Introduction
Rising information heart energy calls for are driving server endequipment producers to achieve larger powerconversion efficiencies as a way to cut back the thermal footprint of their techniques. The transition from a 12V energy distribution bus to a 48V bus creates the necessity for a highefficiency, smallfootprint stepdown converter (48 V to 12 V). Gallium nitride (GaN) area impact transistors (FETs) are the first enablers for the scale reductions and efficiencies wanted in these techniques.
In Energy Tip #122, I supplied an outline of a highefficiency 1kW bus converter design that addresses this want utilizing highperformance GaN switches [1]. That design makes use of a matrix transformerbased inductorinductorcapacitor (LLC) converter and an builtin printed circuit board (PCB) transformer.
On this energy tip, I wish to unpack the customized design of the transformer and clarify how I derived it. Particularly, I wish to present how one can analytically predict the transformer dimensions that may yield the transformer with the smallest footprint and highest converter effectivity, which would require equations for some currents within the system together with estimates of the winding resistances as a operate of the geometry, each shared in shared in Energy Tip #122. With this information, I’ll clarify how one can make this prediction utilizing a device reminiscent of Mathcad.
LLC converter energy losses
Determine 1 is a highlevel schematic for the LLC converter that’s the focus of this text. Desk 1 lists the corresponding specs. The builtin matrix transformer that I’m going to optimize is proven in grey in Determine 1.
Determine 1 LLC converter with the builtin matrix transformer that will probably be optimized on this article (proven in grey). Supply: Texas Devices
Parameter 
Minimal 
Typical 
Most 
V_{in} 
40 V 
48 V 
60 V 
V_{out} 
9.5 V 
12 V 
15 V 
P_{out} 
1 kW 

Peak effectivity 
98 % 

Transformer turns ratio 
4to1 

f_{s} 
1 MHz 

L_{m}, magnetizing inductance 
2 µH 

L_{r}, resonant inductance 
16 nH 

C_{r}, resonant capacitance 
3.52 µF 

Type issue 
Oneeighth brick 

Main GaN FETs 
LMG2100R044 

Secondary GaN FETs 
EPC2066 

Controller 
F2800157QRHBRQ1 or UCD3138ARJAT 
Desk 1 Working specs for the bus converter proven in Determine 1.
The mathematical prediction of the minimal dimension and most effectivity would require equations for the losses within the system. These losses must be parameterized in such a approach as to be a operate of the transformer geometry. In actuality, you’ll must accommodate losses from many various sources; nonetheless, in an effort to make this text digestible, I’m solely going to cowl 4 loss parts. Desk 2 lists the loss parameters of those parts, together with an outline of every.
Parameter 
System 
Description 
P_{core} 
Transformer core loss. ok, α, and β are materials constants from the fabric information sheet. V_{e} is the amount of the core materials and is a operate of the core geometry dimensions. 

P_{cu} 
Transformer winding loss. I_{lr,rms }and I_{sec,rms} are supplied in Energy Tip #122 together with the AC resistance time period. 

P_{fet,pri} 
Main and secondary GaN FET losses. Because the system is zero voltage switched, solely the R_{ds,on}related losses are required. The currents may be derived as described in Energy Tip #122 and are listed as (1) and (2) beneath. 

