How to include frequency dependent complex permeability Into SPICE models to improve EMI filters design?

Electromagnetic interference (EMI) ﬁlters design is a rather difﬁcult task where engineers have to choose adequate magnetic materials, design the magnetic circuit and choose the size and number of turns. The ﬁnal design must achieve the attenuation requirements (constraints) and has to be as compact as possible (goal). Alternating current (AC) analysis is a powerful tool to predict global impedance or attenuation of any ﬁlter. However, AC analysis are generally performed without taking into account the frequency-dependent complex permeability behaviour of soft magnetic materials. That’s why, we developed two frequency-dependent complex permeability models able to be included into SPICE models. After an identiﬁcation process, the perfor-mances of each model are compared to measurements made on a realistic EMI ﬁlter prototype in common mode (CM) and differential mode (DM) to see the beneﬁt of the approach. Simulation results are in good agreement with the measured ones especially in the middle frequency range. © 2017 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

Electromagnetic interference (EMI) filters design is a rather difficult task where engineers have to choose adequate magnetic materials, design the magnetic circuit and choose the size and number of turns. The final design must achieve the attenuation requirements (constraints) and has to be as compact as possible (goal). Alternating current (AC) analysis is a powerful tool to predict global impedance or attenuation of any filter. However, AC analysis are generally performed without taking into account the frequency-dependent complex permeability behaviour of soft magnetic materials. That's why, we developed two frequency-dependent complex permeability models able to be included into SPICE models. After an identification process, the performances of each model are compared to measurements made on a realistic EMI filter prototype in common mode (CM) and differential mode (DM) to see the benefit of the approach. Simulation results are in good agreement with the measured ones especially in the middle frequency range. © 2017 Author(s). All

I. INTRODUCTION
Static converters have made a lot of progress these last decades thanks to a better knowledge of semiconductor power switches and the increase of switching frequencies which allows to reduce drastically the volume of passive components (magnetic, and capacitors). Unfortunately, increasing the switching frequency leads to generate a large amount of electromagnetic interferences (EMI). Two types of disturbances exist: common mode (CM) and differential mode (DM). Usually, each type of disturbance requires a specific magnetic component (an inductance designed with winding modes and materials well suited to every need) and specific capacitors (Cx(DM),Cy(CM)). Scientists and engineers are then involved in the development of EMI filters design softwares. 1,2 These approaches generally neglect the frequency dependent complex permeability (µ) of the soft magnetic materials (with µ( f ) = µ ( f ) − j µ ( f )). The real part µ gives an image of how much energy can be stored whereas µ gives an image of the losses. Two frequency-dependent complex permeability models able to be included into SPICE models for AC analysis are presented here. The first model is inspired by the magnetic field diffusion equation solution 3 and the second one is inspired by Nomura's work. 4 Both models are implemented thanks to behavioural sources and Laplace equations available in Spice softwares. Their implementation in LTSpice®will be explained in the second section. Results concerning the ability of each model to predict complex permeability versus frequency on two different materials will be shown on the third section as well as all simulation cases of the EMI filter to see the benefits of the approach on a realistic device. Discussion of all the results will be made in the fourth section. The last section will summarize the a Electronic mail: fabien.sixdenier@univ-lyon1.fr results and propose some prospects about using this approach in a future complete filter design tool.

II. COMPLEX PERMEABILITY MODELS AND IMPLEMENTATION IN SPICE
Due to their excellent magnetic properties (low losses, high µ r and high B s ), nanocrystalline materials became more suitable for common mode inductors than laminated or powder iron and ferrites. 5 In our case, we chose, for our EMI filter prototype, Nanophy 6 cores (µ r ≈ 30000) for CM and Kµ 6 (µ r ≈ 200) cores for DM. Real and imaginary parts of complex permeability were extracted from impedance measurements made on each core thanks to an HP4294A impedance analyser. 7 These measurements were, then, taken as reference to fit both models.

Diffusion's like model
As nanocrystalline alloys have a relatively high electrical conductivity (σ ≈ 833 kS/m), skin effect will necessary occur, even if the layers thickness is very low (≈ 18 µm). Starting from the diffusion equation, the complex permeability µ can be expressed like (1), 3 where ω 0 is the pulsation where the skin depth is equal to the half thickness of the strip and µ DC is the low frequency value of µ . Only 2 parameters per material have to be identified with this model. Nomura 4 proposed a model that used two straight lines on a log-log plot via a generic equation (2) able to describe µ or µ versus frequency.

Nomura's model
10 a 2 log 10 ( f )+b 2 1 + 10 (a 2 −a 1 ) log 10 ( f )+(b 2 −b 1 ) (2) a 1 , a 2 are the slopes and b 1 and b 2 are y intercepts of each line respectively. Thus, for a given material, we had to identify 4 parameters per curve (µ and µ ) so 8 parameters in total. These parameters were fitted numerically for better accuracy.

B. LTSPICE implementation
For both models implementation, a gyrator 8,9 was used in order to separate electrical and magnetic physical domains. With this analogy, magneto-motive forces and flux time derivative (in AC analysis the flux time derivative dφ dt becomes jωφ) became equivalent to voltages and currents respectively. Both models implementation required current controlled voltage sources, behavioural voltage sources, spice directives and mathematical functions expressed in function of the Laplace operator (s = jω) instead of the frequency f.

