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Simulating Reluctance in a Xfmr Model

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Enceladus

Tue Feb 21 2017, 03:10PM

Registered Member #61428 Joined: Sat Jan 14 2017, 12:39PM
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Posts: 50

Let me just start by saying I simulate everything in EveryCircuit 2.18 for Android. It is simple, intuitive, and simulates the live action evolution of circuits. It is useful and fun at every skill level. I simply cannot recommend it highly enough.

That said, you don't need to use it to help me, but it would help. Don't talk to me about PTSDspice.

So in EveryCircuit, for those of you who don't use it, the transformer has 4 parameters you can define: Primary turns (Np), Secondary Turns(Ns), Coupling coefficient(k) and Primary Inductance(Lp). (Secondary inductance depends on Lp and Ns:Np.)

In order to define the series resistance, shunt conductance, parasitic capacitance, leakage inductance, and any other non-ideal properties of the windings, you must use separate resistors, inductors and capacitors in an equivalent circuit and then define them that way. Besides those elements you can also simulate ideal VS, CS, VCVS, VCCS, CCVS and CCCS. (And a great many other functional elements not relevant here)

That's fine for transformers that have cores with constant permeability, but I'm having trouble trying to derive equivalent electrical circuit models to represent a transformer having a saturable high-permeability core with no air gap and what happens as a gap is introduced.

I did some reading on flyback transformer regulator design just left me more confused. It's often said that gapped cores used in LOPT flybacks store energy in their air gaps, but are designed to have low leakage inductance. So how can the gap store magnetic energy exactly like an inductor without appearing to the circuit AS leakage inductance? If the effect of added reluctance does not appear as leakage inductance in series with the mutual inductance, where DOES it appear? Granted, I only skimmed the mathematical descriptions of the FBT regulator design guide. I was hoping for schematics, not equations. For me, understanding the math comes later.

The trouble I'm having is with what the relationship is between reluctance, and leakage inductance. Can the magnetic reluctance in a transformer be modeled by adding leakage inductance to one or both of the windings? Can you have one without the other? Is it even possible to construct a useful model for these things with the degree of freedom I have at my disposal? If the problem is with my simulator, how would the model manifest in a more sophisticated simulation environment?

It's quite a tangled mess of half-digested factoids and unknown unknowns.

I'm only here to find out what you know.
So let me know.

Sulaiman

Tue Feb 21 2017, 05:17PM

Registered Member #162 Joined: Mon Feb 13 2006, 10:25AM
Location: United Kingdom
Posts: 3140

An airgap in a magnetic circuit causes an increase in reluctance, regardless of coupling.

if you have two windings on one continuous magnetic core section
then (almost) all of the magnetic flux passes through both coils - the coils are tightly coupled
there may or may not be air gaps in the core elsewhere.

if you have two windings, each on a separate magnetic core, connected by a magnetic circuit with gaps, then not all of the flux in one coil flows through the other - loose coupling
there may or may not be other gaps in the core elsewhere

So, coupling is a measure of how much of the magnetic flux in two windings is the same
1.0 = all of the flux passing through one coil passes through the other
<1.0 = some of the flux takes a short-circuit bypassing one of the coils.

Enceladus

Tue Feb 21 2017, 06:06PM

Registered Member #61428 Joined: Sat Jan 14 2017, 12:39PM
Location:
Posts: 50

Right, but I wasn't really asking about coupling. In fact, I'm most interested in modeling the behavior of modern CRT FBT's, and common MOT's with a 2 part "E-I" core that is easy to separate. Both have tightly coupled windings, wound adjacent or concentric in the same leg of the magnetic circuit. I would set the coupling of both at >0.9.

I thought maybe since an increase in reluctance decreases the permeability of the magnetic circuit and thus the inductance of both windings, maybe the an air gap acts more like an inductor in parallel with both windings, but that's just a guess. Even a small air gap in a high permeability core can reduce the inductance of all windings by a factor of 10 or even more when compared to without the gap. If I had my equipment I would figure this out experimentally.

