Electromagnetics and Radiation

Transcription

1 Electromagnetics and Radiation

2 This book is a part of the course by Jaipur National University, Jaipur. This book contains the course content for Electromagnetics and Radiation. JNU, Jaipur First Edition 2013 The content in the book is copyright of JNU. All rights reserved. No part of the content may in any form or by any electronic, mechanical, photocopying, recording, or any other means be reproduced, stored in a retrieval system or be broadcast or transmitted without the prior permission of the publisher. JNU makes reasonable endeavours to ensure content is current and accurate. JNU reserves the right to alter the content whenever the need arises, and to vary it at any time without prior notice.

3 Index I. Content...II II. List of Figures... V III. List of Tables...VI IV. Abbreviations... VII V. Application VI. Bibliography VII. Self Assessment Answers Book at a Glance I/JNU OLE

4 Content Chapter I... 1 Coulomb s Law and Electric Field Intensity... 1 Aim... 1 Objectives... 1 Learning outcome Introduction Coulomb s Law Principle of Superposition Electric Field Intensity Field due to Continuous Volume Distribution Field due to Line Charge Field of Sheet Electric Flux Density Gauss s Law Gauss s Law and the Divergence Theorem Divergence Divergence and Maxwell s First Equation in Point Form The Divergence Theorem Summary References Recommended Reading Self Assessment Chapter II Energy and Potential Aim Objectives Learning outcome Introduction to Energy and Potential Energy Expended in Moving a Point Charge in an Electric Field The Line Integral Definition of Potential Difference and Potential The Potential Field of a Point Charge The Potential Field of a System of Charges: Conservative Property Potential Gradient Summary References Recommended Reading Self Assessment Chapter III Poisson s and Laplace s Equations Aim Objectives Learning outcome Introduction Poisson s and Laplace s Equations Uniqueness Theorem Procedure for Solving Laplace s Equation Summary References Recommended Reading Operation in Different Co-ordinate Systems II/JNU OLE

5 Self Assessment Chapter IV Magnetic Forces, Materials and Inductance Aim Objectives Learning outcome Introduction to Magnetic Flux Magnetic Flux and Magnetic Flux Density Magnetic Force on a Moving Charge Magnetic Force on Current Torque on a Current Loop Magnetisation Magnetic Materials Magnetic Boundary Conditions Inductors and Inductance Mutual Inductance Calculations Internal and External Inductance Magnetic Forces on Magnetic Materials Magnetic Circuits Summary References Recommended Reading Self Assessment Chapter V Time Varying Fields And Maxwell s Equations Aim Objectives Learning outcome Faraday s Law Displacement Current Maxwell s Equations in Point Form Maxwell s Equations in Integral Form Summary References Recommended Reading Self Assessment Chapter VI Antenna Fundamentals Aim Objectives Learning outcome Introduction How an Antenna Radiates Near and Far Field Regions Antenna Performance Parameters Radiation Pattern Directivity Input Impedance Voltage Standing Wave Ratio (VSWR) Return Loss (RL) Antenna Efficiency Antenna Gain III/JNU OLE

6 6.4.8 Polarisation Bandwidth Types of Antennas Half Wave Dipole Monopole Antenna Loop Antenna Helical Antennas Horn Antennas Summary References Recommended Reading Self Assessment IV/JNU OLE

7 List of Figures Fig. 1.1 Coulomb interaction between two charges... 2 Fig. 1.2 Evaluation of the E field due to a volume charge distribution... 4 Fig. 1.3 Evaluation of the E field due to a line charge... 6 Fig. 1.4 Evaluation of the E field due to an infinite sheet of charge... 7 Fig. 1.5 Flux lines... 9 Fig. 1.6 The vector differential area, Fig. 1.7 Example of interpretation of the divergence Fig. 2.1 A graphical interpretation of line integral in a uniform field Fig. 2.2 (a) Circular path (b) A radial path Fig. 2.3 A general path between general points B and A in the field of a point charge Q at the origin Fig. 2.4 A potential field is shown by its equipotential surfaces Fig. 3.1 Concentric right cylinders Fig. 4.1 Direction of the charge in an electric and magnetic field Fig. 4.2 Magnetic force on current Fig. 4.3 Example of force between line currents Fig. 4.4 x-y plane Fig. 4.5 Forces due to B z Fig. 4.6 Forces due to B y Fig. 4.7 Loop area Fig. 4.8 Loop area Fig. 4.9 Magnetic moments Fig Magnetic flux density Fig Magnetic moments Fig B-H curve Fig Tangential magnetic field Fig Tangential magnetic field Fig Normal magnetic flux density Fig Example mutual inductance between coaxial loops Fig Magnetic forces on magnetic materials Fig Magnetic field Fig Electric and magnetic circuits Fig. 5.1 A filamentary conductor forms a loop connecting the two plates of a parallel-plate capacitor Fig. 6.1 Radiation from an antenna Fig. 6.2 Field regions around an antenna Fig. 6.3 Radiation pattern of a generic directional antenna Fig. 6.4 Equivalent circuit of transmitting antenna Fig. 6.5 A linearly (vertically) polarised wave Fig. 6.6 Commonly used polarisation schemes Fig. 6.7 Measuring bandwidth from the plot of the reflection coefficient Fig. 6.8 Half wave dipole Fig. 6.9 Radiation pattern for half wave dipole Fig Monopole antenna Fig Radiation pattern for the monopole antenna Fig Loop antenna Fig Radiation pattern of small and large loop antenna Fig Helix antenna Fig Radiation pattern of helix antenna Fig Types of horn antenna V/JNU OLE

8 List of Tables Table 4.1 Difference between magnetisation and polarisation Table 4.2 Difference between inductors and capacitors Table 4.3 Difference between electric and magnetic circuits VI/JNU OLE

9 Abbreviations Emf Electromotive force RL Return Loss VSWR Voltage Standing Wave Ratio VII/JNU OLE

10 Chapter I Coulomb s Law and Electric Field Intensity Aim The aim of this chapter is to: describe Coulomb s law explain electric flux intensity and electric flux density illustrate Gauss s law and divergence theorem Objectives The objective of this chapter is to: examine various fields of intensity learn Gauss s law calculate divergence and Maxwell s first equation Learning outcome At the end of this chapter, the students will be able to: recall Coulomb s law understand electric flux intensity and electric flux density categorise Gauss s law, divergence theorem and Maxwell s equation 1/JNU OLE

11 Electromagnetics and Radiation 1.1 Introduction It is a common experience that if we comb our hair, and subsequently bring the comb close to tiny bits of paper, the comb attracts pieces of paper. The phenomenon involved is the production of static electric charges, which has been known to mankind as long back as 600 BC. If you rub a glass rod with a piece of silk, the rod develops an electric charge. When another glass rod, similarly rubbed, is brought near it, the two repel each other. However, if the glass rod is brought towards an amber rod rubbed with fur, the two rods attract each other. Rods treated in this fashion are said to have developed electric charge, and, the force between such charged bodies is called electric force. There are two types of observed electric charge, which we designate as positive and negative. The convention was derived from Benjamin Franklin s experiments. He rubbed a glass rod with silk and called the charges on the glass rod positive. He rubbed sealing wax with fur and called the charge on the sealing wax negative. Like charges repel and opposite charges attract each other. The unit of charge is called the Coulomb (C). The smallest unit of free charge known in nature is the charge of an electron or proton, which has a magnitude of, (1) Charge of any ordinary matter is quantized in integral multiples of e. An electron carries one unit of negative charge, -e, while a proton carries one unit of positive charge, +e. In a closed system, the total amount of charge is conserved since charge can neither be created nor destroyed. A charge can, however, be transferred from one body to another. 1.2 Coulomb s Law Consider a system of two point charges, q 1 and q 2, separated by a distance r in vacuum. Coulomb s law states that the force F between two point charges q 1 and q 2 is: Along the line joining them Directly proportional to the product q to q of the charges 1 2 Inversely proportional to the square of the distance r between them. The force exerted by q 1 on q 2 is given by Coulomb s law, 12 = (2) Where k e is the Coulomb constant, and is a unit vector directed from q 1 to q 2, as illustrated in fig. below Fig. 1.1 Coulomb interaction between two charges (Source: Note that electric force is a vector which has both magnitude and direction. In SI units, the Coulomb constant k e is given by (3) 2/JNU OLE

12 Where, This is known as the permittivity of free space. Similarly, the force on q 1 due to q 2 is given by 21 = 12 as illustrated in fig 1.1 (b). This is consistent with Newton s third law. 1.3 Principle of Superposition Coulomb s law applies to any pair of point charges. When more than two charges are present, the net force on any one charge is simply the vector sum of the forces exerted on it by the other charges. For example, if three charges are present, the resultant force experienced by q 3 due to q 1 and q 2 will be (5) 1.4 Electric Field Intensity If we now consider one charge fixed in position, say q 1. and move a second charge slowly around, we note that there exists everywhere a force on this second charge; in other words, this second charge is displaying the existence of a force field. Call this second charge a test charge q t. The force on it is given by Coulomb s law, Writing this force as force per unit charge The quantity is a function of only q 1 and the directed line segment from q 1 to the position of the test charge. This describes a vector field and is called the electric field intensity. The electric field intensity is defined as the vector force on a unit positive test charge. Electric field intensity must be measured by the unit newtons per coulomb the force per unit charge. Using capital letter E.for electric field intensity, we have (4) (6) (7) (8) (9) The above equation defines electric field intensity due to a single point charge q 1 in a vacuum Field due to Continuous Volume Distribution Let the volume charge distribution with uniform charge density p v be as shown in fig 4.8. The charge dq associated with the elemental volume dv is (10) 3/JNU OLE

13 Electromagnetics and Radiation Fig. 1.2 Evaluation of the E field due to a volume charge distribution (Source: And hence the total charge in a sphere of radius a is (11) (12) The electric field de at P(0, 0, z) due to the elementary volume charge is Where Due to the symmetry of the charge distribution, the contributions to E x or E y. We are left with only E z, given by, Again, we need to derive expressions for dv, R 2, and α Applying the cosine rule to fig. 1.2, we have (13) (14) (15) (16) 4/JNU OLE

14 (17) It is convenient to evaluate the integral in eq (14) in terms of R and r. Hence we express cosθ, cosα, and sinθ dθin terms of R and r', that is (18) (19) Differentiating equation (19) with respect to keeping z and r fixed, we obtain (20) Substituting equation (15) to equation (20) into equation (14) yields (21) (22) (23) or This result is obtained for E at P (0, 0, z). Due to the symmetry of the charge distribution, the electric field at P(r,θ, φ) is readily obtained from equation (25) as (24) (25) (26) This is identical to the electric field at the same point due to a point charge Q located at the origin or the center of the spherical charge distribution Field due to Line Charge Consider a line charge with uniform charge density ρ L extending from A to B along the z-axis as shown in fig The charge element dq associated with element dl = dz of the line is (27) 5/JNU OLE

15 Electromagnetics and Radiation And hence, the total charge Q is Fig. 1.3 Evaluation of the E field due to a line charge (Source: The electric field intensity due to each of the charge distributions p L, p s, and p v may be regarded as the summation of the field contributed by the numerous point charges making up the charge distribution. The electric field intensity E at an arbitrary point P(x, y, z) can be found using equation (29). Thus from fig. 1.3 or Substituting this in equation(29), we get To evaluate this, it is convenient that we define α, α 1 and α 2 as in fig 1.3. (28) (29) (30) (31) (32) (33) (34) (35) (36) (37) 6/JNU OLE

16 (38) Hence, equation (35) becomes (39) (40) Thus, for a finite line charge, As a special case, for an infinite line charge, point B is at (0, 0, ) and A at (0, 0, ) so that α 1 =π/2, α 2 = π/2; the z-component vanishes and equation (41) becomes Bear in mind that equation (42) is obtained for an infinite line charge along the z-axis so that p and a p have their usual meaning. If the line is not along the z-axis, p is the perpendicular distance from the line to the point of interest and a p is a unit vector along that distance directed from the line charge to the field point Field of Sheet Consider an infinite sheet of charge in the xy-plane with uniform charge density p s. The charge associated with elemental area ds is (42) (41) And hence the total charge is, (43) The contribution of E field at point P(0,0,h) by the elemental surface shown in Fig. 1.4 is (44) Fig. 1.4 Evaluation of the E field due to an infinite sheet of charge (Source: 7/JNU OLE

17 Electromagnetics and Radiation Now from fig. 1.4, Substituting these terms in equation (44) gives (45) Due to the symmetry of the charge distribution, for every element 1, there is a corresponding element 2 whose contribution along a p cancels that of element 1, as illustrated in fig Thus the contributions to Ep add up to zero so that E has only z-component. This can also be shown mathematically by replacing a p with cos φa x + sin φa y Integration of cosφ or sinφ over 0 < φ < 2π gives zero. Therefore That is, E has only z-component if the charge is in the xy-plane. In general, for an infinite sheet of charge (46) (47) Where a n is a unit vector normal to the sheet. From equation (46) or (47), we notice that the electric field is normal to the sheet and it is surprisingly independent of the distance between the sheet and the point of observation P. In a parallel plate capacitor, the electric field existing between the two plates having equal and opposite charges is given by (48) 1.5 Electric Flux Density In (approximately) 1837, Michael Faraday, being interested in static electric fields and the effects which various insulating materials (or dielectrics) had on these fields, devised the following experiment: 8/JNU OLE

18 Faraday had two concentric spheres constructed in such a way that the outer one could be dismantled into two hemispheres. With the equipment taken apart, the inner sphere was given a known positive charge. Then, using about 2cm of perfect (ideal) dielectric material in the intervening space, the outer shell was clamped around the inner. Next, the outer shell was discharged by connecting it momentarily to ground. The outer shell was then carefully separated and the negative charge induced on each hemisphere was measured. Faraday found that the magnitude of the charge induced on the outer sphere was equal to that of the charge on the inner sphere, irrespective of the dielectric used. He concluded that there was some kind of displacement from the inner to the outer sphere which was independent of the medium. This is now referred to severally as displacement, displacement flux, or, as we shall use, electric flux. (of course, the idea of electric flux lines as entities streaming away from electric charge (i.e., streamlines) is simply an invention to aid our conceptualisation of the presence of an electric field). This flux, denoted by Ψ, is in SI units related to the charge, Q, producing it via a dimensionless proportionality constant of unity; i.e., the electric flux in coulombs is given by Ψ = Q Thus, the flux is independent of the properties of the medium between the spheres and its magnitude depends only on the size of the charge producing it. The path of the flux lines is radially away from the inner sphere as shown in fig. 1.5 Fig. 1.5 Flux lines (Source: Biblio2/chapt03.pdf) An important entity in electromagnetics is the idea of electric flux density, at the surface of the inner sphere,. For example, in the above illustration, While at the outer surface, Clearly, is measured in coulombs/metre 2 (abbreviated C/m 2 ). If we think of the inner sphere as shrinking to a point charge, Q, then at a distance r from the charge, (49) when considering Q to be at the origin so that it may be seen that, for free space, (50) 9/JNU OLE

