sadiku_ch14_probs - PROBLEMS 751 SEBTIDN 14.1 FIELD...

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Unformatted text preview: PROBLEMS 751 SEBTIDN 14.1 FIELD PLDTTING 14.1 Use the program developed in Example 14.1 or your own equivalent code to plot the elec- tric field lines and equipotential lines for the following cases: (a) Three point charges —-1 C, 2 C, and 1 C placed at (—l, 0), (0,2), and (1, 0), respectively. (b) Five identical point charges of 1 C located at (-— 1, -1), (-1, 1), (1, —1), (1, 1), and (0, 0), respectively. SECTION 14.2 THE FINITE DIFFERENCE METHOD 14.2 Consider the one-dimensional differential equation dzy 3+4y=0 subject to y(0) = 0, y(1) = 10. Use the finite difference (iterative) method to find y(0.25). You may take A = 0.25 and perform five iterations. , dV dZV . 14.3 (3) Obtain :1— and d—i— at x = 0.15 from the followmg table. x x x 0.1 0.15 0.2 0.25 0.3 V 1.0017 1.5056 2.0134 2.5261 3.0452 M (b) The data in the table are obtained from V = 10 sinh x. Compare your result in part (a) with the exact values. 14.4 Show that the finite difference equation for Laplace’s equation in cylindrical coordinates, V = V(p, z), is 1. h V(po,zo) = Z [V000, z. + h) + V(10.,,z0 - h) + (1 + 2p ) h V(po + h,Zo) + — 2 Po ) V(po —' ha zo)] whereh = A2 = Ap. 14.5 Using the finite difference representation in cylindrical coordinates (p, (b) at a grid point P shown in Figure 14.42, let p = m Ap and d) = n AqS so that V(p, (Mp = V(mAp, nA¢) = V2. Show that 2 1K 1>' ( ‘) =—~—— ——— _— + ~— VV'mm Apz 1 2m V’ml 1+ m an+1 1 + 0» AW (V; 752 CHAPTER 14 NUMERICAL METHODS Figure 14.42 Finite difference grid in cylindri- cal coordinates; for Problem 14.5. 14.6 The four sides of a square trough are maintained at potentials 10 V, ~40 V, 50 V, and 80 V. Determine the potential at the center of the trough. 14.7 Use the finite difference method to calculate the potentials at nodes 1 and 2 in the poten- tial system shown in Figure 14.43. 1 14.8 Rework Problem 14.7 if pv = —%9 nC/m3, h = 0.1 m, and e = 80, where h is the mesh size. 14.9 Use the finite difference technique to find the potentials at nodes 1 to 4 in the potential system shown in Figure 14.44. Five iterations are sufficient. 14.10 For the potential system shown in Figure 14.45 (divided into a square grid), write down the finite differential equations of nodes 1 to 3. Solve for potential for nodes 1, 2, and 3. 14.11 For the rectangular region shown in Figure 14.46, the electric potential is zero on the boundaries and the charge distribution pv is 50 nC/m3. Although there are six free nodes, there are only four unknown potentials (V1 — V4) because of symmetry. Solve for the un- known potentials. Figure 14.43 For Problem 14.7. PROBLEMS 753 Figure 14.44 For Problem 14.9. 100V Figure 14.45 For Problem 14.10. Figure 14.46 For Problem 14.11. 754 CHAPTER 14 NUMERICAL METHDDS 100 V Figure 14.47 For Problem 14.12. 14.12 Consider the potential system shown in Figure 14.47. (a) Set the initial values at the free nodes equal to zero and calculate the potential at the free nodes for five iterations. (b) Solve the problem by the band matrix method and compare result with part (a). 14.13 Apply the band matrix technique to set up a system of simultaneous difference equations for each of the problems in Figure 14.48. Obtain matrices [A] and [B]. 14.14 (a) How would you modify matrices [A] and [B] of Example 14.3 if the solution region had charge density pv? (b) Write a program to solve for the potentials at the grid points shown in Figure 14.49 assuming a charge density pv = x(y - l) nC/m3. Use the iterative finite difference method and take a, = 1.0. 