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**Chapter 4 ****Transformer Design: Current Sharing of Parallel Layers**

**Introduction**

To minimize the transformer conduction loss, it is important that the current distribution is uniform across all the windings so that all the copper are well utilized. The merged winding method proposed in chapter 2 reduces the current crowding and ensures a good current distribution on every single layer of winding. The interleaved termination and via structure proposed in chapter 3 not only reduces the conduction loss and leakage inductance created by the terminations and vias but also achieve a better utilization of the vias. However, as discussed before, there are four paralleled layers for each winding. To well utilize these layers, good current sharing among them is also critical. In the parallel winding structure, the conventional 1-D analysis [30] which assumes serial connections is no longer valid. The layer arrangement is found to have a very important effect on the current distribution and loss. It can be demonstrated with our design with the original layer arrangement. First, we still use one elemental transformer for simplicity. The 3D model, the layer arrangement (cross-section view) and its magneto-motive force (MMF) is shown in Figure 4.1. Here only the middle twelve winding layers (layer number 2 to layer number 13) are presented. Note that this is a fully interleaved arrangement. Ideal windings are used in the model with no termination or vias. An external circuit is used for the current excitation of windings. Figure 4.2 shows the concept drawing of the external circuit used. We simulate the positive half cycle where secondary winding #1 conducts. Ignoring the magnetizing inductance, the current source on the primary *I** _{prim}* and secondary side

*I*

*has the same value. By using ideal windings and parallel layers through the external circuit, the impact of terminations and vias are decoupled. The simulation results are shown in Figure, where the percentage on each layer of winding indicates the current distribution ratio. Similarly, we can do the simulation for the negative half cycle when the secondary winding #2 conducts. The results are shown in Figure 4.3. It is clear that even with a fully interleaved layer arrangement, the current distribution is not uniform. It means that even we have good terminations and vias design so that the via impedance is negligible, the current still tend to have an uneven distribution, which causes higher conduction loss and uneven thermal stress. If we go back to the equivalent circuit, there are two things we ignored. First, although the DC winding resistances of each layer are identical due to the same winding structure, the AC resistance, which is determined by the skin effect and proximity effect, may not be the same. Second, the leakage inductance of each layer, which is influenced by the electromagnetic field distribution, can also be different. Therefore, we could have different impedance on each layer and hence a non-balanced current distribution. Since these two factors are both related to the electromagnetic field distribution, the circuit model is not adequate for the analysis. Based on the simulation, we learn that the current distribution is related to frequency, layer copper thickness, layer distance (FR-4 thickness), and layer arrangement. So now the question is whether there is a simplified model that can be used to predict the electromagnetic field and current distribution, like the well-known 1-Dimension (1-D) model [30, 31, 32] which is used in transformer structures where both primary and secondary windings are in series. This problem was noticed in planar magnetics in [33]. Some basic analysis are per-formed on this issue in [33, 34, 35] and some experimental guidelines according to a number of simulation examples are summarized. But they are completely simulation-based, which is time-consuming and non-convenient for iterative design and analysis. It also lacks physical meanings or intuition. [36] proposed application of the extremum co-energy principle for calculation and prediction of the current distribution and resulting AC resistance. It was verified by the experiments. However, it did not consider the frequency effect on the current distribution results. An analytic model based on Faraday’s law and voltage balance was pro-posed in [37]. This model well predicts the AC winding resistance of a wide frequency range. An example of 8:1 transformer with primary windings in series and two secondary windings in parallel is presented to verify the calculation. However, the problem with this approach is that the equations become very complicated when applied to transformer structure with more number of parallel layers, e.g., four parallel primary windings and four parallel sec-ondary windings in our case. Also [37] only showed the concept and two key equations. More detailed works are required if we want to apply this model to a general transformer winding structure. To study this issue in detail, we will first use the concept proposed in [37] to analyze the problem. And then a simplified method using superposition concept is proposed. It makes some assumptions during the simplification, but these assumptions are satisfied in the practical design of PCB-winding planar transformer. The method can be easily extended to a larger number of parallel windings and is verified by FEA simulation with two examples.*

