#6 🚀 Bonus Edition: Complete Article about Rac and Dowell
The Newsletter of High Frequency folks
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the jungle my Newsletter about everything I’m learning during my journey at Frenetic, and my favorite topic, High-Frequency Magnetics 🚀.
This is a Bonus edition, unifying the three parts of the article from Alfonso Martinez, CTO of Frenetic.
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It is 1966. The Vietnam War is raging while the hippie movement grows. My beloved Doors record their first album, while Sinatra and Cash are collecting everything not taken by the Beach Boys. Walt Disney
gets cryogeni c dies. Electric appliances are arriving in American kitchens, and a weird series called Star Trek is debuting on TV. Europe looks worried at the 23rd Congress of the Communist Party of the Soviet Union, while in Berlin people can cross the Wall to celebrate Christmas.
But for us Engineers, something else happened in Denmark in 1966: P. L. Dowell published his famous paper “Effects of eddy currents in transformer windings”. And I can assure you I see more people infatuated as of today with his equations than with The Doors.
So I have decided to dedicate my first
and only number in a series of Alf’s Musings to P. L. Dowell. Concretely to why you shouldn’t use Dowell’s method to design your transformers
Let’s start at the beginning, we don’t know much about P.L. Dowell apart from his
being a time-traveller papers, so let’s focus on it. In the introduction, he clarifies how preceding engineers have developed equations for slot-wound-armature conductors, so he decides to apply the same principles to the windings of a transformer, a valiant effort, and the first stone for all the work done in the decades. Since then it has been cited 1091 times (as of December 2021), included in any good book about magnetics, and used for most medium-high advanced magnetic designs.
Dowell's method consists of an easy-to-implement equation, that given a frequency, number of layers, and conductor width it will give you the AC factor of a given inductor, which represents the proportion of the resistance the winding has at a certain frequency divided by its resistance at DC.
At the time it was really an amazing solution, and the author even provided a series of graphs (Dowell’s curves) that quickly allow you to point out your AC factor depending on the diameter of your wire and the skin depth at your operating frequency. At a time when there were no calculators, no excel sheets, no personal computers, these curves were really good! So good that there are still people using them instead of the equations, which is quite
stupid anachronic since they are just adding the thick point error to Dowell’s error, and just gaining more dioptres.
I have divided my explanation into several pitfalls, let’s begin with the first!
The first pitfall: singular, dual, plural
Dowell’s method can be readily summarized: you calculate your DC resistance, calculate the AC factor given by his equation, you multiply and there you have your AC resistance and your winding losses. Easy peasy. And since the AC resistance is extremely difficult to measure and extremely important at high frequencies, nobody questioned a method that gives you a simple solution to a difficult problem.
And here comes the first pitfall: since you have one DC resistance per winding, people expect to have one AC resistance per winding too. So how do they do it? They apply Dowell to each winding, with its own number of layers. And now they can multiple by the square of each current and there you have the winding losses! And they told me magnetics were difficult…
Maybe this gets a bit on the realms of opinion, but the AC resistance is a virtual construct we engineers to have an equivalent model for the AC losses as we have for the DC losses. And this virtual AC resistance should be either a matrix or a unique value, not a value per winding. And the reason is simple, though the explanation is complex:
The losses happening in an individual turn inside a transformer winding come from three sources:
The ohmic losses due to its own DC resistance and its DC current
The skin losses, which consist also of ohmic losses due to the component of the AC current of a given harmonic and the skin resistance at the frequency of the harmonic. This skin resistance is similar to the DC resistance but takes into account the reduction in the area produced by the skin effect at the harmonic frequency.
The proximity losses, which are similarly ohmic losses, but in this case the current that produces them is induced from the H field surrounding the turn, which is produced by the rest of the conducting turns in the magnetic. And reciprocally, our individual turn is also inducing current in the rest.
This means the losses in a turn come from two main excitations: the current circulating through it, DC and AC; and the AC current circulating in other turns. From this we can extract certain implications:
The AC resistance is not a fixed value depending only on the construction of the magnetic.
The AC losses of wiring are not uniquely dependent on its current.
The AC depends on the current waveform of every winding, including all its harmonic amplitudes, frequencies, and phases.
My favorite one in this: If we are using Litz wire, every strand behaves like a turn, and is producing proximity losses on the rest, additionally of the losses produced from turn to turn. This is commonly referred to as Internal Proximity Effect and was studied by Profesor Albach.
Adding more parallels to wiring to decrease the DC resistance might increase our total losses.