19 Electromagnetics and Radiation Equation (50) is one of the important so-called constitutive relations which are essential in solving electromagnetics problems and it is not restricted to point charges If the charge is distributed within a volume, such that the charge density leads to a volume integral (51) Here, as usual, R = r r is the distance from the differential volume, dv, under consideration to the point of observation, and is the unit vector ( r r )/ r r in that direction. If the region of interest is NOT effectively free space, then the permittivity, ε 0, must be replaced with the permittivity, ε, of the region. 1.6 Gauss s Law Thinking back to Faraday s experiment, it could be observed that the shape of the source charge inside the outer sphere would not be the critical factor in inducing a Q charge on the outer sphere. In fact, the inner charged body could take any shape and if it had a charge of +Q in total, this would induce a Q charge on the outer sphere. The total amount of flux in the dielectric at any distance that completely enclosed the inner charged object would thus be the same irrespective of the object s shape it could be a cubical charge or even a charge on an irregularly shaped object. Of course, the distribution of the flux lines (i.e., the shape of the field or equivalently the distribution of the flux density) in the dielectric would be affected, but not the total flux. The generalisation of Faraday s experiments led to the following formalisation known as Gauss s Law - The electric flux passing through any closed surface is equal to the total free charge enclosed by that surface. In general, the closed surface may take any form we wish to visualize which surface shape will be more convenient to consider for a particular application of Gauss s law will usually depend on the shape of the charge distribution Fig. 1.6 The vector differential area, (Source: Biblio2/chapt03.pdf) Since the differential flux, dψ, crossing the differential area must be the product of the normal component of the differential surface : and Therefore, Gauss s law (52) Since Ψ = Q, measured in coulombs, where Q is the charge enclosed by S. Thus, too, 10/JNU OLE

20 This means, in the statement of Gauss s law, the surface over which the flux is integrated surrounds the charge the charge is enclosed by the surface. This closed surface, used in the context of Gauss s law, is often referred to as a Gaussian surface. Of course, if the charge is due to a linear charge density, ρ L, or a surface charge density, ρ S (surface on which the charge exists is not necessarily closed), the volume integral in equation (53) must be replaced by a line integral or a surface integral as follows: (53) For line charge, where ρ L is the linear charge density in C/m. For surface charge, where ρ s is the surface charge density in C/m 2. As intimated above, facilitating application of Gauss s law is dependent on a suitable choice of the closed surface for integration. In fact, if the charge distribution is known, equation (53) can be used to obtain in an easy manner if it possible to choose a closed surface, S, which satisfies the following two properties: is everywhere either normal or tangential to S so that, respectively. On that portion of S where, the magnitude, is constant. Before considering a few important examples, we note that the flux, Ψ, passing through a non-closed surface is given simply as, 1.7 Gauss s Law and the Divergence Theorem Each concept is discussed below in detail Divergence In the analysis of the vector and scalar fields commonly found in the study of electromagnetics, there are several operations which involve the so-called Del operator. In Cartesian coordinates, this operator takes the form We first consider the dot product, Definition: The divergence of a vector function (i.e., in Cartesian coordinates) is defined as Therefore, (54) Notice that the divergence of a vector is a scalar! The divergence will not have such a simple form in cylindrical and spherical coordinates. Consider the example below 11/JNU OLE

21 Electromagnetics and Radiation The divergence of (sometimes written, div ), is Interpretation of the divergence For the sake of simplicity, consider a vector function,, that has only an component. From (54) With reference to the figure below and noting that the rectangular block is of differential volume, the definition of partial differentiation gives or We next observe that, (55) Fig. 1.7 Example of interpretation of the divergence (Source: In (55), the first term in the numerator is the flow out while the second is the flow into the block through ds x. Therefore, (55) may be written as, (56) While recalling that points away from the block at both the right and left faces. Making identical arguments for and, (56) may be generalised to include all surfaces by writing Where S is now the whole surface of the block. Thus, (57) (58) 12/JNU OLE

22 and the divergence of a vector field represents the net outflowof the vector per unit volume as the volume shrinks to zero. For example, could be the electric flux density,, so that would be the net flux per unit volume leaving a particular region as the volume shrinks to zero. Hence the term divergence; if there is net outflow (i.e., a source region) and if there is a net inflow (i.e., a sink region) Divergence and Maxwell s First Equation in Point Form In equation (58), we now replace the general vector field.this gives with the specific vector field, the electric flux density, Since we immediately recognize from Gauss s law that (59) Thus, equation (59) may be written as Maxwell s first equation (60) This is the first of four equations referred to as Maxwell s equations as they apply to electrostatics and magnetostatics. It is simply Gauss s law rewritten on a per-unit-volume basis. Here it is said to be in point form as it indicates that the flux per unit volume leaving (i.e., diverging from) a vanishingly small volume is equal to the charge per unit volume contained therein. If you simply consider the units on each side of the equation you will see this is a reasonable interpretation The Divergence Theorem Note that equation (53) may be written for any volume in which charge is enclosed and this volume may indeed be taken to have S as the enclosing surface. Thus, we may write where S surrounds v. Then, it can be noticed that equation (60) says, From these two equations, on replacing ρv in the first with the divergence of found in the second, we have Divergence theorem (61) This theorem is true for any vector field (not just ). Its usefulness can be immediately seen when we consider that it is often easier to carry out a surface integral than a volume integral. What does this integral say physically? It says that the flux exiting through a closed surface equals the sum total of the divergence of the flux density throughout the volume which the surface encloses. If we think of the total volume as consisting of small constituent volumes, then as the flux diverges from one of the constituents it converges on another and so on until it reaches a constituent which contains a portion of the outer surface where at least part of the flux may escape i.e., diverge from the volume altogether. It is this escaping flux that corresponds to the net divergence throughout the volume. 13/JNU OLE

23 Electromagnetics and Radiation Summary There are two types of observed electric charge, which are designated as positive and negative, this convention was derived from Benjamin Franklin s experiments. An electron carries one unit of negative charge, -e, while a proton carries one unit of positive charge, +e. Coulomb s law states that the force F between two point charges q and q is: 1 2 along the line joining them directly proportional to the product q to q of the charges 1 2 inversely proportional to the square of the distance r between them Gauss s Law states that The electric flux passing through any closed surface is equal to the total free charge enclosed by that surface. The divergence of a vector function (i.e., in Cartesian coordinates) is defined as, References Coulomb s Law. Available at: < guide02.pdf)> [Accessed 2 February 2011]. Electrostatics. Available at: < Accessed on 2 nd February, Unit 3 - Electric Flux Density, Gauss s Law and Divergence. Available at: < index_files/5812_w10/5812_unit3_2011.pdf> [Accessed 2 February 2011]. Recommended Reading Balanis, C.A, Advanced Engineering Electromagnetics. Wiley, Solution Manual edition. Cheng, D.K., Fundamentals of Engineering Electromagnetics. Prentice Hall, 1st ed. Ida, N., Engineering Electromagnetics. Springer, 2nd ed. 14/JNU OLE

24 Self Assessment The unit of charge is called the. a. Coulomb b. c. d. Faraday Joule Henry Coulomb s law applies to any pair of charges. a. line b. c. d. point vector static The field intensity is defined as the vector force on a unit positive test charge. a. magnetic b. c. d. electronic static electric The electric passing through any closed surface is equal to the total free charge enclosed by that surface. a. curl b. c. d. density volume flux found that the magnitude of the charge induced on the outer sphere was equal to that of the charge on the inner sphere, irrespective of the dielectric used. a. Faraday b. c. d. Coulomb Henry Joule Which of the following is true? a. Divergence theorem is true for any vector field. b. c. d. Gauss s theorem is true for any vector field. Divergence theorem is not true for any vector field. Gauss s theorem is not true for any field. Which of the following is true? a. Faraday s law applies to any pair of point charges. b. c. d. Coulomb s law applies to any pair of line charges. Coulomb s law applies to any pair of point charges. Coulomb s law do not applies to any pair of point charges. 15/JNU OLE

25 Electromagnetics and Radiation 8. Which of the following is true? a. A proton carries one unit of negative charge. b. A neutron carries one unit of negative charge. c. An atom carries one unit of negative charge. d. An electron carries one unit of negative charge. 9. The divergence of a vector function (i.e., in Cartesian coordinates) is given by which of the following? 10. The force exerted by q on q is given in which of the following Coulomb s law? 1 2 a. = 12 b. = 12 c. = 12 d. = 12 16/JNU OLE

26 Chapter II Energy and Potential Aim The aim of this chapter is to: describe energy expanded in moving a point charge in an electric field explain the definition of potential difference and potential describe potential gradient Objectives The objectives of this chapter are to: examine energy expanded in moving a point charge in an electric field elucidate line integral explain potential, potential difference and potential gradient Learning outcome At the end of this chapter, the students will be able to: comprehend line integral recall definition of potential difference and potential understand about potential field of a point charge 17/JNU OLE

27 Electromagnetics and Radiation 2.1 Introduction to Energy and Potential In the previous chapter we became acquainted with Coulomb s law and its use in finding the electric field about several simple distributions of charge, and also with Gauss s law and its application in determining the field about some symmetrical charge arrangements. The use of Gauss s law was invariably easier for these highly symmetrical distributions, because the problem of integration always disappeared when the proper closed surface was chosen. However, if we had attempted to find a slightly more complicated field, such as that of two unlike point charges separated by a small distance, we would have found it impossible to choose a suitable Gaussian surface and obtain an answer. Coulomb s law, however, is more powerful and enables us to solve problems for which Gauss s law is not applicable. The application of Coulomb s law is lengthy, detailed, and often quite complex, the reason for this being precisely the fact that the electric field intensity, a vector field, must be found directly from the charge distribution. Three different integrations are needed in general, one for each component and the resolution of the vector into components usually adds to the complexity of the integrals. Indeed it would be desirable if we could find some as yet undefined scalar function with a single integration and then determine the electric field from this scalar by some simple straightforward procedure, such as differentiation. This scalar function does exist and is known as the potential or potential field. 2.2 Energy Expended in Moving a Point Charge in an Electric Field The electric field intensity is defined as the force on a unit test charge at that point at which we wish to find the value of this vector field. If we attempt to move the test charge against the electric field, we have to exert a force equal and opposite to that exerted by the field, and this requires us to expend energy, or do work. If we wish to move the charge in the direction of the field, our energy expenditure turns out to be negative; the field does this work. Suppose we want to move a charge Q a distance dl in an electric field E. The force on Q due to the electric field is, Where the subscript reminds us that this force is due to the field. The component of this force in the direction dl must overcome as, (1) Where a L = a unit vector in the direction of dl. The force which must be applied is equal and opposite to the force due to the field, and the expenditure of energy is the product of the force and distance. That is, differential work done by external source moving Q or (2) where we have replaced a L dl by the simpler expression dl. 18/JNU OLE

28 This differential amount of work required may be zero under several conditions determined easily from equation(2). There are the trivial conditions for which E, Q, or dl is zero, and a much more important case in which E and dl are perpendicular. Here the charge is moved always in a direction at right angles to the electric field. Here, a good analogy can be drawn between the electric field and the gravitational field, where, again, energy must be expended to move against the field. Sliding a mass around with constant velocity on a frictionless surface is an effortless process if the mass is moved along a constant elevation contour; positive or negative work must be done in moving it to a higher or lower elevation, respectively. Returning to the charge in the electric field, the work required to move the charge a finite distance must be determined by integrating, Where the path must be specified before the integral can be evaluated. The charge is assumed to be at rest at both its initial and final positions. This definite integral is basic to field theory, and we shall devote the following section to its interpretation and evaluation. 2.3 The Line Integral The integral expression for the work done in moving a point charge Q from one position to another, equation (3), is an example of a line integral, which in vector analysis notation always takes the form of the integral along some prescribed path of the dot product of a vector field and a differential vector path length dl. Without using vector analysis we should have to write (3) Where E L = component of E along dl. A line integral is like many other integrals which appear in advanced analysis, including the surface integral appearing in Gauss s law, in that it is essentially descriptive. It tells us to choose a path, break it up into a large number of very small segments, multiply the component of the field along each segment by the length of the segment, and then add the results for all the segments. This is a summation, of course, and the integral is obtained exactly only when the number of segments becomes infinite. This procedure is indicated in fig. 2.1, where a path has been chosen from an initial position B to a final position A and a uniform electric field selected for simplicity. The path is divided into six segments, L 1, L 2,..., L 6 and the components of E along each segment denoted by E L1, E L2,..., E L6. The work involved in moving a charge Q from B to A is then approximately or, using vector notation, and since we have assumed a uniform field, 19/JNU OLE

29 Electromagnetics and Radiation What is this sum of vector segments in the parentheses above? Vectors add by the parallelogram law, and the sum is just the vector directed from the initial point B to the final point A; L BA. Therefore, (uniform E) (4) Fig. 2.1 A graphical interpretation of line integral in a uniform field (Source: Biblio2/chapt04.pdf) Remembering the summation interpretation of the line integral, this result for the uniform field can be obtained rapidly now from the integral expression as applied to a uniform field. (5) where the last integral becomes L BA and (6) (uniform E) (7) For this special case of a uniform electric field intensity, note that the work involved in moving the charge depends only on Q, E, and L BA, a vector drawn from the initial to the final point of the path chosen. It does not depend on the particular path we have selected along which to carry the charge. 2.4 Definition of Potential Difference and Potential We are now ready to define a new concept from the expression for the work done by an external source in moving a charge Q from one point to another in an electric field E, (8) 20/JNU OLE

30 In much the same way as we defined the electric field intensity as the force on a unit test charge, we now define potential difference V as the work done (by an external source) in moving a unit positive charge from one point to another in an electric field, We shall have to agree on the direction of movement, as implied by our language, and we do this by stating that V AB signifies the potential difference between points A and B and is the work done in moving the unit charge from B(last named) to A (first named). Thus, in determining V AB, B is the initial point and A is the final point. The reason for this somewhat peculiar definition will become clearer shortly, when it is seen that the initial point B is often taken at infinity, whereas the final point A represents the fixed position of the charge; point A is thus inherently more important. Potential difference is measured in joules per coulomb, for which the volt is defined as a more common unit, abbreviated as V. Hence the potential difference between points A and B is (9) (10) and V AB is positive if work is done in carrying the positive charge from B to A. Fig. 2.2 (a) Circular path (b) A radial path along which a charge of Q is carried in the field of an infinite line charge (Source: Biblio2/chapt04.pdf) In fig. 2.2, work done in taking charge Q from ρ=b to ρ=a is given by Thus, the potential difference between points at ρ=a and ρ=b is We can try out this definition by finding the potential difference between points A and B at radial distances r A and r B from a point charge Q. Taking origin as Q (11) and We have (12) 21/JNU OLE

