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Assignment 3

NOTE: QUESTION 1 IS WORTH 20 MARKS

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. Test your suggestion and comment on the results.
(d) The magnetic circuit of Figure 1 has a core made of Ml9 steel, with a cross-sectional area of 1 cm2. Lc = 30 cm and La = 0.5 cm. The coil has N = 800 turns and carries a current I = 10 A. Derive a (nonlinear) equation for the flux  in the core, of the form f() = 0.
(e) 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.
(f) 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.

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

2. For the circuit shown in Figure 2 below, the DC voltage E is 200 mV, the resistance R is 512 , the reverse saturation current for diode A is IsA = 0.8 A, the reverse saturation current for diode B is IsB
= 1.1 A, and assume kT/q = 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].

Figure 2

3. Write a program that accepts as input the values for the parameters x0, xN, and N and integrates a function f(x) on the interval x = x0 to x = xN by dividing the interval into N equal segments and using one-point Gauss-Legendre integration for each segment.

(a) Use your program to integrate the function f(x) = sin(x) on the interval x0 = 0 to xN = 1 for N =
1, 2,…, 20. 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 f(x) = ln(x), only this time for N = 10, 20,…, 200. Comment on the result.

(c) Repeat part (b) for the function f(x) = ln(0.2sin(x)). Comment on the result.

(d) 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 f(x) in part (b) and (c) using only 10 segments. Comment on the results.
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