4.    (a) Prove that any vector in a 3-D space is completely defined by specifying its divergence, its curl and its normal component over the boundary.

(b)    Show that the magnetostatic boundary condition across a surface carrying a current of K A/m is given by

Babove - Bbiow⁼ Ho (K x n), where n is the normal unit vector out of the surface.

(c)    Show that electromotive force is not a force but work done by the source to move unit charge around a closed electric circuit.    14+5+2]

5.    (a) A permanent magnet is 120 mm long and 2 sq. cm in cross section and is bent in the shape of the English letter D. An air gap of 5 mm exists in the vertical arm of this D\ Calculate graphically the flux density in the air gap. The demagnetisation curve for the material of the permanent magnet is as follows:

B(T): 0 0.25 0.5 0.75 1.0 1.1 1.25 H(kA/m): -45 -43 -41 -38 -34.5 -28 0 Neglect leakage and fringing.

(b)    The above magnet material is to be used maintain a flux density of 0.7 T across an air gap 2mm long and 15 mm x 25 mm in cross section. Calculate the minimum volume of the material required.

(c)    Prove that

V. (r/r²) = 47t8¹(r), where symbols have their usual significance.    |4+4+3]

N.B. All capital bold letters represent vectors(or vector operators) and lowercase bold letters represent unit vectors. Other symbols have their usual significance unless otherwise mentioned. You can take the following values of the commonly used physical constants.

Po = 4 7i x 10'⁷H/m 8o = 8.854 x 1 O'¹² F/m

SECOND HALF

6. a) Draw the oriented graph of the resistive network shown in Fig.6(a). Hence obtain the following:

i)    Reduced incidence matrix

ii)    Number of possible trees of the graph

iii)    Fundamental tie-set matrix for the tree {1, 2, 4}

iv)    Fundamental cut-set matrix for the tree {1, 2, 4}

b) What do you understand by dual circuit? Draw the dual of the R-L-C network shown in Fig.-6(b).    I⁷⁺⁴1

m

-vWVV-

12V /+

Fig.-6(b)

7.    a) State the necessary conditions of stability of a network function.

b) State the properties of a Hurwitz polynomial. Hence check whether the polynomial P(s) = s⁴ + 1 Is³ + 41s² + 61s + 30 is Hurwitz.    |5+6|

8.    a) Comment and explain the stability of a system with reference to the position of poles in

the s-plane.

b) By means of Routh criteria, determine the stability of the system represented by the following characteristic equation. Wherever necessary find the number of roots in the right half of s-plane and on the imaginary axis.

s⁵ + s⁴ + 3s³ + 9s² + 16s + 10 = 0    [5+6]

9.    a) State the conditions of a network function to be a positive real function. The driving point

impedance function is given by

_ . . s³ + 5s² + 9s + 3

Z(s)= 5-5-

s³ + 4s² + 7s + 9

Check whether the function is a positive real function.

b) Realize the following function in First and Second Foster form in R-C format.

(s + 2) (s + 5)

Z(s) = ------[5+6)

^v ' (s + 1 )(s + 4)    ^{1 1}

10.    a) What do you mean by constant -K prototype filter. Why is it so named?

Derive the expression of the characteristic impedance and the cut-off frequency of a T-section low pass filter.

b) Design a Tt-section of an in-derived high pass filter having a nominal impedance of 600 ohm, cut off frequency of 4 kHz and an infinite attenuation at 3.6 kHz. Clearly state the necessary expressions used in the design procedure.    |5+6|

1

Show that for steady currents

V x B = Ho J .

Justify all assumptions made.

Thereafter make appropriate derivations to introduce Maxwells correction. Clarify why this correction is essential.

Attachment: bengal-engineering-and-science-university-2007-b-e-electrical-engineering-1011-1.pdf

Earning:   Approval pending.