1. Fundamentals of Electromagnetics for Teaching and Learning: A Two-Week Intensive Course for Faculty in Electrical-, Electronics-, Communication-, and Computer- Related Engineering Departments in Engineering Colleges in India by Nannapaneni Narayana Rao Edward C. Jordan Professor Emeritus of Electrical and Computer Engineering University of Illinois at Urbana-Champaign, USA Distinguished Amrita Professor of Engineering Amrita Vishwa Vidyapeetham, India

2. Program for Hyderabad Area and Andhra Pradesh Faculty Sponsored by IEEE Hyderabad Section, IETE Hyderabad Center, and Vasavi College of Engineering IETE Conference Hall, Osmania University Campus Hyderabad, Andhra Pradesh June 3 – June 11, 2009 Workshop for Master Trainer Faculty Sponsored by IUCEE (Indo-US Coalition for Engineering Education) Infosys Campus, Mysore, Karnataka June 22 – July 3, 2009

3. 6-2 Module 6 Statics, Quasistatics, and Transmission Lines 6.1 Gradient and electric potential 6.2 Poisson’s and Laplace’s equations 6.3 Static fields and circuit elements 6.4 Low-frequency behavior via quasistatics 6.5 Condition for the validity of the quasistatic approximation 6.6 The distributed circuit concept and the transmission-line

4. 6-3 Instructional Objectives 42. Understand the geometrical significance of the gradient operation 43. Find the static electric potential due to a specified charge distribution by applying superposition in conjunction with the potential due to a point charge, and further find the electric field from the potential 44. Obtain the solution for the potential between two conductors held at specified potentials, for one- dimensional cases (and the region between which is filled with a dielectric of uniform or nonuniform permittivity, or with multiple dielectrics) by using the Laplace’s equation in one dimension, and further find the capacitance per unit area (Cartesian) or per unit length (cylindrical) or capacitance (spherical) of the arrangement

5. 6-4 Instructional Objectives (Continued) 45. Perform static field analysis of arrangements consisting of two parallel plane conductors for electrostatic, magnetostatic, and electromagnetostatic fields 46. Perform quasistatic field analysis of arrangements consisting of two parallel plane conductors for electroquastatic and magnetoquasistatic fields 47. Understand the condition for the validity of the quasistatic approximation and the input behavior of a physical structure for frequencies beyond the quasistatic approximation 48. Understand the development of the transmission-line (distributed equivalent circuit) from the field solutions for a given physical structure and obtain the transmission- line parameters for a line of arbitrary cross section by using the field mapping technique

6. 6.1 Gradient and Electric Potential (EEE, Secs. 5.1, 5.2; FEME, Sec. 6.1)

7. 6-6 Gradient and the Potential Functions

8. 6-7 B can be expressed as the curl of a vector. Thus A is known as the magnetic vector potential. Then

9. 6-8  is known as the electric scalar potential. is the gradient of 

10. 6-9

11. 6-10 Basic definition of : For a constant  surface, d  = 0. Therefore is normal to the surface.

12. 6-11 Thus, the magnitude of at any point P is the rate of increase of  normal to the surface, which is the maximum rate of increase  at that point. Thus Useful for finding unit normal vector to the surface.

13. 6-12 D5.1 Finding unit normal vectors to the surface at several points:

14. 6-13

15. 6-14 (1) (2) (4) (3) (1) (3)

16. 6-15 Potential function equations (2)

17. 6-16 Laplacian of scalar Laplacian of vector In Cartesian coordinates,

18. 6-17 also known as the potential difference between A and B, for the static case. But, For static fields,

19. 6-18 Given the charge distribution, find V using superposition. Then find E using the above. since agrees with the previously known result. For a point charge at the origin,

20. 6-19 Thus for a point charge at an arbitrary location P Q R P P5.9

21. 6-20 Considering the element of length dz at (0, 0, z), we have Using

22. 6-21

23. 6-22 Magnetic vector potential due to a current element R P Analogous to

24. 6-23 Review Questions 6.1. What is the divergence of the curl of a vector? 6.2. What is the expansion for the gradient of a scalar in Cartesian coordinates? When can a vector be expressed as the gradient of a scalar? 6.3. Discuss the basic definition of the gradient of a scalar. 6.4. Discuss the application of the gradient concept for the determination of unit vector normal to a surface. 6.5. Define electric potential. What is its relationship to the electric field intensity? 6.6. Distinguish between voltage as applied to time-varying fields and potential difference. 6.7. What is the electric potential due to a point charge? Discuss the determination of electric potential due to a charge distribution.

