Purdue University School of Electrical and Computer Engineering

Power and

Magnetic Equivalent Circuit (M

Coordinator:

I. INTRODUCTION The Magnetic Equivalent Circuit (MEC) toolbox

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

Purdue University School of Electrical and Computer Engineering

Power and Energy Devices and System Area

Magnetic Equivalent Circuit (MEC) Toolbox

Users Manual

Version 1.0, February 2010

Coordinator: S.D. Sudhoff (sudhoff@purdue.edu)

The Magnetic Equivalent Circuit (MEC) toolbox is a Matlab toolbox for magnetic analysis ofsuch as inductors, transformers, and electric machinery

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

, denoted as Rbn / Pbn is in series with MMF source FS,bn, and this combination is

,S bnΦ . The flux flowing into the branch is denoted

potential drop across the branch is denoted FB,bn.

Figure 1. Standard Magnetic Branch

Purdue University School of Electrical and Computer Engineering

toolbox for magnetic analysis of electric machinery. It sets up and based analysis with permeability

This toolbox is developed based on the standard particular branch with

, and this combination is he flux flowing into the branch is denoted ,B bnΦ and the

II. MEC CORE ROUTINES In this section, the main routines in the MEC toolbox will be explained. The routines include: 1. mec_init

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

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.

Inputs: Nmesh – Number of mesh loops in the circuit Nbranch – Number of branches in the circuit

maxiter – (Optional) Maximum number of iteration to solve non-linear circuits (default: 50)

rec – (Optional) Relative error criteria (default: 10-12) aec – (Optional) Absolute error criteria (default: 10-14)

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)

MEC.R – Vector of branch reluctances (size: Nbranch x 1) (1/H)

MEC.Fs – Vector of branch MMF sources (size: Nbranch x 1) (A)

MEC.Ps – Vector of flux sources (size: Nbranch x 1) (Wb) MEC.A – Vector of cross sections of ferro/ferrimagnetic branch

(m2) MEC.l – Vector of lengths of ferro/ferrimagnetic material (m)

MEC.pmub – Vector of pointers to permeability functions for ferro / ferrimagnetic branches

MEC.phi_est – Vector of estimated flux into the permeance / reluctance part of the branches

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)

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

circuit. Inputs: MEC – MEC data structure created using mec_init function

bn – Branch number of the linear permeance in the circuit (Note: This number should be unique)

P – Branch permeance (H) ml – Vector of branch mesh list

Outputs: MEC – Updated MEC data structure

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

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)

Inputs: MEC – MEC data structure created using mec_init function bn – Branch number of the permeance and source in the

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

Outputs: MEC – Updated MEC data structure

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

circuit. Inputs: MEC – MEC data structure created using mec_init function

bn – Branch number of the permeance and source in the circuit (Note: This number should be unique)

R – Branch reluctance (1/H) ml – Vector of branch mesh list

Outputs: MEC – Updated MEC data structure

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

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).

Inputs: MEC – MEC data structure created using mec_init function bn – Branch number of the reluctance and source in the

circuit (Note: This number should be unique) R – Branch reluctance (1/H) Fs – Branch MMF source (A)

Ps – Branch flux source (Wb) ml – Vector of branch mesh list

Outputs: MEC – Updated MEC data structure

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

bn – Branch number of the non-linear branch in the circuit (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

and the derivative of the permeability with respect to the flux density (B) as a function of flux density

MMP – Structure of magnetic material parameters ml – Vector of branch mesh list phi_est – Estimated flux going into the reluctance part of the

branch (T) Outputs: MEC – Updated MEC data structure

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

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).

Inputs: MEC – MEC data structure created using mec_init function bn – Branch number of the non-linear branch in the circuit

(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

and the derivative of the permeability with respect to the flux density (B) as a function of flux density

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

branch (T) Outputs: MEC – Updated MEC data structure

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 –

analysis converges, 0 – otherwise)

III. INSTALLATION INSTRUCTIONS The MEC toolbox can be used in Matlab by adding the location of the toolbox to the Matlab

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.

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.

