# "To Gap, or Not to Gap" Inductor/XFMR - Energy POV

This post/question is coming from a question I asked within the comments section of here: How to calculate air gap in flyback transformer?

The great answer given talks about his solution to the question of "How to Calculate [an] air gap in [a] flyback transformer"? He goes about this using the energy point of view, but assumes DCM (Discontinuous Mode), meaning the energy delivered per cycle completely transfers before the end of the switching period.

My Question is how does this approach change when tackling a CCM application?

To essentially quote myself (the comment I made), my guess would be you would need to find the energy required by the load per cycle, translate that back to the primary, and figure out how many cycles you want/require to transfer that amount of energy. 1 cycle being a good entry point into this iterative process, leading to a larger current ripple and potentially core size, but better dynamic response; Or, X number of cycles which would reduce the current ripple and potentially the core size but have worse dynamic performance (comparatively to the entry point case). These are just my thoughts by the way, so whether this is true or not is what I am trying to address.

• Unitrode had some really nice lectures performed by Lloyd H. Dixon, Jr. I attended some before he left us, many years back. At the time, he would hand out published materials for the class. Some of these can now be found here and I definitely recall going through this one as part of the materials handed out then. Lloyd should be your god on this topic. And he discusses DCM and CCM and gapping. There are details and tradeoffs, so not a bright line. Hence a real need for Lloyd teaching the topic. Jul 11 at 15:16
• The gap equation is the same regardless of mode of operation. Jul 11 at 15:17
• @periblepsis that is very cool you got to go through that with him, very awesome! His iterative approach to calculating the gap by guessing, taking the answer and re-entering it a few times to converge I find is hit or miss. I find it does not converge more often than not, do you have that problem as well? Jul 14 at 16:27

how does this approach change when tackling a CCM application?

If DCM operation necessitates a transformer-core air-gap (to avoid excessive magnetic saturation) then, CCM operation might need a bigger air-gap because, the peak current in CCM will inevitably be higher than that seen during DCM operation.

DCM current waveform from my basic website: -

The waveforms are for a flyback converter operating at light loads to heavier loads. On light loads the hold period is long. On heavier loads the hold period might become zero. A load that takes even more current/power will move operation into CCM (if permitted by the control system): -

Given that the slopes of each line also represent the input voltage (red) and output voltage (green), if more output current/power is required, the waveforms have no option but to shift upwards after entering CCM operation. In other words, load current changes are accommodated by the current waveforms rising or falling.

This always means a higher peak current and therefore, a higher susceptibility to saturation hence, if a gap is needed for DCM it will likely be a bigger gap in CCM.

The math is exactly the same as the answer (I gave) that you linked except, that in CCM, not all of the energy is released to the secondary at the end of each cycle (as it is in DCM). And, it may seem odd that CCM can release more energy per switching cycle but, if you remember that energy is proportional to $$\I^2\$$, then it should be clear.

Equivalent circuit to ease the analysis (assumes a 1:1 transformer): -

• $$\\color{red}{S1}\$$ replaces the MOSFET
• $$\\color{blue}{L1}\$$ replaces the transformer
• $$\\color{green}{S2}\$$ replaces diode D1

In your circuit diagram, can you always draw 1:1 XFMR's in that way? Can you extend that easier to understand model but with a turns ratio, and not have an XFMR within the circuit?

It's exactly the same as this so maybe this helps (?): -

But this version provides input to output galvanic isolation (not necessary to understand how flyback regulators work). As for a non 1:1 turns ratio you can add a tap for the output: -

The versions with a single inductor are impractical to build and, are just provided to show how the flyback circuit can be viewed more simplistically in terms of energy transfer from input to output.

• Great answer per usual! thanks Andy Jul 12 at 15:08
• In your circuit diagram, can you always draw 1:1 XFMR's in that way? Can you extend that easier to understand model but with a turns ratio, and not have an XFMR within the circuit? That last question may be looking for magic, but might as well throw it out there... Jul 12 at 15:20
• @RogerDodger I've added a bit to the end of my answer. Jul 12 at 18:23

The same way we calculate airgap for any other kind of inductor.

Terminology:

• Transformer: a multi-winding magnetic component with very high magnetizing inductance, and generally a high coupling factor, so that transformed (instantaneous, induced) current dominates over magnetizing current; the energy storage during a cycle is negligible compared to the power transformed.
• Coupled inductor: a multi-winding magnetic component, intended to store and release a significant amount of energy over one or several cycles, generally through both windings.