25. 6-24 Review Questions (Continued) 6.8. What is the Laplacian of a scalar? What is the expansion for the Laplacian of a scalar in Cartesian coordinates? 6.9. What is the magnetic vector potential? How is it related to the magnetic flux density?

26. 6-25 Problem S6.1. Finding the gradient of a two-dimensional function and associated discussion

27. 6-26 Problem S6.2. Finding the angle between two plane surfaces, by using the gradient concept

28. 6-27 Problem S6.3. Finding the image charge(s) for a point charge in the presence of a conductor

29. 6-28 Problem S6.3. Finding the image charge(s) for a point charge in the presence of a conductor (Continued)

30. 6.2 Poisson’s and Laplace’s Equations (EEE, Sec. 5.3; FEME, Sec. 6.2)

31. 6-30 Poisson’s Equation For static electric field, Then from If  is uniform, Poisson’s equation

32. 6-31 If  is nonuniform, then using Thus Assuming uniform , we have For the one-dimensional case of V(x),

33. 6-32 Anode, x = d V = V 0 Cathode, x = 0 V = 0 Vacuum Diode D5.7 (a)

34. 6-33 (b)

35. 6-34 (c)

36. 6-35 Laplace’s Equation Let us consider uniform  first. E6.1. Parallel-plate capacitor If  Poisson’s equation becomes x = d, V= V 0 x = 0, V = 0

37. 6-36 Neglecting fringing of field at edges, General solution

38. 6-37 Boundary conditions Particular solution

39. 6-38

40. 6-39 area of plates For nonuniform  For

41. 6-40 E6.2 x = d, V = V 0 x = 0, V = 0

42. 6-41

43. 6-42

44. 6-43 Review Questions 6.10. State Poisson’s equation for the electric potential. How is it derived? 6.11. Outline the solution of the Poisson’s equation for the potential in a region of known charge density varying in one dimension. 6.12. State Laplace’s equation for the electric potential. In what regions is it valid? 6.13. Outline the solution of Laplace’s equation in one dimension by considering a parallel-plate arrangement. 6.14. Outline the steps in the determination of the capacitance of a parallel-plate capacitor.

45. 6-44 Problem S6.4. Solution of Poisson’s equation for a space charge distribution in Cartesian coordinates

46. 6-45 Problem S6.5. Finding the capacitance of a spherical capacitor with a dielectric of nonuniform permittivity

47. 6.3 Static Fields and Circuit Elements (EEE, Sec. 5.4; FEME, Sec. 6.3)

48. 6-47 Classification of Fields Static Fields ( No time variation; ) Static electric, or electrostatic fields Static magnetic, or magnetostatic fields Electromagnetostatic fields Dynamic Fields (Time-varying) Quasistatic Fields (Dynamic fields that can be analyzed as though the fields are static) Electroquasistatic fields Magnetoquasistatic fields

49. 6-48 Static Fields For static fields,, and the equations reduce to  Edl  0 C  H  dl  C  J  dS S  D  dS  dv V  S   BdS  0 S  J  dS  0 S 

50. 6-49 Solution for charge distribution Solution for point charge Electric field due to point charge Solution for Potential and Field

51. 6-50 Laplace’s Equation and One-Dimensional Solution Laplace’s equation For  Poission’s equation reduces to

52. 6-51 Example of Parallel-Plate Arrangement: Capacitance   S  S

53. 6-52 Capacitance of the arrangement, F Electrostatic Analysis of Parallel-Plate Arrangement

54. 6-53 Magnetostatic Fields Poisson’s equation for magnetic vector potential H  dl  J  dS S  C  B  dS  0 S 