P_{fet,sec} 
Desk 2 LLC loss parameters and an outline of every.
The whole system losses can then be outlined as P_{complete}(w,r) = P_{core}(w,r)+ P_{cu}(w,r)+P_{fet,pri}+P_{fet,sec}. The P_{core} and P_{cu} parameters are proven as specific features of the transformer winding geometry. The parameters w and r are placeholders in the mean time and will probably be substituted for the related geometric parameters.
Determine 2 exhibits a mockup of the board and core. The sunshine purple area signifies the overall PCB dimension. The inexperienced space is the realm taken up by the transformer windings, and the grey materials is the gapped transformer core.
Determine 2 Board mannequin the sunshine purple area is the overall PCB dimension, inexperienced is the realm taken up by the transformer windings, and darkish grey is the gapped transformer core. Supply: Texas Devices
Determine 3 exhibits essentially the most important geometric parameters for the transformer windings. This drawing is a prime view of 1 copper layer of the inexperienced area proven in Determine 2. For simplicity, Determine 3 doesn’t present any vias or layer cuts, though these will probably be needed for implementation.
Determine 3 Essentially the most important transformer winding geometry. A prime view of the inexperienced area proven in Determine 2. Supply: Texas Devices
The parameter r_{c} is the radius of the transformer core put up. And r_{c,s} is the spacing between the core and the PCB windings. w_{cu,1} and w_{cu,2} are the gap from the PCB gap to the outer fringe of the winding. Utilizing these parameters lets you outline the overall loss as a operate of those parameters as P_{complete}(w_{cu,2},r_{c}). Utilizing Determine 3, it’s also possible to outline the realm of the transformer footprint as a operate of those identical parameters as proven in equation (3).
Optimization
You should utilize P_{complete}(w_{cu,2},r_{c}) and A_{xfmr}(w_{cu,2},r_{c}) to optimize the system for minimal energy loss and minimal dimension by making a contour plot of the effectivity equation (4), after which superimposing on that plot one other contour plot design with a relentless footprint space. See Determine 4.
Determine 4 Optimum transformer dimensions plot with a contour plot of the effectivity equation (4) and one other contour plot with a relentless footprint space superimposed on it. Supply: Texas Devices
In Determine 4, the curved traces signify contours of fixed effectivity, whereas the straight traces sloping downward from left to proper signify designs of fixed space. Be aware of the truth that the smaller footprint designs are those furthest to the left within the plot. As well as, the purpose the place a relentless effectivity contour simply barely touches one among these traces is the purpose the place the design ends in the smallest footprint for that effectivity contour. Primarily based on this, you possibly can visualize a line of small transformers, as proven by the darkish blue line. Any design on this line would be the smallest design doable for the goal effectivity—or, should you favor, the very best effectivity you could obtain for a design of that dimension. The pink dot in Determine 4 exhibits the ultimate design dimensions chosen for the {hardware}.
It’s straightforward to generate contour plots reminiscent of these in Determine 4 in instruments together with Matlab, Mathcad, or Mathematica. This kind of evaluation is what occurs while you resolve a constrained optimization downside utilizing Lagrange multipliers [4] and may be carried out with Equations 5, 6 and seven. Whereas fixing the issue this manner is extra mathematically intensive, the tip result’s an identical to what you possibly can obtain through the use of the contour plots.
Evaluating the loss within the transformer (as produced by the equations) to the transformer loss (produced by an unbiased simulation of the transformer utilizing finite factor evaluation, or FEA) will validate this technique. The outcomes of the 2 fashions are inside 1% of one another. Moreover, the overall losses within the system in comparison with the prediction even have glorious correlation, as proven in Determine 5.
Determine 5 Loss comparability the place the overall losses within the system in comparison with the prediction even have glorious correlation. Supply: Texas Devices
Transformer dimension optimization
On this energy tip, I offered a way for fixing a constrained optimization downside that ends in the transformer parameter needed to realize the smallestsize transformer and highest effectivity converter. The accuracy of the strategy was inside 1%, as demonstrated by FEA simulation. This technique doesn’t want the advanced derivatives to formally resolve a Lagrange multiplier downside, permitting you to exactly zero in on higher options and additional leverage the scale and effectivity advantages of GaN switches.
Brent McDonald works as a system engineer for the Texas Devices Energy Provide Design Companies crew, the place he creates reference designs for a wide range of highpower functions. Brent obtained a bachelor’s diploma in electrical engineering from the College of WisconsinMilwaukee, and a grasp’s diploma, additionally in electrical engineering, from the College of Colorado Boulder.
Associated Content material
 Energy Suggestions #122: Overview of a planar transformer utilized in a 1kW highdensity LLC energy module
 Energy Suggestions #117: Measure your LLC resonant tank earlier than testing at full working circumstances
 Energy Suggestions #84: Assume outdoors the LLC collection resonant converter field
 Energy Suggestions #92: Excessivefrequency resonant converter design issues, Half 2
References
 Texas Devices. n.d. 100V 4.4mΩ halfbridge GaN FET with builtin driver and safety. Accessed July 23, 2024.
 Liu, Ya. 2007. “Excessive Effectivity Optimization of LLC Resonant Converter for Huge Load Vary.” Grasp’s thesis, Virginia Polytechnic Institute and State College.
 Dowell, P.L. “Results of Eddy Currents in Transformer Windings.” Printed in Proceedings IEE (UK) 113, no. 8 (August 1966): pp. 13871394.
 n.d. Lagrange multiplier. Accessed July 23, 2024.
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