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Applicability and Limitations of an M2Spice-assisted Planar-Magnetics-in-the-Circuit Simulation Approach Samantha J. Gunter 1, Minjie Chen 1, Stephanie A. Pavlick 1, Rose A. Abramson 1, Khurram K. Afridi
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Applicability and Limitations of an M2Spice-assisted Planar-Magnetics-in-the-Circuit Simulation Approach Samantha J. Gunter 1, Minjie Chen 1, Stephanie A. Pavlick 1, Rose A. Abramson 1, Khurram K. Afridi 2, and David J. Perreault 1 1 Massachusetts Institute of Technology, Cambridge, MA, 02139, USA 2 University of Colorado Boulder, Boulder, CO, 80309, USA Abstract Planar magnetics design for power electronics naturally involves many tradeoffs, especially in the selection of the core size, winding structure and printed circuit board stackup. Magnetics-in-the-circuit SPICE simulations can facilitate quick magnetics design evaluation and iteration. This paper introduces and evaluates a planar-magnetics-in-the-circuit simulation approach with an M2Spice software tool, which has been developed based on an earlier presented Modular Layer Model (MLM) analysis approach [1]. M2Spice converts the magnetics geometry into a SPICE netlist, which can be simulated with other circuit elements in a power converter under a unified setup. This paper presents an analysis of the applicability and limitations of this approach across wide frequency bands, followed by an evaluation of the accuracy of the SPICE simulation results (by comparing the simulation results to finite-element-modeling (FEM) results and experimental measurements). Multiple planar magnetics prototypes are designed, modeled, simulated, built, and measured, with results reported and discussed. Index Terms planar magnetics, lumped circuit model, 1-D methods, time-domain simulation. I. INTRODUCTION PLANAR magnetics offer low profile, good thermal characteristics, high power density, high repeatability and the ease of realizing complex winding structures [2] [4]. As frequency increases, accurate modeling of planar magnetics becomes both important and increasingly challenging, mostly due to the impact of skin- and proximity-effects. In previous work, efforts have been made to estimate the loss [5] [9], extract parasitics [10], and investigate the current distribution in windings [11] [20]. Numerical methods (e.g., finiteelement-modeling) and discretization-based experimental measurements are widely applicable, but are often difficult to use for design optimizations that involve many tradeoffs. Following earlier modeling work that utilizes modularized sub-circuit cells to represent planar layers [12] [20], an analytical model, named the Modular Layer Model (MLM), that is widely applicable to modeling planar magnetics, has been recently presented in [1]. The MLM is developed based on a minimum set of assumptions - the 1-D and the magnetoquasistatic (MQS) assumptions. It utilizes analytical solutions for field and current relationships and captures the relationship between variables in the electromagnetic domain and variables in the circuit domain using KVL and KCL equations. The current distribution and field strength can be rapidly found S. Gunter and M. Chen have equal contribution to this paper. (a) (b) Fig. 1. Modular Layer Model-based modeling approach presented in [1]. (a) Physical structure consisting of many layers. (b) The full structure broken into many sub-sections. using SPICE simulations, and be visualized in the time-domain when the magnetic device is connected to an external circuit. With additional assumptions and approximations, 1-Dsingle-frequency magnetic models can be extended to cover 2-D or wide-frequency-band cases by curve-fitting methods (linearization), or discretization methods such as those used in [13], [15]; at the price of additional approximation and increased computational requirements. In many cases, it may be preferable to maintain the simplicity of the MLM, while having a clear estimation about inaccuracies owing to violating modeling limits (e.g., non 1-D and non-sinusoidal effects), and thus avoid the computationally-demanding and non-intuitive discretization process, and keep the complexity manageable for analytical analysis. It was shown in [1] that the MLM can provide reasonable accuracy in predicting the port impedances without extensive modeling of non 1-D effects in singlefrequency planar magnetics designs with sinusoidal waveforms. In this paper, we focus on evaluating the applicability and limitations of the proposed method in circuit simulations in the presence of substantial non 1-D and non-sinusoidal conditions, without using linearization or discretization. Fig. 2. Lumped circuit model consisting of many modular sub-circuits, each representing a portion of the physical planar magnetic structure. The remainder of this paper is organized as follows: section II provides a brief overview of the MLM modeling approach and the M2Spice software tool. The frequency dependent behavior of the MLM approach is analyzed