55. 6-54 Solution for current distribution Solution for current element Magnetic field due to current element  2 A = 0 For current-free region Solution for Vector Potential and Field

56. 6-55 Example of Parallel-Plate Arrangement: Inductance

57. 6-56 Magnetostatic Analysis of Parallel-Plate Arrangement

58. 6-57 Inductance of the arrangement, H Magnetostatic Analysis of Parallel-Plate Arrangement (Continued)

59. 6-58 Electromagnetostatic Fields E  dl  0 C  H  dl  C  J c  dS S   E  dS S  D  dS  0 S  B  dS  0 S   x E  0  x H  J c  E

60. 6-59 Example of Parallel-Plate Arrangement

61. 6-60 Electromagnetostatic Analysis of Parallel-Plate Arrangement

62. 6-61 Conductance, S Resistance, ohms Electromagnetostatic Analysis of Parallel-Plate Arrangement (Continued)

63. 6-62 Internal Inductance Electromagnetostatic Analysis of Parallel-Plate Arrangement (Continued) H  H y (z)a y  H y d(dz   1 I c –z l     z  – l 0 

64. 6-63 Electromagnetostatic Analysis of Parallel-Plate Arrangement (Continued) Equivalent Circuit Alternatively, from energy considerations, L i  1 I c 2 (dw)  H y 2 dz z  l 0   1 3  dl w

65. 6-64 Review Questions 6.15. Discuss the classification of fields as static, dynamic, and quasistatic fields. 6.16. State Maxwell’s equations for static fields in (a) integral form, and (b) differential form. 6.17. Outline the steps involved in the electrostatic field analysis of a parallel-plate structure and the determination of its capacitance. 6.18. Outline the steps involved in the magnetostatic field analysis of a parallel-plate structure and the determination of its inductance. 6.19. Outline the steps involved in the electromagnetostatic field analysis of a parallel-plate structure and the determination of its circuit equivalent. 6.20. Explain the term, “internal inductance.”

66. 6-65 Problem S6.6. Finding the internal inductance per unit length of a cylindrical conductor arrangement

67. 6.4 Low Frequency Behavior via Quasistatics (EEE, Sec. 5.5; FEME, Sec. 6.4)

68. 6-67 Quasistatic Fields For quasistatic fields, certain features can be analyzed as though the fields were static. In terms of behavior in the frequency domain, they are low-frequency extensions of static fields present in a physical structure, when the frequency of the source driving the structure is zero, or low-frequency approximations of time-varying fields in the structure that are complete solutions to Maxwell’s equations. Here, we use the approach of low-frequency extensions of static fields. Thus, for a given structure, we begin with a time- varying field having the same spatial characteristics as that of the static field solution for the structure and obtain field solutions containing terms up to and including the first power (which is the lowest power) in  for their amplitudes.

69. 6-68 Electroquasistatic Fields  x y z z=–l z=0 z –    ––– I g (t) + – x  0 x  d  –––– H 1 E 0 J S  S

70. 6-69 Electroquasistatic Analysis of Parallel-Plate Arrangement E 0  V 0 d cos  ta x H 1   V 0 z d sin  ta y

71. 6-70 where Electroquasistatic Analysis of Parallel-Plate Arrangement (Continued) I g (t)  wH y 1  z  l   wl d     V 0 sin  t  C dV g (t) dt

72. 6-71 Electroquasistatic Analysis of Parallel-Plate Arrangement (Continued) P in  wdE x 0 H y 1  z  0   wl d      V 0 2 sin  tcos  t  d dt 1 2 CV g 2    

73. 6-72 Magnetoquasistatic Fields   S

74. 6-73 Magnetoquasistatic Analysis of Parallel-Plate Arrangement

75. 6-74 where Magnetoquasistatic Analysis of Parallel-Plate Arrangement (Continued) V g (t)  dE x 1  z  l   dl w     I 0 sin  t  L dI g (t) dt

76. 6-75 Magnetoquasistatic Analysis of Parallel-Plate Arrangement (Continued) P in  wdE x1 H y 0  z  l   dl w      I 0 2 sin  tcos  t  d dt 1 2 LI g 2    

