# MEC Toolbox Users Manual

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<ul><li><p>Purdue University School of Electrical and Computer Engineering </p><p>Power and </p><p>Magnetic Equivalent Circuit (M</p><p>Coordinator:</p><p>I. INTRODUCTION The Magnetic Equivalent Circuit (MEC) toolbox </p><p>electromagentic devices such as inductors, transformerssolves a nonlinear magnetic equivalent circuitrepresented as a function of fluxmagnetic branch shown in Fig.branch number bn, denoted as in parallel with flux source potential drop across the branch is denoted </p><p>Purdue University School of Electrical and Computer Engineering </p><p>Power and Energy Devices and System Area </p><p>Magnetic Equivalent Circuit (MEC) Toolbox Users Manual </p><p>Version 1.0, February 2010 </p><p>Coordinator: S.D. Sudhoff (sudhoff@purdue.edu) </p><p>The Magnetic Equivalent Circuit (MEC) toolbox is a Matlab toolbox for magnetic analysis ofsuch as inductors, transformers, and electric machinery</p><p>solves a nonlinear magnetic equivalent circuit using a mesh-based analysisrepresented as a function of flux density. This toolbox is developed based on the standard magnetic branch shown in Fig. 1. Therein, the reluctance / permeance of a particular </p><p>, denoted as Rbn / Pbn is in series with MMF source FS,bn, and this combination is ,S bn . The flux flowing into the branch is denoted </p><p>potential drop across the branch is denoted FB,bn. </p><p>Figure 1. Standard Magnetic Branch </p><p>Purdue University School of Electrical and Computer Engineering </p><p>toolbox for magnetic analysis of electric machinery. It sets up and based analysis with permeability </p><p>This toolbox is developed based on the standard particular branch with </p><p>, and this combination is he flux flowing into the branch is denoted </p><p>,B bn and the </p></li><li><p> II. MEC CORE ROUTINES In this section, the main routines in the MEC toolbox will be explained. The routines include: 1. mec_init </p><p>Function call: [MEC] = mec_init(Nmesh, Nbranch) [MEC] = mec_init(Nmesh, Nbranch, maxiter, rec, aec) Purpose: This routine is used for setting up the data structure required to do the </p><p>analysis based on the number of mesh loops and branches of the equivalent circuit. User can also specify the number of iteration to be used in solving non-linear circuits, as well as the relative and absolute error criteria. </p><p>Inputs: Nmesh Number of mesh loops in the circuit Nbranch Number of branches in the circuit </p><p> maxiter (Optional) Maximum number of iteration to solve non-linear circuits (default: 50) </p><p> rec (Optional) Relative error criteria (default: 10-12) aec (Optional) Absolute error criteria (default: 10-14) </p><p>Outputs: MEC Initialized data structure for the MEC MEC.Nbranch Number of branches in the circuit MEC.bn Vector of branch number (size: Nbranch x 1) MEC.type Vector of branch types (1: linear, 2: non-linear) </p><p>MEC.R Vector of branch reluctances (size: Nbranch x 1) (1/H) </p><p>MEC.Fs Vector of branch MMF sources (size: Nbranch x 1) (A) </p><p> MEC.Ps Vector of flux sources (size: Nbranch x 1) (Wb) MEC.A Vector of cross sections of ferro/ferrimagnetic branch </p><p>(m2) MEC.l Vector of lengths of ferro/ferrimagnetic material (m) </p><p>MEC.pmub Vector of pointers to permeability functions for ferro / ferrimagnetic branches </p><p> MEC.phi_est Vector of estimated flux into the permeance / reluctance part of the branches </p><p> MEC.ml Vector of branch mesh lists (size: Nbranch x 1) MEC.pml Vector of postive entries of mesh list (size: Nbranch x 1) MEC.nml Vector of negative entries of mesh list (size: Nbranch x 1) </p><p>2. mec_lp_branch Function call: [MEC] = mec_lp_branch(MEC,bn,P,ml) Purpose: Adds a magnetically linear permeance branch to the magnetic equivalent </p><p>circuit. Inputs: MEC MEC data structure created using mec_init function </p></li><li><p> bn Branch number of the linear permeance in the circuit (Note: This number should be unique) </p><p>P Branch permeance (H) ml Vector of branch mesh list </p><p>Outputs: MEC Updated MEC data structure </p><p>3. mec_lps_branch Function call: [MEC] = mec_lps_branch(MEC,bn,P,Fs,Ps,ml) Purpose: Adds a magnetically linear branch with a permeance and source to the </p><p>magnetic equivalent circuit. Note that the permeance is in series with the MMF source (Fs) and this combination is in parallel with the flux source (Ps) </p><p>Inputs: MEC MEC data structure created using mec_init function bn Branch number of the permeance and source in the </p><p>circuit (Note: This number should be unique) P Branch permeance (H) Fs Branch MMF source (A) Ps Branch flux source (Wb) ml Vector of branch mesh list </p><p>Outputs: MEC Updated MEC data structure </p><p>4. mec_lr_branch Function call: [MEC] = mec_lr_branch(MEC,bn,R,ml) Purpose: Adds a magnetically linear reluctance branch to the magnetic equivalent </p><p>circuit. Inputs: MEC MEC data structure created using mec_init function </p><p>bn Branch number of the permeance and source in the circuit (Note: This number should be