Uniform Plane Waves. Ranga Rodrigo. University of Moratuwa. November 7, 2008

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1 Uniform Plane Waves Ranga Rodrigo University of Moratuwa November 7, 2008 Ranga Rodrigo (University of Moratuwa) Uniform Plane Waves November 7, / 51

2 Summary of Last Week s Lecture Basic Relations D = εe, H = B µ, where ε and µ are the permittivity and permeability, respectively, of the medium. In addition, if the current density J is due to the conduction, then J = J c = σe, where σ is the conductivity of the medium. Ranga Rodrigo (University of Moratuwa) Uniform Plane Waves November 7, / 51

3 Summary of Last Week s Lecture Maxwell s Equations in Integral Form E dl = d B ds. C dt S H dl = J ds + d C S dt D ds = ρdv. S V B ds = 0. S S D ds. Ranga Rodrigo (University of Moratuwa) Uniform Plane Waves November 7, / 51

4 Summary of Last Week s Lecture Boundary Conditions At a point on the boundary, the tangential components of E and the normal components of B are continuous, whereas the tangential components of H are discontinuous by the amount equal to J S at that point, and the normal components of D are discontinuous by the amount equal to ρ S at that point. i n (E 1 E 2 ) = 0 i n (H 1 H 2 ) = J s i n (D 1 D 2 ) = ρ s i n (B 1 B 2 ) = 0 Ranga Rodrigo (University of Moratuwa) Uniform Plane Waves November 7, / 51

5 Summary of Last Week s Lecture Maxwell s Equations in Differential Form E = B t. H = J + D t. D = ρ. B = 0. Ranga Rodrigo (University of Moratuwa) Uniform Plane Waves November 7, / 51

6 Summary of Last Week s Lecture Maxwell s Equations in Free Space E = B t. H = D t. D = 0. B = 0. Ranga Rodrigo (University of Moratuwa) Uniform Plane Waves November 7, / 51

7 Introduction Introduction Now we shall turn our attention to the application of Maxwell s equations to time-varying fields. Many of these applications are based on electromagnetic wave phenomena, and hence it is necessary to gain an understanding of the basic principles of wave propagation. We will consider wave propagation in free space. Ranga Rodrigo (University of Moratuwa) Uniform Plane Waves November 7, / 51

8 Introduction Introduction We shall learn how the coupling between space variations and time variations of electric and magnetic fields, as indicated by Maxwell s equations, results in wave motion and illustrate the basic principles of radiation of waves from an antenna. Ranga Rodrigo (University of Moratuwa) Uniform Plane Waves November 7, / 51

9 Uniform Plane Waves in Time Domain in Free Space Uniform Plane Waves in Time Domain in Free Space Space variations of the electric and magnetic field components are related to the time variations of the magnetic and electric field components, respectively, through Maxwell s equations. This interdependence gives rise to the phenomenon of electromagnetic wave propagation. Ranga Rodrigo (University of Moratuwa) Uniform Plane Waves November 7, / 51

10 Uniform Plane Waves in Time Domain in Free Space In the general case, electromagnetic wave propagation involves electric and magnetic fields having more than one component, each dependent on all three coordinates, in addition to time. However a simple and very useful wave serves as a building block in the study of electromagnetic waves consists of electric and magnetic fields which are perpendicular to each other and to the direction of propagation and are uniform in planes perpendicular to the direction of propagation. Ranga Rodrigo (University of Moratuwa) Uniform Plane Waves November 7, / 51

11 Uniform Plane Waves in Time Domain in Free Space These waves are known as uniform plane waves. Electric field: x-direction Magnetic field: y-direction Direction of propagation: z-direction E = E x (z, t)i x H = H y (z, t)i y (1a) (1b) Uniform plane waves do not exist in practice because they cannot be produced by finite-sized antennas at large distances from physical antennas and ground, however, the waves can be approximated as uniform plane waves. Ranga Rodrigo (University of Moratuwa) Uniform Plane Waves November 7, / 51

12 Uniform Plane Waves in Time Domain in Free Space Let s consider an idealized, hypothetical source consisting of an infinite sheet lying in the xy-plane. J s = J s (t)i x for z = 0. (2) J s (t) is a given function of time. The current crossing the line parallel to y-axis of width w is wj s (t). If J s (t) = J so cos ωt, then the current wj so cos ωt crossing the width w, actually alternates between negative x- and positive x-directions. Ranga Rodrigo (University of Moratuwa) Uniform Plane Waves November 7, / 51

