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[Producing ↑ wound components]
[A guide to the terminology used in the science of magnetism]
[ Power loss in wound components]
[The force produced by a magnetic field]
[Faraday's law]
[Flux in a multilayer coil]
In wound components the magnetic core plays a vital role in raising the level of magnetic flux produced by a given current flow in the windings. That is to say, it increases the inductance.
Unfortunately, the increase in flux provided by ferromagnetic materials is never a simple linear factor. At first, as the field strength is raised, the flux increase is a modest one. At higher currents the increase is more rapid, but at yet higher currents the flux increase slows again. Finally, saturation is reached and any further current increase generates but a minuscule increase in flux.
These changes in inductance with current are a nuisance at best. In high power transformers, however, saturation must be avoided at all costs because the ensuing drop in inductance leads to a sharp increase in magnetizing current, further core overload and a 'runaway' situation that often ends with smoke and flames :-(
Also, ferromagnetic materials all suffer from temperature dependance. In resonant circuits, for example, you normally prefer that the inductance remains constant with temperature and not detune the circuit as the environment changes.
In a high-Q tuned circuit you almost certainly want to run the core well below its saturation level - not through fear of flames but because you cannot afford to loose power to core hysteresis (which increases rapidly with B).
These three problems are usually alleviated by using a gapped core.
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In an ordinary inductor core the magnetic flux lines are guided all the way round the windings by way of the ferromagnetic material in the core. However, in a gapped core a small section of the flux path is replaced by a non-magnetic medium - such as air. The term 'air gapped core' is still used even if the gap is filled not by air but by nylon or some other material immune to saturation.
Some core types may be purchased with a gap prefabricated, such as the
RM7 shown above; but the photo at right shows a 100 mH EC70 core in
which the gap has been created by inserting sheets of cardboard between
the core halves. Adhesive tape can hold the core together
adequately because clamping force requirements are less than for an
ungapped core.
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Your first thought may be that if the core is overloaded then what you need is a larger one, and that the last thing you would want to do is remove part of it! Well, magnetics is a curious business.
Consider an analogy: you have a battery with an EMF of 12 volts and a two ohm load. However, you don't want six amps to flow round this circuit - that's too much; you only want four amps. What's the simplest way to achieve that? The answer is a no-brainer: you insert a one ohm resistor between the battery and the load. You now have a total of three ohms in the circuit and so all that a 12 volt battery can deliver into it is four amps. Job done.
The problem which an air gap solves in a core is the excessive flux produced by a high level of current in the windings. Transferring the previous analogy, suppose that your windings produce an MMF of 12 ampere turns and that your magnetic core (your 'load') has a reluctance of two ampere-turns per weber. However, a flux of six webers is too much; you can only tolerate four webers. To solve the problem you just add in a 'resistor' (call it a 'reluctor') of one ampere-turn per weber. Now you have a total reluctance of three ampere-turns per weber 'in circuit' so that now four webers is all that the MMF can push round the core. Job done.
You may have guessed that the role of this extra reluctor is taken on by the air gap. Its task is to increase the reluctance of the core so that less flux flows for any given level of MMF. The virtue of the air gap is that, because it is free from ferromagnetic material, it does not suffer any change of reluctance with flux level. Its reluctance depends only on its length, l_{g}, and cross sectional area, A_{e}; and both of those parameters you can make very stable.
In summary, the gap reluctance is -
while the ferrite reluctance is -
These results combine in series to give the reluctance of the inductor
as a whole -
From which you see that a large value for μ_{r} leaves the
air gap as the only significant contributor to the overall reluctance.
If you use a spacer then l_{g} will be twice the spacer thickness because flux must cross the spacer twice in a complete circuit of the core.
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Almost. Remember why you use a core in the first place: to raise the inductance of your windings. When you add an air gap to increase the reluctance of the core then it is almost as if you have decreased its permeability, and thereby lowered the inductance of a winding on it. Indeed, when you buy a core with a pre-fabricated gap then the manufacturer may specify what is called the effective permeability of the core, μ_{e}. It may be 20 times less than the 'raw' value for mu of the ferrite. However, you need only substitute this value of mu for that of the actual material and then calculate your inductor design (using μ_{e}) in the same way you would normally.