in section III. Section IV-A investigates the applicability and limitations of the MLM time-domain simulation approach with a single-frequency experimental setup that has substantial non- 1D effects (e.g. fringing effects, edge effects). In section IV-B, the investigation is further extended to a practical magnetics design that is simulated and tested together with a dc-dc converter. Section V summarizes the applicability and limitations of the proposed planar-magnetics-in-the-circuit simulation approach. Finally, section VI concludes the paper. II. MLM AND M2Spice OVERVIEW Here we briefly introduce the modular layer model (MLM) presented in [1]. A multilayer planar magnetic structure as shown in Fig. 1a can be modeled as a combination of multiple circuit blocks as shown in Fig. 1b, and represented by a lumped circuit model comprising many iterative sub-circuit blocks as shown in Fig. 2. Each sub-circuit block represents a portion of the magnetic structure, including the magnetic reluctance on top of the layer stack, the conductor layers, the spacings, and the magnetic reluctance below the layer stack. This model can analytically capture skin- and proximityeffects in the windings under 1-D and MQS assumptions [1]. The cross and through variables in the lumped circuit model, i.e., values of E and H in Fig. 2, represent the current density and the field strength in the magnetic structure. By simulating this lumped circuit model with the external driving circuits and probing the current flowing through the corresponding subcircuit in the SPICE simulation, the current flowing through each layer can be visualized and evaluated in SPICE. A software tool M2Spice that can rapidly compute the element values, and automatically generate a SPICE netlist has been created to minimize the additional effort of using the model. Figure 3 shows its user interface. This tool is open-sourced by the authors 1 and is utilized for the studies here. Fig. 4 shows the information flow in the M2Spiceassisted design approach. The magnetic geometry information is first processed by M2Spice, which produces a netlist 1 M2Spice [Online]. Available: Fig. 3. Screenshot of the user-interface of the M2Spice tool. Fig. 4. Information flow of the M2Spice-based magnetics design approach. for a subcircuit that captures the electrical behavior of the magnetic component. This netlist is combined with a netlist that represents other elements in the circuit (e.g., capacitors, resistors, switching devices, etc.) and fed into a SPICE simulation platform (e.g., LTSpice). Based on the simulated circuit performance, designers can very quickly adjust the geometry of the magnetic component (e.g., layer thickness, interleaving patterns, core shapes, etc.), and iterate the design. III. FREQUENCY DEPENDENT IMPEDANCE VALUES This section analyzes the applicability and limitations of using the MLM in circuit simulations. Assuming the permeability and permitivity of all materials stay constant across the full operating range, many elements in the MLM are Fig. 5. (a) Single frequency T network of the MLM and (b) its Low- Frequency-Limit (LFL) simplification (i.e., when h δ), and (c) its High- Frequency-Limit (HFL) simplification (i.e., when h δ). frequency independent, e.g. those representing the spacing and the magnetic core (Z tf, Z ts, Z S1 Z S(n 1), Z bs, and Z bf in Fig. 2), as well as the wire connections describing how each layer is connected to other layers. There is generally no limitation in using these element values in SPICE simulations. Elements related to the conductor layers are frequency dependent. As derived in [1] and shown in Fig. 5, the accurate netlist representing each conductor layer consists of two impedances, Z a, and Z b. They can be calculated using: { Za = dψ(1 e Ψh ) wσ(1+e Ψh ) (1) Z b = 2dΨe Ψh wσ(1 e 2Ψh ) where d is the conductor length per turn, w is the total width of the copper on this layer, h is the thickness, Ψ = 1+j where δ = 2 ωµσ is the skin depth of the conductor, µ is its permeability, σ is the conductivity, and ω is the angular frequency. In the netlist generated by M2Spice, each complex impedance is represented by a resistor and an inductor, which can have positive or negative values, i.e., Z a = R a + ωl a j, and Z b = R b + ωl b j, as shown in Fig. 5a. Since Z a and Z b are both frequency dependent, R a, R b, L a, and L b are all frequency dependent. As derived in Appendix I, when ω 0, R a approaches zero; L a has a limit of µdh w ; R b d has a limit of σwh ; and L b has a limit of µdh 6w. We name these element values as Low-Frequency-Limit (LFL) element values. And when ω +, R b, L a, and L b all approach zero, and R a = d sinh (h/δ) sin σwδ cosh (h/δ)+cos (h/δ). We name these element values as High-Frequency-Limit (HFL) element values. 