IV. EXAMPLE The use of the MEC toolbox will be demonstrated in this section. Consider the following EI

core inductor shown in Fig. 2. The specific dimensions of the EI core used herein are listed on Table I.

TABLE I

DIMENSIONS OF EXPERIMENTAL EI CORE 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

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.

Figure

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

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.

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,

Figure 2. EI Core Inductor Structure

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

analysis, fringing and leakage fluxes are not considered. A comprehensive MECthe corresponding expression for each permeance have been set forth in [1].

Figure 3. EI Core MEC

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

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

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.

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,

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

MEC of the EI core and

In this analysis, thirteen branches are chosen to describe the magnetic elements of the EI core. blue circles along with the

The direction of the MMF drop may be arbitrarily The first ten branches are associated with the permeances in the magnetic material and the

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,

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 1Φ and negatively coupled with the mesh 2Φ . 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 1Φ . 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 2Φ .

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 (1Φ and 2Φ ) along with their directions are indicated with solid-

red lines. 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 = … .

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.

1 %------------------------------------------------- -- 2 % SCRIPT: example.m 3 % DESCRIPTION: A simple MEC of the EI core induct or 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

25 % Specify the dimensions of the EI Core based on 26 % J.Cale, S.D.Sudhoff, and L.Tan, "Accurately Model ing EI Core Inductors 27 % Using a High-Fidelity Magnetic Equivalent Circuit Approach", 28 % IEEE Transactions on Magnetics, Vol.42, No.1, Jan uary 2006 29 we = 28.7 * mm; 30 wc = 57.4 * mm; 31 ws = 37.0 * mm; 32 ds = 48.8 * mm;

33 wb = 27.3 * mm; 34 wi = 30.5 * mm; 35 g = 2.96 * mm; 36 d = 119.8 * mm; 37 N = 40; 38 i = linspace(0,65,100); 39 Npoints = length(i); 40 41 % Determine the MMF source on the circuit 42 F1 = N*i; 43 44 % Calculate the permeances on the air gap 45 P11 = mu0*(wc*d)/g; 46 P12 = mu0*(we*d)/g; 47 P13 = P12; 48 49 % 3C90 material parameters to calculate permeabilit y as a function of B 50 MP.mur = 22340.9259; 51 MP.muB.a = [ 1.1542 0.049742 0.049644 0.041155]; 52 MP.muB.b = [431.1763 2.29503 15.04824 74.28908]; 53 MP.muB.g = [ 0.4742 2.7955 0.59862 0.43996]; 54 MP.muB.d = MP.muB.a./MP.muB.b; 55 MP.muB.t = exp(-MP.muB.b.*MP.muB.g); 56 MP.muB.h = MP.muB.a.*MP.muB.t; 57 MP.muB.e = MP.muB.t./(MP.muB.t+1); 58 MP.muB.z = 1./(MP.muB.t+1); 59

60 % Initialize the estimated flux on the nonlinear br anches 61 phi_est1 = 0.0; 62 phi_est2 = 0.0; 63 phi_est3 = 0.0; 64 phi_est4 = 0.0; 65 phi_est5 = 0.0; 66 phi_est6 = 0.0; 67 phi_est7 = 0.0; 68 phi_est8 = 0.0; 69 phi_est9 = 0.0; 70 phi_est10 = 0.0; 71 72 % BUILD THE MEC MODEL OF THE EI CORE INDUCTOR 73 for n = 1:Npoints 74 [MEC] = mec_init(2,13); % Initialize the MEC 75 [MEC] = mec_nls_branch(MEC,1,wc*d,(wb/2+d s/2),@muB,MP,F1(n),0.0,

[1 -2],phi_est1); % Branch #1 76 [MEC] = mec_nl_branch(MEC,2,we*d,(wb/ 2+ds/2),@muB,MP,[1], phi_est2); % Branch #2

77 [MEC] = mec_nl_branch(MEC,3,we*d,(wb/2+ds /2),@muB,MP,[2], phi_est3); % Branch #3