13 Uniform Plane Waves in Time Domain in Free Space To find the electromagnetic field due to the time-varying current sheet we shall begin with Faraday s law and Ampère s circuital law. E = B t Faraday s law. H = J + D t Ampère s circuital law. Ranga Rodrigo (University of Moratuwa) Uniform Plane Waves November 7, / 51

14 Uniform Plane Waves in Time Domain in Free Space The process will comprise of four steps: 1 Obtain the particular differential equations. 2 Derive the general solutions without regard to the current on the sheet. 3 Show that the solution obtained in step 2 is superposition of traveling waves propagating in +z- and z-directions. 4 Extend the general solution of step 2 to take into account the current on the sheet and thereby obtain the required solution. Ranga Rodrigo (University of Moratuwa) Uniform Plane Waves November 7, / 51

15 Particular Differential Equations 1. Particular Differential Equations J s = J s (t)i x for z = 0 (current density). Current distribution has only an x-component of the current density which varies only with z. So we can set J y, J z, and all derivatives with respect to x and y in E = B t H = J + D t equal to 0. and E y z = B x t E x z = B y t H y z = J x + D x t H x z = D y t (3) (4) Ranga Rodrigo (University of Moratuwa) Uniform Plane Waves November 7, / 51

16 Particular Differential Equations 0 = B z 0 = D z t t There are only two equations involving J x : (5) E x z = B y, and (6a) t H y z = J x D x. (6b) t These are simplified forms of Faraday s law and Ampère s circuital law, respectively, for the special case of the electric and magnetic fields characterized by 1a and 1b. Ranga Rodrigo (University of Moratuwa) Uniform Plane Waves November 7, / 51

17 Derivation of Wave Equation 2. Derivation of Wave Equation In applying 6a and 6b to 2 we note that J x in 6b is a volume current density, whereas 2 represents a surface current density. Hence we shall solve 6a and 6b by setting J x = 0 and then extend the solution to take into account the current on the sheet. For J x = 0, 6a and 6b E x z = B y t H y = µ 0. (7a) t H y z = D x t = ε 0 E u t. (7b) Ranga Rodrigo (University of Moratuwa) Uniform Plane Waves November 7, / 51

18 We have thus eliminated H y from 7a and 7b and obtained a single second order partial differential equation involving E x only. Equation 8 is known as the wave equation. In particular it is a one-dimensional wave equation in time domain form, that is, for arbitrary time dependence of E x. To obtain a solution for 8 we introduce a change of variable by defining τ = z µ 0 ε 0. Ranga Rodrigo (University of Moratuwa) Uniform Plane Waves November 7, / 51 Derivation of Wave Equation Differentiating 7a with respect to z and then substituting H y z from 7b we obtain 2 E x z 2 = µ 0ε 0 2 E x t 2 (8)

19 Derivation of Wave Equation Substituting for z in 8 in terms of τ, we then have or 2 E x τ 2 = 2 E x t 2, (9) 2 E x τ 2 E x 2 t = 0. ( 2 τ + ) ( t τ ) E x = 0. (10) t Equation 10 is satisfied if ( τ ± ) E x = 0. t Ranga Rodrigo (University of Moratuwa) Uniform Plane Waves November 7, / 51

20 Derivation of Wave Equation or τ E x = E x t. (11) Consider E x τ = E x t. This equation says that the partial derivative of E x (τ, t) with respect to τ is equal to the negative of the partial derivative of E x (τ, t) with respect to t. Ranga Rodrigo (University of Moratuwa) Uniform Plane Waves November 7, / 51

21 Derivation of Wave Equation The simplest function that satisfies this requirement is (t τ). Any arbitrary function of (t τ), say, f(t τ) satisfies the requirement: and t [f(t τ)] = f (t τ) t (t τ) = f (t τ) t [f(t τ)] = f (t τ) τ (t τ) = f (t τ) = [f(t τ)], τ where f (t τ) is the derivative with respect to t τ. Ranga Rodrigo (University of Moratuwa) Uniform Plane Waves November 7, / 51

22 Derivation of Wave Equation In a similar manner, the solution for the equation corresponding to E x τ = E x t can be seen to be any function of (t + τ), say, g(t + τ). Ranga Rodrigo (University of Moratuwa) Uniform Plane Waves November 7, / 51

