up, make your measurement, and turn it back down again. The sampling resistor and the core can get scary hot if you let ’em cook while pondering the ‘scope display. A storage scope is most helpful, which comes pretty much for free with digital scopes these days.

I wrote about all this in more detail in the context of a resistance soldering unit in my February 08 Circuit Cellar column, so I should have known better.

Memo to self: measure first, write later.

And remember to turn off the WordPress Smiley option (Dashboard → Settings → Writing) so equations don’t get cutsy artwork instead of numbers. Feh!