78 79 [MEC] = mec_nl_branch(MEC,4,wc*d,ds/ 2,@muB,MP,[1 -2], phi_est4); % Branch #4 80 [MEC] = mec_nl_branch(MEC,5,we*d,ds/2 ,@muB,MP,[1], phi_est5); % Branch #5 81 [MEC] = mec_nl_branch(MEC,6,we*d,ds/ 2,@muB,MP,[2], phi_est6); % Branch #6

82 83 [MEC] = mec_nl_branch(MEC,7,wb*d,(wc/2+ws +we/2),@muB,MP,[1],

phi_est7); % Branch #7 84 [MEC] = mec_nl_branch(MEC,8,wb*d,(wc/2+ ws+we/2),@muB,MP,[2],

phi_est8); % Branch #8 85

86 [MEC] = mec_nl_branch(MEC,9,wi*d,(wc/2+ws +we/2),@muB,MP,[1], phi_est9); % Branch #9

87 [MEC] = mec_nl_branch(MEC,10,wi*d,(wc/2+w s+we/2),@muB,MP,[2], phi_est10); % Branch #10

88 89 [MEC] = mec_lp_branch(MEC,11,P11,[1 -2] ); % Branch #11

90 [MEC] = mec_lp_branch(MEC,12,P12,[1]); % Branch #12 91 [MEC] = mec_lp_branch(MEC,13,P13,[2]); % Branch #13 92

93 % SOLVE THE MEC 94 [Pm,Pb,PR,Fb,converge] = mec_solve(MEC); 95 96 % Flux flowing in each branch (PR) obtained from th e solution for 97 % present current is used as initial flux guess for the next 98 % current point 99 phi_est1 = PR(1); 100 phi_est2 = PR(2); 101 phi_est3 = PR(3); 102 phi_est4 = PR(4); 103 phi_est5 = PR(5); 104 phi_est6 = PR(6); 105 phi_est7 = PR(7); 106 phi_est8 = PR(8); 107 phi_est9 = PR(9); 108 phi_est10 = PR(10); 109 110 % compute total flux linkage 111 lambda(n) = N * -PR(1); 112 113 % copy convergence 114 c(n)=converge; 115 116 end

117 118 % Generate the lambda-vs-current curve 119 plot(i,lambda) 120 xlabel( 'Current (A)' ) 121 ylabel( '\lambda (V.s)' )

The first part of the code, starting from line 22 to line 39, sets the dimensions of the EI core as

defined in Table I. Line 37 – 42 of the code concerns with the MMF source of the MEC. As

mentioned previously, the MMF source is given as1F Ni= , where N is the number of turns, and i

is the current vector in the windings, varied linearly from 0 A to 65 A with 100 points in between. Line 45 – 47 of the code calculate the linear permeance in the air gap of the EI core, denoted

as P11 – P13 in Fig. 3. P11 is the permeance in the air gap on top of the center post. This quantity can be calculated

011

cw dP

g

µ= (1)

where 70 4 10 H/mµ π −= . Similarly for P12 and P13,

012 13

ew dP P

g

µ= = (2)

Line 50 – 58 of the code define the structure of parameters associated with the material of the EI core. In this particular example, the material of the EI core is ferrite grade 3C90. These parameters are needed for evaluating the permeability function and its derivative as a function of flux density B.

On lines 61 – 70, the estimated fluxes flowing in the material are initialized. Note that only fluxes associated with the nonlinear branches, in this case branches 1–10, need to be estimated. These fluxes, denoted as phi_est1 – phi_est10 , are estimated to be zero initially since the first point of the current vector to be evaluated is 0 A. This initialization is done because of nonlinearity of the material where the relative permeability is now a function of the flux density.

Line 73 – 116 of the code build and solve the MEC. It should be noted that the ‘for loop’ on line 73 is intended for the purpose of generating the λ–i curve by describing and evaluating the MEC at every operating point of the current vector i.

On line 74, the MEC is initialized by first calling the mec_init function. From previous analysis, it has been determined that the circuit has two meshes and thirteen branches. The inputs to the function are therefore Nmesh = 2 and Nbranch = 13.