31 Electromagnetics and Radiation If r B > r A, the potential difference V AB is positive, indicating that energy is expended by the external source in bringing the positive charge from r B to r A.This agrees with the physical picture showing the two like charges repelling each other. It is often convenient to speak of the potential, or absolute potential, of a point, rather than the potential difference between two points, but this means only that we agree to measure every potential difference with respect to a specified reference point which we consider to have zero potential. Common agreement must be reached on the zero reference before a statement of the potential has any significance. A person having one hand on the deflection plates of a cathode-ray tube which are at a potential of 50 V and the other hand on the cathode terminal would probably be too shaken up to understand that the cathode is not the zero reference, but that all potentials in that circuit are customarily measured with respect to the metallic shield about the tube. The cathode may be several thousands of volts negative with respect to the shield. Perhaps the most universal zero reference point in experimental or physical potential measurements is "ground, by which we mean the potential of the surface region of the earth itself. Theoretically, we usually represent this surface by an infinite plane at zero potential, although some large-scale problems, such as those involving propagation across the Atlantic Ocean, require a spherical surface at zero potential. Another widely used reference "point is infinity. This usually appears in theoretical problems approximating a physical situation in which the earth is relatively far removed from the region in which we are interested, such as the static field near the wing tip of an airplane that has acquired a charge in flying through a thunderhead, or the field inside an atom. Working with the gravitational potential field on earth, the zero reference is normally taken at sea level; for an interplanetary mission, however, the zero reference is more conveniently selected at infinity. A cylindrical surface of some definite radius may occasionally be used as a zero reference when cylindrical symmetry is present and infinity proves inconvenient. In a coaxial cable the outer conductor is selected as the zero reference for potential. And, of course, there are numerous special problems, such as those for which a two-sheeted hyperboloid or an oblate spheroid must be selected as the zero-potential reference, but these need not concern us immediately. If the potential at point A is V A and that at B is V B, then (13) where we necessarily agree that V A and V B shall have the point. 2.5 The Potential Field of a Point Charge In the previous section we found an expression (12) for the potential difference between two points located at r=r A and r = r B in the field of a point charge Q placed at the origin, (14) It was assumed that the two points lay on the same radial line or had the same θ and φ coordinate values, allowing us to set up a simple path on this radial line along which to carry our positive charge. We now should ask whether different θ and φ coordinate values for the initial and final position will affect our answer and whether we could choose more complicated paths between the two points without changing the results. Let us answer both questions at once by choosing two general points A and B (fig. 1.3) at radial distances of r A and r B, and any values for the other coordinates. The differential path length dl has r, θ, and φ components, and the electric field has only a radial component. Taking the dot product then leaves us only 22/JNU OLE

32 We obtain the same answer and, therefore the potential difference between two points in the field of a point charge depends only on the distance of each point from the charge and does not depend on the particular path used to carry our unit charge from one point to the other. Fig. 2.3 A general path between general points B and A in the field of a point charge Q at the origin (Source: Biblio2/chapt04.pdf) The simplest possibility is to let V = 0 at infinity. If we let the point at r = r B recede to infinity the potential at r A becomes or, since there is no reason to identify this point with the A subscript, (15) This expression defines the potential at any point distant r from a point charge Q at the origin, the potential at infinite radius being taken as the zero reference. Returning to a physical interpretation, we may say that Q/4πε 0 r joules of work must be done in carrying a 1-C charge from infinity to any point r meters from the charge Q. A convenient method to express the potential without selecting a specific zero reference entails identifying r A as r once again and letting Q/4πε 0 r B be a constant. Then, (16) C 1 may be selected so that V = 0 at any desired value of r. We could also select the zero reference indirectly by electing to let V be V 0 at r = r 0. It should be noted that the potential difference between two points is not a function of C 1 Equation (15) or (16) represents the potential field of a point charge. The potential is a scalar field and does not involve any unit vectors. 23/JNU OLE

33 Electromagnetics and Radiation 2.6 The Potential Field of a System of Charges: Conservative Property The potential at a point has been defined as the work done in bringing a unit positive charge from the zero reference to the point, and we have suspected that this work, and hence the potential, is independent of the path taken. To prove the assertion, begin with the potential field of the single point charge showed, in the last section, the independence with regard to the path, noting that the field is linear with respect to charge so that superposition is applicable. It will then follow that the potential of a system of charges has a value at any point which is independent of the path taken in carrying the test charge to that point. Thus, the potential field of a single point charge, which we shall identify as Q 1 and locate at r 1, involves only the distance r r1 j from Q 1 to the point at r where we are establishing the value of the potential. For a zero reference at infinity, we have The potential due to two charges, Q 1 at r 1 and Q 2 at r 2, is a function only of r r1 and r r 2, the distances from Q 1 and Q 2 to the field point, respectively. Continuing to add charges, we find that the potential due to n point charges is or (17) If each point charge is now represented as a small element of a continuous volume charge distribution ρv v, then As we allow the number of elements to become infinite, we obtain the integral expression We have come quite a distance from the potential field of the single point charge, and it might be helpful to examine (18).The potential V(r) is determined with respect to a zero reference potential at infinity and is an exact measure of the work done in bringing a unit charge from infinity to the field point at r where we are finding the potential. The volume charge density ρ y (r ) and differential volume element dv combine to represent a differential amount of charge ρ v (r )dv located at r. The distance r r is that distance from the source point to the field point. The integral is a multiple (volume) integral. (18) 24/JNU OLE

34 If the charge distribution takes the form of a line charge or a surface charge, the integration is along the line or over the surface: (19) (20) The most general expression for potential is obtained by combining (17), (18), (19), and (20). These integral expressions for potential in terms of the charge distribution should be compared with similar expressions for the electric field intensity, such as (18) The potential again is inverse distance, and the electric field intensity, inverse-square law. 2.7 Potential Gradient We now have two methods of determining potential, one directly from the electric field intensity by means of a line integral, and another from the basic charge distribution itself by a volume integral. Neither method is very helpful in determining the fields in most practical problems, however, for as we shall see later, neither the electric field intensity nor the charge distribution is very often known. Preliminary information is much more apt to consist of a description of two equipotential surfaces, such as the statement that we have two parallel conductors of circular cross section at potentials of 100 and À100 V. Perhaps we wish to find the capacitance between the conductors, or the charge and current distribution on the conductors from which losses may be calculated. These quantities may be easily obtained from the potential field, and our immediate goal will be a simple method of finding the electric field intensity from the potential. We already have the general line-integral relationship between these quantities, (21) But this is much easier to use in the reverse direction: given E, find V. However, equation (21) may be applied to a very short element of length L along which E is essentially constant, leading to an incremental potential difference V, (22) Let us see first if we can determine any new information about the relation of V to E from this equation. Consider a general region of space, as shown in fig. 4.6, in which E and V both change as we move from point to point. Equation (22) tells us to choose an incremental vector element of length L = L a L and multiply its magnitude by the component of E in the direction of a L (one interpretation of the dot product) to obtain the small potential difference between the final and initial points of VL. If we designate the angle between L and E as θ, then 25/JNU OLE

35 Electromagnetics and Radiation We now wish to pass to the limit and consider the derivative dv/dl. To do this, we need to show that V may be interpreted as a function V(x, y, z). So far, V is merely the result of the line integral equation (21). If we assume a specified starting point or zero reference and then let our end point be (x, y, z) we know that the result of the integration is a unique function of the end point (x, y, z) because E is a conservative field. Therefore V is a singlevalued function V(x, y, z). We may then pass to the limit and obtain In which direction should ÁL be placed to obtain a maximum value of V? Remember that E is a definite value at the point at which we are working and is independent of the direction of L. The magnitude L is also constant, and our variable is a L, the unit vector showing the direction of L. It is obvious that the maximum positive increment of potential, V max, will occur when cosθ is 1, or L points in the direction opposite to E. For this condition, This little exercise shows us two characteristics of the relationship between E and V at any point: The magnitude of the electric field intensity is given by the maximum value of the rate of change of potential with distance. This maximum value is obtained when the direction of the distance increment is opposite to E or, in other words, the direction of E is opposite to the direction in which the potential is increasing the most rapidly. Let us now illustrate these relationships in terms of potential. Fig. 1.4 is intended to show the information about some potential field. Fig. 2.4 A potential field is shown by its equipotential surfaces (Source: Biblio2/chapt04.pdf) It is done by showing the equipotential surfaces (shown as lines in the two-dimensional sketch). We desire information about the electric field intensity at point P. Starting at P; we lay off a small incremental distance L in various directions, hunting for that direction in which the potential is changing (increasing) the most rapidly. From the sketch, this direction appears to be left and slightly upward. From the second characteristic above, the electric field intensity is therefore oppositely directed, or to the right and slightly downward at P. Its magnitude is given by dividing the small increase in potential by the small element of length. It seems likely that the direction in which the potential is increasing most rapidly, is perpendicular to the equipotentials (in the direction of increasing potential), and this is correct, for if L is directed along an equipotential, V = 0 by our definition of an equipotential surface. But then 26/JNU OLE

36 and since neither E nor L is zero, E must be perpendicular to this ÁL or perpendicular to the equipotential. Since the potential field information is more likely to be determined first, let us describe the direction of L which leads to a maximum increase in potential mathematically in terms of the potential field rather than the electric field intensity. We do this by letting a N be a unit vector normal to the equipotential surface and directed toward the higher potentials. The electric field intensity is then expressed in terms of the potential, this shows that the magnitude of E is given by the maximum space rate of change of V and the direction of E is normal to the equipotential surface (in the direction of decreasing potential) Since dv/dl max occurs when L is in the direction of a N, we may remind ourselves of this fact by letting (23) and (24) Equation (23) or (24) serve to provide a physical interpretation of the process of finding the electric field intensity from the potential. Both are descriptive of a general procedure, and we do not intend to use them directly to obtain quantitative information. This procedure leading from V to E is not unique to this pair of quantities, however, but has appeared as the relationship between a scalar and a vector field in hydraulics, thermodynamics, and magnetics, and indeed in almost every field to which vector analysis has been applied. The operation on V by which E is obtained is known as the gradient, and the gradient of a scalar field T is defined as where a N is a unit vector normal to the equipotential surfaces, and that normal is chosen which points in the direction of increasing values of T. (25) Using this new term, we now may write the relationship between V and E as (26) 27/JNU OLE

37 Electromagnetics and Radiation Summary Coulomb s law is more powerful and enables us to solve problems for which Gauss s law is not applicable. The application of Coulomb s law is lengthy, detailed, and often quite complex, the reason for this being precisely the fact that the electric field intensity, a vector field, must be found directly from the charge distribution. The electric field intensity is defined as the force on a unit test charge at that point at which we wish to find the value of this vector field. A line integral is like many other integrals which appear in advanced analysis, including the surface integral appearing in Gauss s law, in that it is essentially descriptive. The potential at a point has been defined as the work done in bringing a unit positive charge from the zero reference to the point, and we have suspected that this work, and hence the potential, is independent of the path taken. We have two methods of determining potential, one directly from the electric field intensity by means of a line integral, and another from the basic charge distribution itself by a volume integral. References Energy and Potential. Available at: < Comunicacion_II/Material/Biblio2/chapt04.pdf> [Accessed 13 February, 2011]. Potential Energy. Available at: < [Accessed 9 February 2011]. Recommended Reading Ryaben kii, V.S., Kulman, N.K., (Translator), Method of Difference Potentials and its Applications. Springer, 1st ed. Rao, N.N., Elements of Engineering Electromagnetics. Prentice Hall, 6th ed. Ulaby, F.T., Fundamentals of Applied Electromagnetics. Prentice Hall, 5th ed. 28/JNU OLE

38 Self Assessment 1. Coulomb s law, is more powerful and enables us to solve problems for which is not applicable. a. Divergence law b. Faraday s law c. Stoke s theorem d. Gauss s law 2. The magnitude of the electric field intensity is given by the value of the rate of change of potential with distance. a. minimum b. c. d. maximum fixed variable 3. The electric field was defined as the force on a unit test charge at that point at which we wish to find the value of this vector field. a. density b. c. d. magnitude intensity area 4. Potential difference is given by which of the following equations? a. b. c. d. 5. The gradient of a scalar field T is given by which equation? a. b. c. d. 29/JNU OLE

39 Electromagnetics and Radiation Which of the following are the two methods to determine potential? a. Directly from the electric field intensity by means of a point integral and another from the basic charge distribution itself by a point integral. b. Directly from the electric field intensity by means of a line integral and another from the basic charge distribution itself by a volume integral. c. One directly from the electric field intensity by means of a volume integral and another from the basic charge distribution itself by a line integral. d. One directly from the electric field intensity by means of a line integral and another from the basic charge distribution itself by a line integral. Which of the following is true? a. The potential at a point has been defined as the displacement in bringing a unit positive charge from the zero reference to the point. b. The potential at a point has been defined as the work done in bringing a unit negative charge from the zero reference to the point. c. The potential at a point has been defined as the work done in bringing a unit positive charge from the zero reference to the point. d. The potential at a point has been defined as the work done in bringing a unit positive charge from the final point to the initial point. 8. Which of the following is true? a. Potential difference is measured in ohms per Coulomb. b. Potential difference is measured in joules per Henry. c. Potential difference is measured in volts per Coulomb. d. Potential difference is measured in joules per Coulomb. 9. Which of the following is true? a. The application of Coulomb s law is lengthy, detailed and often quite complex. b. The application of Faraday s law is lengthy, detailed and often quite complex. c. The application of Gauss s law is lengthy, detailed and often quite complex. d. The application of Coulomb s law is not lengthy, detailed and often quite easy. 10. Which of the following is true? a. The potential field intensity is defined as the force on a unit test charge at that point at which we wish to find the value of this vector field. b. The electric field intensity is defined as the force on a unit test charge at that point at which we wish to find the value of this vector field. c. The electric field intensity is defined as the energy on a unit test charge at that point at which we wish to find the value of this vector field. d. The electric field intensity is defined as the force on a unit test charge at that point at which we wish to find the value of this variable field. 30/JNU OLE

40 Chapter III Poisson s and Laplace s Equations Aim The aim of this chapter is to: explain Laplace s equation illustrate Poisson s equation state uniqueness theorem Objectives The objectives of this chapter are to: derive Poisson s and Laplace s equations examine uniqueness theorem give examples on Poisson s and Laplace s equations Learning outcome At the end of this chapter, the students will be able to: understand Poisson s and Laplace s equations determine uniqueness theorem identify applications of these equations 31/JNU OLE

41 Electromagnetics and Radiation 3.1 Introduction In earlier chapters, the and in the given region are obtained using Coulomb s law and Gauss s law. Using these laws is easy, if the charge distribution or potential throughout the region is known. Practically it is not possible in many situations, to know the charge distribution or potential variation throughout the region. Practically, charge and potential may be known at same boundaries of the region, only. From those values it is necessary to obtain potential and throughout the region. Such electrostatic problems are called boundary value problems. To solve such problems, Poisson s and Laplace s equations must be known. This chapter derives the Poisson s and Laplace s equations and explains its use in few practical situations. 3.2 Poisson s and Laplace s Equations From the Gauss s law in the point form, Poisson s equation can be derived. Consider the Gauss s law in the point form is given as, (1) Where, = Flux density = Volume charge density It is known that for a homogeneous, isotropic and linear medium, flux density and electric field intensity are directly proportional. Thus Therefore, From the gradient relationship Substituting (4) in (3), Taking e outside as constant, (2) (3) (4) (5) (6) Now Therefore, 2 operation is called del squared operation and denoted as. (7) This equation (7) is called Poisson s equation If in a certain region, volume charge density is zero ( r v = 0 ), which is true for dielectric medium then the Poisson s equation takes the form, 32/JNU OLE

