лабораторная работа / мэмс / 9lw

Ministry of science and education, youth and sport of Ukraine

Zaporozhie National Technical University

Department EEA

Report

Laboratory work#9

Done L.A.Askerova

Checked T.M.Kornus

Zaporozhie

2012

TOPIC : Modeling of electromagnetic processes in magnetic circuits of a direct magnetic flux in MATLAB environment. Part 1.

PURPOSE OF THE WORK: Problem statement, development of the program of calculation of a nonlinear magnetic circuit by the Newton method using discrete models of nonlinear resistance and their spline-interpolation.

Mathematical model

Let's make mathematical model of the magnetic circuit (Fig. 9.1) in MATLAB environment if all nonlinear components are given by discrete models, and their curves of magnetization as reference points.

Figure 9.1 - Modeled magnetic circuit

Let parameters of a circuit are given:

Length of average magnetic lines according to fig. 9.1:

l1=0.3 m; l2=0.3 m; l3=0.1 m;

Breadth of magnetic limbs:

D1=0.1 m; D2=0.1 m; D3=0.1 m;

Thickness of the core:

dlt=0.1 m;

Magnitude of the air-gap:

Luft=0.00002 m;

Current and loops of windings:

I=1 A; N=20.

Curve of magnetization of core B (H) is shown in figure 9.2.

а).

b).

а) – general view;

b) – an initial section of the magnetization curve.

Figure 9.2 – Magnetization curve B(H) of the core steel

An equivalent circuit of the given magnetic circuit is represented in fig. 9.3.

Figure 9.3 – The equivalent circuit

According to the second Kirchhoff law it is possible to transfer MMF source Fm into a branch containing conductance G1 and then to transform it to a source of magnetic flux FmG1. The equivalent circuit according to these transformations is offered in fig. 9.4.

Figure 9.4 – The equivalent transformed circuit with magnetic flux source FmG1

The system of the potential equations looks as:

Where

, ,

The node magnetic fluxes:

The Block-diagram

The program of calculation in MATLAB environment

The main program:

% Newt6

% CALCULATION OF THE NONLINEAR MAGNETIC CIRCUIT BY THE NEWTON %METHOD

n=20; % number of iterations of the Newton method

h=0.001; % a step for realization of numerical derivation

I=5; N=50; % Current and loops of windings

mu0=4*pi*1e-7;

l1=0.3; l2=0.3; l3=0.1; % length of magnetic branches

D1=0.1; D2=0.1; D3=0.1; % breadth of magnetic branches

dlt=0.1; % thickness of the core

luft=0.00002; % thickness of the air-gap

Fm=N*I; % MMF

S1=D1*dlt; S2=D2*dlt; S3=D3*dlt; % cross-section area of magnetic branches of the core

Rm3=luft/(mu0*S3); % magnetic resistance of the air-gap

Gm3=1/Rm3; % Magnetization curve of steel

B = [-2.1,-2,-1.98,-1.96,-1.94,-1.92,-1.9,-1.86,-1.84,-1.8,-1.7,-1.6,-1.4, -1.2,-1, -0.8, -0.4, -0.2, -0.1, 0, 0.1, 0.2, 0.4, 0.8, 1, 1.2, 1.4, 1.6, 1.7, 1.8, 1.84, 1.86, 1.9, 1.92, 1.94, 1.96, 1.98, 2, 2.1];

H = [-1000000,-11820,-5331,-3191,-2249,-1625,-1218,-660,-520,-295,-97,-51.4,-31.9,-26.2, -23.2,-20.8,-15.2,-10.8,-7.58, 0,7.58, 10.8, 15.2, 20.8, 23.2, 26.2, 31.9, 51.4, 97, 295, 520, 660, 1218, 1625, 2249, 3197, 5331, 11820, 1000000];

Flux (1,1)= 0.02; % the initial approximation of magnetic fluxes

Flux (2,1)= 0.015;

Flux (3,1)= 0.01;

% Iterative calculation of magnetic fluxes in the linear network containing

% discrete flux models by node potentials method

eps1f=0;

for k=2:n

% calculation of discrete flux models conductivity for K-1 iteration

U3=l3*spline (B, H, Flux (3, k-1)/S3);

G3=S3/l3*der7 (H, B, U3/l3, h);

U1=l1*spline (B, H, Flux (1, k-1)/S1);

G1=S1/l1*der7 (H, B, U1/l1, h);

U2=l2*spline (B, H, Flux (2, k-1)/S2);

G2=S2/l2*der7 (H, B, U2/l2, h);

% the fluxes of discrete sources for K-1 iteration

Flux1=Flux (1, k-1)-U1*G1;

Flux2=Flux (2, k-1)-U2*G2;

Flux3=Flux (3, k-1)-U3*G3;

% node conductivities for K-1 iteration

G22=G3+G1+G2;

G11=G3+Gm3;

G12=G3;

G = [G11, -G12;

-G12, G22];

% the node fluxes for K-1 iteration

Fluxnode = [-Flux3; Flux3+Flux2-Flux1-Fm*G1];

phi=G\Fluxnode; % calculation of the node potentials for K iteration

% magnetic fluxes for K iteration

Flux (3, k) = (-phi (1) *Gm3+Flux (3, k-1) *2)/3;

Flux (1, k) = (Flux1+Fm*G1+phi (2) *G1+Flux (1, k-1) *2)/3;

Flux (2, k) = (Flux2-phi (2) *G2+Flux (2, k-1) *2)/3;

eps1=Flux(3,k)+Flux(2,k)-Flux(1,k);

if abs(eps1)>abs(eps1f)

eps1f=eps1;

end

end

% Construction of the graphics of convergence of iterative processes of the magnetic fluxes calculation through the core

F1=Flux(1,n)/S1

F2=Flux(2,n)/S2

F3=Flux(3,n)/S3

epsi=eps1f

p=1:n;

plot (p, Flux (1, p)/S1, p, Flux (2, p)/S2, p, Flux (3, p)/S3);

legend('B1','B2','B3',-1)

The subprogram -function der7 of calculation of derivative in the given point is performed in methodical directions to laboratory work # 8.

Listing of the program

F1 =

1.8740

F2 =

1.3857

F3 =

0.4884

epsi =

0.0033

In figure 9.5 graphical functional dependences of calculated values of induction in the magnetic circuit (fig. 9.1) accord to each step of iteration are represented.

Figure 9.5 – Calculated curves of magnetic induction in limbs of the core according to iteration step

Conclusion: The magnetic induction in limbs of the core according to iteration step was found. The graph is presented. Accuracy is ok.