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

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

4-2 Module 4 Wave Propagation in Free Space 4.1 Uniform Plane Waves in Time Domain 4.2 Sinusoidally Time-Varying Uniform Plane Waves 4.3 Polarization 4.4 Poynting Vector and Energy Storage

4-3 Instructional Objectives 23. Write the expression for a traveling wave function for a set of specified characteristics of the wave 24. Obtain the electric and magnetic fields due to an infinite plane current sheet of an arbitrarily time-varying uniform current density, at a location away from it as a function of time, and at an instant of time as a function of distance, in free space 25. Find the parameters, frequency, wavelength, direction of propagation of the wave, and the associated magnetic (or electric) field, for a specified sinusoidal uniform plane wave electric (or magnetic) field in free space 26. Write expressions for the electric and magnetic fields of a uniform plane wave propagating away from an infinite plane sheet of a specified sinusoidal current density, in free space

4-4 Instructional Objectives (Continued) 27. Obtain the expressions for the fields due to an array of infinite plane sheets of specified spacings and sinusoidal current densities, in free space 28. Write the expressions for the fields of a uniform plane wave in free space, having a specified set of characteristics, including polarization 29. Express linear polarization and circular polarization as superpositions of clockwise and counterclockwise circular polarizations 30. Find the power flow and the electric and magnetic stored energies associated with electric and magnetic fields

4.1 Uniform Plane Waves in Time Domain (EEE, Sec. 3.4; FEME, Secs. 4.1, 4.2, 4.4, 4.5)

4-6 Infinite Plane Current Sheet Source Example:

4-7 For a current distribution having only an x-component of current density that varies only with z,

4-8 The only relevant equations are: Thus,

4-9 In the free space on either side of the sheet, J x = 0 Combining, we get Wave Equation

4-10 Solution to the Wave Equation

4-11

4-12

4-13 represents a traveling wave propagating in the +z-direction. represents a traveling wave propagating in the –z-direction. Where velocity of light

4-14 E4.1: Examples of Traveling Waves

4-15

4-16

4-17 Thus, the general solution is For the particular case of the infinite plane current sheet in the z = 0 plane, there can only be a (  ) wave for z > 0 and a (  ) wave for z < 0. Therefore,

4-18 Applying Faraday’s law in integral form to the rectangular closed path abcda in the limit that the sides bc and da  0,

4-19 Therefore, Now, applying Ampere’s circuital law in integral form to the rectangular closed path efgha in the limit that the sides fg and he  0,

4-20 Uniform plane waves propagating away from the sheet to either side with velocity v p = c. Thus, the solution is

4-21 In practice, there are no uniform plane waves. However, many practical situations can be studied based on uniform plane waves. For example, at large distances from physical antennas and ground, the waves can be approximated as uniform plane waves.

4-22 x y z z = 0

4-23 x y z z = 0 z > 0z < 0  z z E4.2

4-24

4-25

4-26 Review Questions 4.1. Outline the procedure for obtaining from the two Maxwell’s equations the particular differential equations for the special case of J = J x (z, t)a x State the wave equation for the case of E = E x (z, t)a x. Describe the procedure for its solution What is a uniform plane wave? Why is the study of uniform plane waves important? 4.4. Discuss by means of an example how a function f(t – z/v p ) represents a traveling wave propagating in the positive z-direction with velocity v p Discuss by means of an example how a function g(t + z/v p ) represents a traveling wave propagating in the negative z-direction with velocity v p.

4-27 Review Questions (Continued) 4.6. What is the significance of the intrinsic impedance of free space? What is its value? 4.7. Summarize the procedure for obtaining the solution for the electromagnetic field due to the infinite plane sheet of uniform time-varying current density State and discuss the solution for the electromagnetic field due to the infinite plane sheet of current density J s (t) = – J s (t)a x for z = 0.

4-28 Problem S4.1. Writing expressions for traveling wave functions for specified time and distance variations

4-29 Problem S4.2. Plotting field variations for a specified infinite plane-sheet current source

4-30 Problem S4.3. Source and more field variations from a given field variation of a uniform plane wave

4.2 Sinusoidally Time-Varying Uniform Plane Waves (EEE, Sec. 3.5; FEME, Secs. 4.1, 4.2, 4.4, 4.5)

4-32 Sinusoidal function of time

4-33 Sinusoidal Traveling Waves

4-34

4-35

4-36 The solution for the electromagnetic field is

4-37 Three-dimensional depiction of wave propagation

4-38 Parameters and Properties

4-39

4-40

4-41

4-42 E4.3 Direction of propagation is –z. Then

4-43 E4.4 Array of Two Infinite Plane Current Sheets

4-44

4-45 For both sheets, No radiation to one side of the array. “Endfire” radiation pattern.

4-46 Depiction of superposition of the two waves

4-47 Review Questions 4.9. Why is it important to give special consideration for sinusoidal functions of time and hence sinusoidal waves? Discuss the quantities ω, β, and v p associated with sinusoidally time-varying uniform plane waves Define wavelength. What is the relationship among wavelength, frequency, and phase velocity? What is the wavelength in free space for a frequency of 15 MHz? How is the direction of propagation of a uniform plane wave related to the directions of its fields? What is the direction of the magnetic field of a uniform plane wave having its electric field in the positive z- direction and propagating in the positive x-direction?

4-48 Review Questions (Continued) Discuss the principle of antenna array, with the aid of an example What should be the spacing and the relative phase angle of the current densities for an array of two infinite, plane, parallel current sheets of uniform densities, equal in amplitude, to confine the radiation to the region between the two sheets?