There are a couple of extra implications, but I am reserving those for the next section.
If we apply these implications to a common case like a Phase-Shifted Full Bridge, they mean:
The current of secondary is affecting the AC losses of the primary and vice versa.
The energy-producing these losses is just not created or extracted from Hell, it comes from your source, and it also increases the current in the primary, which also increases losses…
TL;DR To wrap all this up, the AC resistance of a winding depends on the current circulating through the winding, and the current circulating through the rest of the windings. This is why in advanced literature it is referred to as an NxN matrix, with N being the number of windings. But since common engineers get nervous with complex math (Dilbert was wrong!), we tend to prefer working with parameters that we can calculate with our CA53W-1 or excel. Which is the reason for the (in my opinion) wrong initial approach. So to be so bold as to propose something:
Let’s accept the nature of the AC resistance, and start working with only ONE value for the whole magnetic, a global AC resistance value.
And what about if I want to calculate the losses in each winding? That’s fine, there are an amount of losses per winding, but its calculation means working with a matrix. Deal with it.
Wait, I said I was talking about Dowell. Does it mean Dowell’s method is wrong? No, and yes.
His initial approach, where he calculates the leakage inductance and resistance, was done for a magnetic with any number of windings and is a valid interpretation.
The problem comes in the second part when he continues his reasoning trying to calculate the AC resistance of a magnetic. All the calculations in this section are based on a continuously increasing MMF (as in an inductor or non-interleaved transformer) to extract his famous equations. So the moment you have a different case, e. g. an uneven number of layers in one winding and even in the other, applying this method would be wrong.
In the previous part of this Musing, especially in the last paragraph, it was stated that the first part of Dowell’s paper was focused on finding the leakage inductance and resistance of a magnetic with any number of windings. That was erred. Dowell’s explanation is done for any number and combination of layers, which can have equal or different currents, belonging to the same or different windings.
The second part though, where he developed his famous equations is, nevertheless, focussed on windings, or more concretely in all the layers of an individual and concrete winding, as does its equations; which maintains the affirmation of their being incorrect.
And this is totally my opinion (well, the rest too), but the reason why the whole confusion started is because in the paper itself, in the conclusions, he commented how he has validated his equations with one-layer windings.
The second pitfall: the temporal partition
In the previous number we were talking about how the losses in a turn come from two main excitations: the current circulating through it, DC and AC; and the AC current circulating in other turns. We extracted a few conclusions from this, but I promised there were a few more, and the trick was in the adjective “conducting” (7th paragraph, second to the last sentence in my previous number, go and check it, true believer!).
Dowell’s dissertation is all based on windings conducting the whole period, as was common at the time. And with sinusoidal waveforms, but I am reserving this point for the next section.
Why is it important that they are conducting the whole time for Dowell’s magnetomotive force (MMF) analysis, especially in transformer/coupled inductors? Because it is based on being able to calculate the resistance leakage for any layer parting from a point of zero MMF. In a magnetic with windings not conducting at some point what we really have is two or more magnetics superposed in time, with layers in between of conducting material, where only induced currents are circulating and generating losses.
The reason for my rant about applying Dowell’s method on these kinds of magnetics is not so much a philosophical reason as much as an engineering one. When we ignore the conducting state of each winding and apply Dowell’s equations, the values extracted from the MMF graphs are calculated assuming the H field of the primary is being canceled by the secondary. The reality is that it is not.
Let me put an example: We have a magnetic with four layers: primary, secondary, primary, secondary (PSPS), each with 10 turns.
If we connect it to a circuit that will feed the primary 1A, we get the following mmf diagram:
This is a classical example, where Dowell’s method is totally applicable. The primary increases the MMF in its layers, and it is afterward counteracted by the current flowing in the opposite direction in the secondary.
But, if now we grab this magnetic and connect it to a Flyback converter, where the primary is conducting half the period (let’s assume 0.5 duty cycle, for clarity) and the secondary is conducting the other half. The MMF diagram we get is the following:
We can observe how the MMF in the first half grows from the first layer and is not counteracted by any opposing current. And equally happens in the second half of the period with the secondary.
I know all this can be a bit dry, but stay with me a bit longer: Supposing we apply Dowell’s method for calculating the AC resistance to this magnetic as proposed by him, not even directly his equations, but the method that is summarized by the equations; the first step would be to calculate the current density “at the tops of conductors” in the diagram, and from it derive the magnetic flux through the previous layers.
But lo and behold! In the first case, we would be parting from half the MMF than in the second case, and this would derive a smaller AC factor for the first case than for the second case. Same magnetic, with the same number of turns and interleaved layers arrangement, and with the same peak current circulating through it, will have a different AC resistance.