42 (For a charge free region) (8) 2 This is special case of Poisson s equation and is called Laplace s equation. The operation is called the Laplacian of V The equation (7) is for homogeneous medium for which e is constant. But if e is not constant and the medium is inhomogeneous, the equation (5) must be used as Poisson s equation for inhomogeneous medium Operation in Different Co-ordinate Systems The potential V can be expressed in any of the three co-ordinate systems as V(x, y, z), V(r,θ,φ) or V(r,φ, z). Depending 2 upon it, the operation required for Laplace s equation must be used. In Cartesian co-ordinate systems, The equation (9) is Laplace s equation in Cartesian form In cylindrical co-ordinate system (9) The equation (10) is Laplace s equation in cylindrical form. In spherical co-ordinate system, (10) The equation (11) is Laplace s equation in spherical form. 3.3 Uniqueness Theorem The boundary value problems can be solved by number of methods such as analytical, graphical, experimental etc. Thus there is a question that is the solution of Laplace s equation solved by any method, unique? The answer to this question is the uniqueness theorem, which is proved by contradiction method. Assume that the Laplace s equation has two solutions say V 1 and V 2, both are function of the co-ordinates of the system used. These solutions must satisfy Laplace s equation. So we can write, 2 V 1 =0 and 2 V 2 =0 (1) (11) 33/JNU OLE

43 Electromagnetics and Radiation Both the solutions must satisfy the boundary conditions as well. At the boundary, the potentials at the different points are same due to equipotential surface then, V 1 =V 2 (2) Let the difference between the two solutions is V d. V d =V 2 V 1 (3) Using Laplace s equation for the difference V d, Therefore, (4) (5) On the boundary V d =0 from the equations (2) and (3) Now the divergence theorem states that, Adv = Ad S (6) vol s Let A =V d V d and from vector algebra, Now use this for (V d V d ) with a = Vd and V d =. But Hence, = 2 (7) Using (4) (8) To use this in (6), let hence V V = V V ds (9) vol d d d d S But V d =0 on the boundary, hence right hand side of equation (9) is zero 34/JNU OLE

44 Vd Vddv = 0 (10) vol This is volume integral to be evaluated on the volume enclosed by the boundary. It is known that,, 2 Vd dv = 0 as V d is vector (11) vol Now integration can be zero under two conditions, The quantity under integral sign is zero The quantity is positive in some regions and negative in other regions by equal amount and hence zero But square term cannot be negative in any region hence; quantity under integral must be zero. i.e., (12) As the gradient of V d =V 2 -V 1 is zero means V 2 -V 1 is constant and not changing with any co-ordinates. But considering boundary it can be proved that V 2 -V 1 =constant=zero. V 2 =V 1 (13) This proves that both the solutions are equal and cannot be different. Thus, Uniqueness Theorem can be stated as If the solution of Laplace s equation satisfies the boundary condition then this solution is unique, by whatever method it is obtained. The solution of Laplace s equation gives the field which is unique, satisfying the same boundary conditions, in a given region. 3.4 Procedure for Solving Laplace s Equation The procedure to solve a problem involving Laplace s equation can be described as follows, Step 1: Solve the Laplace s equation using the method of integration. Assume constants of integration as per the requirement. Step 2: Determine the constants applying the boundary conditions given or known for the region. The solution obtained in step 1 with constants obtained using boundary conditions is a unique solution. Step 3: Then can be obtained for the potential field V obtained, using gradient operation V. Step 4: For homogeneous medium, can be obtained as. Step 5: At the surface, ρs=d N hence once is known, the normal component D N to the surface is known. Hence the charge induced on the conductor surface can be obtained as. Step 6: One the charge induced Q is known and potential V is known then the capacitance C of the system can be obtained. Let us have a look on few examples Example 3.1: Verify that the potential field given below satisfies the Laplace s equation. Solution: Given field is in Cartesian system, 35/JNU OLE

45 Electromagnetics and Radiation = = 4 6+2=0 As =0, the field satisfies the Laplace s equation. Example 3.2: The region between two concentric right cylinders contains a uniform charge densityρ. Solve the Poisson s equation for the potential in the region. Solution: The cylinders are shown in the fig.3.1 Select the cylindrical co-ordinate system. In co-axial cable like structure, the electric field intensity direction from inner to outer cylinder. is in radial Fig. 3.1 Concentric right cylinders Hence and V both are function of only r and not of φ and z. Therefore, is existing while and are zero. According to Poisson s equation,, here ρv=ρ given Integrating both sides Where C 1 = Constant of integration 36/JNU OLE

46 Integrating both sides, Where C 2 =Constant of integration Knowing the boundary conditions, C 1 and C 2 can be obtained. 37/JNU OLE

47 Electromagnetics and Radiation Summary Gauss s law in the point form is given as. For a homogeneous, isotropic and linear medium, flux density and electric field intensity are directly proportional. Poisson s equation is given by. Laplace s equation in Cartesian form is given by. Laplace s equation in cylindrical form is given by. Laplace s equation in spherical form is given by Uniqueness Theorem can be stated as, If the solution of Laplace s equation satisfies the boundary condition then this solution is unique, by whatever method it is obtained. References Laplace s and Poisson s Equations. Available at: < [Accessed 14 February 2010]. Bakshi, U.A., Bakshi, A.V., Electromagnetic Field Theory, Technical Publications. Recommended Reading Kraus, J.D., Fleisch, D., Electromagnetics, McGraw Hill Higher Education, 5th ed. Edminister, J., Schaum s Outline of Electromagnetics, McGraw-Hill. 3rd ed. Ida, N., Engineering Electromagnetics. Springer, 2nd ed.. 38/JNU OLE

48 Self Assessment If the solution of Laplace s equation satisfies the boundary condition then this solution is, by whatever method it is obtained. a. unique b. c. d. one similar different For a homogeneous, isotropic and linear medium, flux density and electric field intensity are proportional. a. directly b. c. d. indirectly inversely laterally Gauss s law in the point form is given by which of the following? a. b. c. d. 4. Poisson s equation is given by which of the following? 5. Laplace s equation in Cartesian form is given by which of the following? a. b. c. d. 39/JNU OLE

49 Electromagnetics and Radiation 6. Laplace s equation in cylindrical form is given by which of the following? a. b. c. d Coulomb s law and Gauss s law can be used easily, if the charge distribution or potential throughout the region are. a. similar b. c. d. different known unknown State which of the following statements is true. 2 a. The operation is called the Del operator. b. The operation is called the Laplacian of V. 2 c. The operation is called the operator of V. 2 d. The operation is called the Laplacian of V Which of the following is true? a. If the solution of Laplace s equation satisfies the boundary condition then this solution is unique, by whatever method it is obtained. b. If the solution of Laplace s equation does not satisfy the boundary condition then this solution is unique, by whatever method it is obtained. c. If the solution of Laplace s equation satisfies the boundary condition then this solution is fixed, by whatever method it is obtained. d. If the solution of Poisson s equation satisfies the boundary condition then this solution is unique, by whatever method it is obtained. Which of the following is true? a. For a homogeneous, isotropic and linear medium, flux density and electric field intensity are indirectly proportional. b. For a homogeneous, isotropic and linear medium, flux density and electric field intensity are directly proportional. c. For a homogeneous, isotropic and linear medium, flux density and electric field intensity are inversely proportional. d. For a homogeneous, isotropic and linear medium, flux density and electric field intensity are not proportional. 40/JNU OLE

50 Chapter IV Magnetic Forces, Materials and Inductance Aim The aim of this chapter is to: analyse magnetic force on a moving charge explore the concept of magnetisation and permeability explain the role of potential energy and forces on magnetic materials Objectives The objectives of this chapter are to: examine force between differential current element evaluate force and torque on a closed circuit give an overview of inductance and mutual inductance Learning outcome At the end of this chapter students will be able to: recall magnetisation and permeability comprehend the concept of force and torque on a closed circuit understand inductance and mutual inductance 41/JNU OLE

51 Electromagnetics and Radiation 4.1 Introduction to Magnetic Flux In this chapter, we shall study the magnetic forces. We will discuss the concepts of magnetic torque, concepts of magnetisation along with permeability. Similar to the boundary conditions in electrostatic fields, we shall study the boundary conditions for the magneto static fields. We shall also discuss different magnetic materials. 4.2 Magnetic Flux and Magnetic Flux Density Magnetic flux is the amount of magnetic field (or number of lines of force) produced by magnetic source. The symbol for magnetic flux is Φ(Greek letter phi ). The unit of magnetic flux is Weber, Wb. Magnetic flux density is the amount of flux passing through a defined area that is perpendicular to the direction of the flux. The symbol for magnetic flux density is B. The unit of magnetic flux density is the tesla, T. Where 1T=1Wb/m 2 Hence, Where A(m 2 ) is the area. Example A magnetic pole face has a rectangular section having dimension 200mm by 100mm. If the total flux emerging from the pole is 150µwb. Calculate the flux density. Solution: Flux Φ=150µwb= Wb Cross-sectional area A= = 20,000mm 2 = m 2 Flux density= =0.0075T or 7.5mT 4.3 Magnetic Force on a Moving Charge Electric charges moving in a magnetic field experience a force due to the magnetic field. Given a charge Q moving with velocity u in a magnetic flux density B, the vector magnetic force F m on the charge is given by Note that the force is normal to the plane containing the velocity vector and the magnetic flux density vector. Also note that the force is zero if the charge is stationary (u=0). Example Determine the vector magnetic force on a point charge +Q moving at a uniform velocity u=u o a y in a uniform magnetic flux density B=B o a z. Solution: Given a charge moving in an electric field and a magnetic field, the total force on the charge is the superposition of the forces due to the electric field and the magnetic field. This total force equation is known as the Lorentz force equation. The vector force component due to the electric 42/JNU OLE

52 z B F m + u y x Fig. 4.1 Direction of the charge in an electric and magnetic field field (F e ) is given by The total vector force (Lorentz force- F) is The Lorentz force can also be written in terms of Newton s law such that where, m is the mass of the charged particle. 4.4 Magnetic Force on Current Given that charge moving in a magnetic field experiences a force, a current carrying conductor in a magnetic field also experiences a force. The current carrying conductor (line, surface or volume current) can be subdivided into current elements (differential lengths, differential surfaces or differential volumes). The charge-velocity product for a moving point charge can be related to an equivalent differential length of line current. Fig. 4.2 Magnetic force on current (Source: The equivalence of the moving point charge and the differential length of line current yield the equivalent magnetic force equation 43/JNU OLE

53 Electromagnetics and Radiation The equivalence of the moving point charge and the differential length of line current yield the equivalent magnetic force equation Such that (A-m) (A-m) The overall force on a line, surface and volume current is found by integrating over the current distribution (Line current) (Surface current) (Volume current) Example (Force between line currents) Determine the force/unit length on a line current I 1 due to the magnetic flux produced by a parallel line current I 2 (separation distance =d) flowing in the opposite direction. z ) x I I 1 I 2 µ (x 0,y 0) (x 0,y d) y d Fig. 4.3 Example of force between line currents The magnetic flux produced on the z-axis (I 1 ) due to the current I 2 is. The force on a length l of current I 1 due to the flux produced by I 2 is 44/JNU OLE

54 The force per unit length on the current I 1 is The force on a length l of current I 2 due to the flux produced by I 1 is The force per unit length on the current I 2 is Note that the currents repel each other given the currents flowing in opposite directions. If the currents flow in the same direction, they attract each another 4.5 Torque on a Current Loop Given the change in current directions around a closed current loop, the magnetic forces on different portions of the loop vary in direction. Using the Lorentz force equation, we can show that the net force on a simple circular or rectangular loop is a torque which forces the loop to align its magnetic moment with the applied magnetic field. Consider the rectangular current loop shown below. The loop lies in the x-y plane and carries a DC current I. The loop lies in a uniform magnetic flux density B given by The loop consists of four current segments carrying distinct vector current components I 1 =Ia y I 2 =-Ia x I 3 =-Ia y I 4 =Ia x. 45/JNU OLE

55 Electromagnetics and Radiation Given a uniform flux density and a DC current along straight current segments, the magnetic force on each conductor segment can be simplified to the following equation Fig. 4.4 x-y plane (Source: The forces on the current segments can be determined for each component of the magnetic flux density. Forces due to B z Fig. 4.5 Forces due to B z (Source: Forces due to B y 46/JNU OLE

56 Fig. 4.6 Forces due to B y (Source: The vector torque on the loop is defined in terms of the force magnitude (IByl 2 ), the torque moment arm distance (l 1 /2), and the torque direction (defined by the right hand rule): Where A=l 1 l 2 is the loop area. The vector torque can be written compactly by defining the magnetic moment (m) of the loop m=ia (magnetic moment magnitude) m=iaa n (vector magnetic moment) where an is the unit normal to the loop (defined by the right hand rule as applied to the current direction) Fig. 4.7 Loop area (Source: The magnitude of the torque in terms of the magnetic moment is 47/JNU OLE

57 Electromagnetics and Radiation The vector torque is then Note that the torque on the loop tends to align the loop magnetic moment with the direction of the applied magnetic field. 4.6 Magnetisation Just as dielectric materials are polarised under the influence of an applied electric field, certain materials can be magnetised under the influence of an applied magnetic field. Magnetisation for magnetic fields is the dual process to polarisation for electric fields. The magnetisation process may be defined using the magnetic moments of the electron orbits within the atoms of the material. Each orbiting electron can be viewed as a small current loop with an associated magnetic moment. Fig. 4.8 Loop area (Source: An unmagnetised material can be characterized by a random distribution of the magnetic moments associated with the electron orbits. These randomly oriented magnetic moments produce magnetic field components that tend to cancel one another (net H=0). Under the influence of an applied magnetic field, many of the current loops align their magnetic moments in the direction of the applied magnetic field. Fig. 4.9 Magnetic moments (Source: If most of the magnetic moments stay aligned after the applied magnetic field is removed, a permanent magnet is formed. 48/JNU OLE

58 The bar magnet can be viewed as the magnetic analogy to the electric dipole. The poles of the bar magnet can be represented as equivalent magnetic charges separated by a distance l (the length of the magnet). Fig Magnetic flux density (Source: The magnetic flux density produced by the magnetic dipole is equivalent to the electric field produced by the electric dipole The preceding equations assume the dipole is centred at the coordinate origin and oriented with its dipole moment along the z-axis. A current loop and a solenoid produce the same B as the bar magnet at large distances (in the far field) if the magnetic moments of these three devices are equivalent. 49/JNU OLE

59 Electromagnetics and Radiation Fig Magnetic moments (Source: If the magnetic moments of these three devices are equal they produce essentially the same magnetic field at large distances (equivalent sources). Assuming the same vector magnetic moment, all three of these devices would experience the same torque when placed in a given magnetic field The parameters associated with the magnetisation process are duals to those of the polarisation process. The magnetisation vector M is the dual of the polarisation vector P. Let us see the difference between magnetisation and polarisation. Magnetisation Polarisation Table 4.1 Difference between magnetisation and polarisation 50/JNU OLE