An analysis of the core and gap as a two component series circuit gives
Adding the gap therefore spoils the inductance. However the inductance
depends also on the number of turns, so what you normally need to do is
wind on more turns. At this point you may raise the following
objection. With more turns the magneto-motive force will increase
which will lead to higher flux once more, and you are hit up again by
core
saturation. This argument is only partially correct because
inductance increases according to the square of the number of turns
whereas mmf is linearly proportional to N. Eventually,
therefore, your additional turns strategy will always win through -
assuming you don't run out of space for the winding.
Looked at another way, what you are doing is to shift the performance burden away from the core and onto the copper wire - after all, an air cored inductor can never saturate. Unfortunately, all those extra turns take up extra space in the winding aperture, and they can't be made any thinner because they still need to carry the same current. So you may find that you need a larger inductor anyway. There's no free lunch :-(
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Because air has a permeability in the order of 10^{3} times lower than ferrite it so happens that the gap often accounts for almost all the reluctance seen by the magnetic field, leading to an interesting property of practical gapped cores: the gap contains nearly all of the field energy.
You may think it strange that energy should prefer to reside in thin air rather than the core material. If so, consider again the current flow analogy developed to explain the method of soothing an overloaded core. Energy dissipation (ie power) in a resistor is given by P = R × I^{2}. The current flow, I, is the same all round the circuit: the battery, the load and the added ballast resistor all carry it. Clearly, then, the power dissipation in the two ohm resistor is twice what it is in the one ohm ballast.
Although the reluctor analogues dissipate no power they do store it -
To a reasonable approximation, you have the same flux, Φ,
'flowing' round all parts of the inductor: the windings, the core and
the air gap. This means that the greatest energy storage takes place in
the larger 'reluctor'. For all but the shortest of gaps it is the air
which has more reluctance.
The reason this is worth stressing is that you may usually equate the energy in the gap with the total energy possessed by the inductor -
Also, you should not be surprised to learn, there is, for every core
type, a limit to the amount of energy that may be stored. In
the next section you will discover that there is an optimum air gap
length, l_{O}, which will deliver that energy. Table AGT shows
l_{O} for a range of core types, under the following conditions -
Basic type |
Size | A_{e} / mm^{2} |
l_{e} / mm |
l_{O} / mm |
A_{l} / nH |
F_{m} / A-t |
B_{pk} / T |
W_{L} / mJ |
---|---|---|---|---|---|---|---|---|
ER | 9.5/2.5/5 | 8.47 | 14.2 | 0.025 | 312 | 5.88 | 0.3 | 0.0103 |
RM | 7 | 44.1 | 30 | 0.18 | 274 | 43.5 | 0.3 | 0.319 |
RM | 10 | 96.6 | 44.6 | 0.35 | 318 | 84.0 | 0.3 | 1.32 |
ETD | 44 | 173 | 103 | 1.83 | 115 | 437 | 0.3 | 11.8 |
EC | 70 | 279 | 144 | 3.88 | 87 | 927 | 0.3 | 38.8 |
E | 80/38/20 | 392 | 184 | 7.69 | 63 | 1840 | 0.3 | 110 |
U+U | 93 | 840 | 354 | 25.7 | 38 | 6130 | 0.3 | 772 |
U+U | 141 | 1350 | 377 | 26.1 | 72 | 6240 | 0.3 | 1260 |
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Param symbol | Value | Unit symbol |
---|---|---|
A_{e} | 44.1E-6 | m^{2} |
l_{e} | 30.0E-3 | m |
B_{pk} | 0.3 | T |
μ_{r} at B_{pk} | 1500 | 1 |
A_{w} | 21.7E-6 | m^{2} |
As an inductor designer, then, you appear to be caught between a rock and a hard place. Make the air gap too short and you end up with core saturation. Make the gap too long and you run out of space for the windings. It's time to calculate a course which avoids these monsters. Your best approach to this problem is to consider how the inductor energy, W, varies with gap length.
Consider first what might be dubbed the 'aperture limited case'. This presumes that, although the total magneto motive force is fixed by the area of the winding aperture, the core flux can assume whatever value is then determined by the gap. The inductor energy is then
Clinging on to the current flow analogy, this is akin to finding the
power dissipation to be P = V^{2} / R. Substituting
equation AGD into equation AGK -
The term in l_{e} may usually be ignored. The energy becomes
an inverse function of the gap length (see figure AGO).
Next, consider the 'flux limited case'. Under this assumption the core flux, Φ_{pk}, is set to the peak value which the ferrite can support; but you don't worry about the MMF necessary to produce it. The variation of energy with gap length is now different
The current flow analogy finds the power dissipation to be P =
I^{2} × R. Substituting equation AGH into equation
AGK -
The term in l_{e} may again be ignored. This time, core energy
is linearly related to the the gap.