( I know, I know, I already reached this implication in my third point before, but this is a more extreme case)
This example shows why you should NOT apply Dowell’s formulas to magnetics used in topologies with windings that are conducting part of the period. And hey, you can believe me, or you can believe Dowell, who literally starts his paper with two principles on which his following reasoning is based, and the second is:
Do you still want to use Dowell’s for your Flyback? Really? Let me point out one more ridiculous consequence of applying Dowell’s to a Flyback.
The MMF diagram is a rough one-dimensional representation of the H field distribution in the winding. If we translate this to practical engineering terms, it means that the bigger the area of the diagram over (or under) the x-axis at a winding layer, the proportionally bigger the losses due to induced currents (proximity effect) in that layer.
So… does that mean that in the second half of the period there are no losses due to the proximity effect in the first (left most) layer of the primary, where the MMF is zero? Wait wait, that is just because we decided to draw the diagram from the left, if we do it from the right, that layer gets the maximum MMF. Which one is correct? The answer is neither, you cannot apply Dowell’s to a Flyback and keep all your sanity points.
And my conscience would let me finish the dense part without a little addendum, especially because I always hated those university professors who explained a simple case and then asked for a complex one in the final exam. Let me go for the final exam question: what happens with a weird, partial interleaving like a PPSPSS? Happens that we will have a local minimum in the third layer, and following Dowell’s own principle we should NOT use his equations. Poor Dowell, we are running out of applications.
TL;DR From the preceding text we can extract certain implications:
The AC resistance is a continuous vector that varies through the period, though we engineers find it more comfortable to average it, as Dowell did. But
In the case of topologies in which one or more windings are not conducting the whole period, the AC resistance is drastically different depending on which windings are conducting at a given point in time.
We should be able to apply Dowell to multi-secondaries magnetics, as long as all the windings are conducting all the period and the MMF has no local minima.
We shouldn't use Dowell's method for Flyback, Forward, Push-Pull, Weinberg, some Half and Full bridges, and probably some more I am forgetting. No can do
If we apply these implications to a common case like Flyback, they mean:
The AC resistance of the inductor depends not only on the voltages, switching frequency, and power but also on the duty cycle.
The current of secondary is affecting the AC losses of the primary and vice versa.
During the conducting time of the primary (t1), the losses are:
Primary: DC, skin, and proximity due to the current through itself.
Secondary: Proximity due to the current through the primary.
During the conducting time of the secondary(t2), the losses are:
Primary: Proximity due to the current through the secondary.
Secondary: DC, skin, and proximity due to the current through itself.
In the previous number, we commented how having windings that are not conducted the whole time can invalidate Dowell’s assumptions and how we should not use his method for topologies like Flyback or Forward. But we left a few details out of the discussion, especially regarding non-sinusoidal waveforms. Let's discuss them.
The third pitfall: Frequency is legion
In 1822, Joseph Fourier presented his work, and maybe the most important contribution to engineering was his investigations in the Fourier Series. I don’t want to get too mathematical, so let’s quickly review the power electronics side of his works.
When we are working in a converter, the signals produced are almost never sinusoidal, or ideal, especially for the case of the currents. Commonly they are in an intermediate point between triangular and sinusoidal, but they can get as crazy as the Flyback. Yes, I know that between primary and secondary there add up a triangular, but the reality is that an individual winding suffers just a chunk of it.
So, applying Fourier Transform, we can transform any of these weird signals into an infinite number of superposed sinusoidal waveforms, and these waveforms are what we call the harmonics of a signal.
These harmonics represent the distortion from an ideal, pure sinusoidal waveform, and as a rule, the more harmonics in your signals the more losses you will have. Their value can range from quite moderate, as in an LLC converter:
To quite high, as in a Flyback:
But I am digressing, we were talking about Dowell. At the time of Dowell's work, the technology was not as advanced as today, especially active components, so the waveforms they were working with were generally at lower frequencies and closer to sinusoidal. This is why in his paper Dowell never worries about harmonics, on non-conducting windings.
As a reference to this topic, we could check a small extract from Dowell’s paper:
In it, he argues that some of his assumptions are justifiable and refers to another document for this justification, which is totally legit. But if we check the source (Design of audio frequency input and intervalve transformers) we observe how the frequencies Dowell refers to are not greater than 10kHz (or as they write it 10 kc/s). More than one engineer should remember this the next time they apply Dowell’s equation to a 200kHz transformer.