60 Note that the magnetic susceptibility Pm is defined somewhat differently than the electric susceptibility Pe. However, just as the electric susceptibility and relative permittivity are a measure of how much polarisation occurs in the material, the magnetic susceptibility and relative permeability are a measure of how much magnetisation occurs in the material. 4.7 Magnetic Materials Magnetic materials can be classified based on the magnitude of the relative permeability. Materials with a relative permeability of just under one (a small negative magnetic susceptibility) are defined as diamagnetic. In diamagnetic materials, the magnetic moments due to electron orbits and electron spin are very nearly equal and opposite such that they cancel each other. Thus, in diamagnetic materials, the response to an applied magnetic field is a slight magnetic field in the opposite direction. Superconductors exhibit perfect diamagnetism (Xm= 1) at temperatures near absolute zero such that magnetic fields cannot exist inside these materials. Materials with a relative permeability of just greater than one are defined as paramagnetic. In paramagnetic materials, the magnetic moments due to electron orbit and spin are unequal, resulting in a small positive magnetic susceptibility. Magnetisation is not significant in paramagnetic materials. Both diamagnetic and paramagnetic materials are typically linear media. Materials with a relative permeability much greater than one are defined as ferromagnetic. Ferromagnetic materials are always nonlinear. As such, these materials cannot be described by a single value of relative permeability. If a single number is given for the relative permeability of any ferromagnetic material, this number represents an average value of µ r Ferromagnetic materials lose their ferromagnetic properties at very high temperatures (above a temperature known as the Curie temperature). The characteristics of ferromagnetic materials are typically presented using the B-H curve, a plot of the magnetic flux density B in the material due to a given applied magnetic field H. The B-H curve shows the initial magnetisation curve along with a curve known as a hysteresis loop. The initial magnetisation curve shows the magnetic flux density that would result when an increasing magnetic field is applied to an initially unmagnetized material. An unmagnetized material is defined by the B=H=0 point on the B-H curve (no net magnetic flux given no applied field). As the magnetic field increases, at some point, all of the magnetic moments (current loops) within the material align themselves with the applied field and the magnetic flux density saturates (B m ). If the magnetic field is then cycled between the saturation magnetic field value in the forward and reverse directions (±H m ), the hysteresis loop results. The response of the material to any applied field depends on the initial state of the material magnetisation at that instant. 51/JNU OLE

61 Electromagnetics and Radiation 4.8 Magnetic Boundary Conditions Fig B-H curve (Source: The fundamental boundary conditions involving magnetic fields relate the tangential components of magnetic field and the normal components of magnetic flux density on either side of the media interface. The same techniques used to determine the electric field boundary conditions can be used to determine the magnetic field boundary conditions Tangential magnetic field In order to determine the boundary condition on the tangential magnetic field at a media interface, Ampere s law is evaluated around a closed incremental path that extends into both regions as shown below. According to Ampere s law, the line integral of the magnetic field around the closed loop yeilds the current enclosed. (µ 1,Є 1,σ 1 ) Δ y (µ 2,Є 2,σ 2 ) Fig Tangential magnetic field (Source: Δ x If we take the limit of this integral as y = 0, the integral contributions on the vertical paths goes to zero. 52/JNU OLE

62 Assuming the magnetic field and surface current components along the incremental length )x are uniform, the integrals above reduce to Dividing this result by x gives For this example, if the surface current flows in the opposite direction, we obtain Or Thus, the general boundary condition on the tangential magnetic field is This is the tangential components of magnetic field are discontinuous across a media interface by an amount equal to the surface current density on the interface. Note that the previous boundary condition relates only scalar quantities. The vector magnetic field and surface current at the media interface can be shown to satisfy a vector boundary condition defined by The above equation indicate vector boundary condition relating the magnetic field and surface current at a media interface. Where n is a unit normal to the interface pointing into region 1 53/JNU OLE

63 Electromagnetics and Radiation (µ 1,Є 1,σ 1 ) (µ 2,Є 2,σ 2 ) Fig Tangential magnetic field (Source: Given a surface current on the interface, the tangential magnetic field components on either side of the interface point in opposite directions. Normal magnetic flux density In order to determine the boundary condition on the normal magnetic flux density at a media interface, we apply Gauss s law for magnetic fields to an incremental volume that extends into both regions as shown below Fig Normal magnetic flux density (Source: The application of Gauss s law for magnetic fields to the closed surface above gives If we take the limit as the height of the volume )z approaches 0, the integral contributions on the four sides of the volume vanishes. 54/JNU OLE

64 The integrals over the upper and lower surfaces on either side of the interface reduce to where the magnetic flux density is assumed to be constant over the upper and lower incremental surfaces. Evaluation of the surface integrals yields Dividing by x y gives such that the general boundary condition on the normal component of magnetic flux density becomes This is the normal components of magnetic flux density are continuous across a media interface. 4.9 Inductors and Inductance An inductor is an energy storage device that stores energy in a magnetic field. An inductor typically consists of some configuration of conductor coils (an efficient way of concentrating the magnetic field). Yet, even straight conductors contain inductance. The parameters that define inductors and inductance can be defined as parallel quantities to those of capacitors and capacitance Inductors Stores energy in magnetic field Capacitors Stores energy in an electric field Inductance L= Inductance (H) λ= Flux linkage (Wb) I=current Capacitance C=Capacitance(F) Q=Charge (C) V=Voltage (V) Table 4.2 Difference between inductors and capacitors 55/JNU OLE

65 Electromagnetics and Radiation The flux linkage of an inductor defines the total magnetic flux that links the current. If the magnetic flux produced by a given current links that same current, the resulting inductance is defined as a self inductance. If the magnetic flux produced by a given current links the current in another circuit, the resulting inductance is defined as a mutual inductance Mutual Inductance Calculations The mutual inductance between two distinct circuits can be determined by assuming a current in one circuit and determining the flux linkage to the opposite circuit. Example (Mutual inductance between coaxial loops) Determine the mutual inductance between two coaxial loops as shown below. Assume that the loop separation distance h is large relative to the radii of both loops (a, b). Also assume that loop (2) is much smaller than loop (1) (b<<a). Fig Example mutual inductance between coaxial loops If we assume that a current flows in loop (1), the magnetic flux density produced by loop (1) along its axis is Given that loop (2) is much smaller than loop (1) we may assume that the flux produced by loop (1) is nearly uniform over the area of loop (2) and is approximately equal to that at the loop center. The flux produced by loop (1) can also be assumed to be approximately normal to loop (2) over its area. Thus, the total flux produced by loop (1) linking loop (2) is approximately. From the definition of inductance, the mutual inductance between the loops is Note that the calculations required by assuming the current in loop (1) are much easier than those required by assuming the current in loop (2). The calculations are easier based on the approximations employed. If the current is assumed in loop (2), the resulting magnetic flux density over loop (1) is a rather complicated function of position 56/JNU OLE

66 which requires a complex integration to arrive at the same approximate 4.11 Internal And External Inductance In general, a current carrying conductor has magnetic flux internal and external to the conductor. Thus, the magnetic flux inside the conductor can link portions of the conductor current which produces a component of inductance designated as internal inductance. The magnetic flux outside the conductor that links the conductor current is designated as external inductance The most efficient technique in determining the internal and external components of inductance is the energy method. The energy method for determining inductance is based on the total magnetic energy expression for an inductor given by Solving this equation for the inductance yields The internal and external components of the inductance are found by integrating the internal and external magnetic flux density expressions, respectively Magnetic Forces on Magnetic Materials Magnetic materials experience a force when placed in an applied magnetic field. This type of force is seen when a bar magnet is placed in a magnetic field, such as that of another bar magnet. The like poles repel each other while unlike poles attract each other. Note that the bar magnets behave in the same way as current loops in an applied magnetic field as each magnet tends to align its magnetic moment in the direction of the applied field of the opposite magnet. Fig.4.17 Magnetic forces on magnetic materials (Source: The magnetised needle on a compass obeys the same principle as the compass needle aligns itself with the Earth s magnetic field. The magnetic field on the Earth s surface has a horizontal component that points toward the magnetic South Pole (near the geographic North Pole). The horizontal component of magnetic field on the Earth s surface is 57/JNU OLE

67 Electromagnetics and Radiation maximum near the equator (approximately 35µT) and falls to zero at the magnetic poles. Fig Magnetic field (Source: An electromagnet can be used to lift ferromagnetic materials (such as scrap iron). The lifting force F of the electromagnet can be determined by considering how much energy is stored in the air gaps when the ferromagnetic materials are separated. The magnitude of the force necessary to separate the two pieces of ferromagnetic material equals the total amount of magnetic energy stored in the air gaps after separation. The total energy required to move the lower magnetic piece a distance l from the electromagnet is the product of the force magnitude F and the distance l. The total magnetic energy in the two air gaps is found by integrating the magnetic energy densities in the air gaps. Assuming short air gaps, the magnetic field in the air gaps can be assumed to be uniform and confined to the volume below the electromagnet poles (the fringing effect of the magnetic field is negligible for small air gaps). The uniform magnetic field in the air gap produces a uniform energy density so that the total magnetic energy stored in each air gap is a simply the product of the energy density and the volume of the air gap. Solving this equation for the force magnitude gives The air gap magnetic field can be determined according to the boundary condition for the normal component of magnetic flux across the air gap. The magnetic flux density is normal to the interfaces between the air and the ferromagnetic core. According to the boundary condition on the normal component of magnetic flux density, the magnetic flux density must be continuous across the air gap Rewriting this boundary condition in terms of the magnetic fields gives Or 58/JNU OLE

68 Given either the magnetic field or the magnetic flux density in the core, the corresponding quantity in the air gap can be found according to the previous equations. Note that the magnetic field in the air gap is much larger than that in the core given the large relative permeability of the ferromagnetic core Magnetic Circuits Magnetic field problems involving components such as current coils, ferromagnetic cores and air gaps can be solved as magnetic circuits according to the analogous behaviour of the magnetic quantities to the corresponding electric quantities in an electric circuit. Fig Electric and magnetic circuits (Source: Electric circuit V=IR V=Electromotive Force (V)[emf] I=total current (A) R=resistance(Ω) Magnetic circuit F=magnetomotive force (A-turns)[mmf] =total magnetic flux (Wb) R=reluctance(H -1 ) G=conductance (S) =permeance (H) Table 4.3 Difference between electric and magnetic circuits Given that reluctance in a magnetic circuit is analogous to resistance in an electric circuit, and permeability in a magnetic circuit is analogous to conductivity in an electric circuit, we may interpret the permeability of a medium as a measure of the resistance of the material to magnetic flux. Just as current in an electric circuit follows the path of least resistance; the magnetic flux in a magnetic circuit follows the path of least reluctance 59/JNU OLE

69 Electromagnetics and Radiation Summary Electric charges moving in a magnetic field experience a force due to the magnetic field. Given a charge. The equivalence of the moving point charge and the differential length of line current yield the equivalent magnetic force equation (A-m) (A-m) The vector torque on the loop is defined in terms of the force magnitude (IByl ), the torque moment arm distance 2 (l 1 /2), and the torque direction. Electric charges moving in a magnetic field experience a force due to the magnetic field. The charge-velocity product for a moving point charge can be related to an equivalent differential length of line current. The vector torque on the loop is defined in terms of the force magnitude. Magnetisation for magnetic fields is the dual process to polarisation for electric fields. An unmagnetized material can be characterized by a random distribution of the magnetic moments associated with the electron orbits. If most of the magnetic moments stay aligned after the applied magnetic field is removed, a permanent magnet is formed. The bar magnet can be viewed as the magnetic analogy to the electric dipole. Magnetic materials can be classified based on the magnitude of the relative permeability. Materials with a relative permeability of just greater than one are defined as paramagnetic. Ferromagnetic materials lose their ferromagnetic properties at very high temperatures (above a temperature known as the Curie temperature). References Bakshi, A.V., Bakshi, U.A., Field Theory. Technical Publications. ece3313notes8.pdf Bird, J., Electrical circuit theory and technology. Newnes. Recommended Reading Goldman, A., Modern Ferrite Technology. Springer, 2 ed. Cullity, B.D., Graham, C.D., Introduction to Magnetic Materials, Wiley-IEEE Press, 2 ed. Yamaguchi, M., Tanimoto, Y., Magneto-Science: Magnetic Field Effects on Materials: Fundamentals and Applications. Springer, 1 ed. 60/JNU OLE

70 Self Assessment charges moving in a magnetic field experience a force due to the magnetic field. a. Electromotive b. c. d. Electronic Electric Fixed The charge-velocity product for a moving point charge can be related to an equivalent length of line current. a. fixed b. c. d. differential varying time varied The torque on the loop is defined in terms of the force magnitude. a. point b. c. d. line vector static for magnetic fields is the dual process to polarisation for electric fields. a. Standarisation b. c. d. Fragmentation Divergence Magnetisation A/an material can be characterized by a random distribution of the magnetic moments associated with the electron orbits. a. unmagnetised b. c. d. magnetised polarised electrified If most of the magnetic moments stay aligned after the applied magnetic field is removed, a magnet is formed. a. temporary b. c. d. permanent fixed stable State which of the following is true. a. The bar magnet can be viewed as the magnetic analogy to the magnetic dipole. b. c. d. The pie magnet can be viewed as the magnetic analogy to the electric dipole. The magnet can be viewed as the magnetic analogy to the electric dipole. The ferro magnet can be viewed as the magnetic analogy to the electric dipole. 61/JNU OLE

71 Electromagnetics and Radiation State which of the following is true. a. Static materials can be classified based on the magnitude of the relative permeability. b. Magnetic materials can be classified based on the displacement of the relative permeability. c. Magnetic materials can be classified based on the magnitude of the relative velocity. d. Magnetic materials can be classified based on the magnitude of the relative permeability. Materials with a relative permeability of just greater than one are defined as which of the following? a. Bar magnetic b. Ferromagnetic c. Electromagnetic d. Paramagnetic 10. Ferromagnetic materials lose their ferromagnetic properties at which of the following? a. Low temperature b. Very high temperatures c. Room temperature d. Cool temperature 62/JNU OLE

72 Chapter V Time-Varying Fields And Maxwell s Equations Aim The aim of the chapter is to: analyse Faraday s law explore displacement current highlight Maxwell s equations Objectives The objectives of this chapter are to: explain Faraday s law explain Maxwell s equations in point form examine Maxwell s equations in integral form Learning outcome At the end of this chapter students will be able to: comprehend Faraday s law determine displacement current understand Maxwell s equations in point form formulate Maxwell s equations in integral form 63/JNU OLE