2 R a, L a, R b, and L b are normalized to their LFL values to investigate their frequency dependent characteristics. We define the dc resistance of a copper layer with thickness h, d width w, and length d, as R dc equals wσh ; and the inductance of a spacing with thickness h, width w, and length d, as L dc equals µdh w. Defining = h/δ, then δ, 2 The LFL and HFL values can be utilized in further simplified MLM modeling (but still capture a major amount of information, especially the loss) as illustrated in Fig. 6: if h δ for the frequency range of interest, HFL element values should be used; if h δ, LFL element values should be used. Intuitively, at low frequencies, the inductive behavior and the dc resistance dominates the conductor behavior; and at high frequencies, the reactive energy stored in the conductor diminishes, leaving only the ac resistance dominating the conductor behavior. Fig. 6. Normalized impedance values of the elements in the T network of Fig. 5a as functions of. ( ) r a = Ra R dc = Re (1+j) (1 e (1+j) ) ( 1+e (1+j) ) l a = La j(1 e L dc = Im (1+j) ) ((1+j) (1+e (1+j) ) ) r b = R b R dc = Re 2(1+j) e (1+j) ( 1 e 2(1+j) ) l b = L b = Im L dc 2je (1+j) (1+j) (1 e 2(1+j) ) Re represents the real part of a complex value, and Im represents the imaginary part of a complex value. r a, l a, r b, and l b are plotted as functions of in Fig. 6. Note = 1 indicates the frequency at which the layer thickness equals the skin depth (h=δ). Figure 6 indicates that for non-sinusoidal waveforms with harmonics distributed in multiple frequencies, if a majority of its harmonic components are in the low frequency ( 1) range (including dc), using the netlist generated at the fundamental frequency of this circuit is accurate, because the element values stay relatively constant across the whole frequency range. If a majority of its harmonic components are distributed at multiple frequencies in the high frequency ( 3) range, then a single frequency netlist cannot capture the wide-band behaviors of the magnetic components. As discussed in Appendix II, under this situation, concepts of effective frequency [21] can be utilized, which offers conservation of loss according to Fourier Analysis. IV. EXPERIMENTAL VERIFICATION It is clear that the MLM approach with layer stacking and frequency dependent impedances included can always provide more information than many conventional magnetic models. An interesting question is how accurate an MLMbased time-domain simulation is in predicting the current distribution and loss in the magnetics (when the magnetics are simulated together with the circuit). To answer this question, we investigate the applicability and limitations of the MLM-based time-domain simulation method by comparing the SPICE simulation results with experimental measurements (2) (a) Fig. 7. Cross-section view of the layer stack of the two coupled inductors (a) Design #1 and (b) Design #2. Note that the dimension of this stackup is not to scale. (b) Fig. 10. Measured waveforms for Design #1 at 400 khz. Fig. 8. Circuit structure of the test setup. The leakage and magnetizing inductances are included in the dashed circuit block. (a) Fig. 11. Simulation and experimental waveforms showing the current sharing between the two primary windings at 400 khz. (a) Design #1: PCB #2 (4 layer) on top. (b) Design #2: PCB #1 (8 layer) on top. Solid line: experimental waveform. Dashed line: simulated waveform. (b) (a) Fig. 9. (a) Photo of the prototype. (b) Experimental setup. in two setups. The first setup investigates a coupled inductor design with sophisticated winding patterns, but only with a single frequency component. The second setup is the design of multiple magnetic components in a newly presented circuit architecture that has switching harmonics [22]. A. Design Evaluation in a Purely Sinusoidal Setup In this setup, the simulation and experimental waveforms of two 10:1 coupled inductors with identical cores but different layer stack-ups are compared. Figure 7 shows the cross-section view of the two coupled inductors. The winding structure of the two coupled inductors are manufactured with two 72 mil printed-circuit boards (PCB #1 and #2), which are linked by a single EPCOS ELP43 core (N49 material) with a distributed air gap on top of both the center and side legs. PCB #1 has eight copper layers and PCB #2 has four copper layers. All layers are 4 oz copper layers. In Design #1, the 4-layer board is placed on top of the 8-layer board, and is closer to the air gap (Fig. 7a); in Design #2, the 8-layer board is placed on top of the 4-layer board, and is closer to the air gap (Fig. 7b). The (b) odd layers (from the top) of both PCBs each have five spiral series-connected turns. Layers 1&3 and 5&7 of PCB #1, and layers 1&3 of PCB #2 are each connected into a series-tied pair to formulate three 10-turn windings (red layers of Fig. 7). These three 10-turn windings are then connected in parallel as primary windings. The even layers of both PCBs each have a single turn and are connected in parallel to form a six layer 1-turn secondary winding (yellow layers of Fig. 7). Figure 8 shows the schematic of the test circuit. A film capacitor (not shown) is connected in series with the input source to block the dc component. The current in the 4-layer PCB, in the 8-layer PCB and the total