All this is to say that Dowell’s method should be applied only to sinusoidal waveforms. No wonder my wife tells me I am exasperating.
But there is hope! We were talking at the beginning about the Fourier Transform and how it can convert a weird waveform into an (infinite) sum of sinusoidal ones. So, could we apply Dowell’s equation to each of these harmonics, extracting the winding losses of each harmonic and adding them? Yes, but with care.
First, mathematically talking, in a Fourier Transform there are infinite terms, but we engineers cannot add infinite losses. Usually, this is solved by neglecting any harmonic with a power smaller than a threshold we find ourselves comfortable with. The only dangerous part would be finding the first negligible harmonics and ignoring from that frequency upward. That’s a bad idea, because the power in the harmonics is not monotonously decreasing, and many times there are high-frequency harmonics with non-negligible power, even if the previous ones are.
The second issue to be careful with is discussed in the previous section, the non-continuous waveforms. When you transform a Flyback primary waveform into its Fourier Transform, each harmonic becomes continuous, and that would mean that they are generating skin effect losses in the primary and proximity effect losses in the rest of turns the whole period. But as we discussed in the previous section, the reality is that in the off-time of the period, there are no skin effect losses in the primary, and no proximity effect on the secondary turns due to primary turns. If you apply Dowell blindly to a Flyback using harmonic decomposition and without taking care of the conducting times, do it at your own risk.
I don’t want to end this series of Musings dedicated to Dowell without explicitly saying that I believe he was one of the foundations of magnetic design as we know it today. His method was really innovative and created a line of investigation that has not finished as of 2021. My only intention with this (constructive, I hope) criticism was helping people understand how and why Dowell’s method can be used and when they are doing a disservice to them as engineers and to the memory of a great Scientist.
The second-derivative pitfalls: the design phase
Many of the points discussed in the previous sections were (or tried to be at least) focused on the physics and effects happening in and around the turns of a magnetic, but a few more fat-fetched consequences were mentioned, and I would like to remark those in this extra section.
When we talked about the harmonics before, we commented how ideally we could calculate the losses associated with each harmonic and add them to have the total losses in the wire.
This has a wider consequence: according to this interpretation, we have currents of many successive frequencies circulating through our conductor, each one a different frequency and each one with a different skin depth.
So, we, being good and pragmatic engineers, chose a conducting width for our wires based on our switching frequency, in order to minimize the losses due to the skin effect. But did we take into account the harmonics?
If we didn’t, what happened is that we chose the thinner wire to avoid the skin effect at our switching frequency, but neglected the skin effect losses of the harmonics. In most cases, this means that we have a non-negligible amount of current running through a much-reduced area (and higher resistance) and therefore producing unnecessary losses.
This is a personal opinion of mine, but when I choose the conductor diameter of the wires for a transformer I like to use what is called “effective frequency”, and it represents the weighted frequencies of all harmonics in a waveform (it can be found here and here, for curious ones).
Using the effective frequency we can calculate the effective skin depth and with it, we can choose a wire taking into account also the harmonics.
But this method is no panacea, as it only takes into account the skin effect losses, and forgets about the proximity effect.
This negligence neglect helps us go to the next design pitfall: over dimensioning the number of parallels. And with parallel, I also mean the number of strands in a Litz wire.
If we go to an extreme case of high-frequency design (be it switching or effective frequency) we can reason, following what we have learned, that we should just find the bigger conductor diameter that allows us to neglect skin effect losses and add parallels (or strands) until we can drive all of our currents at a safe density.
Sure, that works nicely at lower frequencies. But as the frequency of the harmonic grows so grows the proximity effect losses from each turn into the other turns.
So, let's say we find ourselves in a case where we have a magnetic component with many turns, but only one parallel, and we are a bit uncomfortable with the high current density we have and the temperature rise it produces in the winding. So, because we have enough window area, we decide to add another parallel, doubling the turns and halving the current density, and suddenly a magnetic that was in the limit of burning up becomes a supernova.
How is that possible? Well, we doubled the number of turns, decreasing the DC losses by half, but the number of proximity interaction between the wires were squared, so it is perfectly possible that the increase in proximity effect losses due to the new turns is higher than the decrease in DC losses. Especially if we have harmonics with high frequency and non-negligible power.
This effect is especially accentuated in the case of Litz wire, where decreasing the strand diameter and adding more strands to the bundle not only increases the cost of the wire, but it can also increase the winding losses, as we are drastically increasing the number of proximity interactions between all the strands in the winding window.
And that was all folks!!
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