73 Electromagnetics and Radiation 5.1 Faraday s Law After Oersted demonstrated in 1820 that an electric current affected a compass needle, Faraday professed his belief that if a current could produce a magnetic field, then a magnetic field should be able to produce a current. The concept of the "field" was not available at that time, and Faraday s goal was to show that a current could be produced by "magnetism". He worked on this problem intermittently over a period of ten years, until he was finally successful in He wound two separate windings on an iron toroid and placed a galvanometer in one circuit and a battery in the other. Upon closing the battery circuit, he noted a momentary deflection of the galvanometer; a similar deflection in the opposite direction occurred when the battery was disconnected. This, of course, was the first experiment he made involving a changing magnetic field, and he followed it with a demonstration that either a moving magnetic field or a moving coil could also produce a galvanometer deflection. In terms of fields, a time-varying magnetic field produces an electromotive force (emf) which may establish a current in a suitable closed circuit. An electromotive force is merely a voltage that arises from conductors moving in a magnetic field or from changing magnetic fields, and we shall define it below. Faraday s law is customarily stated as (1) Equation (1) implies a closed path, although not necessarily a closed conducting path; the closed path, for example, might include a capacitor, or it might be a purely imaginary line in space. The magnetic flux is that flux which passes through any and every surface whose perimeter is the closed path, and dφ/dt is the time rate of change of this flux. A nonzero value of dφ/dt may result from any of the following situations: A time-changing flux linking a stationary closed path Relative motion between a steady flux and a closed path A combination of the two The minus sign is an indication that the emf is in such a direction as to produce a current whose flux, if added to the original flux, would reduce the magnitude of the emf. This statement that the induced voltage acts to produce an opposing flux is known as Lenz s law. If the closed path is that taken by an N-turn filamentary conductor, it is often sufficiently accurate to consider the turns as coincident and let (2) Where Φ is now interpreted as the flux passing through any one of N coincident paths. We need to define emf as used in eq (1) or eq (2). The emf is obviously a scalar, and (perhaps not so obviously) a dimensional check shows that it is measured in volts. We define the emf as (3) and note that it is the voltage about a specific closed path. If any part of the path is changed, generally the emf changes. The departure from static results is clearly shown by (3), for electric field intensity resulting from a static charge distribution must lead to zero potential difference about a closed path. In electrostatics, the line integral leads to a potential difference; with time-varying fields, the result is an emf or a voltage. Replacing Φ in eq (1) by the surface integral of B, we have 64/JNU OLE

74 (4) Where the fingers of our right hand indicate the direction of the closed path, and our thumb indicates the direction of ds. A flux density B in the direction of ds and increasing with time thus produces an average value of E which is opposite to the positive direction about the closed path. The right-handed relationship between the surface integral and the closed line integral in (4) should always be kept in mind during flux integrations and emf determinations. Let us divide our investigation into two parts by first finding the contribution to the total emf made by a changing field within a stationary path (transformer emf), and then we will consider a moving path within a constant (motional, or generator, emf). We first consider a stationary path. The magnetic flux is the only time varying quantity on the right side of (4), and a partial derivative may be taken under the integral sign, (5) Before we apply this simple result to an example, let us obtain the point form of this integral equation. Applying Stokes theorem to the closed line integral, we have where the surface integrals may be taken over identical surfaces. The surfaces are perfectly general and may be chosen as differentials, (6) Eq (5) is the integral form of this equation and is equivalent to Faraday s law as applied to a fixed path. If B is not a function of time, (5) and (6) evidently reduce to the electrostatic equations, and (electrostatics) (electrostatics) 5.2 Displacement Current Faraday s experimental law has been used to obtain one of Maxwell s equations in differential form, (7) which shows us that a time-changing magnetic field produces an electric field. We see that this electric field has the special property of circulation; its line integral about a general closed path is not zero. Now let us turn our attention to the time-changing electric field. We should first look at the point form of Ampere s circuital law as it applies to steady magnetic fields, (8) and show its inadequacy for time-varying conditions by taking the divergence of each side, 65/JNU OLE

75 Electromagnetics and Radiation The divergence of the curl is identically zero, so J is also zero. However, the equation of continuity, then shows us that (8) can be true only if. This is an unrealistic limitation, and (8) must be amended before we can accept it for time-varying fields. Suppose we add an unknown term G to (8), Again taking the divergence, we have Thus, Replacing by D from which we obtain the simplest solution for G, Ampere s circuital law in point form therefore becomes (9) Eq (9) has not been derived. It is merely a form we have obtained which does not disagree with the continuity equation. It is also consistent with all our other results, and we accept it as we did each experimental law and the equations derived from it. We now have a second one of Maxwell s equations and shall investigate its significance. The additional term has the dimensions of current density, amperes per square meter. Since it results from a time-varying electric flux density (or displacement density), Maxwell termed it a displacement current density. We sometimes denote it by J d This is the third type of current density we have met. Conduction current density, is the motion of charge (usually electrons) in a region of zero net charge density, and convection current density, is the motion of volume charge density. Both are represented by J in eq (9). Bound current density is, of course, included in H. In a non conducting medium in which no volume charge density is present, J = 0, and then 66/JNU OLE

76 (if J=0) (10) Notice the symmetry between (10) and (7): Again the analogy between the intensity vectors E and H and the flux density vectors D and B is apparent. Too much faith cannot be placed in this analogy, however, for it fails when we investigate forces on particles. The force on a charge is related to E and to B, and some good arguments may be presented showing an analogy between E and B and between D and H. We shall omit them, however, and merely say that the concept of displacement current was probably suggested to Maxwell by the symmetry first mentioned above. The total displacement current crossing any given surface is expressed by the surface integral, and we may obtain the time-varying version of Ampere s circuital law by integrating eq (9) over the surface S, and applying Stokes theorem What is the nature of displacement current density? Let us study the simple circuit of Fig. 5.1, containing a filamentary loop and a parallel-plate capacitor. Within the loop a magnetic field varying sinusoidal with time is applied to produce an emf about the closed path (the filament plus the dashed portion between the capacitor plates) which we shall take as (11) Using elementary circuit theory and assuming the loop has negligible resistance and inductance, we may obtain the current in the loop as Where the quantities, S, and d pertain to the capacitor. Let us apply Ampere s circuital law about the smaller closed circular path k and neglect displacement current for the moment: 67/JNU OLE

77 Electromagnetics and Radiation Fig. 5.1 A filamentary conductor forms a loop connecting the two plates of a parallel-plate capacitor (Source: Biblio2/chapt10.pdf) The path and the value of H along the path are both definite quantities (although difficult to determine), and k, H dl is a definite quantity. The current I k is that current through every surface whose perimeter is the path k. If we choose a simple surface punctured by the filament, such as the plane circular surface defined by the circular path k, the current is evidently the conduction current. Suppose now we consider the closed path k as the mouth of a paper bag whose bottom passes between the capacitor plates. The bag is not pierced by the filament, and the conductor current is zero. Now we need to consider displacement current, for within the capacitor and therefore This is the same value as that of the conduction current in the filamentary loop. Therefore the application of Ampere s circuital law including displacement current to the path k leads to a definite value for the line integral of H. This value must be equal to the total current crossing the chosen surface. For some surfaces the current is almost entirely conduction current, but for those surfaces passing between the capacitor plates, the conduction current is zero, and it is the displacement current which is now equal to the closed line integral of H. Physically, we should note that a capacitor stores charge and that the electric field between the capacitor plates is much greater than the small leakage fields outside. We therefore introduce little error when we neglect displacement current on all those surfaces which do not pass between the plates. Displacement current is associated with timevarying electric fields and therefore exists in all imperfect conductors carrying a time-varying conduction current. 5.3 Maxwell s Equations in Point Form We have already obtained two of Maxwell s equations for time-varying fields, (12) and (13) The remaining two equations are unchanged from their non-time-varying form: 68/JNU OLE

78 (14) (15) Eq (14) essentially states that charge density is a source (or sink) of electric flux lines. Note that we can no longer say that all electric flux begins and terminates on charge, because the point form of Faraday s law (12) shows that E, and hence D, may have circulation if a changing magnetic field is present. Thus the lines of electric flux may form closed loops. However, the converse is still true, and every coulomb of charge must have one coulomb of electric flux diverging from it. Equation (15) again acknowledges the fact that ``magnetic charges, or poles, are not known to exist. Magnetic flux is always found in closed loops and never diverges from a point source. These for equations form the basis of all electromagnetic theory. They are partial differential equations and relate the electric and magnetic fields to each other and to their sources, charge and current density. The auxiliary equations relating D and E are relating B and H defining conduction current density, (16) (17) (18) and defining convection current density in terms of the volume charge density, (19) are also required to define and relate the quantities appearing in Maxwell s equations. The potentials V and A have not been included above because they are not strictly necessary, although they are extremely useful. 5.4 Maxwell s Equations in Integral Form The integral forms of Maxwell s equations are usually easier to recognize in terms of the experimental laws from which they have been obtained by a generalization process. Experiments must treat physical macroscopic quantities, and their results therefore are expressed in terms of integral relationships. A differential equation always represents a theory. Let us now collect the integral forms of Maxwell s equations of the previous section. Integrating eq (12) over a surface and applying Stokes theorem, we obtain Faraday s law, (20) and the same process applied to (13) yields Ampere s circuital law, (21) Gauss s laws for the electric and magnetic fields are obtained by integrating eq (14) and (15) throughout a volume and using the divergence theorem: 69/JNU OLE

79 Electromagnetics and Radiation (22) (23) These four integral equations enable us to find the boundary conditions on B, D, H, and E which are necessary to evaluate the constants obtained in solving Maxwell s equations in partial differential form. These boundary conditions are in general unchanged from their forms for static or steady fields, and the same methods may be used to obtain them. Between any two real physical media (where K must be zero on the boundary surface), (20) enables us to relate the tangential E-field components, E t1 =E t2 (24) and from (21) H t1 =H t2 (25) The surface integrals produce the boundary conditions on the normal components, D N1 D N2 = (26) and B N1 =B N2 (27) It is often desirable to idealize a physical problem by assuming a perfect conductor for which σ is infinite but J is finite. From Ohm's law, in a perfect conductor, and it follows from the point form of Faraday's law that for time-varying fields. The point form of Ampere's circuital law then shows that the finite value of J is E=0 H=0 J=0 and current must be carried on the conductor surface as a surface current K. Thus, if region 2 is a perfect conductor, (24) to (27) become, respectively, E t1 =0 (28) H t1 =K (H t1 =k a N ) (29) D N1 = (30) B N1 =0 (31) Where a N is an outward normal at the conductor surface. Note that surface charge density is considered a physical possibility for dielectrics, perfect conductors, or imperfect conductors, but that surface current density is assumed only in conjunction with perfect conductors. 70/JNU OLE

80 The boundary conditions stated above are a very necessary part of Maxwell s equations. All real physical problems have boundaries and require the solution of Maxwell s equations in two or more regions and the matching of these solutions at the boundaries. In the case of perfect conductors, the solution of the equations within the conductor is trivial (all time-varying fields are zero), but the application of the boundary conditions (28) to (31) may be very difficult. Certain fundamental properties of wave propagation are evident when Maxwell s equations are solved for an unbounded region. It represents the simplest application of Maxwell s equations, because it is the only problem which does not require the application of any boundary conditions. 71/JNU OLE

81 Electromagnetics and Radiation Summary Faraday s law is customarily stated as. Ampere s circuital law in point form is given as. The analogy between the intensity vectors E and H and the flux density vectors D and B is apparent Two of Maxwell s equations for time-varying fields are and. Certain fundamental properties of wave propagation are evident when Maxwell s equations are solved for an unbounded region. References Time Varying Fields and Maxwell s Equations. Available at < Computacion/Sistemas_de_Comunicacion_II/Material/Biblio2/chapt10.pdf>. [Accessed 11 February 2011]. Time Varying Fields and Maxwell s Equations. Available at < maxwell.pdf> [Accessed 11 February 2011]. Recommended Reading Fleisch, D., A Student s Guide to Maxwell s Equations. Cambridge University Press, 1 ed. Huray, P.G., Maxwell s Equations. Wiley-IEEE Press. Karmel, P.R., Colef, G.D., Camisa, R.L., Introduction to Electromagnetic and Microwave Engineering. Wiley-Interscience, 1 ed. 72/JNU OLE

82 Self Assessment 1. professed his belief that if a current could produce a magnetic field, then a magnetic field should be able to produce a current. a. Faraday b. c. d. Coulomb Henry Joule 2. In terms of fields, a magnetic field produces an electromotive force (emf) which may establish a current in a suitable closed circuit. a. fixed b. c. d. time-varying variable speed-varying 3. A is merely a voltage that arises from conductors moving in a magnetic field or from changing magnetic fields. a. electromotive force b. c. d. unit force electric force electronic force 4. Certain fundamental properties of wave propagation are evident when Maxwell s equations are solved for a/ an region. a. bounded b. c. d. fixed variable unbounded 5. current is associated with time-varying electric fields and therefore exists in all imperfect conductors carrying a time-varying conduction current a. Displacement b. c. d. Variable Varying Fixed 6. in 1820, who demonstrated that an electric current affected a compass needle? a. Oersted b. Faraday c. Henry d. Joule 73/JNU OLE

83 Electromagnetics and Radiation 7. Which of the following equation states Faraday s law? a. b. c. d. 8. Ampere s circuital law in point form is given by which of the following equation? a. b. c. d. 9. One of Maxwell s equations for time-varying fields is given by which of the following equation? a. b. c. d. 10. State which of the following is true. a. Magnetic flux is always found in closed loops and never diverges from a point source. b. Magnetic flux is always found in open loops and never diverges from a point source. c. Magnetic flux is always found in closed loops and diverges from a point source. d. Magnetic flux is not found in closed loops and never diverges from a point source. 74/JNU OLE

84 Chapter VI Antenna Fundamentals Aim The aim of this chapter is to: analyse how antenna radiates explore radiation pattern of antenna classify different types of antennas Objectives The objectives of this chapter are to: examine directivity of an antenna formulate antenna efficiency enlist various types of antenna Learning outcome At the end of this chapter, the students will be able to: recall how antenna radiates identify various performance parameters of antenna categorise the types of antenna 75/JNU OLE

85 Electromagnetics and Radiation 6.1 Introduction Antennas are metallic structures designed for radiating and receiving electromagnetic energy. An antenna acts as a transitional structure between the guiding devices (e.g., waveguide, transmission line) and the free space. The official IEEE definition of an antenna as given by Stutzman and Thiele [4] follows the concept: That part of a transmitting or receiving system that is designed to radiate or receive electromagnetic waves. 6.2 How an Antenna Radiates In order to know how an antenna radiates, let us first consider how radiation occurs. A conducting wire radiates mainly because of time-varying current or an acceleration (or deceleration) of charge. If there is no motion of charges in a wire, no radiation takes place, since no flow of current occurs. Radiation will not occur even if charges are moving with uniform velocity along a straight wire. However, charges moving with uniform velocity along a curved or bent wire will produce radiation. If the charge is oscillating with time, then radiation occurs even along a straight wire as explained by Balanis. The radiation from an antenna can be explained with the help of fig.6.1 which shows a voltage source connected to a two conductor transmission line. When a sinusoidal voltage is applied across the transmission line, an electric field is created which is sinusoidal in nature and these results in the creation of electric lines of force which are tangential to the electric field. The magnitude of the electric field is indicated by the bunching of the electric lines of force. The free electrons on the conductors are forcibly displaced by the electric lines of force and the movement of these charges causes the flow of current which in turn leads to the creation of a magnetic field. Fig. 6.1 Radiation from an antenna (Source: 76/JNU OLE

