Introduction

Rising information heart energy calls for are driving server end-equipment producers to succeed in larger power-conversion efficiencies with a view to cut back the thermal footprint of their techniques. The transition from a 12-V energy distribution bus to a 48-V bus creates the necessity for a high-efficiency, small-footprint step-down converter (48 V to 12 V). Gallium nitride (GaN) subject impact transistors (FETs) are the first enablers for the dimensions reductions and efficiencies wanted in these techniques.

In Energy Tip #122, I offered an outline of a high-efficiency 1kW bus converter design that addresses this want utilizing high-performance GaN switches [1]. That design makes use of a matrix transformer-based inductor-inductor-capacitor (LLC) converter and an built-in printed circuit board (PCB) transformer.

On this energy tip, I need to unpack the customized design of the transformer and clarify how I derived it. Particularly, I need to present the best way to analytically predict the transformer dimensions that can 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 perform of the geometry, each shared in shared in Energy Tip #122. With this information, I’ll clarify the best way to make this prediction utilizing a device similar to Mathcad.

LLC converter energy losses

Determine 1 is a high-level schematic for the LLC converter that’s the focus of this text. Desk 1 lists the corresponding specs. The built-in matrix transformer that I’m going to optimize is proven in grey in Determine 1.

Determine 1 LLC converter with the built-in matrix transformer that can be optimized on this article (proven in grey). Supply: Texas Devices

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 have to be parameterized in such a means as to be a perform of the transformer geometry. In actuality, you’ll must accommodate losses from many alternative sources; nevertheless, in an effort to make this text digestible, I’m solely going to cowl 4 loss components. Desk 2 lists the loss parameters of those components, together with an outline of every.

Desk 2 LLC loss parameters and an outline of every.

The full system losses can then be outlined as P_whole(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 express features of the transformer winding geometry. The parameters w and r are placeholders in the intervening time and can be substituted for the related geometric parameters.

Determine 2 reveals a mockup of the board and core. The sunshine purple area signifies the full 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 full 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 reveals probably the most vital geometric parameters for the transformer windings. This drawing is a high 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 can be needed for implementation.

Determine 3 Probably the most vital transformer winding geometry. A high view of the inexperienced area proven in Determine 2. Supply: Texas Devices

The parameter r_c is the radius of the transformer core submit. 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 means that you can outline the full loss as a perform of those parameters as P_whole(w_cu,2,r_c). Utilizing Determine 3, you may also outline the realm of the transformer footprint as a perform of those identical parameters as proven in equation (3).

Optimization

You should utilize P_whole(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 continuing 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 continuing footprint space superimposed on it. Supply: Texas Devices

In Determine 4, the curved traces characterize contours of fixed effectivity, whereas the straight traces sloping downward from left to proper characterize 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 continuing effectivity contour simply barely touches considered one of these traces is the purpose the place the design leads to 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 attainable for the goal effectivity—or, for those who want, the very best effectivity you can obtain for a design of that dimension. The crimson dot in Determine 4 reveals the ultimate design dimensions chosen for the {hardware}.

It’s simple to generate contour plots similar to these in Determine 4 in instruments together with Matlab, Mathcad, or Mathematica. One of these 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 fashion is extra mathematically intensive, the tip result’s equivalent to what you possibly can obtain by utilizing 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 full losses within the system in comparison with the prediction even have wonderful correlation, as proven in Determine 5.

Determine 5 Loss comparability the place the full losses within the system in comparison with the prediction even have wonderful correlation. Supply: Texas Devices

Transformer dimension optimization

On this energy tip, I introduced a technique for fixing a constrained optimization downside that leads to the transformer parameter needed to attain the smallest-size transformer and highest effectivity converter. The accuracy of the strategy was inside 1%, as demonstrated by FEA simulation. This technique doesn’t want the complicated derivatives to formally resolve a Lagrange multiplier downside, permitting you to exactly zero in on higher options and additional leverage the dimensions and effectivity advantages of GaN switches.

Brent McDonald works as a system engineer for the Texas Devices Energy Provide Design Companies staff, the place he creates reference designs for quite a lot of high-power purposes. Brent obtained a bachelor’s diploma in electrical engineering from the College of Wisconsin-Milwaukee, 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 1-kW high-density LLC energy module
• Energy Suggestions #117: Measure your LLC resonant tank earlier than testing at full working situations
• Energy Suggestions #84: Assume exterior the LLC sequence resonant converter field
• Energy Suggestions #92: Excessive-frequency resonant converter design issues, Half 2

References

1. Texas Devices. n.d. 100-V 4.4-mΩ half-bridge GaN FET with built-in driver and safety. Accessed July 23, 2024.
2. Liu, Ya. 2007. “Excessive Effectivity Optimization of LLC Resonant Converter for Large Load Vary.” Grasp’s thesis, Virginia Polytechnic Institute and State College.
3. Dowell, P.L. “Results of Eddy Currents in Transformer Windings.” Printed in Proceedings IEE (UK) 113, no. 8 (August 1966): pp. 1387-1394.
4. n.d. Lagrange multiplier. Accessed July 23, 2024.

The submit Energy Suggestions #131: Planar transformer dimension and effectivity optimization algorithm for a 1 kW high-density LLC energy module appeared first on EDN.

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