Description

1. 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.

Continued on reverse …

NOTE: ANSWER ONLY ONE OF THE TWO FOLLOWING QUESTIONS (EACH IS WORTH 10 MARKS)

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.