77. 6-76 Quasistatic Fields in a Conductor 

78. 6-77 Quasistatic Analysis of Parallel-Plate Arrangement with Conductor

79. 6-78 Quasistatic Analysis of Parallel-Plate Arrangement with Conductor (Continued)

80. 6-79 Quasistatic Analysis of Parallel-Plate Arrangement with Conductor (Continued) E x  V 0 d cos  t   V 0 2d z 2  l 2  sin  t

81. 6-80 Quasistatic Analysis of Parallel-Plate Arrangement with Conductor (Continued) Y in  I g V g  j   wl d   d 1  j   l 2 3        j   d  1 d  1  j   l 2 3  I g  wH y  z  l   d  j   d  j   2 3 3d       V g

82. 6-81 Equivalent Circuit Quasistatic Analysis of Parallel-Plate Arrangement with Conductor (Continued)

83. 6-82 Review Questions 6.21. What is meant by the quasistatic extension of the static field in a physical structure? 6.22. Outline the steps involved in the electroquasistatic field analysis of a parallel-plate structure and the determination of its input behavior. Compare the input behavior with the electrostatic case. 6.23. Outline the steps involved in the magnetoquasistatic field analysis of a parallel-plate structure and the determination of its input behavior. Compare the input behavior with the magnetostatic case. 6.24. Outline the steps involved in the quasistatic field analysis of a parallel-plate structure with a conducting slab between the plates and the determination of its input behavior. Compare the input behavior with the electromagnetostatic case.

84. 6-83 Problem S6.7. Frequency behavior of a capacitor beyond the quasistatic approximation

85. 6-84 Problem S6.7. Frequency behavior of a capacitor beyond the quasistatic approximation (Continued)

86. 6-85 6.5 Condition for the validity of the quasistatic approximation (EEE, Sec. 5.5; FEME, Secs. 6.5, 7.1)

87. 6-86 We have seen that quasistatic field analysis of a physical structure provides information concerning the low-frequency input behavior of the structure. As the frequency is increased beyond that for which the quasistatic approximation is valid, terms in the infinite series solutions for the fields beyond the first-order terms need to be included. While one can obtain equivalent circuits for frequencies beyond the range of validity of the quasistatic approximation by evaluating the higher order terms, we shall here obtain the exact solution by resorting to simultaneous solution of Maxwell’s equations to find the condition for the validity of the quasistatic approximation, and further investigate the behavior for frequencies beyond the quasistatic approximation. We shall do this by considering the parallel-plate structure, and obtaining the wave solutions, which will then lead us to the distributed circuit concept and the transmission-line.

88. 6-87 Wave Equation One-dimensional wave equation For the one-dimensional case of

89. 6-88 Solution to the One-Dimensional Wave Equation Traveling wave propagating in the +z direction

90. 6-89 Solution to the One-Dimensional Wave Equation Traveling wave propagating in the –z direction t   4 

91. 6-90 Phase constant Phase velocity Intrinsic impedance General Solution in Phasor Form A  Ae j  ,B  Be j  –,,,

92. 6-91  ss Example of Parallel-Plate Structure Open-Circuited at the Far End H y  0 atz=0 E x  V g d z =  l      B.C.

93. 6-92 Standing Wave Patterns (Complete Standing Waves)

94. 6-93 Complete Standing Waves Complete standing waves are characterized by pure half-sinusoidal variations for the amplitudes of the fields. For values of z at which the electric field amplitude is a maximum, the magnetic field amplitude is zero, and for values of z at which the electric field amplitude is zero, the magnetic field amplitude is a maximum. The fields are also out of phase in time, such that at any value of z, the magnetic field and the electric field differ in phase by t =  / 2 .