Diffusion's like model
The figure 1 shows an example of a single winding and a single magnetic circuit. In figure 1, B1 and B2 are current controlled voltage sources that represent the winding and the coupling between electrical and magnetic physical domains (gyrator). The behavioural source B Re mu represents the magnetic circuit (A e is the magnetic circuit cross section, MPL is the average magnetic path length and µ 0 is the vacuum permeability). One can also see in figure 1, how (1) is programmed with a .func SPICE directive. The other equation calculates the complex reluctance as (3). It has to be mentioned that resistors R3 and R1 are fictitious resistors inserted here, in order to avoid to have only pure voltage sources linked in a current loop.  Nomura 4 used behavioural voltage sources (bvs) to reproduce the resistive and inductive behaviour of the total impedance of a common mode choke. In our case, the electrical inductive and resistive behaviours are modeled by using in the magnetic domain a capacitor (C µ ) and a resistor (R µ ) respectively 10 linked in series.

Nomura's model implementation
To sum up, the Nomura's model needs two coupled current controlled voltage sources to represent the winding through a gyrator) and two behavioural sources to represent R µ and C µ . Calculations of (2), (4), (5) are made via SPICE directives as explained before.

A. Material models results
The Nanophy and Kµ materials were characterized and the parameters of the both models were identified by numerically fitting the simulated curves with respect to the measured ones. Nomura and diffusion's like model parameters are listed in Table I.
The figure 2 shows the real and imaginary parts of complex permeability computed by both models compared to measurements for high (Nanophy) and low (Kµ) permeability cores. The shape of each curve is globally retrieved by both models. Nevertheless, one can see that (1) is limited in terms of accuracy compared to (2) especially for Nanophy cores in the whole frequency range. Other tests (not shown) made on materials with permeabilities µ r ≈3000, µ r ≈30000 and µ r ≈500000 confirmed this tendency. So far, the only drawback of Nomura's model is that it has more parameters to identify than the diffusion's like model (8 versus 2 respectively). The identification process is then a bit more difficult and takes more time.

B. EMI filter prototype
The figure 3 shows the EMI filter scheme with common mode and differential excitation sources connections to test one or other mode. EMI filters nominal values for each component are listed in the Table II The input impedance of this filter will be measured and the results will be be compared to 4 simulation cases, listed in the Table III, in each mode (common and differential) for each model. Equivalent serial R, L, C values for C x and C y capacitors (used for simulation cases B to D) are listed in the Table IV. In the same table are reported the dimensions and number of turns of each inductor (simulation cases C and D).
Impedance modulus and argument have been measured and simulated (4 cases) with the proposed models. Impedance modulus in common and differential modes are plotted in figures 4 and 5 respectively. Discussion about the results is on the next section.

IV. DISCUSSION
First, one can observe that cases "A" and "B" are quite the same. Indeed, self-resonance frequencies of C x and C y are very high ( f C x ≈ 2.2 MHz, f C y ≈10 MHz) and ESL values are very low compared to L CM or L DM , so their influence on the impedance modulus in any mode is negligible. The benefits of our approach can be clearly seen in figure 4, when 40 kHz < f < 10 MHz because simulated cases "C" and "D" are in far better agreement with measurements than the cases "A" and "B". This is mainly due to Nanophy cores µ and µ large variations in this frequency range (see figure 2). As the complex permeability was better predicted with Nomura's model (2) than with the diffusion's like model (1) (see figure 2), the results obtained on the EMI filter (see figures 4) of cases "C" or "D" are better when using (2) than (1) in the filter SPICE model. In differential mode (figure 5), the K µ (µ r = 200) core is solicited. For this core µ and µ are relatively stable until 2 MHz. That's why all simulation cases (even ideal ones) are in good agreement with experimental results. No model seems better than the other for DM mode. By connecting on each winding, of each core, a parasitic capacitance C p = 8 pF (fitted after a complete EMI filter measurement), the simulated second resonance peaks (curves of Cases "D") in each mode are in good agreement with the measurements. The third and fourth measured resonance peaks ( f > 10 MHz) are certainly due to turn to shield FIG. 4. Impedance modulus results in common mode excitation for all simulation case and both models. capacitances 11 that are resonating with stray inductances (connections mainly). To predict these very high resonance peaks, accurate numerical tools should be employed and are therefore out of the scope of this study.

V. CONCLUSION
Two analytical models able to predict the frequency dependent magnetic complex permeability behaviour were analysed in this paper. After identification, Nomura's model showed that it has a better accuracy in the whole frequency range and is more robust (better adaptability to various materials). Nomura's model seems to be a better choice even if it is a bit more difficult to identify. This tendency has to be confirmed by further work on other materials like ferrite or iron powders. Both models have been successfully implemented in LTSpice® thanks to the tools available in the software, to simulate a realistic EMI filter. Both models showed that it is crucial to take into account the frequency dependant properties of magnetic material in the medium frequency range especially in common mode. Adding a parasitic capacitance on each winding allowed to retrieve the second resonance peak in both modes. Unfortunately, its value is very hard to predict analytically. 11 That's why Future work will focus on this task in order to have a full design tool accurate enough to do EMI filter virtual prototyping.

ACKNOWLEDGMENTS
This project has received funding from the European Union's Horizon 2020 research and innovation programme under grant agreement N • 636170.