23 Derivation of Wave Equation Combining the two solutions for 11 E x (τ, t) = Af(t τ) + Bg(t + τ) (12) where A and B are arbitrary constants. Substituting for τ in 12 in terms of z, we obtain the solution for 8 to be E x (z, t) = Af(t z µ 0 ε 0 )+Bg(t +z µ 0 ε 0 ). (13) Ranga Rodrigo (University of Moratuwa) Uniform Plane Waves November 7, / 51

24 Derivation of Wave Equation The corresponding solution for H y (z, t) can be obtained by substituting 13 into 7a and 7b H y ε0 [ = Af (t z µ 0 ε 0 ) Bg (t + z µ 0 ε 0 ) ]. t µ 0 (14) H y (z, t) = 1 [ Af(t z µ0 ε 0 ) Bg(t + z µ 0 ε 0 ) ]. µ0 ε 0 Ranga Rodrigo (University of Moratuwa) Uniform Plane Waves November 7, / 51

25 Traveling Wave Function Traveling Wave Function Let s consider, for example, f(t z µ 0 ε 0 ) = ( t z µ 0 ε 0 ) 2. As time increases from 0 to µ 0 ε 0, every point in the plot for t = 0 moves by one unit in the +z direction, thereby making the plot for t = µ 0 ε 0, an exact replica of the plot for t = 0, except displaced by one unit in the +z direction. The function f is therefore said to represent a traveling wave propagation in the +z direction or simply a (+) wave. In particular, it is a uniform plane wave since its value does not vary within a constant z plane. Ranga Rodrigo (University of Moratuwa) Uniform Plane Waves November 7, / 51

26 Traveling Wave Function By dividing the distance traveled by the time taken, the velocity of propagation of the wave can be obtained by v p = 1 µ0 ε 0 = m/s, (15) which is equal to c, the velocity of light in free space. Function g is said to represent a traveling wave propagating in the z- direction or simply a ( ) wave Ranga Rodrigo (University of Moratuwa) Uniform Plane Waves November 7, / 51

27 Traveling Wave Function We can define the intrinsic impedance of free space,η 0, to be η 0 = µ0 ε 0 120πΩ = 377Ω (16) From 13 and 14 we see that η 0 is the ratio of E x to H y for the (+) wave or the negative of the same ratio for the ( ) wave. E x V/m H y A/m E x /H y V/A = Ω Ranga Rodrigo (University of Moratuwa) Uniform Plane Waves November 7, / 51

28 Traveling Wave Function Using η 0 = E x (z, t) = Af µ0 and v p = 1, ε 0 µ0 ε 0 ) ) (t (t zvp + zvp + Bg H y (z, t) = 1 η 0 [Af(t z v p ) Bg(t + z v p ), (17a) ]. (17b) Ranga Rodrigo (University of Moratuwa) Uniform Plane Waves November 7, / 51

29 Electromagnetic Field due to the Current Sheet Electromagnetic Field due to the Current Sheet Let s now consider the current density given by J s = J s (t)i x for z = 0. Since the current sheet, which is the source of waves, in the z = 0 plane, then can be only a (+) wave in the region z > 0 and only a ( ) wave in the region z < 0. Ranga Rodrigo (University of Moratuwa) Uniform Plane Waves November 7, / 51

30 Electromagnetic Field due to the Current Sheet Thus Af ( ) t z v ix E(z, t) = p for z > 0, Bg ( ) t + z v p t x for z < 0. A η H(z, t) = 0 f ( ) t z v iy p for z > 0, B η 0 g ( ) t + z v iy p for z < 0. (18a) (18b) Ranga Rodrigo (University of Moratuwa) Uniform Plane Waves November 7, / 51

31 Electromagnetic Field due to the Current Sheet From the boundary condition applied to the surface z = 0, we have [E x ] z=0 + [E x ] z=0 = 0 (19) or Af(t) = Bg(t). Thus 18a and 18b reduce to ) (t zvp E(z, t) = F H(z, t) = ± 1 η 0 F (t zvp ) where Af(t) = Bg(t) = F(t) i x for z 0 (20a) i y for z 0 (20b) Ranga Rodrigo (University of Moratuwa) Uniform Plane Waves November 7, / 51