Line 75 describes the first branch of the MEC. This branch consists of a nonlinear permeance with an MMF source and no flux source. The cross sectional area and the effective length of this

branch can be calculated as ( )cw d and( ) 2b sw d+ , respectively. The permeability of the material

is determined by evaluating the function muB.m, which is included in the MEC Toolbox package. This function determines material permeability and its derivative as a function of flux density B using the material parameters specified in the structure MP and the estimated flux flowing in this particular branch, denoted as phi_est1 . As indicated in the previous analysis, this particular

branch is positively coupled with the mesh 1Φ and negatively coupled with the mesh 2Φ . Thus,

the function mec_nls_branch is used to describe this branch with the following inputs: data

structure of the MEC created using the mec_init function denoted as MEC, branch number bn =

1, branch cross sectional area A = wc*d , effective branch length l = (wb/2 + ws/2) , pointer

to permeability function pmub = @muB, structure of material parameters MMP = MP, MMF

source at the operating point Fs = F1(n) , flux source Ps = 0.0 , the vector of mesh list ml = [1

-2] , and the estimated flux flowing in the branch phi_est1 . Line 76 – 77 describe the second and third branch of the MEC. These two branches consist of

a nonlinear permeance in the material with no MMF source or flux source. The cross sectional

area and the effective length of these two branches can be calculated as ( )ew d and ( ) 2b sw d+ ,

respectively. Similar to the first branch, the permeability of the material as a function of flux density needs to be determined by evaluating the function muB.m using material parameters MP and the estimated flux flowing in each branch. The estimated flux flowing in the second and third branch is denoted phi_est2 and phi_est3 , respectively. From Fig. 3, it can be observed that

the second branch is coupled positively with the mesh 1Φ , while the third branch is coupled

positively with the mesh 2Φ . The function mec_nl_branch is therefore used to model these

two branches. For the second branch, the inputs to the function are the data structure MEC, bn =

2, A = we*d , l = (wb/2 + ws/2) , pmub = @muB, MMP = MP, ml = [1] , phi_est =

phi_est2 . For the third branch, the inputs to the function are bn = 3 , A = we*d , l = (wb/2

+ ws/2) , pmub = @muB, MMP = MP, ml = [2] , phi_est = phi_est3 . Using similar technique as the second and third branch, the other nonlinear branches of the EI

core are modeled using the mec_nl_branch function. The differences on the inputs to the function to describe each branch are the branch number, the corresponding cross sectional area,

effective length of the branch, how the branch is coupled with the mesh 1Φ and or 2Φ , and the

corresponding estimated flux flowing in each branch. These calls are made in line 79 – 87 of the code.

Line 89 – 91 of the code concern with the modeling of the permeances in the air gap of the EI core. Since these branches are associated with linear permeances, the function mec_lp_branch

are used. The permeances for these branches, denoted P11 – P13, are given in (1) and (2), and they have been calculated on line 45 – 47 of the code, respectively. Note that branch 11 is positively

coupled with the mesh 1Φ and negatively coupled with the mesh 2Φ , while branch 12 is only

coupled positively with the mesh 1Φ , and branch 13 is coupled positively with the mesh 2Φ . And

so, the inputs to the function mec_lp_branch to model branch 11 are the structure MEC, bn =

11 , permeance P = P11 , and ml = [1 -2] . For branch 12, the inputs to the function are the

structure MEC, bn = 12 , P = P12 , and ml = [1] . For branch 13, the inputs are the structure

MEC, bn = 13 , P = P13 , and ml = [2] .

After describing the MEC, the circuit can now be solved using the mec_solve function,

shown on line 94. The mesh fluxes are now contained in Pm, branch fluxes in Pb, flux in each

reluctance / permeance in PR, potential drop in each branch in Fb, and the convergence of

analysis is indicated by converge .

On line 99 – 108 of the code, the elements of the vector PR, which correspond to the flux in each reluctance, are used as the estimated flux for the next point of iteration. This is done in order to help the solution to converge faster on every iteration.