unique) </p><p>R Branch reluctance (1/H) ml Vector of branch mesh list </p><p>Outputs: MEC Updated MEC data structure </p><p>5. mec_lrs_branch Function call: [MEC] = mec_lrs_branch(MEC,bn,R,Fs,Ps,ml) Purpose: Adds a magnetically linear branch with reluctance and source to the </p><p>magnetic equivalent circuit. Note that the reluctance (R) and MMF source (Fs) are in series. This combination is in parallel with the flux source (Ps). </p><p>Inputs: MEC MEC data structure created using mec_init function bn Branch number of the reluctance and source in the </p><p>circuit (Note: This number should be unique) R Branch reluctance (1/H) Fs Branch MMF source (A) </p></li><li><p>Ps Branch flux source (Wb) ml Vector of branch mesh list </p><p>Outputs: MEC Updated MEC data structure </p><p>6. mec_nl_branch Function call: [MEC] = mec_nl_branch(MEC,bn,A,l,pmub,MMP,ml,phi_est) Purpose: Adds a magnetically non-linear branch to the magnetic equivalent circuit. Inputs: MEC MEC data structure created using mec_init function </p><p>bn Branch number of the non-linear branch in the circuit (Note: This number should be unique) </p><p>A Effective cross-sectional area of the branch (m2) l Effective length of the branch (m) pmub Pointer to a function which returns the permeability </p><p>and the derivative of the permeability with respect to the flux density (B) as a function of flux density </p><p>MMP Structure of magnetic material parameters ml Vector of branch mesh list phi_est Estimated flux going into the reluctance part of the </p><p>branch (T) Outputs: MEC Updated MEC data structure </p><p>7. mec_nls_branch Function call: [MEC] = mec_nls_branch(MEC,bn,A,l,pmub,MMP,Fs,Ps,ml,phi_est) Purpose: Adds a magnetically non-linear branch with a source to the magnetic </p><p>equivalent circuit. Note that the reluctance part is in series with the MMF source (Fs) and this combination is in parallel with the flux source (Ps). </p><p>Inputs: MEC MEC data structure created using mec_init function bn Branch number of the non-linear branch in the circuit </p><p>(Note: This number should be unique) A Effective cross-sectional area of the branch (m2) l Effective length of the branch (m) pmub Pointer to a function which returns the permeability </p><p>and the derivative of the permeability with respect to the flux density (B) as a function of flux density </p><p>MMP Structure of magnetic material parameters Fs Branch MMF source (A) Ps Branch flux source (Wb) ml Vector of branch mesh list phi_est Estimated flux going into the reluctance part of the </p><p>branch (T) Outputs: MEC Updated MEC data structure </p></li><li><p>8. mec_solve Function call: [Pm,Pb,PR,Fb,converge] = mec_solve(MEC) Purpose: Solves the magnetic equivalent circuit Inputs: MEC MEC data structure created using mec_init function Outputs: Pm Vector of mesh fluxes (Wb) (size: Nmesh x 1) Pb Vector of branch fluxes (Wb) (size: Nbranch x 1) PR Vector of reluctances fluxes (Wb) (size: Nbranch x 1) Fb Vector of branch potential drops (size: Nbranch x 1) converge Flag denoting convergence of the analysis (1 </p><p>analysis converges, 0 otherwise) </p><p>III. INSTALLATION INSTRUCTIONS The MEC toolbox can be used in Matlab by adding the location of the toolbox to the Matlab </p><p>path. It should be noted that the MEC Toolbox consists of three separate folders, namely MEC Core, MEC Models and Supplemental, and License, Documentation, Example. The MEC Core folder contains all core routines mentioned in the previous section, while MEC Models and Supplemental folder contains additional routines that will help users to build and solve the MEC for several inductor and transformer arrangements. The folder License, Documentation, and Example includes license information, this manual, and the example discussed below. </p><p>To add the MEC Toolbox to the Matlab path, click on File drop down menu from Matlab startup screen then click on Set Path. From there a new dialogue window will appear. Click on the Add Folder button. The specific location of the folder in the computer can now be selected. It should be noted that these steps need to be done for both folders in order to use the MEC Toolbox. </p><p>IV. EXAMPLE The use of the MEC toolbox will be demonstrated in this section. Consider the following EI </p><p>core inductor shown in Fig. 2. The specific dimensions of the EI core used herein are listed on Table I. </p><p>TABLE I DIMENSIONS OF EXPERIMENTAL EI CORE </p><p>Parameter Value (mm) Parameter Value (mm) we 28.7 wb 27.3 wc 57.4 wi 30.5 ws 37.0 g 2.96 ds 48.8 d 119.8 </p><p>Therein, wi is the width of the I-core, g is the air gap length, ds is the depth of the slot, wb is the width of the bottom part of the E-core, we is the width of the end post of the E-core, ws, is the width of the slot, wc is the width of the center post of the E-core, and d denotes the depth of the EI core into the page. The core material used was a ferrite material grade 3C90. </p></li><li><p>Figure </p><p>The MEC model of the EI core used herein is shown in Fig. is a very simple MEC of the EI core designed to