32 Electromagnetic Field due to the Current Sheet From boundary condition applied to the surface z = 0 we have i z { [H] z=0 + [H] z=0 } = Js (t)i x (21) or 2 η 0 F(t) = J s (t). Thus F(t) = ( ) η 0 2 Js (t) and 20a and 20b become E(z, t) = η 0 z J s H(z, t) = ± 1 z J s (t zvp ) (t zvp ) i x for z 0. (22a) i y for z 0. (22b) Ranga Rodrigo (University of Moratuwa) Uniform Plane Waves November 7, / 51

33 Electromagnetic Field due to the Current Sheet Equation 22a and 22b represent the complete solution for the electromagnetic field due to infinite plane current sheet of surface current density given by J s (t) = J s (t)i x for z = 0. (23) The solution corresponds to uniform plane waves having their field components uniform in planes parallel to the current sheet and propagating to either side of the current sheet with velocity v p (= c). Ranga Rodrigo (University of Moratuwa) Uniform Plane Waves November 7, / 51

34 Electromagnetic Field due to the Current Sheet The time variation of the electric field component E x in a given z = constant plane is the same as the current density variation delayed by the time z v p and multiplied by η 0 2. The time variation of the magnetic field component in a z =constant plane is the same as the current density variation delayed by z v p and multiplied by ± 1 z depending on z 0. Using these properties we can construct plots of the field components versus time for fixed values of z and versus z for fixed values of t. Ranga Rodrigo (University of Moratuwa) Uniform Plane Waves November 7, / 51

35 Sinusoidally Time-Varying Uniform Plane Waves in Free Space Sinusoidally Time-Varying Uniform Plane Waves in Free Space Fields that vary sinusoidally with time are produced by sources whose current densities vary sinusoidally with time. Thus assuming the current density on the infinite plane sheet to be J s = J s0 cos ωt i x for z = 0 (24) where J s0 is the amplitude and ω is the radian frequency, we obtain the corresponding solution for the electromagnetic field by substituting J s (t) = J s0 cos ωt in 22a and 22b. Ranga Rodrigo (University of Moratuwa) Uniform Plane Waves November 7, / 51

36 Sinusoidally Time-Varying Uniform Plane Waves in Free Space E = η 0J s0 z cos(ωt β z )i x for z 0 (25a) H = ± J s0 z cos(ωt β z)i y for z 0 (25b) β = ω v p (26) Equations 25a and 25b represent sinusoidally time-varying uniform plane waves propagating away from the current sheet. Ranga Rodrigo (University of Moratuwa) Uniform Plane Waves November 7, / 51

37 Sinusoidally Time-Varying Uniform Plane Waves in Free Space It should be understood that in these sketches the field variation depicted along the z-axis hold also for any other line parallel to the z-axis. The argument (ωt β z ) of the cosine function is the phase of the fields φ = ωt β z (27) φ is a function of t and z. Since φ = ω (28) t the rate of change of phase with time for a fixed value of z is equal to ω, the radian frequency of the wave. Ranga Rodrigo (University of Moratuwa) Uniform Plane Waves November 7, / 51

38 Sinusoidally Time-Varying Uniform Plane Waves in Free Space The linear frequency is given by f = ω 2π. (29) Since φ = β, (30) z the magnitude of the rate of change of phase with distance is equal to β, known as the phase constant. It follows that the distance, along z-direction, in which the phase changes by 2π radian for a fixed value of time is equal to 2π β. Ranga Rodrigo (University of Moratuwa) Uniform Plane Waves November 7, / 51

39 Sinusoidally Time-Varying Uniform Plane Waves in Free Space This length is known as the wavelength denoted by the symbol λ. Thus λ = 2π β. (31) From 26 we note that the velocity of propagation of the wave is given by v p = ω β. (32) Here it is known as the phase velocity, since a constant value of phase progresses with that velocity along the z-direction Ranga Rodrigo (University of Moratuwa) Uniform Plane Waves November 7, / 51

40 Sinusoidally Time-Varying Uniform Plane Waves in Free Space It is the velocity with which an observer has to move along the direction of propagation of the wave to be associated with a particular phase point on the moving sinusoid. Thus it follows from 27 that which gives ωdt βdz = 0 d(ωt βdz) = 0 (33) dz dt = ±ω β where + and signs correspond to (+) and ( ) waves, respectively. Ranga Rodrigo (University of Moratuwa) Uniform Plane Waves November 7, / 51