In order to generate the λ–i curve, the total flux linking the winding needs to be calculated. This quantity is given as

1PNλ = Φ (3)

where N denotes the number of turns and 1PΦ denotes the flux flowing in permeance P1, which is

the first element in the vector PR. This calculation is shown on line 111 of the code. It should be noted that the direction of flux in permance P1 (positive flux flowing out from the positive node of the branch) is opposite to the direction of flux as defined in the standard branch definition

(positive flux flowing into the positive node of the branch). 1PΦ is therefore multiplied with a

negative sign to take into account for this difference in direction.

On line 114, the convergence flag at each operating point, denoted converge , is copied to

the vector c . This is done in order to check for the convergence of the MEC analysis. The vector c

should contain 1’s to indicate that the analysis at each operating point converges.

After the total flux linkage at every current has been calculated, the λ–i curve can now be built using the plot command in Matlab, shown on line 119 – 121 of the code. The curve is shown in Fig. 4 as the result of this analysis.

Figure 4. λ–i Characteristic of the EI Core Inductor

V. ADDITIONAL ROUTINES In addition to the MEC core routines mentioned on section II, the MEC Toolbox also provides

additional routines corresponding to several physical models. 1. BE_core

Description: Sets up data structure associated with a bobbin E core

2. BEE_transformer_mec Description: MEC analysis of a bobbin EE core transformer

3. bob_wind Description: Sets up data structure associated with a bobbin winding

4. draw_BEEtrans Description: Draws a bobbin EE core transformer

5. draw_coil_profileview Description: Draws a profile view of a coil

6. draw_coil_topview Description: Draws a top view of a coil

7. draw_U_profileview Description: Draws a profile view of a U core

0 10 20 30 40 50 60 700

0.02

0.04

0.06

0.08

0.1

0.12

0.14

Current (A)

λ (V

.s)

8. draw_U_topview

Description: Draws a top view of a U core

9. draw_UUtrans Description: Draws a UU core transformer

10. ei_inductor_mec Description: Comprehensive MEC analysis of an EI core inductor

11. ei_inductor_mec2 Description: Comprehensive MEC analysis of an EI core inductor with a saturation

permeance

12. P_rct_bndle_fm Description: Calculate the internal and external permeance associated with a rectangular

bundle of conductor above a ferro- or ferri-magnetic material

13. Pint_rct_bndle_air Description: Calculates the internal permeance of a rectangular wire bundle in air based on

method set forth in [3]

14. rect_coil Description: Calculates the dimensions of a rectangular coil

15. resistance Description: Determines the ac and dc resistance of the wire given wire cross sectional

area, length, and frequency

16. u_volume Description: Calculates the volume of a U core

17. ui_inductor_mec Description: MEC analysis of a UI core inductor

18. uu_transformer_mec Description: MEC analysis of a UU core transformer

VI. CONTRIBUTORS The contributors for this MEC Toolbox include Purdue University faculties Prof. Steve

Pekarek and Prof. Scott D. Sudhoff; Purdue University graduate students Michelle Bash, Cahya Harianto, and Grant Shane; James Cale; Brandon Cassimere; and Hamza W. Derbas.

VII. ACKNOWLEDGEMENTS Sponsors for this work include the Office of Naval Research, grants N00014-08-1-0080

(Electric Ship Research and Development Consortium) and N00014-08-1-0397.

VIII. REFERENCES [1] J.Cale, S.D.Sudhoff, L.Tan, “Accurately Modeling EI Core Inductors using a High-Fidelity

Magnetic Equivalent Circuit Approach”, in IEEE Transactions on Magnetics, Vol. 42, No. 1, January 2006, pp. 40–46.

[2] H.W.Derbas, J.M.Williams, A.C.Koenig, S.D.Pekarek, “A Comparison of Nodal- and Mesh-Based Magnetic Equivalent Circuit Models”, in IEEE Transactions on Energy Conversion, Vol. 24, No. 2, June 2009, pp. 388–396.

[3] B.N.Cassimere, “Modeling and Evolutionary Design of Permanent Magnet Synchronous Machines”, Ph.D. Dissertation, Purdue University, May 1998.

IX. LICENSE MEC Toolbox is free software: it be redistributed and/or modified under the terms of the GNU Lesser General Public License as published by the Free Software Foundation, either version 3 of the License, or (at your option) any later version.