Inductor Q optimization

[TimFox]

In replies to another post   https://www.eevblog.com/forum/rf-microwave/energymagnetic-field-of-planar-vs-solenoid-inductor/msg4913938/#msg4913938 
mention was made of obtaining optimal Q for an inductor by making length and diameter of the coil equal.
I have seen similar rules of thumb, but decided to do a calculation for a specific set of assumptions.
I don't expect to calculate the Q itself, but wanted to see how it scales with the ratio of length/radius of the coil.
1.  Start with a popular formula for the inductance of a solenoid, which is more accurate than it looks.  I have used this formula to obtain the inductance of lengths of B+W "Miniductor" coil stock with good results.
L = (a² n²)/(9a + 10b), where a is the coil radius (in inches), b is the coil length (in inches), n is the number of turns, and L is the inductance (in uH).
2.  Specific constraints:  keep the L value constant, keep the wire diameter constant, always space the windings by one diameter so that the winding pitch is 2x wire diameter.
3.  Assumption:  assume the series resistance is proportional (at a fixed frequency) to the total winding length 2pi x a x n .  The proportionality constant depends on the wire diameter and a host of skin-effect and resistivity concerns that are assumed to be constant for the calculation.
I then calculated a "normalized" Q value for this inductance, as a function of the ratio of b/a = length/radius.
I was surprised to get this result:

Over a range of 0.1 to 10 for length/radius, the peak is not strong, but the maximum occurs near length = radius, rather than length = diameter.
The algebra is too complicated to post.
Has anyone seen a real treatment of this specific case, either calculated or measured?

[Conrad Hoffman]

I've seen something on the topic, but can't remember where. It would have been important to lower frequency ham stuff. I remember the Brooks coil formula, but that optimizes inductance, not Q. I've also improved Q with the right Litz wire.

[TimFox]

Litz wire is good for Q in the low MHz and below range.
In my calculation, I assume that the same wire is used over the range of length/radius ratio, with an RF resistance proportional to winding length.

[mawyatt]

Would think if you started with the self inductance of a straight wire, length/radius = infinity, then Grover's inductance formula should apply, so would expect a relatively low Q of a straight wire. As you bring the length back from infinity within a given radius, the resistance will reduce by length change while the inductance will decrease less as the fields begin to see other sections of the wire due to curvature an impede current flow, thus more inductance. At a length/radius of 1, seems the field coupling is maximum and thus maximum inductance per unit length, and as you get below one the field coupling decreases as the wire diameter approaches the radius and the Q drops.

Anyway, not surprised this looks as it does and peaks around length/radius of 1, length/diameter would reduce coupling vs wire "bend" it seems with a more stretched coil, thus reduce Q. A 3D field solver like Sonnet could be employed to study this more and other similar coil geometries.

Recall in 60~70s using tiny single & multi turn inductors implemented with wire bonds in hybrids and tuning them with tweezers by spreading or compressing the coil length while in Corning liquid clear uncured potting material.

Anyway, interesting topic!!

Best,

[trobbins]

I used Wheeler's formula (your #1 step) based on single layer coil radius and length, and included skin and proximity effects (after Medhurst) for a 2MHz class E converter.  The theoretical optimum Q for a given b/a was somewhat dependent on d/c (wire diameter to pitch).  Theoretical max Q had a shallow optimum for b/a varying from 1.5 to 7.5.  Measurements on 7 coils showed moderate agreement, given the measurement conditions at the time in early 1980's.  The project then went on with using a 4C6 pot-core, due to volume reduction.

[RoGeorge]

The project then went on with using a 4C6 pot-core, due to volume reduction.

Does this means that a magnetic core coil can achieve similar Q with an air coil, or that that project was not so demanding in terms of high Q?

Asking because I was under the impression that air coils would always have the biggest possible Q (biggest as in ten times bigger or so), though I'm a total noob when it comes to RF.

[Conrad Hoffman]

Air cores lose out because you need a lot more wire and thus have higher resistance. The right iron or ferrite core will almost always be better. OTOH, for best linearity for audio and such, an air core is often preferred.

[T3sl4co1l]

Simplified formulas will not help you here, unfortunately -- the mechanisms of loss are not simply current flow, but the current distribution within each turn, and varying along the coil itself.  The field close in between adjacent turns is much more inhomogeneous than the expected (naive?) overall solenoid field, with the effect of increasing eddy currents.

There is a solution however -- this exquisite calculator:
http://hamwaves.com/antennas/inductance.html

Given the discussion below, it seems quite accurate, and inductance at least has been within measurement error on every part I've inputted.  Q might expect higher error, given the nature of things, but I think can still be considered high confidence for most any application (say within 10%).

