🫢 I started this Newsletter ambitiously thinking that I could fit it into a single article, but here I am several pages later, starting a two-part series. I’ve said previously that I would have liked to talk about Optimal Operating Frequencies in Converters, and the time has now come. Brace yourselves and follow me for this first part.
🤌 But, where does one start? Unfortunately, there isn’t a universal rule that guides us to the best operating frequency for all converters because:
All converters behave differently and have differing relationships with frequency.
There are many conflicting design parameters. The balance of which relies on the requirements of the unique application.
🔎 So really, where does one start?
Well, it must begin with a good, high-level understanding of the topology you are working with. It’s first important to understand the areas of the topology that vary significantly with operating frequency, and more specifically how the top-level priorities you have will be affected by their varying.
💡 Today I’m going to talk about a Phase-Shifted Full-Bridge, a popular topology for its wide power handling range, isolation, control simplicity and its ability to take advantage of Zero Voltage Switching (ZVS). To that end you’ll find it in many modern applications.
I decided on the arbitrary and simple specification shown below, the less detail the better at this point.
Table 1. High-Level Converter Specification
⚙️ I’ve discussed in my previous Newsletter for Chema Molina, how varying the operating frequency of a power Converter affects its major components, both the switching elements and the reactive components. It’s clear however, that one of the main challenges facing power engineers in design is the non-linearity of the magnetics relationship with frequency. This complexity warrants a section of its own, so today I will only talk about the Transformer.
⚠️ You may have read in one place or another that in order to design a Transformer for efficiency, you need to find a balance between the core and winding losses. To prove this point, I’m going to run a thought exercise. However, it requires me to assert a simplification and you should bear in mind that this only works because it’s a PSFB topology.
If the magnetizing inductance is sufficiently large enough, the operation of the converter will be consistent over a range of frequencies, thus allowing me to observe the optimal frequency of a fixed transformer design.
I started my design on Frenetic, deliberately choosing to start big and simple with an E71/33/32, un-gapped 3C95 core.
Figure 2. Frenetic Online Core Configuration Page
Using the simulator, I was able to calculate the distribution of core and copper losses between 20 kHz and 150 kHz.
Figure 3. Transformer Optimal Operating Frequency E71/33/32
📈 As the frequency increases, we can see that the core losses decrease. This is because, whilst all other parameters stay the same (or Ceteris Paribus for those who still use Latin), the increase in frequency reduces the volts-seconds imposed across the transformer primary which reduces the flux density.
📉 Following that, when calculating the core losses, over this frequency range (and with this material) the Steinmetz coefficient for flux density has a more dominant effect on the overall equation than the coefficient for frequency. So as the frequency increase adds to the core losses, the drop in flux density has the larger net effect and therefore the core loss tends to reduce. If I was to go beyond 150 kHz and upwards, we would see the gap between the coefficients close, and the dominance of the flux density reduce. I think this is a topic for another day… Oh my.
💔 Regarding the copper losses, the best outcome is that these losses are only caused by the DC resistance of the wires and the RMS currents through them (Irms2Rdc). As you increase the frequency however, this reality slowly fades away along with all your hopes and dreams.
So, you have on one side the core losses decreasing with frequency and on the other side the copper losses increasing. It becomes quite clear then why the intersection of these two loss mechanisms provides you with the most efficient magnetic design. For us, our magnetic here is the most efficient at around 50 kHz.
💭 Is this the best design I could have achieved having started knowing I would use 50 kHz? I suspect I could have done a little better. But let’s continue this journey together and see where it takes us.
☝️ What happens if we change the core?
I conducted a similar analysis for two other cores E65/32/27 and E55/28/25, see the data below. Before reading further though, what conclusions can you draw?
Figure 4. Transformer Optimal Operating Frequency E65/32/27
Figure 5. Transformer Optimal Operating Frequency E55/28/25
From my initial case, I have gone through three consecutive E cores reducing in size each time. I kept the number of turns the same following my rule: that I don’t particularly care about the magnetizing inductance providing it is sufficiently large. However, to make it work I have reduced the number of strands in the Litz wire to ensure there is always a comparable fit.
🔥 The first thing I notice is that the loss intersection point moves to a higher frequency as I reduce the size of the core. Why is that? Look a little deeper, you might notice that the core losses are getting higher as we reduce the size of the core and the copper losses are getting lower, so that explains why the intersection point is moving forwards.
Regarding the core losses, the only thing that I’ve changed is the effective area Ae of the core. Reducing this will increase my flux density at any given frequency. As we’ve discovered, the flux density is currently dominant for us in the core loss calculation and therefore we would expect to see an increase, this all makes sense.
🤔 Regarding the copper losses, when I reduce the size of the core, I’m also reducing the size of the window and increasing the current density. Why aren’t the copper losses increasing?
Well, I’m also reducing the amount of wire used to rap around the smaller core and therefore the DC resistance. This also has the net effect of reducing the aggressiveness of the AC resistance as we go up in frequency and overall, it softens the copper loss gradient we observe in the graphs.
💪 I’ve managed to find an optimal operating point for a fixed transformer design using Frenetic, by simply moving the frequency around until the core and copper losses arrive at a balance. Playing with different cores whilst keeping some other parameters fixed, we saw that the optimal frequency point moved around, only on the horizontal axis though...
👀 I wonder then what levers (if any) I can pull to make significant changes in the vertical axis? In the next part to this series, I’m going to look at how the rest of the converter is affected by frequency changes. Can I justify an optimal transformer design for a specific frequency, or are there bigger things at play?
Meanwhile, I encourage you to use Frenetic, not just as a design tool but as a learning platform and a place to be curious. Look for novel and interesting ways to optimize your designs whilst becoming a better engineer in the process.
😎 See you next time for the second episode!
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