Estimation of open-circuit inductance from the leakage inductance

Question

The title may not reflect the problem I explained below. Please feel free to update.

For a power supply application, I designed a transformer with a primary turns of 15 and a secondary turns of 4+4 (centre-tapped):

^{simulate this circuit – Schematic created using CircuitLab}

The core is PQ3225, the material is DMR44 (almost equivalent to N87). The winding structure is like this:

Here are the specs:

• \$L_{OC}\$, open-circuit inductance: 105 μH
• \$L_{PX}\$, primary leakage inductance: 2.8~3.0 μH
• \$R_{DC-p}\$, primary winding resistance: 36 mΩ
• \$R_{DC-s}\$, very low to measure, should be a few tens of μΩ
• \$l_g\$, centre-leg air gap length (calculated): 0.4 mm approx. (the manufacturer may adjust to meet the \$L_{OC}\$ spec).

And here's my question:

The transformer does its job really well, and the power supply works absolutely fine. But the leakage inductance of the primary winding plays a role on general operation and I may need a little bit higher leakage inductance, around 4 μH, for a slightly lower switching frequency (if needed). At first I thought of increasing the gap length as this will increase the \$L_{PX}\$, and decrease the \$L_{OC}\$. But I can't estimate the \$L_{OC}\$ for the required \$L_{PX}\$. Is there way to estimate the open-circuit inductance from some of the existing parameters? I'm too lazy to prepare quite a few number of samples to measure. Instead, I'd like to put a rough number into the spec sheet as a starting point for the manufacturer.

Answer 1

After reading Andy's comment and thinking about the problem a bit more, I realised that increasing the gap length will not change the leakage much.

Instead, the coupling between the primary and the secondary should get worse by, for example, increased effective distance between the primary and the secondary (e.g. more layers of tape or having the windings side by side instead of stacked) in order to have a higher leakage.

This also answers my main question:

Is there way to estimate the open-circuit inductance from some of the existing parameters?

The open-circuit inductance, \$L_{OC}\$, can be kept under control via the gap length therefore it can be estimated with an acceptable accuracy. However, the initial leakage inductance stays the same (pretty much) but since the open-circuit inductance decreases with higher gap (because of the reduction of the permeability), the percentage of leakage in open-circuit inductance increases.

Answer 2

In conventional (layered or banked) windings, leakage inductance is dominated by the distance between windings, or more particularly between wires of the windings. This is -- quite literally -- the flux that isn't shared by the core, that "leaks out" from it, but it also means the flux that never got there in the first place, flux close in, around/between the wires themselves.

As a result, leakage generally varies little with core size, permeability, air gap, etc.

I find that, understanding transformers as transmission line components, is very illuminating in this regard: for the case of a 1:1 transformer wound from bifilar wire (or twin-lead or twisted pair, whatever), there is some transmission line impedance and velocity, and thus inductance and capacitance (leakage in the differential mode, isolation capacitance in the common mode) are determined wholly by the cross-sectional geometry of the windings as a transmission line, and the wire length. In such a geometry, very little leaks out to the core

For multilayer windings, banked windings, etc., this understanding becomes more tenuous; more flux spills out from the windings into nearby materials. But a different but similarly deep lesson remains true: transformers are characteristic-impedance, band-limiting networks. The details of where (in space) their network components manifest (like leakage inductance) may vary, but it will always be down to the distance between conductors, and their geometry; this might not help with calculating leakage from first principles (say because the geometry of multi-layer banks is nontrivial), but it is encouraging to know you can always measure values for the same basic 2nd-order equivalent model and expect that model to apply over a reasonable frequency range.

Arrangements where leakage does depend on air gap or core geometry (or, are at least more sensitive), include a split-bobbin winding where primary and secondary are wound on respective core halves: in that case, the flux leakage around the air gap (between center peg and the legs) can act as a magnetic shunt (or more specifically, a flux divider, between the air gap between core halves, and the air gap between legs of a given core half), and as air gap increases, magnetizing inductance decreases while leakage increases.

In the extreme case (two coils merely nearby, with no core between them), leakage is only due to mutual inductance, which might be quite small (we wouldn't usually express it as leakage at that point, as the LL+Lm model doesn't apply, and a pi/tee equivalent must be used, or just the definition of mutual inductance itself).

Of course, altering air gap (between cores) affects the magnetizing inductance too -- an important parameter for an LLC, so there may not be much design freedom here. Explicit shunts could be introduced to such a design (i.e., pieces of ferrite added to the winding window, between primary and secondary banks), but this probably isn't worth the complexity and cost, unless perhaps the design must be very highly optimized for size, or is produced in large enough quantity to solve the issues of customization.

As the diagram in question appears to be a more conventional vertical-layered ("shell") winding, the above doesn't apply here, or at least won't be very significant.

There are other construction details that are relevant. Solid heavy wire is opaque to magnetic fields, i.e. field is shielded from the middle/core/bulk of the wire at most [SMPS-relevant] frequencies, thus leakage includes only the space between wires; whereas if litz cable is used, it is transparent to transverse fields, and leakage is increased. (This has probably already been done, for ampacity reasons, so inductance is already maximum by way of this effect.)

If there is space to spare on the bobbin, more layers of tape (or margin tape) could be added between P/S, or if the winding window is wider than it is tall (this is typically the case for PQ family cores, except for very short ones), divided into left and right banks so that the facing area between windings is significantly reduced.