DerAlbi

Tue Feb 21 2017, 06:38PM

Registered Member #2906 Joined: Sun Jun 06 2010, 02:20AM
Location: Dresden, Germany
Posts: 727

Aww your problem is not easy to solve. Honestly the only thing that comes to mind is using FEM based modeling at this point, but thats hard to couple to a netlist.
The problem with saturation is that not only that all the inductances change but also the coupling factor between them. (Since an air core == highly saturated core will yield a different k than an unsaturated core.)
If you have arbitrary behavioral sources like Spice has then you may build yourself a mathematical model.
This means you keep track of the integral of the applied voltage to get the current, derive the saturation level from that, estimate the differential inductance from that and feed it back to the integral.
That alone only simulates a saturating inductor (which spice has built in). To get the coupling done .. its a whole different story, since you have 4 time-varying inductances (stray and mutual on both sides) to keep track of and to conserve energy in...
i have done this before however i never verified the results.. because its hard to do.

Steve Ward

Fri Mar 03 2017, 07:47PM

Registered Member #146 Joined: Sun Feb 12 2006, 04:21AM
Location: Austin Tx
Posts: 1055

Transformers can be confusing... especially the non-linear stuff. Here's my thoughts, not sure they are actually "correct", but i do get what i believe to be useful approximations of the real world.

I like to model transformers as a series leakage inductance (Ll) feeding a shunting "magnetizing" inductance (Lm), which is shunting the primary inductance (Lp) that has perfect coupling (K=1) to the secondary inductance (Ls). The inductance of Lp is fictitious and is so large that it doesn't consume appreciable current (assuming secondary is open circuit), lets say Lp = 100*Lm. That is, all the current should flow through Lm, except any current that flows through Ls via magnetic shielding (aka, transformer action). Ls is of course Lp*(Ns/Np)^2. To measure Lm you'd measure the primary inductance, then subtract out the leakage Ll (measured by shorting the other winding).

The only way i know of simulating saturable materials is in LTspice, but you don't wanna talk about it

. In LTspice you can assign the flux of an inductor (which has units of volt*seconds or inductance*current) but you can make the permeability part (the inductance) take on some function like tanh(current/saturation_current) which will sorta give you an idea of the effect. Basically, flux = Lm*Isat*tanh(I/Isat) or something to that effect (add coefficients to get the shape how you'd like). This saturating inductance is the Lm in my model described above. The theory is that a saturating core loses magnetizing inductance and draws more current for a given volt-seconds applied. Maybe your simulator allows some definition like this rather than simply assigning a fixed inductance?

When introducing an air gap to a core (no shunts): The leakage inductance does not (or should not?) change as it is indeed from the flux that doesn't pass through the core/other coil (by the way, you can model leakage inductance on either side of the transformer, typically it's only put on the primary side as it's easier to think of it as a series impedance with the "driving" source). The "magnetizing" inductance does drop as the gap increases and so consequently the coupling coefficient drops too (that is, Ll stayed the same, Lm got smaller, so the ratio of Ll/Lm went up). Gapping a core allows the magnetising force from the coils (H) to increase (more current) because the permeability of the core material is essentially reduced by the added reluctance of the air gap and we still obey that Bcore = u*H. This is why they gap flyback transformers, so that they can hold sufficient energy. The L drops but the current can go up and energy goes with I^2, so its a win overall.

Microwave oven transformers are more complicated with the shunt flux path (which, id assume is saturating as its cross section is much smaller than the rest of the core). The shunts look like an increased leakage inductance, or at least has that effect, so current is limited when the secondary is shorted. So as the shunts saturate (at some amount of volt-seconds applied to primary) their effect should reduce some, and so relatively more flux will get to the secondary. This would be modeled, perhaps, as an additional series leakage inductance that saturates. Also, the main core saturates on MOTs at about 80% of their rated voltage from what i remember, so this looks like a sudden loss in magnetizing inductance at enough volt-seconds. You could model this as a saturating Lm in parallel with the primary of the ideal (K=1) transformer.

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