86 Due to the time varying electric and magnetic fields, electromagnetic waves are created and these travel between the conductors. As these waves approach open space, free space waves are formed by connecting the open ends of the electric lines. Since the sinusoidal source continuously creates the electric disturbance, electromagnetic waves are created continuously and these travel through the transmission line, through the antenna and are radiated into the free space. Inside the transmission line and the antenna, the electromagnetic waves are sustained due to the charges, but as soon as they enter the free space, they form closed loops and are radiated. 6.3 Near and Far Field Regions The field patterns, associated with an antenna, change with distance and are associated with two types of energy: - radiating energy and reactive energy. Hence, the space surrounding an antenna can be divided into three regions Fig. 6.2 Field regions around an antenna (Source: The three regions shown in fig 6.2 are: Reactive near-field region: In this region, the reactive field dominates. The reactive energy oscillates towards and away from the antenna, thus appearing as reactance. In this region, energy is only stored and no energy is dissipated. The outermost boundary for this region is at a distance where R 1 is the distance from the antenna surface, D is the largest dimension of the antenna and λ is the wavelength. Radiating near-field region (also called Fresnel region): This is the region which lies between the reactive near-field region and the far field region. Reactive fields are smaller in this field as compared to the reactive near-field region and the radiation fields dominate. In this region, the angular field distribution is a function of the distance from the antenna. The outermost boundary for this region is at a distance where R 2 is the distance from the antenna surface. Far-field region (also called Fraunhofer region): The region beyond is the far field region. In this region, the reactive fields are absent and only the radiation fields exist. The angular field distribution is not dependent on the distance from the antenna in this region and the power density varies as the inverse square of the radial distance in this region. 6.4 Antenna Performance Parameters The performance of an antenna can be gauged from a number of parameters. Certain critical parameters are listed below: 77/JNU OLE

87 Electromagnetics and Radiation Radiation Pattern The radiation pattern of an antenna is a plot of the far-field radiation properties of an antenna as a function of the spatial co-ordinates which are specified by the elevation angle θ and the azimuth angle φ. More specifically it is a plot of the power radiated from an antenna per unit solid angle which is nothing but the radiation intensity. Let us consider the case of an isotropic antenna. An isotropic antenna is one which radiates equally in all directions. If the total power radiated by the isotropic antenna is P, then the power is spread over a sphere of radius r, so that the power density S at this distance in any direction is given as (1) Then the radiation intensity for this isotropic antenna Ui can be written as (2) An isotropic antenna is not possible to realize in practice and is useful only for comparison purposes. A more practical type is the directional antenna which radiates more power in some directions and less power in other directions. A special case of the directional antenna is the omnidirectional antenna whose radiation pattern may be constant in one plane (e.g. E-plane) and varies in an orthogonal plane (e.g. H-plane). The radiation pattern plot of a generic directional antenna is shown in fig.6.3 Fig. 6.3 Radiation pattern of a generic directional antenna (Source: Figure 6.3 shows the following HPBW: The half power beam width (HPBW) can be defined as the angle subtended by the half power points of the main lobe. Main Lobe: This is the radiation lobe containing the direction of maximum radiation. Minor Lobe: All the lobes other then the main lobe are called the minor lobes. These lobes represent the radiation in undesired directions. The level of minor lobes is usually expressed as a ratio of the power density in the lobe in question to that of the major lobe. This ratio is called as the side lobe level (expressed in decibels). Back Lobe: This is the minor lobe diametrically opposite the main lobe. Side Lobes: These are the minor lobes adjacent to the main lobe and are separated by various nulls. Side lobes are generally the largest among the minor lobes In most wireless systems, minor lobes are undesired. Hence a good antenna design should minimize the minor lobes. 78/JNU OLE

88 6.4.2 Directivity The directivity of an antenna is defined as the ratio of the radiation intensity in a given direction from the antenna to the radiation intensity averaged over all directions. In other words, the directivity of a nonisotropic source is equal to the ratio of its radiation intensity in a given direction, over that of an isotropic source, (3) Where, D is the directivity of the antenna U is the radiation intensity of the antenna U i is the radiation intensity of an isotropic source P is the total power radiated Sometimes, the direction of the directivity is not specified. In this case, the direction of the maximum radiation intensity is implied and the maximum directivity is given by (4) Where D max = maximum directivity U max = maximum radiation intensity. Directivity is a dimensionless quantity, since it is the ratio of two radiation intensities. Hence, it is generally expressed in dbi. The directivity of an antenna can be easily estimated from the radiation pattern of the antenna. An antenna that has a narrow main lobe would have better directivity, then the one which has a broad main lobe, hence it is more directive Input Impedance The input impedance of an antenna is defined by Balanis as the impedance presented by an antenna at its terminals or the ratio of the voltage to the current at the pair of terminals or the ratio of the appropriate components of the electric to magnetic fields at a point. Hence the impedance of the antenna can be written as Where Z in = antenna impedance at the terminals R in = antenna resistance at the terminals X in = antenna reactance at the terminals (5) The imaginary part, X in of the input impedance represents the power stored in the near field of the antenna. The resistive part, R in of the input impedance consists of two components, the radiation resistance R r and the loss resistance R L. The power associated with the radiation resistance is the power actually radiated by the antenna, while the power dissipated in the loss resistance is lost as heat in the antenna itself due to dielectric or conducting losses. 79/JNU OLE

89 Electromagnetics and Radiation Voltage Standing Wave Ratio (VSWR) Fig. 6.4 Equivalent circuit of transmitting antenna (Source: In order for the antenna to operate efficiently, maximum transfer of power must take place between the transmitter and the antenna. Maximum power transfer can take place only when the impedance of the antenna (Z in ) is matched to that of the transmitter (Z S ). According to the maximum power transfer theorem, maximum power can be transferred only if the impedance of the transmitter is a complex conjugate of the impedance of the antenna under consideration and vice-versa. Thus, the condition for matching is Where (6) Z in = R in + jx in Z S =R S + jx S If the condition for matching is not satisfied, then some of the power maybe reflected back and this leads to the creation of standing waves, which can be characterized by a parameter called as the Voltage Standing Wave Ratio (VSWR). The VSWR is given by Makarov as (7) (8) Where Γ V r V i is called the reflection coefficient is the amplitude of the reflected wave is the amplitude of the incident wave 80/JNU OLE

90 The VSWR is basically a measure of the impedance mismatch between the transmitter and the antenna. The higher the VSWR, the greater is the mismatch. The minimum VSWR which corresponds to a perfect match is unity. A practical antenna design should have an input impedance of either 50 Ω or 75 Ω since most radio equipment is built for this impedance Return Loss (RL) The Return Loss (RL) is a parameter which indicates the amount of power that is lost to the load and does not return as a reflection. As explained in the preceding section, waves are reflected leading to the formation of standing waves, when the transmitter and antenna impedance do not match. Hence the RL is a parameter similar to the VSWR to indicate how well the matching between the transmitter and antenna has taken place. The RL is given as by Makarov as (db) (9) For perfect matching between the transmitter and the antenna, Γ = 0 and RL = which means no power would be reflected back, whereas a Γ = 1 has a RL = 0 db, which implies that all incident power is reflected. For practical applications, a VSWR of 2 is acceptable, since this corresponds to a RL of db Antenna Efficiency The antenna efficiency is a parameter which takes into account the amount of losses at the terminals of the antenna and within the structure of the antenna. These losses are given as Reflections because of mismatch between the transmitter and the antenna I 2 R losses (conduction and dielectric) Hence the total antenna efficiency can be written as Where e t = total antenna efficiency e r = (1 ) = reflection (mismatch) efficiency e c = conduction efficiency e d = dielectric efficiency (10) Since e c and e d are difficult to separate, they are lumped together to form the e cd efficiency which is given as (11) e cd is called as the antenna radiation efficiency and is defined as the ratio of the power delivered to the radiation resistance R r, to the power delivered to R r and R L Antenna Gain Antenna gain is a parameter which is closely related to the directivity of the antenna. We know that the directivity is how much an antenna concentrates energy in one direction in preference to radiation in other directions. Hence, if the antenna is 100% efficient, then the directivity would be equal to the antenna gain and the antenna would be an isotropic radiator. Since all antennas will radiate more in some direction that in others, therefore the gain is the amount of power that can be achieved in one direction at the expense of the power lost in the others as explained by Ulaby [7]. The gain is always related to the main lobe and is specified in the direction of maximum radiation unless indicated. It is given as 81/JNU OLE

91 Electromagnetics and Radiation Polarisation (dbi) (12) Polarisation of a radiated wave is defined by Balaris as that property of an electromagnetic wave describing the time varying direction and relative magnitude of the electric field vector. The polarisation of an antenna refers to the polarisation of the electric field vector of the radiated wave. In other words, the position and direction of the electric field with reference to the earth s surface or ground determines the wave polarisation. The most common types of polarisation include the linear (horizontal or vertical) and circular (right hand polarisation or the left hand polarisation). Fig. 6.5 A linearly (vertically) polarised wave (Source: If the path of the electric field vector is back and forth along a line, it is said to be linearly polarised. Fig. 6.5 shows a linearly polarised wave. In a circularly polarised wave, the electric field vector remains constant in length but rotates around in a circular path. A left hand circular polarised wave is one in which the wave rotates counter clockwise whereas right hand circular polarised wave exhibits clockwise motion as shown in fig 6.6 Fig. 6.6 Commonly used polarisation schemes (Source: 82/JNU OLE

92 6.4.9 Bandwidth The bandwidth of an antenna is defined as the range of usable frequencies within which the performance of the antenna, with respect to some characteristic, conforms to a specified standard. The bandwidth can be the range of frequencies on either side of the center frequency where the antenna characteristics like input impedance, radiation pattern, beam width, polarisation, side lobe level or gain, are close to those values which have been obtained at the center frequency. The bandwidth of a broadband antenna can be defined as the ratio of the upper to lower frequencies of acceptable operation. The bandwidth of a narrowband antenna can be defined as the percentage of the frequency difference over the center frequency Balaris. These definitions can be written in terms of equations as follows (13) (14) Where f H =upper frequency f L =lower frequency f C =center frequency An antenna is said to be broadband if. One method of judging how efficiently an antenna is operating over the required range of frequencies is by measuring its VSWR. A VSWR 2 ( RL 9.5dB ) ensures good performance Fig. 6.7 Measuring bandwidth from the plot of the reflection coefficient (Source: Types of Antennas Antennas come in different shapes and sizes to suit different types of wireless applications. The characteristics of an antenna are very much determined by its shape, size and the type of material that it is made of. Some of the commonly used antennas are briefly described below Half Wave Dipole 83/JNU OLE

93 Electromagnetics and Radiation The length of this antenna is equal to half of its wavelength as the name itself suggests. Dipoles can be shorter or longer than half the wavelength, but a trade off exists in the performance and hence the half wavelength dipole is widely used Fig. 6.8 Half wave dipole (Source: The dipole antenna is fed by a two wire transmission line, where the two currents in the conductors are of sinusoidal distribution and equal in amplitude, but opposite in direction. Hence, due to cancelling effects, no radiation occurs from the transmission line. As shown in fig.6.8 the currents in the arms of the dipole are in the same direction and they produce radiation in the horizontal direction. Thus, for a vertical orientation, the dipole radiates in the horizontal direction. The typical gain of the dipole is 2dB and it has a bandwidth of about 10%. The half power beam width is about 78 degrees in the E plane and its directivity is 1.64 (2.15dB) with a radiation resistance of 73 Ω [4]. Fig.6.9 shows the radiation pattern for the half wave dipole Fig. 6.9 Radiation pattern for half wave dipole (Source: 84/JNU OLE

94 6.5.2 Monopole Antenna The monopole antenna, shown in fig.6.10, results from applying the image theory to the dipole. According to this theory, if a conducting plane is placed below a single element of length L/2 carrying a current, then the combination of the element and its image acts identically to a dipole of length L except that the radiation occurs only in the space above the plane as discussed by Saunders [8]. Fig Monopole antenna (Source: For this type of antenna, the directivity is doubled and the radiation resistance is halved when compared to the dipole. Thus, a half wave dipole can be approximated by a quarter wave monopole (L/2 = λ/4). The monopole is very useful in mobile antennas where the conducting plane can be the car body or the handset case. The typical gain for the quarter wavelength monopole is 2-6dB and it has a bandwidth of about 10%. Its radiation resistance is 36.5 Ω and its directivity is 3.28 (5.16dB) [4]. The radiation pattern for the monopole is shown below in fig Fig Radiation pattern for the monopole antenna (Source: Loop Antenna The loop antenna is a conductor bent into the shape of a closed curve such as a circle or a square with a gap in the conductor to form the terminals as shown in fig There are two types of loop antennas-electrically small loop antennas and electrically large loop antennas. If the total loop circumference is very small as compared to the wavelength (L <<< λ), then the loop antenna is said to be electrically small. An electrically large loop antenna typically has its circumference close to a wavelength. The far-field radiation patterns of the small loop antenna are insensitive to shape [4]. 85/JNU OLE

95 Electromagnetics and Radiation Fig Loop antenna (Source: As shown in fig.6.12, the radiation patterns are identical to that of a dipole despite the fact that the dipole is vertically polarised whereas the small circular loop is horizontally polarised. Fig Radiation pattern of small and large loop antenna (Source: The performance of the loop antenna can be increased by filling the core with ferrite. This helps in increasing the radiation resistance. When the perimeter or circumference of the loop antenna is close to a wavelength, then the antenna is said to be a large loop antenna. The radiation pattern of the large loop antenna is different then that of the small loop antenna. For a one wavelength square loop antenna, radiation is maximum normal to the plane of the loop (along the z axis). In the plane of the loop, there is a null in the direction parallel to the side containing the feed (along the x axis), and there is a lobe in a direction perpendicular to the side containing the feed (along the y axis). Loop antennas generally have a gain from -2dB to 3dB and a bandwidth of around 10%.. The small loop antenna is very popular as a receiving antenna. Single turn loop antennas are used in pagers and multiturn loop antennas are used in AM broadcast receivers. 86/JNU OLE

96 6.5.4 Helical Antennas A helical antenna or helix is one in which a conductor connected to a ground plane, is wound into a helical shape. Fig 6.14 illustrates a helix antenna. The antenna can operate in a number of modes, however the two principal modes are the normal mode (broadside radiation) and the axial mode (end fire radiation). When the helix diameter is very small as compared to the wavelength, then the antenna operates in the normal mode. However, when the circumference of the helix is of the order of a wavelength, then the helical antenna is said to be operating in the axial mode Fig Helix antenna (Source: In the normal mode of operation, the antenna field is maximum in a plane normal to the helix axis and minimum along its axis. This mode provides low bandwidth and is generally used for hand-portable mobile applications Fig Radiation pattern of helix antenna (Source: 87/JNU OLE