14.15 The two—dimensional wave equation is given by i 6245 _ 3% + 8245 CZ at2 6x2 6Z2 15V 100V (b) Figure 14.48 For Problem 14.13. PROBLEMS 755 Figure 14.49 For Problem 14.14. By letting 455,1,” denote the finite difference approximation of 45(xm, z", tj), show that the finite difference scheme for the wave equation is . l . ._l . . . (pin—tn = 2 4,711”: — (ping: + 0‘ (q)rjrz+1,n + ¢{71—1,}: '- 2 (prjnm) + a ((pljn,n+l + ¢{u,n—l — 2 d’rjnm) whereh = Ax = A2 anda = (cAt/h)2. 14.16 Write a program that uses the finite difference scheme to solve the one-dimensional wave equation 62V ~ 62V 6x2 7972—, Ost1, t>0 given boundary conditions V(0, t) = 0, V(1, t) = 0, t> 0 and the initial condition BV/at (x, 0) = 0, V(x, 0) = sin me, 0 < x < 1. Take Ax = At = 0.1. Compare your solution with the exact solution V(x, t) = sin 7rx cos M for 0 < t < 4. 14.17 (a) Show that the finite difference representation of Laplace’s equation using the nine- node molecule of Figure 14.50 is V0: (b) Using this scheme, rework Example 14.4. Figure 14.50 For Problem 14.17. 756 CHAPTER 14 NUMERICAL METHODS Figure 14.51 For Problem 14.18. EEBTIDN 14.4 THE MDMENT METHDD 14.18 A transmission line consists of two identical wires of radius a, separated by distance d as shown in Figure 14.51. Maintain one wire at 1 V and the other at -1 V and use the method of moments to find the capacitance per unit length. Compare your result with exact formula for C in Table 11.1. Take a = 5 mm, d = 5 cm, 6 = 5 m, and e = 80. 14.19 Determine the potential and electric field at point (- 1, 4, 5) due to the thin conducting wire of Figure 14.18. Take Vo = l V, L = 1 m, a = 1 mm. 14.20 Two conducting wires of equal length L and radius a are separated by a small gap and in- clined at an angle 0 as shown in Figure 14.52. Find the capacitance between the wires by using the method of moments for cases 0 = 10°, 20°, . . . , 180°. Take the gap as 2 mm, a =1mm,L= 2m,s,= 1. 14.21 Given an infinitely long thin strip transmission line shown in Figure 14.53(a), use the moment method to determine the characteristic impedance of the line. We divide each strip into N subareas as in Figure 14.53(b) so that on subarea i, 2N Vi = 2 AU Pi j=l where _ M In R,-, i 9e j _ 2’1r80 1 AU- “MunAe- 15] i=‘ 27reo ' ’ J Figure 14.52 For Problem 14.20. L ~ 6 W Gap PROBLEMS 757 (b) Figure 14.53 Analysis of strip transmission line using moment method; for Problem 14.21. R; is the distance between the ith and jth subareas, and V,- = l or —1 depending on whether the ith subarea is on strip 1 or 2, respectively. Write a program to find the char- acteristic impedance of the line using the fact that V #030 C where C is the capacitance per unit length and N iAlf Q=i=zlp Zo= and Vd = 2 V is the potential difference between strips. Take H = 2 m, W = 5 m, and N = 20. 14.22 Consider an L-shaped thin wire of radius 1 mm as shown in Figure 14.54. If the wire is held at a potential V = 10 V, use the method of moments to find the charge distribution on the wire. Take A = 0.1. y Figure 14.54 For Problem 14.22. 758 CHAPTER 14 NUMERICAL METHODS (b) Figure 14.55 For Problem 14.23; coaxial line of (a) arbitrary cross section, (b) elliptical cylindrical cross section. 14.23 Consider the coaxial line of arbitrary cross section shown in Figure l4.55(a). Using the moment method to find the capacitance C per length involves dividing each conductor into N strips so that the potential on the jth strip is given by 2N VJ" : 2 Pi Aij i=1 where __ R.. M In J, i ¢ j A” = 21m r0 ‘1 — . M [In-[Lei -1.5], i =j 21w ro and VJ- = —l or 1 depending on whether A€i lies on the inner or outer conductor respec— tively. Write a Matlab program to determine the total charge per length on a coaxial cable of elliptical cylindrical cross section shown in Figure l4.55(b) by using 1v Q=§pi and the capacitance per unit length with C = Q/2. (a) As a way of checking your program, take A = B = 2 cm and a = b = 1 cm (coaxial line with circular cross section), and compare your result with the exact value of C = 27re/ln (A/a). (b) TakeA = 2cm,B = 4cm,a = lcm,andb = 2cm. Him: For the inner ellipse of Figure l4.55(b), for example, a V sin2 (75 + vzcos2 d) where v = a/b, d6 = rd¢>. Take r0 = 1 cm. r: PROBLEMS 759 Figure 1456» For Problem 14.24. 