_{sec}**Analytic Model for Parallel-Winding Transformer**

The method proposed in [37] is still based on the basic 1-D model [30, 32]. So we will first go through the 1-D electromagnetic analysis of the transformer and then introduce the parallel-winding model. In the following analysis, we assume that the electromagnetic field is magneto-quasi-static (MQS), which means the displacement current can be ignored. MQS assumption always holds for most of the power electronics applications [38]. We also assume the field is one dimensional, which means the field strength is constant across the breadth of the winding and changes only along the thickness of the winding [30]. In Dowell’s 1-D model [30], an additional assumption is that the primary and secondary windings are both in series. So the current in windings is known and can be used to solve for field distributions. The first step is to solve the J (current density) field in windings. Based on MQS and Maxwell equations, the J field satisfies the following second-order differential equation in the frequency domain where *ω* is the frequency, *µ*_{0} is the magnetic permeability of air, and *η* is the compensation factor for the ”layer porosity”. For our case where each layer has one turn of winding and the winding fills the whole window area of the core, this factor equals to one. *ρ* is the resistivity of the conductor, which is copper in this case. X-axis is along the thickness of the winding. To solve this equation, two boundary conditions are needed to solve for P and Q. The first boundary condition can be easily obtained based on the knowledge that the integration of J upon the cross-section area of the winding is the current flowing through the winding.where *b* is the winding width, *h* is the thickness of the winding layer and N is the number of turns per layer, which is one in our case. Note that despite the fact that this relationship is always valid, we cannot directly use this equation for boundary condition in parallel-winding case, where the current in each parallel winding is unknown. The other boundary condition used is a Neumann condition If we start from the zero MMF position to count layers. The *p*th layer is the layer where we calculate the J distribution. *p* always equals zero in a fully interleaved winding structure. The MMF only accumulates within one layer. This equation can be derived by the integral form of Faraday’s law, as presented in [30]. After J field is solved, the voltage can be obtained by Ohm’s law and Faraday’s law. From the voltage across the winding, the AC resistance and leakage inductance information can be extracted. Derivation details can be found in [30] and are not discussed here. From the previous analysis, we learn that the key to solving the field is the two boundary conditions. In Dowell’s model, they are readily obtained because the current *I* is a known parameter under its series assumption. In parallel-winding case, however, this assumption is no longer valid. Now all the current term in the two boundary equations becomes unknown variables. As the number of unknowns is increased, more equations are required. One apparent equation is based on Kirchhoff’s current law, the summation of the current is the total current on primary or secondary. But one additional relationship is not enough. Consider a simple case where primary windings are in series, and two layers of secondary windings are in parallel, as shown in Figure 4.4. Each winding has one turn on each layer. For the primary windings, we know the current *I** _{p}* due to the serial connection. For the secondary windings, the current on layer 1

*I*

_{1}and layer 3

*I*

_{3}are two unknowns. Ignoring the magnetizing inductance, we have For another equation, [37] select a loop between the two secondary windings (layer 1 and 3). The loop is shown in Figure 4.5 by the dashed circle. The blue and red boxes are partial cut of the secondary and primary windings. Due the parallel connection of layer 1 and 3, the loop is electrically connected. Therefore, the voltage induced by the flux through the loop plus the resistive voltage drop along the where

*J*

_{1}(

*h*) and

*J*

_{3}(0) are current density at the bottom position of layer 1 and top of layer 3, respectively. Φ

*is the total magnetic flux passing through the loop.*

_{tot}*l*is the length of the winding. Relationships developed in previous Dowell’s model can be applied here. To use Dowell’s model, first we set the top of layer 1 to MMF starting point. From equations (4.3), (4.4), and (4.5).

**1 Introduction**

1.1 History of Intermediate Bus Architecture

1.2 Topology of Intermediate Bus Converter

1.3 Transformer Design of Intermediate Bus Converter .

1.4 Thesis Outline

**2 Two-Stage Solution for Regulated Intermediate Bus Converter **

2.1 Design of theTwo-Stage Architecture

2.2 Design of theLLC-DCX Converter

**3 Transformer Design:Terminations and Vias**

3.1 Secondary Termination Design

3.2 Primary Termination Design

**4 Transformer Design: Current Sharing of Parallel Layers**

4.1 Introduction

4.2 Analytic Model for Parallel-Winding Transformer

4.3 Simplified Model Using Superposition Method

4.4 Symmetrical Layer Arrangement

**5 Transformer Optimization and Experimental Results**

5.1 Transformer Design Optimization Procedure

5.2 Hardware and Experimental Results

**6 Conclusions**

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Thesis submitted to the Faculty of the Virginia Polytechnic Institute and State University in partial fulfillment of the requirements for the degree of