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EXAMPLE OF CUBIC B-SPLINE REPRESENTING A MAGNETIC HYSTERESIS LOOP

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B is the flux density in Tesla H is the magnetic field strength in Amps/meter BACKGROUND

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Given a ferromagnetic core wrapped in wire Apply a square wave voltage Results in a magnetic hysteresis loop Increasing voltage Decreasing voltage

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MODELING THIS LOOP Accurate hysteresis loop model can save time Use a cubic spline approximation based on experimental data B can be determined for any H value

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Paint application is very useful Use the cursor to find pixel points* Select as many points as we wish for the model Be careful! Pixel origin is in upper left corner GET TO THE POINT S *

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SEE IT IN ACTION

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Find y of any point (x-hat) on the spline Binary search algorithm to find polynomial Specify end conditions Solve system of equations for coefficients: a,b,c,d ONCE YOU HAVE THE POINTS P 3 (x hat ) = a 3 + b 3 (x hat - x 3 ) + c 3 (x hat - x 3 ) 2 + d 3 (x hat - x 3 ) 3

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The binSearch() function assumes all points are given with increasing x values Two sets of points are defined One for positive voltage One for negative voltage Splint() function does the rest Use zero slopes at endpoints for this case Plot or display values The binSearch() function assumes all points are given with increasing x values Two sets of points are defined One for positive voltage One for negative voltage Splint() function does the rest Use zero slopes at endpoints for this case Plot or display values MATLAB AND SOME LOGIC

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DIFFERENT END CONDITIONS

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% A and B must be entered with ascending x’s A = [100,333;129,307;167,264;210,192;254,143; ,111;345,87;377,76;423,62;478,53]; B = [100,333;128,330;196,314;262,287;302,262; ,223;373,181;413,120;455,75;478,53]; % Enter the origin pixel point O = [290,194]; % Enter the points at the first positive tic marks tic_x = [377,194]; % 50 tic mark to right of origin tic_x_label = 50; tic_y = [290,164]; % 0.1 tic mark up from origin tic_y_label = 0.1; n = 100; % number of xhats to compute % (higher means smoother curve) scx=tic_x_label/abs(tic_x(1)-O(1)); scy=tic_y_label/abs(tic_y(2)-O(2)); AA=(A-ones(length(A),1)*O)*[scx 0;0 -scy]; BB=(B-ones(length(B),1)*O)*[scx 0;0 -scy]; plot(AA(:,1),AA(:,2),'b*') hold on; grid on; plot(BB(:,1),BB(:,2),'r*') AAintx = linspace(min(AA(:,1)),max(AA(:,1)),n); BBintx = AAintx; AAinty=splint(AA(:,1),AA(:,2),AAintx,0,0); BBinty=splint(BB(:,1),BB(:,2),BBintx,0,0); plot(AAintx,AAinty,'b') plot(BBintx,BBinty,'r') THE CODE

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splint Cubic-spline interpolation with various end conditions Synopsis: yhat = splint(x,y,xhat) yhat = splint(x,y,xhat,endType) yhat = splint(x,y,xhat,fp1,fpn) [yhat,a,b,c,d] = splint(x,y,xhat) [yhat,a,b,c,d] = splint(x,y,xhat,endType) [yhat,a,b,c,d] = splint(x,y,xhat,fp1,fpn) Input: x,y = vectors of discrete x and y = f(x) xhat = (scalar or vector) x value(s) where interpolant is evaluated endType = (string, optional) either 'natural' or 'notaKnot'; used to select either type of end conditions. End conditions must be same on both ends. Default: endType='notaKnot'. For fixed slope end conditions, values of f'(x) are specified, not endType fp1 = (optional) slope at x(1), i,e., fp1 = f'(x(1)) fpn = (optional) slope at x(n), i,e., fpn = f'(x(n)); Output: yhat = (vector or scalar) value(s) of the cubic spline interpolant evaluated at xhat. size(yhat) = size(xhat) a,b,c,d = (optional) coefficients of cubic spline interpolants CODE: SPLINT() PARAMETERS

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