14.24 A conducting bar of rectangular cross section is shown in Figure 14.56. By dividing the bar into N equal segments, we obtain the potential at the jth segment as N Vj = inij where 1 . . ---— z #1 47r8 R,~’ AU z 01] . . 2—“. r——’ 1:] ‘90 7rhA and A is the length of the segment. If we maintain the bar at 10 V, we obtain [AHq] = IOU] Where[1] =[111. . .,1]Tandq,-= pvt/1A. (a) Write a program to find the charge distribution pv on the bar and take 6 = 2 m, h = 2cm,t= lcm,andN= 20. (b) Compute the capacitance of the isolated conductor by using C=%=(q1 +612+"~+qN)/10 $EE3TEUN 14.353 THE FlNETE ELEMENT METHDD 14.25 Another way of defining the shape functions at an arbitrary point (x, y) in a finite element is using the areas A1, A2, and A3 shown in Figure 14.57. Show that A =-, k=1,2,3 ozk A where A = A1 + A2 + A3 is the total area of the triangular element. 14.26 For each of the triangular elements of Figure 14.58, (a) Calculate the shape functions. (b) Determine the coefficient matrix. 760 CHAPTER 14 NUMERICAL METHODS l , (x1, Y1) Figure 14.57 For Problem 14.25. 2 _ (x2! .YZ) 3’ Figure 14.58 Triangular elements of Problem 14.26. (I. 2) (2.5. 2) (l, 0.25) (0,0) (3) (b) 14.27 The nodal potential values for the triangular element of Figure 14.59 are VI = 100 V, ' V2 = 50 V, and V3 = 30 V. (a) Determine where the 80 V equipotential line intersects the boundaries of the element. (b) Calculate the potential at (2, 1). 14.28 The triangular element shown in Figure 14.60 is part of a finite element mesh. If V1 = 8 V, V2 = 12 V, and V3 = 10 V, find the potential at (a) (1, 2) and (b) the center of the element. 14.29 Determine the global coefficient matrix for the two-element region shown in Figure 14.61. 14.30 Calculate the global coefficient matrix for the two-element region shown in Fig- ure 14.62. Figure 14.59 For Problem 14.27. PROBLEMS 761 (1,4) Figure 14.60 For Problem 14.28. Figure 14.61 For Problem 14.29. y 3 Figure 14.62 For Problem 14.30. (0, 2) (0. 0) (4, 0) 14.31 Find the global coefficient matrix of the two—element mesh of Figure 14.63. 14.32 For the two-element mesh of Figure 14.63, let V1 = 10 V and V3 = 30 V. Find V2 and V4. 14.33 The mesh in Figure 14.64 is part of a large mesh. The shaded region is conducting and has no elements. Find C5; and C5,]. 762 Figure 14.64 For Problem 14.33. 14.34 Use the program in Figure 14.33 to solve L aplace’s equation in the problem shown in Figure 14.65, where V0 = 100 V. Comparet he finite element solution to the exact solu- tion in Example 6.5; that is, W9C,” = 4Vo °° sinmnxsinhmry’ n = 2k+1 1r k=0 insmh mr 14.35 Repeat Problem 14.34 for V0 = 100 sin 1rx. Com the theoretical solution [simil pare the finite element solution with ar to Example 6.6(a)]; that is, PROBLEMS 763 Figure rams" For Problem 14.34. 14.38 The cross section of a transmission line is shown in Figure 14.67. Use the finite differ— ence method to compute the characteristic impedance of the line. 14.39 Half a solution region is shown in Figure 14.68 so that the y—axis is a line of symmetry. Use finite difference to find the potential at nodes 1 to 9. Five iterations are sufficient if you use an iterative method. Figure 142.66 For Problem 14.37. 764 CHAPTER 14 NUMERICAL METHODS Figure 14.67 For Problem 14.38. 3cm { Figure 14.68 For Problem 14.39. I I ...
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This note was uploaded on 02/04/2012 for the course ECE 311 taught by Professor Peroulis during the Fall '08 term at Purdue University-West Lafayette.

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sadiku_ch14_probs - PROBLEMS 751 SEBTIDN 14.1 FIELD...

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