95. 6-94 Input Admittance For  l << 1, I g  wH y  z  l  jwV g  d tan  l  Y in  j w  d  l  (  l) 3 3  2(  l) 5 15        

96. 6-95 Condition for the Validity of the Quasistatic Approximation The condition  l << 1 dictates the range of validity for the quasistatic approximation for the input behavior of the structure. In terms of the frequency f of the source, this condition means that f << v p /2  l, or in terms of the period T = 1/f, it means that T >> 2  (l/v p ). Thus, quasistatic fields are low-frequency approximations of time-varying fields that are complete solutions to Maxwell’s equations, which represent wave propagation phenomena and can be approximated to the quasistatic character only when the times of interest are much greater than the propagation time, l/v p, corresponding to the length of the structure. In terms of space variations of the fields at a fixed time, the wavelength  ( = 2  ), which is the distance between two consecutive points along the direction of propagation between which the phase difference is 2 , must be such that l <<  /2  ; thus, the physical length of the structure must be a small fraction of the wavelength.

97. 6-96 For frequencies slightly beyond the approximation  l <<1,

98. 6-97 In general, f 3v p 4l 5v p 4l 0 v p 2l v p l 3v p 2l v p 4l Y in Y in capacitive inductive

99. 6-98 ss  Example of Parallel-Plate Structure Short-Circuited at the Far End E x  0 atz  0 H y  I g w z  l l      B.C.

100. 6-99 Standing Wave Patterns (Complete Standing Waves)  

101. 6-100 Input Impedance For  l << 1,  Z in  j  d w  l  (  l) 3 3  2(  l) 5 15        V g  dE x  z  l  j  dI g w tan  l 

102. 6-101 For frequencies slightly beyond the approximation  l <<1,

103. 6-102 In general, f

104. 6-103 Review Questions 6.25. Outline the steps in the solution for the electromagnetic field in a parallel-plate structure open-circuited at the far end. 6.26. What are complete standing waves? Discuss their characteristics. 6.27. What is the input admittance of a a parallel-plate structure open-circuited at the far end? Discuss its variation with frequency. 6.28. State and discuss the condition for the validity of the quasistatic approximation. 6.29. Outline the steps in the solution for the electromagnetic field in a parallel-plate structure short-circuited at the far end. 6.30. What is the input impedance of a a parallel-plate structure short-circuited at the far end? Discuss its variation with frequency.

105. 6-104 Problem S6.8. Frequency behavior of a parallel-plate structure from input impedance considerations

106. 6-105 Problem S6.8. Frequency behavior of a parallel-plate structure from input impedance considerations (Continued)

107. 6.6 The Distributed Circuit Concept and the Transmission Line (EEE, Secs. 6.1, 11.5; FEME, Secs. 6.5, 6.6)

108. 6-107

109. 6-108

110. 6-109

111. 6-110

112. 6-111

113. 6-112

114. 6-113

115. 6-114

116. 6-115

117. 6-116

118. 6-117

119. 6-118

120. 6-119

121. 6-120

122. 6-121

123. 6-122

124. 6-123

125. 6-124

126. 6-125

127. 6-126

128. 6-127

129. 6-128

130. 6-129

131. 6-130

132. 6-131

133. 6-132

134. 6-133 Review Questions 6.31. Discuss the phenomenon taking place in a parallel-plate structure at any arbitrary frequency. 6.32. How is the voltage between the two conductors in a given cross-sectional plane of a parallel-plate transmission line related to the electric field in that plane? 6.33. How is the current flowing on the plates across a given cross-sectional plane of a parallel-plate transmission line related to the magnetic field in that plane? 6.35. Discuss transverse electromagnetic waves. 6.36. What are transmission-line equations? How are they derived from Maxwell’s equations?

135. 6-134 Review Questions 6.37. Discuss the concept of the distributed equivalent circuit. How is it obtained from the transmission-line equations? 6.38. Discuss the solutions for the transmission-line equations for the voltage and current along a line. 6.39. Explain the “characteristic impedance” of a transmission line. 6.40. Discuss the relationship between the transmission-line parameters. 6.41. What are the transmission-line parameters for a parallel- plate line? 6.42. Describe the curvilinear squares technique of finding the line parameters for a line with an arbitrary cross section.

136. 6-135 Problem S6.9. Transmission-line equations and power flow from the geometry of a coaxial cable

137. 6-136 Problem S6.10. Application of the curvilinear squares technique for an eccentric coaxial cable

138. The End