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Fundamentals of Electromagnetics: A Two-Week, 8-Day, Intensive Course for Training Faculty in Electrical-, Electronics-, Communication-, and Computer-

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Presentation on theme: "Fundamentals of Electromagnetics: A Two-Week, 8-Day, Intensive Course for Training Faculty in Electrical-, Electronics-, Communication-, and Computer-"— Presentation transcript:

1 Fundamentals of Electromagnetics: A Two-Week, 8-Day, Intensive Course for Training Faculty in Electrical-, Electronics-, Communication-, and Computer- Related Engineering Departments 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 Amrita Viswa Vidya Peetham, Coimbatore August 11, 12, 13, 14, 18, 19, 20, and 21, 2008

2 Statics, Quasistatics, and Transmission Lines
Module 6 Statics, Quasistatics, and Transmission Lines Gradient and electric potential Poisson’s and Laplace’s equations Static fields and circuit elements Low-frequency behavior via quasistatics The distributed circuit concept and the transmission line

3 Instructional Objectives
25. 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 26. Obtain the solution for the potential between two conductors held at specified potentials, for one-dimensional cases in the Cartesian coordinate system (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 length (or capacitance in the case of spherical conductors) of the arrangement

4 Instructional Objectives (Continued)
27. Perform static field analysis of arrangements consisting of two parallel plane conductors for electrostaic, magnetostatic, and electromagnetostatic fields 28. Perform quasistatic static field analysis of arrangements consisting of two parallel plane conductors for electroquastatic and magnetoquasistatic fields 29. Understand the development of the transmission-line (distributed equivalent circuit) from the field solutions for a given physical structure

5 Gradient and Electric Potential (FEME, Sec. 6. 1; EEE6E, Secs. 5. 1, 5

6 Gradient and the Potential Functions

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

8 F is known as the electric scalar potential.
is the gradient of F.

9

10 Basic definition of From this, we get

11 Potential function equations

12 Laplacian of scalar Laplacian of vector In Cartesian coordinates,

13 For static fields,

14 But, also known as the potential difference between A and B, for the static case.

15 Given the charge distribution, find V using superposition.
Then find E using the above. For a point charge at the origin, Since agrees with the previously known result.

16 Thus for a point charge at an arbitrary location P
Q P5.9

17 Considering the element of length dz at (0, 0, z), we have
Using

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19 Magnetic vector potential due to a current element
Analogous to

20 Poisson’s and Laplace’s Equations (FEME, Sec. 6.2; EEE6E, Sec. 5.3)

21 Poisson’s Equation For static electric field, Then from
If e is uniform, Poisson’s equation

22 If e is nonuniform, then using
Thus Assuming uniform e, we have For the one-dimensional case of V(x),

23 D5.7 Anode, x = d V = V0 Vacuum Diode Cathode, x = 0 V = 0 (a)

24 (b)

25 (c)

26 Laplace’s Equation If r = 0, Poisson’s equation becomes
Let us consider uniform e first Parallel-plate capacitor x = d, V= V0 x = 0, V = 0

27 Neglecting fringing of field at edges,
General solution

28 Boundary conditions Particular solution

29

30 area of plates For nonuniform e, For

31 Example x = d, V = V0 x = 0, V = 0

32

33

34 Static Fields and Circuit Elements (FEME, Sec. 6.3; EEE6E, Sec. 5.4)

35 Classification of Fields
Static Fields ( No time variation; t = 0) 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

36 × ò × ò ò × × ò ò × ò × ò Ñ x E = E d l = Ñ x H = J Static Fields H d
For static fields, t = 0, and the equations reduce to × ò E d l = Ñ x E = C × × ò H d l = ò J d S Ñ x H = J C S × ò D d S = ò r dv Equations (1.64 a, b, c, d, e) (1.65a, b, c, d, e) S v × ò B d S = S × ò J d S = S

37 Solution for Potential and Field
charge distribution Solution for point charge Equations (1. 69)-(1.71) Electric field due to point charge

38 Laplace’s Equation and One-Dimensional Solution
For Poission’s equation reduces to Laplace’s equation Equation (1.72)(1.73)

39 Example of Parallel-Plate Arrangement; Capacitance
 r S Figure 1.13, Eq. (1.74) r S

40 Electrostatic Analysis of Parallel-Plate Arrangement
Capacitance of the arrangement, F Equations (1.75)-(1.79)

41 × × ò ò × ò Magnetostatic Fields H d l = J d S Ñ x H = J
B d S = S Poisson’s equation for magnetic vector potential Eq. (1.80)(1.65 b, d)(1.64b, d)

42 Solution for Vector Potential and Field
current distribution Solution for current element B ( r ) = m I d l x - 4 p 3 Magnetic field due to current element Equations (1.81)-(1.84) 2A = 0 For current-free region