97 Electromagnetics and Radiation In the axial mode of operation, the antenna radiates as an end fire radiator with a single beam along the helix axis. This mode provides better gain (upto 15dB) [4] and high bandwidth ratio (1.78:1) as compared to the normal mode of operation. For this mode of operation, the beam becomes narrower as the number of turns on the helix is increased. Due to its broadband nature of operation, the antenna in the axial mode is used mainly for satellite communications. Fig.7.15 above shows the radiation patterns for the normal mode as well as the axial mode of operations Horn Antennas Horn antennas are used typically in the microwave region (gigahertz range) where waveguides are the standard feed method, since horn antennas essentially consist of a waveguide whose end walls are flared outwards to form a megaphone like structure Fig Types of horn antenna (Source: Horns provide high gain, low VSWR, relatively wide bandwidth, low weight, and are easy to construct [4]. The aperture of the horn can be rectangular, circular or elliptical. However, rectangular horns are widely used. The three basic types of horn antennas that utilize a rectangular geometry are shown in fig These horns are fed by a rectangular waveguide which have a broad horizontal wall as shown in the figure. For dominant waveguide mode excitation, the E-plane is vertical and H-plane horizontal. If the broad wall dimension of the horn is flared with the narrow wall of the waveguide being left as it is, then it is called an H-plane sectoral horn antenna as shown in the figure. If the flaring occurs only in the E-plane dimension, it is called an E-plane sectoral horn antenna. A pyramidal horn antenna is obtained when flaring occurs along both the dimensions. The horn basically acts as a transition from the waveguide mode to the free-space mode and this transition reduces the reflected waves and emphasizes the travelling waves which lead to low VSWR and wide bandwidth [4]. The horn is widely used as a feed element for large radio astronomy, satellite tracking, and communication dishes. 88/JNU OLE

98 Summary Antennas are metallic structures designed for radiating and receiving electromagnetic energy. A conducting wire radiates mainly because of time-varying current or an acceleration (or deceleration) of charge. Due to the time varying electric and magnetic fields, electromagnetic waves are created and these travel between the conductors. The field patterns, associated with an antenna, change with distance and are associated with two types of energy: - radiating energy and reactive energy. The radiation pattern of an antenna is a plot of the far-field radiation properties of an antenna as a function of the spatial co-ordinates which are specified by the elevation angle θ and the azimuth angle φ. An isotropic antenna is not possible to realize in practice and is useful only for comparison purposes. In most wireless systems, minor lobes are undesired. Hence a good antenna design should minimize the minor lobes. The directivity of an antenna is defined as the ratio of the radiation intensity in a given direction from the antenna to the radiation intensity averaged over all directions. The Return Loss (RL) is a parameter which indicates the amount of power that is lost to the load and does not return as a reflection. The antenna efficiency is a parameter which takes into account the amount of losses at the terminals of the antenna and within the structure of the antenna. Antenna gain is a parameter which is closely related to the directivity of the antenna. Polarisation of a radiated wave is defined by as that property of an electromagnetic wave describing the time varying direction and relative magnitude of the electric field vector. Antennas come in different shapes and sizes to suit different types of wireless applications. The characteristics of an antenna are very much determined by its shape, size and the type of material that it is made of. The loop antenna is a conductor bent into the shape of a closed curve such as a circle or a square with a gap in the conductor to form the terminals. A helical antenna or helix is one in which a conductor connected to a ground plane, is wound into a helical shape. Horn antennas are used typically in the microwave region (gigahertz range) where waveguides are the standard feed method. References Antenna Basics. Available at: < [Accessed 17 February 2011]. Antenna Fundamentals. Available at: < Chapter2.pdf> [Accessed 17 February 2011]. Recommended Reading Balanis, C.A., Antenna Theory: Analysis and Design, Wiley-Interscience, 3 ed. Minin, I.V., Minin, O.V., Basic Principles of Fresnel Antenna Arrays. Springer, 1 ed. Kraus, J.D., Marhefka, R.J., Antennas for All Applications. McGraw-Hill Science/Engineering/Math, 3 ed. 89/JNU OLE

99 Electromagnetics and Radiation Self assessment Antennas are structures designed for radiating and receiving electromagnetic energy. a. metallic b. c. d. plastic oxide silver A conducting wire radiates mainly because of time-varying current or a/an (or deceleration) of charge. a. acceleration b. c. d. speed displacement volume An antenna is not possible to realize in practice and is useful only for comparison purposes. a. bidirectional b. c. d. rhombic isotropic unidirectional The is a parameter which indicates the amount of power that is lost to the load and does not return as a reflection. a. load loss b. c. d. reflection loss loss factor return loss The antenna is a parameter which takes into account the amount of losses at the terminals of the antenna and within the structure of the antenna. a. gain b. c. d. resolution directivity efficiency State which of the following statement is true. a. Antenna efficiency is a parameter which is closely related to the directivity of the antenna. b. c. d. Antenna gain is a parameter which is closely related to the directivity of the antenna. Antenna loss is a parameter which is closely related to the directivity of the antenna. Antenna power factor is a parameter which is closely related to the directivity of the antenna. State which of the following statement is true. a. Antennas do not come in different shapes and sizes to suit different types of wireless applications. b. c. d. Antennas come only in rectangular shape to suit different types of wireless applications. Antennas come in different shapes and sizes to suit different types of wireless applications. Antennas come in different shapes and sizes to suit all types of non-wireless applications. 90/JNU OLE

100 8. 9. State which of the following statement is false. a. The characteristics of an antenna are very much determined by its shape, size and the type of material that it is made of. b. c. d. Due to the time varying electric and magnetic fields, electromagnetic waves are created and these travel between the conductors. Horn antennas are used typically in the microwave region (gigahertz range) where waveguides are the standard feed method. The loop antennas are used typically in the microwave region (gigahertz range) where waveguides are the standard feed method. Which of the following antenna is the one in which a conductor connected to a ground plane, is wound into a helical shape? a. Helical antenna b. c. d. Loop antenna Rhombic antenna Isotropic antenna 10. Which of the following antennas are used typically in the microwave region (gigahertz range) where waveguides are the standard feed method? a. Helical antenna b. c. d. Horn antenna Rhombic Pie antenna 91/JNU OLE

101 Electromagnetics and Radiation Application I Helical Antenna for Medical Applications Introduction An increased interest for thermotherapy using microwaves has been observed in the last decade. A large number of devices have been designed and tested in order to produce therapeutic heating for medical applications and more particularly microwave hyperthermia (for the treatment of tumors having different sizes and located in various places of the human body). Among these devices, there is large interest in the study of interstitial coaxial applicators and more particularly of endocavitary applicators. They are generally used in urology for the heating of tumors or for the improvement of medical treatments like radiotherapy or chemotherapy. In this paper, we present the theoretical study and the experimental verifications concerning a generation of applicators of helical type. They have been designed as to reduce the heating zone along the cable in order to avoid possible thermal necrosis. Materials and method used The microwave antenna is realized from a flexible coaxial cable of 50 W characteristic impedance. Previous antennas were realized by removing the outer conductor of the cable on a length h. The improvement consists in making a double helical antenna, the first helix is the inner conductor which is rolled up around the teflon sheath. The second helix is soldered at the outer conductor and is rolled up around the cable (figure 1). The thermotherapy system consists of a microwave generator (heating frequency 915 MHz and maximum power 100 W) and a microwave radiometer centered around 3 GHz for the measurement of the temperatures. In order to take into account the heterogeneousness of the volume surrounding the antenna, but also the exact shape of tissues and applicator, a complete 3D model based on the well known FDTD method [3] has been developed. With this model, it is possible to know how the electromagnetic energy is deposited inside lossy media and, so to obtain the specific absorption rate (SAR). We can also determine the matching of the applicator inside the surrounding media at the heating frequency, but also in the radiometric frequency bandwidth. The heating pattern is then deduced from the resolution of the bio-heat transfer equation. As to verify the theoretical results, experimental measurements have been carried out on phantom model of human tissues (polyacrylamide gel). First, the return loss (S11 parameter) has been measured as a function of frequency by means of a network analyzer HP 8510 in order to obtain the level of adaptation of the applicator at the heating frequency and in the radiometric bandwidth. The next part of the experiment consists in the determination of the energy distribution. The method is based on the temperature increase in a polyacrylamide gel, induced by microwave power for a short time (about one minute) in order to avoid thermal conduction phenomena inside the gel. The thermal performances of the applicator are obtained from temperatures measurement on a polyacrylamide gel after a heating session of about forty five minutes using an automatic experimental system Results and discussion The comparison between theoretical results and experimental measurements concerning the S11 parameter as a function of frequency is shown on figure 2 : we can observe that the matching is quite good. The reflection coefficient is below -10 db at the heating frequency, that is to say that at least 90 % of the incident power is delivered to the surrounding media. We can observe a maximum of power behind the junction plane of the two helix (the junction plane is the plane where the two helix begin). The 40 % isopower line spreads on a length nearly equal to the total antenna length. A succession of power peaks appears in the vicinity of each metallic element corresponding to the helix. If we compare these results to the ones obtained with the previous urethral antenna, we can observe that the maximum of the SAR extends in the front of the junction plane of the applicator. So, we can conclude that the power deposition spreads on a less extensive zone for the helical applicator. 92/JNU OLE

102 Conclusion We have studied a new kind of endocavitary applicator of helical type. The theoretical results (obtained from the FDTD method) and confirmed by experimental measurements show clearly an improvement of the power deposition along the coaxial cable, which will make possible to avoid potential burns near the bladder neck. Fig. 1 Scheme of helical applicator Fig. 2 Comparison between experimental measurements (dotted line) and theoretical results (full line) for the reflection coefficient (S11 parameter) as a function of frequency obtained for the helical applicator dived in a polyacrylamid gel Questions: 1. Which material is used to realize helical antenna? Answer The helical antenna is realized from a flexible coaxial cable with 50 W characteristic impedance According to the above study what improvements have been made in previous antenna? Answer Previous antennas were realized by removing the outer conductor of the cable on a length h. The improvement consists in making a double helical antenna, the first helix is the inner conductor which is rolled up around the teflon sheath. The second helix is soldered at the outer conductor and is rolled up around the cable. The thermotherapy system consists of which devices? Answer The thermotherapy system consists of a microwave generator (heating frequency 915 MHz and maximum power 100 W) and a microwave radiometer centered around 3 GHz for the measurement of the temperatures. 93/JNU OLE

103 Electromagnetics and Radiation Application II Design And Analysis of Uwb Tem Horn Antenna for Ground Penetrating Radar Applications Introduction A TEM horn antenna is usually applied to the air-launching GPR system. It is made from two tapered metal plates, including exponentially tapered or linearly tapered. There are a narrow feed point and a wide open end at both sides of TEM horn antennas. Traditionally, the variation of characteristic impedance of a TEM horn antenna is usually set to range from 50 Ω (characteristic impedance of a coaxial cable) to Ω (free space wave impedance). However, a difference regularly exists between transmission-line characteristic impedance and free space wave impedance. Since there will be a large reflection from the aperture of the antenna, the concept of matching the impedance at the antenna aperture to that of free space was proved not the best case. Therefore, in this paper, we designed several TEM horn antennas with different aperture impedances to better understand which impedance is better to be used in antenna aperture. From the simulated result, it shows that the performance of the designed TEM horn antenna matching with impedance of 200 Ω is better than that with free space impedance of 376.7Ω, but there is no huge difference of performance with the different aperture impedance from 200 Ω to 400 Ω of the antenna. It demonstrated that the matching impedance at the antenna aperture is not a critical factor to manipulate the TEM horn antenna performance. Design approach Because TEM horn antenna is composed of two tapered metal plates, the current flows on these two plates and the TEM wave propagates between these two plates simultaneously. The current flowing on the two plates leads to the generation of the magnetic fields of TEM mode wave. The voltage difference between two plates leads to the generation of the electric fields of TEM mode wave. A TEM horn antenna can be considered to be a transformer from the impedance of a transmission line to the impedance of the free space and the variation of characteristic impedance is usually designed to be between 50 ohms and ohms. Since the characteristic impedance variation can be adjusted with the difference of the width of the plates and distance between two plates, the variation of characteristic impedance between two plates has to be calculated carefully in order to make reflection coefficient as small as possible over a large frequency range. In general, there are four main steps in designing a TEM horn antenna. Antenna geometry Following this design procedure, we can obtain the shape of the conductor plate and make the plates to be a linear TEM horn antenna. Since difference between the transmission-line wave characteristic impedance and the free space wave characteristic impedance usually exists, such difference should be taken into consideration in the design of TEM horn antenna. In order to determine the best matching impedance between the antenna aperture and the free space, different shapes of TEM horn antennas are designed based on different characteristic impedance at the antenna aperture. These virtual antenna models are run in the finite difference time domain (FDTD) method based software XFDTD. The antenna aperture characteristic impedance is designed separately to be 200Ω, 250Ω, 300Ω, and up to 400Ω. There may be a little difference in the lengths of the antennas because of the requirements of matching section. Generally speaking, when matching 50 Ω to 100Ω, we get the shortest antenna length; when matching 50Ω to 400Ω, we get the longest antenna length. The flare angles between two conductor plates are all fixed at 20. FDTD simulated results The most important factor to the pulse antenna is wide bandwidth with low reflection coefficient (S 11 ). As for a usable GPR antenna, S 11 should be at least less than -10 db over a wide frequency range. Large S 11 usually causes by the mismatch of the impedance between the feed cable and the antenna or the antenna and free space. The FDTD simulation of the TEM horn antennas can help us to understand better what the best design aperture impedance is. Fig.1 show the simulated results of the reflection coefficient (S 11 ) with different aperture impedances are from 200 Ω to 400Ω. We find that there is no obvious difference of performances of the TEM horn antennas as aperture impedances ranging from 200Ω to 400Ω. Setting the antenna aperture impedances between 200Ω and 300Ω will result 94/JNU OLE

104 in better performance. Their S 11 can be less than -15 db from 1.2 GHz to 3 GHz. It seems that the best performance happens when aperture impedance is set to be 200Ω. If aperture impedance is set over than 300Ω, we can see that the performance of S11 becomes worse. With impedances ranging over 300Ω as shown in Fig. 1, the S 11 is greater than -15dB; and the performance of low frequency band (below 1 GHz) is worse than that with impedances ranging from 200Ω to 300Ω It is obvious that the best matching impedance between the free space and the antenna aperture is not the free space wave impedance. To get the best performances of S 11, the aperture impedance should be set within the range from 200Ω to 300Ω. For aperture impedance of 200Ω, the S 11 is less than -15 db from 1.0 GHz to 3 GHz, even close to -20 db from 1.2 GHz up to 3 GHz. Conclusion From this research, it demonstrates that for general TEM horn antennas, there is no significant difference of performance with the different characteristic impedance of the antenna aperture. By comparing the simulated results with different aperture impedances, we can find out that matching the aperture impedance to free space impedance does not guarantee the best matching performance. The antenna can obtain the best performance with aperture impedance of 200Ω.. Questions: What is the antenna geometry used in the given application What is the design approach used in the above application? State the applications of TEM horn antenna. 95/JNU OLE