## Description

ECSE 543A NUMERICAL METHODS IN ELECTRICAL ENGINEERING

Assignment 3

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 NewtonRaphson. 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.