This is far from a rigorous definition of course; we could have a component with some windings closely coupled to each other, but loosely to others, with high and low magnetizing or leakage inductances all around. The point is more to give

This terminology is not in widespread use. In commerce, "transformer" and "coupled inductor" have no specific meaning and are loosely interchangeable. You often see "flyback transformer" (a component intended to store energy), whereas a "common mode choke" is not intended to store energy. (A CMC is better described as a transformer; the choking action can be understood as a 1:1 current transformer, enforcing equal and opposite currents in the lines.) We might also call a two-winding component, where the secondary is very small (for purposes of auxiliary power output or voltage sensing), a choke or inductor, because the main (primary) winding is of, well, primary importance.

Language is a very squishy thing, so I add these details to be specific.

A "flyback transformer" is therefore better described as a "coupled inductor", and it also happens to need a high coupling factor. The fact that it needs two windings, is immaterial to the magnetic design -- the core sees the same total field whether it's one or many windings doing the work!

So, we can reduce this problem to that of inductor design. We do need to reserve double the winding area, as we need two windings which carry current alternately.

The basics of inductor design are:

• Pick an initial core. Doesn't have to be a good guess, but guesses will improve with experience. An ETD29 core is good for ballpark 100-200W at 100kHz, for example. Power scales with volume.
• Pick a flux density. Probably this will be from the core loss curves (look up the core material datasheet), or you can take typical values, for example in the ETD 29/16/10 | TDK datasheet they suggest N87 material at 200mT and 100kHz, and 2.8W is a reasonable amount for a core that size. (Permissible power dissipation depends on available cooling, and desired efficiency; if you need very high efficiency, the required core size can be dramatically larger than usual!)
• Pick an effective permeability, typically in the 20-100 range. The rationale for this is too lengthy/complicated to get into here, but in short, this figure derives from the fact that copper has the resistivity that it does. (Consequently, if you're using a higher resistance material like aluminum, or the part must operate at high temperatures (most metals have a strong positive tempco), aim for a higher permeability. Conversely, if we had significantly better conductors than copper, we could get by with very low permeability; as it happens, superconducting magnets are quite effective with ludicrous numbers of turns, of very fine wire, though they're also air-core by necessity as they can operate far above the saturation level of any ferromagnet.)
• Notice the winding area ($$\A_N\$$, or often $$\W_A\$$ or etc.) and inductivity ($$\A_L\$$). The latter determines how many turns you need, and the former determines what wire size can fit those turns onto the bobbin. Don't forget to include a winding factor: at the very least, round wire has a maximum packing density, but it's common to be able to use merely 70%, or 50% or even less, of the total winding area. Triple-insulated wire may also help out in smaller transformers.
• If you find the winding would dissipate too much power, or after checking flux density at design parameters you find the core losses would be too high, then restart from the top with a larger core, and repeat. Or conversely if it's very low and you wish to save some cost and space.

This should give you a basic explanation, albeit stopping short of the full equations written out.

And yes, the iterative process is something of a PITA; fortunately there are design tools available on most manufacturers' sites, and core data can be compiled into a spreadsheet where the calculations can be ran instantly in parallel, finding the best core among them for a given application.

There also exists an analytical solution, to find the maximum-Q point of a given core geometry and wire resistance, assuming the core material can be reasonably described by a generalized Steinmetz loss relation; see Fundamentals of Power Electronics, Erickson and Maksimovic (2001), §15.4.2 for example. I have found this approach to give a reasonable starting point, though it consistently overestimates Q factor by 20-80%.

As for DCM vs. CCM, the AC losses only depend on the AC component of course, so core loss and winding ACR are weighted lower for CCM than DCM. DC winding resistance is present in both cases, of course. From perspective of the magnetics design: CCM permits higher power levels from a given core size, or higher loss materials to be used (such as powdered iron). For example, at 100kHz, a 40mm toroid in #52 powdered iron might be good for 30VA of reactive power, or about 30W at 100% ripple fraction (BCM); at 10% ripple (CCM), you get the same AC component but 300W of DC output!

I'm afraid I don't have a feel, offhand, for what DCM vs. CCM would do to the air gap length in a fully optimized (all variables solved for) design. It seems like a question sideways to the correct design route; not that it would be a meaningless question, but that it's not the most important or meaningful one.

The choice of DCM/CCM is typically defined more by the controller, or overall economics of the design, than the magnetics. (Of note: peak current mode controllers require a high ripple fraction, indeed ≥ 100% (DCM) without slope compensation.)

• Really nice answer, I'm curious to understand more about how you CHOOSE the effective permeability, since it's normally a parameter I calculate based on the relative permeability, gap length (chosen or iteratively calculated) and magnetic mean path length. Jul 12 at 15:17
• @RogerDodger Right. It's equivalent to choosing gap length (given that mu_r and l_e are fixed by a given core), but this measure is relatively independent of core parameters. Jul 12 at 20:08