Estimating ESR due to hysteresis in nail-cored inductor

[rs20]

If I use a standard steel nail as the core for my inductor, and I know the dimensions of that nail, the number of windings, the frequency I intend to operate it at (100 kHz or so for the record), and <insert properties of steel here>, how do I go about figuring out the expected ESR of the resulting inductor due to hysteresis? Intended use is a Colpitts oscillator.

Alternatively, does there exist an easily obtained material that comes in wire form, is high-permeability and low-hysteresis, yet physically malleable/formable into a loop?

[ Don't worry, I have a backup plan that involves cutting a slot in a proper toroidal core rather than bending a nail into an almost-complete circle, but I want to properly weigh up my options ]

[MrAl]

Hi,

Where did you get the idea that the hysteresis was that related to the ESR of the inductor?
Are you sure you want to know the ESR and not the core loss?
The core loss is an equivalent parallel resistor, not series.

The nail will have very low permeability and high core loss compared to metal made for inductor use.

Also, gapping a toroid by cutting slot in it is not something you usually want to do, because even a small gap lowers the permeability by a huge factor.  The kind of gap you would need is on the order of a few thousandths of an inch, not 1/16 inch for example.
One way to get a an equivalent some what small gap is to place the core in a napkin or clean rag, then place it between the jaws of a vise and crank the vise slowly until the core just cracks.  It should crack into about 4 pieces which can be glued back together with say super glue.  Even the thin glue changes the permeability by a lot so you get the idea.  The core wont be perfect but it will work that way if there are no extreme temperatures.

I dont know what kind of permeability you are looking for here though, or even why you care concerned with the ESR that you want to calculate it.  If you buy an inductor it will tell you the ESR on the data sheet or at least give you some idea what to expect.

The ESR comes from DC and AC resistance, and the AC resistance comes from the skin effect.  The AC resistance can be estimated.

[HackedFridgeMagnet]

The ESR comes from DC and AC resistance, and the AC resistance comes from the skin effect.  The AC resistance can be estimated.

I realise you are probably talking about impedance here, but we should be clear.

There is resistance, impedance, reactance and many ways to model components but no such thing as "AC resistance".


Put some known DC and AC voltages (near the required frequency) into your inductor and measure the currents. You should be able to characterise the inductor.

[macboy]

The ESR comes from DC and AC resistance, and the AC resistance comes from the skin effect.  The AC resistance can be estimated.

I realise you are probably talking about impedance here, but we should be clear.

There is resistance, impedance, reactance and many ways to model components but no such thing as "AC resistance".

As you so sure that there is no such thing as "AC resistance"? There sure is. As frequency goes up, the proportion of the conductor that actually conducts goes down, due to the skin effect. This increases the resistance as a function of frequency, and this effect is completely separate from the reactance due to inductance.

[T3sl4co1l]

You can exactly model any arbitrary impedance network as one R and one L or C, in series or parallel.  At exactly one frequency.

Whether that model is at all useful at other frequencies is beside the point.

Most often, a wider frequency range IS the point, and therefore there are a few typical models used to express things like inductors and capacitors of traditional construction.

The simplest is the series model, where you assume a resistance in series with an inductance.  At DC, the reactance is zero, so the resistance must be the DC resistance (DCR).  At some range of frequencies, the reactance is nonzero, and roughly proportional to F, therefore represented by an inductance.

The second and probably most useful model is the same, but with a resistor connected in parallel with the inductance.  Therefore reactance rises with frequency, until it's on the order of the equivalent parallel resistance (EPR), at which point the network resembles a small resistor in series with a large resistor.  EPR represents eddy currents in the wire and core, and approximates hysteresis loss too.

More and more accurate models can be constructed using more RLC components, often using ones which are frequency dependent, representing physics like diffusion and skin effect.  These can be quite complex.

As for your nail core, at frequencies more than maybe 100Hz, eddy currents will dominate.  You can calculate this based on the skin depth, which depends on permeability, resistivity and frequency.  When skin depth is less than a third of the diameter, magnetic field is no longer penetrating most of the depth, and the core ceases to act as a core but rather a resistor.  The EPR seen at the winding corresponds to the turns ratio: obviously, the core acts as one turn of whatever resistance it has, transformed up by the impedance ratio (which is the square of the turns ratio).

At 100kHz, I can guarantee the effective permeability of your core will be quite small, in the 10's, and the effective area even tinier (only a thin skin on the surface, which will be true even for very fine nails).

Tim

[rs20]

Thank you all for the great info! It's very interesting to learn that eddy currents are probably my main issue; makes a lot of sense only once it's pointed at to me! Clearly I need to re-think my core material, and possibly my entire approach -- I've posted a new thread explaining my actual project.

[MrAl]

Hi,

Sounds good.

Just to add a few notes (as others have explained all this very well already), the model for the impedance of a pure inductance in the frequency domain is:
ZL=s*L

and add some DC resistance and we have:
ZL=s*L+Rdc

and add some AC resistance and we have:
ZL=s*L+Rdc+Rac

and note that the Rac is a true resistance not the reactance although it does change with frequency:
Rac=f(w)

so it is considered real not complex.

With parallel resistance Rp (which can be caused by several things) we end up with:
ZL=(Rp*s*L)/(Rp+s*L)+Rdc+Rac