Well, as figure AGO shows, there can be just one gap length which yields both Φ_{pk} together with the maximum MMF -
Then substitute equation TMS to obtain
the notably concise result -
The term in
l_{e} may usually be ignored.
So, broadly speaking, when you select a specific core then you have set
a limit on the amount of energy you are able to store, and on the gap
length that delivers it. Manufacturers sometimes specify this information
in the form of Hanna curves, which show how inductor energy varies
with gap length.
Consider a concrete example for the case of the RM7 core, for which
typical parameters are given in table AGN. This gives the energy
characteristics shown in figure AGO. A value of 2×10^{6}
Am^{-2} is taken for J_{W}, which is a realistic
practical value. That leads to a value for
F_{m} of 43.5 ampere-turns.
Only when the gap length is about 0.18 millimetres will the flux produced
by a fully wound aperture drive exactly 0.3 teslas round the core. One
manufacturer supplies RM7s ready gapped at 0.24, 0.41 and 0.73
millimetres. This spread suggests that the above result is
at least of the right order.
The narrow outer limbs of RM cores are especialy prone to fringeing, so a ready built centre limb gap will conform better than a gap fabricated with a spacer. Reliable estimates of the reluctance of large air gaps is possible with the assistance of magnetics modeling software. Alternatively, you should just increase the gap until the inductance factor, as measured, drops to the 'theoretical' linear model value.
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The advantages of an air gap can be summarised
The disadvantages of the air gap are -
The most serious of these is usually leakage. Values as high as 20% are seen in practice (particularly if a spacer has been used for the gap) and this can have a disastrous effect upon the efficiency of switching supplies. Those using the 'flyback' principle with gapped transformers are especially vulnerable to leakage effects.
As the gap size increases, flux will start to 'spray' into the area of the windings in proximity to the gap. This will generate high levels of eddy current losses in the wire. To avoid this problem you can wind on a few turns of insulating tape around the middle of the coil former to exclude wire from the region.
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Let's suppose you need to design a relatively large inductor: 470μH at 40 amps. Straight away, equation AGC gives the stored energy -
That's a whopper. Well, table AGT
shows that a U93 core pair is the smallest which can store that much
energy. In fact, since the U93 can store 772 mJ with the optimum gap
(about 18 mm) there should be a range of air gaps that might permit the
basic specification to be met. The lower limit for the gap will be
decided by the flux limited condition. Putting the datasheet values
into equation AGU -
The term in l_{e} is of little significance.
The upper limit for the gap will be decided by the aperture limited
condition. Substituting into equation AGQ -
So, which gap length in the range 12.3 to 25 mm do you choose? The
answer depends on what other constraints you have. For example, are you
more worried by core losses than copper losses? If this inductor is
intended for use as an output filter then the flux swing will be small,
and a short gap will be best. OTOH, for use as a flywheel inductor in
the main switching circuit then the flux swing will be large. Even at
frequencies as low as 25 kHz a swing of +/- 0.3 tesla will likely
generate excessive core losses. This is reducible through a larger gap
to cut down B_{pk}.
Any of these gaps will be more than 10% of the limb cross section (2.8 mm), so you will need to experiment a bit. Assume that core loss is your enemy, leading you to favour a gap of 25 mm. That will mean a spacer thickness of 12 or 13 mm (half an inch will be fine). Substituting this into equation AGK -
and this yields an inductance factor factor of 42 nH, so
for 470 μH you need 106 turns. Unfortunately, such a huge gap will
lead to high leakage flux, and the simple analysis above will not
be too accurate. Still, it's a start.
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The photograph below shows an EC30 core being ground to provide air gaps
in the outer limbs. Although it is normal instead to grind the centre
limb, this technique may provide some reduction in leakage reactance.
It should be possible, by means of an annular abrasive pad, to machine
the centre limb instead.
The bath is constructed from folded up tin plate, and is filled with
water to cool and lubricate the core. The drill press is rotated at just
86 RPM. A fresh sheet of emery paper will remove a millimetre of core
in about three minutes, but will become less effective quite rapidly.
The quill feed needs to be adjusted at intervals to maintain the core in
a horizontal position.
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E-mail:
R.Clarke@surrey.ac.uk
Last modified: 2011 May 26^{th}.