43 Example of Parallel-Plate Arrangement; Inductance
 Figure 1.14

44 Magnetostatic Analysis of Parallel-Plate Arrangement
Equations (1.85)=(1.88)

45 Magnetostatic Analysis of Parallel-Plate Arrangement
Inductance of the arrangement, H Equations (1. 89)-(1.92)

46 Electromagnetostatic Fields
× ò E d l = Ñ x E = C × × × ò H d l = ò J d S = s ò E d S Ñ x H = J c s E c C S S × ò D d S = Equations (1.93 a, b, c, d) (1.94 a, b, c, d) S × ò B d S = S

47 Example of Parallel-Plate Arrangement
 r S Figure 1.15 r S

48 Electromagnetostatic Analysis of Parallel-Plate Arrangement
Equations (1.95)-(1.97)

49 Electromagnetostatic Analysis of Parallel-Plate Arrangement
Conductance, S Resistance, ohms Equations (1.98)-(1.00)

50 ò Electromagnetostatic Analysis of Parallel-Plate Arrangement [m H
= y ( z ) a 1 æ z ö = ò [m H y d(dz¢)] è ø I l = z l c Equations (1.101a, b, c)(1.102) Internal Inductance

51 ò Alternatively, Electromagnetostatic Analysis of
Parallel-Plate Arrangement Alternatively, L i = 1 I c 2 ( dw ) m H y dz z - l ò 3 dl w Equivalent Circuit Figure 1.16 Eq. (1.103)

52 Low Frequency Behavior via Quasistatics (FEME, Sec. 6.4; EEE6E, Sec. 5.5)

53 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 w for their amplitudes. 63

54 Electroquasistatic Fields
J r S H S I ( t ) 1 + + + + + + g + + x = y z E +  x = d x Figure 1.17 z z = l z = 64

55 Electroquasistatic Analysis of Parallel-Plate Arrangement
= V d cos w t a x Equations (1.104)(1.105)(1.106) H 1 = w e V z d sin t a y

56 [ ] Electroquasistatic Analysis of Parallel-Plate Arrangement I ( t )
= w H g y 1 z = - l æ e w l ö = - w V sin w t è ø d dV ( t ) g = C d t Equations (1.107)(1.108) where

57 [ ] Electroquasistatic Analysis of Parallel-Plate Arrangement P = wd E
H in x y 1 z = æ e wl ö 2 = - w V sin w t cos w t è ø d d 1 æ 2 ö Equation (1.109) = CV è ø dt 2 g

58 Magnetoquasistatic Fields
r S  Figure 1.18

59 Magnetoquasistatic Analysis of Parallel-Plate Arrangement
Eq. (1.110)(1.111)(1.112)

60 [ ] Magnetoquasistatic Analysis of Parallel-Plate Arrangement V ( t )
= d E g x 1 z = - l æ m dl ö = - w I sin w t è w ø dI ( t ) g = L dt Eq. (1.113)(1.114) where

61 [ ] Magnetoquasistatic Analysis of Parallel-Plate Arrangement P = wd E
H in x 1 y z = - l æ m d l ö 2 = - w I sin w t cos w t è w ø d 1 æ 2 ö Eq. (1.115) = LI è ø dt 2 g

62 Quasistatic Fields in a Conductor
  Figure 1.19

63 Quasistatic Analysis of Parallel-Plate Arrangement with Conductor
Eq. (1.116)-(1.120)

64 ( ) Quasistatic Analysis of Parallel-Plate Arrangement with Conductor
H E y 1 = - s E - e x z x 1 t 2 w m s V ( ) w e V 2 2 = z - l sin w t + sin w t 2 d d Eq. (1.121), (1.122)

65 ( ) Quasistatic Analysis of Parallel-Plate Arrangement with Conductor
V w m s V ( ) 2 2 E = cos w t - z - l sin w t x d 2 d Eqs. (1.123a,b) & (1.124a,b)

66 ( ) [ ] Quasistatic Analysis of Parallel-Plate
Arrangement with Conductor [ ] I = w H g y z = - l 2 3 æ ö s wl e wl m s wl = + j w - j w V ç ÷ è d d 3 d g ø I 2 æ ö e wl s wl m s l g Y = = j w + 1 - j w in ç ÷ Eqs. (1.125) & (1.126) V d d è 3 ø g e wl 1 j w + ( ) d m s l 2 d 1 + j w s 3 wl

67 Equivalent Circuit Quasistatic Analysis of Parallel-Plate
Arrangement with Conductor Equivalent Circuit Eq. (1.127), Fig. 1.16

68 The Distributed Circuit Concept and the Transmission Line (FEME, Secs
The Distributed Circuit Concept and the Transmission Line (FEME, Secs. 6.5, 6.6; EEE6E, Secs. 6.1)

69 Waves and the Distributed Circuit Concept
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, no further insight is gained through that process, and it is more straight- forward to obtain the exact solution by resorting to simultaneous solution of Maxwell’s equations when a closed form solution is possible. 78

70 Wave Equation ¶ E ¶ E = m e ¶ z ¶ t ¶ D ¶ E Ñ x H = = e ¶ t ¶ t ¶ B ¶
- = - m Ñ x H = = e t t t t For the one-dimensional case of 2 2 E E x = m e x One-dimensional wave equation 2 2 z t 79

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91 The End


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