illustrate the use of the MEC toolboxanalysis, fringing and leakage fluxes arthe corresponding expression for each permeance </p><p>In this analysis, thirteen branches are chosen to describe the magnetic elements of the EI core. In Fig. 3, each branch of the MEC (direction of the MMF drop in each branch. chosen. The first ten branches are associated with the permeances in the magnetic materiallast three branches are associated with the permeances in the air gap. </p><p>It should be noted that by using the definition of standard magnetic branch shown in Fig. MMF source is included as part of the first branch of the MEC. The branch number can be chosen arbitrarily, however the corresponding number for each branch must be unique. In other words, </p><p>Figure 2. EI Core Inductor Structure </p><p>MEC model of the EI core used herein is shown in Fig. 3. It should be noted here that this a very simple MEC of the EI core designed to illustrate the use of the MEC toolbox</p><p>analysis, fringing and leakage fluxes are not considered. A comprehensive MECthe corresponding expression for each permeance have been set forth in [1]. </p><p>Figure 3. EI Core MEC </p><p>In this analysis, thirteen branches are chosen to describe the magnetic elements of the EI core. of the MEC (P1 P13) is indicated with dashed-blue circles along with the </p><p>direction of the MMF drop in each branch. The direction of the MMF drop may be arbitrarily The first ten branches are associated with the permeances in the magnetic material</p><p>last three branches are associated with the permeances in the air gap. It should be noted that by using the definition of standard magnetic branch shown in Fig. </p><p>MMF source is included as part of the first branch of the MEC. The branch number can be chosen arbitrarily, however the corresponding number for each branch must be unique. In other words, </p><p>It should be noted here that this a very simple MEC of the EI core designed to illustrate the use of the MEC toolbox. In this </p><p>MEC of the EI core and </p><p>In this analysis, thirteen branches are chosen to describe the magnetic elements of the EI core. blue circles along with the </p><p>The direction of the MMF drop may be arbitrarily The first ten branches are associated with the permeances in the magnetic material and the </p><p>It should be noted that by using the definition of standard magnetic branch shown in Fig. 1, the MMF source is included as part of the first branch of the MEC. The branch number can be chosen arbitrarily, however the corresponding number for each branch must be unique. In other words, </p></li><li><p>two separate branches may not share the same branch number. For branch 1, 4, and 11, the direction of the MMF drop is chosen such that the flux flowing through the permeance is positively coupled with the mesh 1 and negatively coupled with the mesh 2 . For branch 2, 5, 7, 9, and 12, the direction of the MMF drop is chosen such that each permeance is positively coupled with the mesh 1 . Similarly for branch 3, 6, 8, 10, and 13, the direction of the MMF drop is such that each permeance is positively coupled with mesh 2 . </p><p>As mentioned in the previous section, the MEC toolbox is developed using a mesh-based analysis. It is interesting to note that the computational performance of the mesh-based MEC far exceeds the nodal-based MEC for nonlinear operating conditions [2]. Each mesh corresponds to a possible flux path in the MEC where the direction of each flux can be chosen arbitrarily. For this particular analysis, two meshes ( 1 and 2 ) along with their directions are indicated with solid-red lines. </p><p>Thus, there are thirteen branches and two meshes in this MEC model of the EI core. From [1], the MMF source is given as 1F Ni= , where N is the number of wire turn, wound around the center post, and i is the current through the winding. For the study conducted herein, N has been set to 40 turns and [ ]0 65 Ai = . </p><p>The complete script to do the MEC analysis of the EI core in Fig. 3 is shown below. This script is named example.m and is located in the same folder as this document. </p><p>1 %--------------------------------------------------- 2 % SCRIPT: example.m 3 % DESCRIPTION: A simple MEC of the EI core inductor 4 % to demonstrate how to use the MEC 5 % toolbox 6 % AUTHOR: Cahya Harianto for Scott D. Sudhoff 7 % Purdue University 8 % Electrical Engineering Building 9 % 465 Northwestern Avenue 10 % West Lafayette, IN 47907-2035 11 % sudhoff@ecn.purdue.edu 12 % 765-497-7648 13 %---------------------------------------------------- 14 15 16 % Initialize 17 clear all 18 close all 19 clc 20 21 % Define some handy numbers 22 mm = 1e-3; 23 mu0 = 4*pi*10^-7; 24 </p><p> 25 % Specify the dimensions of the EI Core based on 26 % J.Cale, S.D.Sudhoff, and L.Tan, "Accurately Modeling EI Core Inductors 27 % Using a High-Fidelity Magnetic Equivalent Circuit Approach", 28 % IEEE Transactions on Magnetics, Vol.42, No.1, January 2006 29 we = 28.7 * mm; 30 wc = 57.4 * mm; 31 ws = 37.0 * mm; 32 ds = 48.8 * mm;...</p></li></ul>