ECSE543 - NUMERICAL METHODS IN ELECTRICAL ENGINEERING  - Assignment 3  - Solved

You are given a list of measured BH points for M19 steel (Table 1), with which to construct a continuous graph of B          versus H.  

 

(a)           Interpolate the first 6 points using full-domain 

 

Lagrange polynomials. Is the result plausible, i.e. do you think it lies close to the true B versus H graph  over this range?  

 

(b)          Now use the same type of interpolation for the 6 points at B = 0, 1.3, 1.4, 1.7, 1.8, 1.9. Is this result  plausible?  

 

(c)           An alternative to full-domain Lagrange polynomials is to interpolate using cubic Hermite polynomials in  each of the 5 subdomains between the 6 points given  in (b). With this approach, there remain 6 degrees of freedom - the slopes at the 6 points. Suggest ways  of fixing the 6 slopes to get a good interpolation of the points.  

                 

2.       The magnetic circuit of Figure 1 has a core made of Ml9 steel, with a cross-sectional area 1 cm2. Lc = 30 cm and    La  = 0.5 cm. The coil has N = 1000 turns and carries a current 

 

1 = 8 A. 

                 

(a)           Derive a (nonlinear) equation for the flux  in the core, of the form f() = 0.     

 

(b)          Solve the nonlinear equation using Newton-               

Raphson.  Use a piecewise-linear interpolation of the data in Table 1.  Start with zero flux and finish  when 

 

 

                                               | f() / f() | < 10−6                                                              

                                   Record the final flux, and the number of steps taken.                

                 

(c)           Try solving the same problem with successive  substitution.  If the method does not converge, 

 suggest and test a modification of the method that does converge.              


 

3.       In the circuit shown below, the DC voltage E is 220 mV, the resistance R is 500 , the diode A reverse saturation current IsA is 0.6 A, the diode B reverse saturation current IsB is 1.2 A, and assume kT/q to be 25 mV. 

 

(a)           Derive nonlinear equations for a vector of nodal voltages, vn, in the form f(vn) = 0.  Give f explicitly in terms of the variables IsA , IsB , E, R and  vn. 

 

(b)          Solve the equation f = 0 by the Newton-Raphson method.  At each step, record f and the voltage across each diode.  Is the convergence quadratic?  [Hint: define a suitable error measure k].   

 
4.  

(a)         Integrate the function cos(x) on the interval x=0 to x=1, by dividing the interval into N equal segments and using one-point Gauss-Legendre integration for each segment.  Plot log10(E) versus log10(N) for N=1, 2,…, 20, where E is the absolute error in the computed integral.  Comment on the result. 

 

(b)         Repeat part (a) for the function loge(x), only this time plot for N=10, 20,…200. Comment on the result. 

 

(c)         An alternative to dividing the interval into equal segments is to use smaller segments in more difficult parts of the interval. Experiment with a scheme of this kind, and see how accurately you can integrate loge(x) using only 10 segments.