4-49 Problem S4.4. Finding parameters and the electric field for a specified sinusoidal uniform plane wave magnetic field

4-50 Problem S4.5. Apparent wavelengths of a uniform plane wave propagating in an arbitrary direction

4-51 Problem S4.5. Apparent wavelengths of a uniform plane wave propagating in an arbitrary direction (Continued)

4-52 Problem S4.5. Apparent wavelengths of a uniform plane wave propagating in an arbitrary direction (Continued)

4-53 Problem S4.6. Ratio of amplitudes of the electric field on either side of an array of two infinite plane current sheets

Polarization (EEE, Sec. 3.6; FEME, Sec. 1.4, 4.5)

4-55 Sinusoidal function of time

4-56 Polarization is the characteristic which describes how the position of the tip of the vector varies with time. Linear Polarization: Tip of the vector describes a line. Circular Polarization: Tip of the vector describes a circle.

4-57 Elliptical Polarization: Tip of the vector describes an ellipse. (i)Linear Polarization Linearly polarized in the x direction. Direction remains along the x axis Magnitude varies sinusoidally with time

4-58 Linear polarization

4-59 Direction remains along the y axis Magnitude varies sinusoidally with time Linearly polarized in the y direction. If two (or more) component linearly polarized vectors are in phase, (or in phase opposition), then their sum vector is also linearly polarized. Ex:

4-60 Sum of two linearly polarized vectors in phase is a linearly polarized vector

4-61 (ii) Circular Polarization If two component linearly polarized vectors are (a) equal to amplitude (b) differ in direction by 90˚ (c) differ in phase by 90˚, then their sum vector is circularly polarized.

4-62 Circular Polarization

4-63 Example:

4-64 (iii) Elliptical Polarization In the general case in which either of (i) or (ii) is not satisfied, then the sum of the two component linearly polarized vectors is an elliptically polarized vector. Example:

4-65 Example:

4-66 D3.17 F 1 and F 2 are equal in amplitude (= F 0 ) and differ in direction by 90˚. The phase difference (say  ) depends on z in the manner –2  z – (–3  z) =  z. (a)At (3, 4, 0),  =  (0) = 0. (b)At (3, –2, 0.5),  =  (0.5) = 0.5 .

4-67 (c) At (–2, 1, 1),  =  (1) = . (d)At (–1, –3, 0.2) =  =  (0.2) = 0.2 .

4-68 Clockwise and Counterclockwise Polarizations In the case of circular and elliptical polarizations for the field of a propagating wave, one can distinguish between clockwise (cw) and counterclockwise (ccw) polarizations. If the field vector in a constant phase plane rotates with time in the cw sense, as viewed along the direction of propagation of the wave, it is said to be cw- or right-circularly (or elliptically) polarized. If it rotates in the ccw sense, it is said to be ccw- or left- circularly (or elliptically) polarized.

4-69 For example, consider the circularly polarized electric field of a wave propagating in the +z-direction, given by Then, considering the time variation of the field vector in the z = 0 plane, we note that for and for

4-70 Review Questions A sinusoidally time-varying vector is expressed in terms of its components along the x-, y-, and z- axes. What is the polarization of each of the components? What are the conditions for the sum of two linearly polarized sinusoidally time-varying vectors to be circularly polarized? What is the polarization for the general case of the sum of two sinusoidally time-varying linearly polarized vectors having arbitrary amplitudes, phase angles, and directions? Discuss clockwise and counterclockwise circular and elliptical polarizations associated with sinusoidally time-varying uniform plane waves.

4-71 Problem S4.7. Expressing uniform plane wave field in terms of right- and left- circularly polarized components

4-72 Problem S4.8. Finding the polarization parameters for an elliptically polarized uniform plane wave field

Power Flow and Energy Storage (EEE, Sec. 3.7; FEME, Sec. 4.6)

4-74 Consider the quantity. Then, from a vector identity, Substituting

4-75 Performing volume integration on both sides, and using the divergence theorem for the last term on the right side, we get where we have defined, known as the Poynting vector. The equation is known as the Poynting’s Theorem.

4-76 Poynting’s Theorem Source power density, (power per unit volume), W/m 3 Electric stored energy density, J/m 3 Magnetic stored energy density, J/m 3 Power flow out of S

4-77 Interpretation of Poynting’s Theorem Poynting’s Theorem says that the power delivered to the volume V by the current source J 0 is accounted for by the sum of the time rates of increase of the energies stored in the electric and magnetic fields in the volume, plus another term, which we must interpret as the power carried by the electromagnetic field out of the volume V, for conservation of energy to be satisfied. It then follows that the Poynting vector P has the meaning of power flow density vector associated with the electromagnetic field. We note that the units of E x H are volts per meter times amperes per meter, or watts per square meter (W/m 2 ) and do indeed represent power flow density.

4-78 In the case of the infinite plane sheet of current, note that the electric field adjacent to and on either side of it is directed opposite to the current density. Hence, some work has to be done by an external agent (source) for the current to flow, and represents the power density (per unit volume) associated with this work.

4-79 Review Questions What is the Poynting vector? What is the physical interpretation of the Poynting vector over a closed surface? State Poynting’s theorem. How is it derived from Maxwell’s curl equations? Discuss the interpretation of Poynting’s theorem What are the energy densities associated with electric and magnetic fields? Discuss how fields far from a physical antenna vary inversely with distance from the antenna.

4-80 Problem S4.9. Finding the Poynting vector and power radiated for specified radiation fields of an antenna

4-81 Problem S4.10. Finding the electric field and magnetic field energies stored in a parallel-plate resonator

4-82 Problem S4.10. (Continued)

4-83 Problem S4.11. Finding the work associated with rearranging a charge distribution

The End