input current (I 4L, I 8L and I in ) are observed using TCP202 and TCP0030 current probes. Since the two PCBs are stacked and are linked with a single core, they ideally have identical flux linkage and identical voltage-drop per turn. The different ac impedance of the two boards will create unbalanced current sharing between their copper layers. Intuitively, at direct current (dc) or low frequency alternate current (ac), the 8-layer board has a lower resistance and will always carry more current. However, high frequency ac current will tend to flow in the board that is closer to the gap, redistributing the current and creating interesting phenomena that we seek to visualize in time-domain simulations. Figure 9a shows the prototype planar coupled inductor with the blocking capacitor and terminating resistor. Fig. 9b shows the experimental bench setup. The gain of the power amplifier (a) Fig. 13. Simulated current distribution in all 12 layers of Design #1 at 400 khz using time-domain LTSpice simulations based on the M2Spice subcircuit. The layers are numbered from top to bottom with layer 1 closest to the gap. Each primary winding consists of two layers. (b) Fig. 12. Comparison of the amplitude and phase of the currents for the experimental (solid lines) and simulated results (dashed lines). (a) Design #1: PCB #2 (4 layer) on top. (b) Design #2: PCB #1 (8 layer) on top. The amplitude and phase of each PCB is normalized to the input current (200 ma). The phase is positive if the PCB current leads the input current. is tuned to make the total input current 200 ma (peak). The 1 ohm load resistor has a constant resistance across the full frequency range. Using the MLM-based approach, the geometry of the magnetic circuit including the two planar winding structures is processed by M2Spice to generate the SPICE netlist. The cross winding capacitance is modeled separately with EQS methods [6]. The layer-to-layer capacitance of the prototype lumped to the secondary is calculated to be 150 nf. The netlist is then imported into LTSpice and simulated. The three currents are then modeled using time-domain simulations. In LTSpice, the device is constantly driven by a 200 ma ac current source and the frequency is swept from 10 khz to 700 khz (with a netlist generated for each frequency). We intentionally isolated the modeling and the measurement process, i.e., all modeling efforts are rigorously developed based on a priori design information (the known system geometry only), without information gathered after the prototype was built. Figure 10 shows the measured waveforms of Design #1 at 400 khz. Figure 11 compares the simulated and measured current waveforms of the two layer stacks operating at 400 khz. The simulated waveforms match well with the experimental waveforms. The time-domain SPICE simulation based on the MLM as generated by M2Spice can accurately predict the current distribution between two parallel windings at different frequencies. The time-domain simulation is capable of accurately quantifying the phenomenon that the PCB closer to the air gap carries more current regardless of its dc resistance. Figure 12 compares the simulated and measured amplitudes and phases of the current flowing through the two layer stacks across a frequency range of 10 khz 700 khz (with netlists separately generated for each frequency). As frequency increases, it can be seen that the phase shift between the two PCB currents also increases, especially when the frequency is above 100 khz. As a result of the increased phase shift, the amplitude increases in the bottom PCB, indicating circulating currents and higher overall rms current. The experimental measurement can only reveal the current sharing between the two PCBs (grouping multiple layers). An experimental effort to determine the current distribution in all 12 layers is extremely challenging, because any additional current measurement infrastructure needs to be accurately modeled and calibrated, and will involve new approximations. While hard to do experimentally, the current distribution in the 12 layers can be easily visualized using SPICE timedomain simulations. Fig. 13 shows the current distribution in the 12 layers in Design #1 of Fig. 7a. As expected, the seriesconnected pairs have identical current distribution (layers 1&3, layers 5&7 and layers 9&11), and the parallel layers unevenly share the current with dramatic phase shift. To validate the current distribution shown in simulation, we predict the current distribution using Ansoft Maxwell 2D simulations. To emulate the condition that the magnetic component is connected to the circuit, the external drive Fig. 14. FEM simulation results for an instantaneous current and field distribution of Design #1 with the 4-layer board on top when the total primary side current is 200 ma at 0 degrees. Fig. 17. Comparing the simulation results of Design #2 using singlefrequency netlists (square: 40 khz, or circle: 400 khz) to visualize the current distribution across a
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