As a result, given the motivation in the other other thread --
https://www.eevblog.com/forum/renewable-energy/high-frequency-smps-what-are-drawbacks/
you're pretty much stuck either building it, or full-field simulating it; planar windings have considerable current crowding, proximity and edge effects.  Distance is your friend -- the further apart turns are from each other, the less of their tight close-in fields they experience -- but this is at odds with the desire (compact efficient design), not to mention the restriction (windings kept in-plane).  I suspect some combination of either very finely stranded litz, or round wire coils, and low-mu powdered iron, or widely gapped NiZn ferrite, would be the angles to pursue.

Also, if the converter is a simple buck/boost, CCM can be targeted, and inductor Q becomes asymptotically irrelevant.  This is already apparent with the use of #26 or #52 powder at ordinary switching frequencies (~100kHz), where Q is pitiful (5 or 10?), but because the material is cheap, excessive inductance can be used, pushing deep CCM (preferably using an average current mode control), thus reducing the circulating (reactive) power -- power that, by Q factor, gives core loss.  (This doesn't work so well for resonant and isolated configurations, where a transformer is required.)

Tim

[T3sl4co1l]

Does this means that a magnetic core coil can achieve similar Q with an air coil, or that that project was not so demanding in terms of high Q?

Asking because I was under the impression that air coils would always have the biggest possible Q (biggest as in ten times bigger or so), though I'm a total noob when it comes to RF.

Ultimate (maximum possible) Q, I'm not sure about, but for practical purposes, that's rarely the issue; and it'll be of considerable size and cost.

I've made and measured ferrite cored inductors at around Q = 400 (PQ32/20 sized core, ~100kHz).  From my calculations, ~1000 should be possible, but it'll be significantly larger, like PQ50/30 maybe, and that's a lot of litz -- and even finer at that (i.e., >38AWG at 100kHz) due to proximity effect.

At radio frequencies, air core is effective enough, and litz value declines; litz should still be usable say in SW band, but in configurations where proximity effect is minimal (single-layer solenoids, toroids?), and preferably with the strands themselves held apart (e.g. insulating rope core surrounded by strands, maybe strands alternating with fibers as well).  By VHF, round wire does good enough (Q in the 100s even in small coils -- see ceramic core chip inductors by most major brands), and by UHF+ you may be considering transmission line or cavity structures instead; also with skin depth being so small up here, silver plating is quite economical to eke out that little bit lower insertion loss or whatever.

Tim

[TimFox]

@T3sl4co1l:
Thanks for the link to that inductor calculator.
Many years ago (ca. 1990), I also used the 1947 Medhurst reference to look at effective resistance (including skin effect and proximity effect) of air-core coils, but I no longer have the paper.
As I remember, Medhurst had just acquired a new General Radio 821A Twin-Tee bridge to measure the empirical values of Q.
I was only doing my simple optimization as a thought experiment:
For the optimization, I assumed constant wire diameter and winding pitch, varying the solenoid's radius and length to obtain a fixed value of inductance.
The important assumption was that the wire resistance (including high-frequency effects) would scale with the winding length, but the calculation did not give actual Q values;  I did not vary the pitch as part of the optimization.
Apparently, Medhurst shows us that if we vary the coil dimensions, the HF resistance per unit wire length at constant pitch is not constant.

[trobbins]

Medhurst paper: https://hamwaves.com/inductance/doc/medhurst.1947.pdf

Does this means that a magnetic core coil can achieve similar Q with an air coil, or that that project was not so demanding in terms of high Q?

An aim was to make a relatively compact dc/dc, such that the spacing around an air-core inductor (if it was to be shielded from egress, and insensitive to placement) was not going to allow a similar outcome to a pot-core inductor solution.  Of course that then has ramifications to the core losses and how much inductor current can be allowed, and another round of optimisation of lowering total inductor loss due to core material and gapping and padding any coil away from a core gap and the winding configuration used within the potcore (there were no RX or core varients back then, and only 4C6 was available).

[mawyatt]

.... and by UHF+ you may be considering transmission line or cavity structures instead; also with skin depth being so small up here, silver plating is quite economical to eke out that little bit lower insertion loss or whatever.

Tim

Tunable helical coil based filters sometimes have silver coatings, some waveguides do also.

BTW thanks for the link to the coil calculator, revels the complexity of these devices we call "coils"!!

Best,

[k6sti]

In the checks I've made against coils measured with an HP 4342A Q meter, this coil calculator is more accurate than the online calculator mentioned:

http://ham-radio.com/k6sti/coil.htm

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Brian