MEHRAN UNIVERSITY OF ENGINEERING AND TECHNOLOGY, JAMSHORO. FIRST TERM THIRD YEAR (5th TERM) BACHELOR OF ELECTRONIC ENGINEERING REGULAR EXAMINATION 2013 OF 11-BATCH

NUMERICAL METHODS Dated: 31 – 05 – 2013 Time Allowed: 03 Hours. Max.Marks 80 NOTE: ATTEMPT ANY FIVE QUESTIONS. MARKS ARE SHOWN AGAINST EACH QUESTION. Q.No. 1 (a) A total charge Q is uniformly distributed around a ring – shaped conductor with radius a. A charge q is located at a distance x from the center of the ring. The force exerted on the charge by the ring is given by

F where e0  8.85  10

1 qQx  3 4e0 x 2  a 2 2

[09]

C 2 / Nm 2  Find the distance x (using Newton – Raphson or Modified Newton – 5 Raphson Method) where the force is 1.25N if q and Q are 2  10 C for the ring with a radius of 0.9m. 12

(b) Prove that the order of convergence of Newton – Raphson Method is two. Also prove that for f  x   0 , the number of iterations

n

required by Bisection Method to attain the error tolerance 

ba log      , where a, b is the closed interval that contains a root of the function f x  0 . n     log 2

  0 

is

[07]

2 (a) The electrical network shown can be viewed as consisting of three loops. Apply Kirchhoff’s law  voltage drops   voltage sources  to each loop to yield the equations for the loop currents i1 , i2 , and

i3 

Determine the three loop currents for R  5  Continue the iterations until absolute percentile error in any one of the currents becomes  5%  (Use either Jacobi or Gauss-Seidal Method). [07] (b) Show that the coefficient matrix of the following linear system is not strictly diagonally dominant and yet the method of Gauss – Seidal Method converges to the solution x1  x2  1 . (Apply minimum 4 iterations). 4 x1  5 x2  1 [04] x1  2 x2  3 (c) Determine the smallest eigenvalue in magnitude of the following matrix by Power Method; where inv  A  8 1 3; 5 1 2; 10 1 4 is the multiplicative inverse of the given matrix. Use suitable initial approximation to the eigenvector and apply 3 iterations:  2 1 1 A   0 2 1  [05]  5 2 3

i  v (current - voltage) relation of a non – linear electrical device is given by: i  0.1e0.2 v t   1 where v is in volts and i in milliamperes. Construct the table for v  2, 0, 3, and 5  Use Newton’s Divided Difference Interpolation Formula to compute i at v  1.265 volts. Compare the interpolated value 3 (a) The

with the exact result and find absolute error?

[08]

(b) Suppose you use a zener diode for a voltage regulator circuit (to filter out the small sinusoidal ripple voltage and to refine the constant power signal). You need to use the voltage-current characteristic i  i v in order to compute the steady-state voltage drop across the electric network. However, this function is not amenable to representation with a simple analytical expression. Instead, measurements are available only for following data points.



v 0 i

10

20

15

22.5

0 227.04 517.35 362.78 602.97

[08]

Use Linear Spline Interpolation to connect the data values and to reproduce a simple analytical representation of the voltage-current characteristic? Estimate the value of i 16 

 

4 (a) What do you think about curve fitting? How curve fitting is different from interpolation? Fitting a straight line to data is also known as linear regression. In this case; a function to be minimized is n

S  a, b     yi  a  bxi  . Determine the formulae for computing a and b. 2

[07]

i 1

(b) An experiment is performed to determine the percent elongation of electrical conducting material as a function of temperature. The resulting data are listed below. Predict the percent elongation for a temperature of b 750 C by fitting a power function curve that is y  ax 

Temp, C 100  4 % elongation

5

6

7

8

8 12.5 18 24.5 32

[09]

5 (a) The following data were taken from an experiment that measured the current in a wire for various imposed voltages: V ,  volts  0.60 0.65 0.70 0.75

i,  amperes  0.6221 0.6155 0.6138 0.6170 Determine for what value of V; i attains the minimum value and also find that value?

[07]

(b) An electron has a 1.6  10 negative charge. How much work is done in separating two electrons from 12 1.0pm to 4.0pm? Use The Composite Trapezoidal rule taking step size 0.5  10 and compare the result with exact answer to estimate the error. [05] 19

(c) Derive the general formula of The Composite Simpson’s 1 3 Rule for Numerical Integration.

[04]

6 (a) The variation of resistance, R ohms, of an aluminum conductor with temperature  C is given by dR [08] R d where   38 104C is the temperature coefficient of resistance of aluminum. Determine the resistance of the aluminum conductor at 30C , when its resistance at 0C is 24.0 ohms? (Use Heun’s or Midpoint Euler’s Method with h = 15). (b) Use the Classical Runge-Kutta method to solve the differential equation:

dy = 3(1 + x) – y dx given the initial conditions that x =1 when y = 4, for the range x =1.0 to x =1.5 with interval of 0.5?

[08]

7 (a) The potential difference, V, between the plates of a capacitor C charged by a steady voltage E through a resistor R is given by the equation dV [07] CR V  E dt Calculate V using RK – Method of order 1 given that at t =0, V =0, correct to 3 significant figures, when E =25 V, C =20× 106 F, R=200× 103 and t =3.0 s. Take h  1.5 . dy 1  xy , y  0   1, y  0.1  1.01, y  0.2   1.022, y  0.3  1.023 to find y  0.4  using Adam’s – dx 2 Bashforth Method as predictor and Adam’s – Moulton Method as corrector? [09]

(b) Solve

8 (a) Solve the Poisson Equation

 2u  2u   8x2 y 2 x 2 y 2

[08]

for the square mesh given below:

with u  x, y   0 on the all the boundaries and mesh length  1 . (Use finite difference approximation). (b) Compute u for one time step by Crank – Nicolson (C-N) Method if ut  uxx

[08]

for 0  x  5, t  0; u  x,0   20; u  0, t   0 and u  5, t   100  Solve the obtained tridiagonal system using any direct or indirect method.

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