41 Sinusoidally Time-Varying Uniform Plane Waves in Free Space We recall that for free space, v p = 1 µ0 ε 0 = c = m/s From 31, 29 and 32 we note that ( ) ( ) 2π ω λf = = ω β 2π β. (34) λf = v p. The electric and magnetic fields are such that Amplitude of E Amplitude of H = η 0 (35) we recall that η 0, the intrinsic impedance of free space, has a value approximately equal to 120π or 377Ω. Ranga Rodrigo (University of Moratuwa) Uniform Plane Waves November 7, / 51

42 Sinusoidally Time-Varying Uniform Plane Waves in Free Space The electric and magnetic fields have components lying in the places of constant phase (z= constant planes) and perpendicular to each other and to the direction of propagation. In fact, the cross product of E and H results in a vector that is directed along the direction of propagation, as can be seen by noting that E H = ± η 2 0J s0 cos 2 (ωt β z )i z for z 0. 4 (36) The fields given in 25a and 25b are linearly polarized. Hence the wave is said to be linearly polarized. Ranga Rodrigo (University of Moratuwa) Uniform Plane Waves November 7, / 51

43 Summary We obtained the uniform plane wave solution to Maxwell s equations in time domain in free space by considering an infinite plane current sheet in the xy-plane with uniform current density, given by J s = J s (t)i x A/m. Ranga Rodrigo (University of Moratuwa) Uniform Plane Waves November 7, / 51

44 Summary We found that the electromagnetic field due to the current sheet to be considering an infinite plane current sheet in the xy-plane with uniform current density, given by E = η 0 2 J s H = ± 1 2 J s (t zvp ) (t zvp ) v p = 1 µ0 ε 0. i x for z 0, i y for z 0. η 0 = µ0 ε 0. Ranga Rodrigo (University of Moratuwa) Uniform Plane Waves November 7, / 51

45 Summary Uniform Plane Waves Due to Sinusoidally Varying Current Density on Sheet We extended the solution to sinusoidally time-varying uniform plane waves. J s = J s0 cos ωt i x A/m. Corresponding field will be E = η 0J s0 2 cos (ωt βz) i x for z 0, H = ± J s0 2 cos (ωt βz) i y for z 0. Ranga Rodrigo (University of Moratuwa) Uniform Plane Waves November 7, / 51

46 Summary Phase constant: β = ω v p = ω µ 0 ε 0. Phase velocity: Wavelength: v p = ω β. λ = 2π β. v p = λf. Ranga Rodrigo (University of Moratuwa) Uniform Plane Waves November 7, / 51

47 Summary Uniform Plane Waves in a Material Medium Maxwell s equations for a material medium: E = B t H = J c + D = µ H t. = σe + ε E t. t Infinite plane current sheet of uniform surface current density J s = J s0 cos ωt i x A/m. in the xy-plane and embedded in the material medium. Ranga Rodrigo (University of Moratuwa) Uniform Plane Waves November 7, / 51

48 Summary Maxwell s equations for a material medium: E = B t H = J c + D t = µ H t. = σe + ε E t. Infinite plane current sheet of uniform surface current density J s = J s0 cos ωt i x A/m. in the xy-plane and embedded in the material medium and obtained. Ranga Rodrigo (University of Moratuwa) Uniform Plane Waves November 7, / 51

49 Summary Electromagnetic field due to it is E = η J s0 2 e αz cos(ωt βz + τ)i x forz 0. H = ± J s0 2 e αz cos(ωt βz)i y forz 0. Here α and β are the attenuation and phase constants given, respectively, by the real and imaginary parts of the propagation constant, γ. Thus γ = α + jβ = jωµ(σ + jωε). η = η e jτ = jωµ σ + jωε. Ranga Rodrigo (University of Moratuwa) Uniform Plane Waves November 7, / 51

50 Summary Above solutions indicate to us that the wave propagation in the material medium is characterized by attenuation as indicated by e αz and the phase difference between E and H by the amount τ. Special cases: 1 Perfect dielectrics: σ = 0 2 Imperfect dielectrics: σ ωε 3 Good conductors: σ ωε 4 Perfect conductors: σ Ranga Rodrigo (University of Moratuwa) Uniform Plane Waves November 7, / 51

51 Reference Summary Nannapaneni Narayana Rao. Elements of Engineering Electromaganetics. Prentice Hall, 4th edition, Ranga Rodrigo (University of Moratuwa) Uniform Plane Waves November 7, / 51

Antennas and Propagation

Antennas and Propagation Ranga Rodrigo University of Moratuwa October 20, 2008 Compiled based on Lectures of Prof. (Mrs.) Indra Dayawansa. Ranga Rodrigo (University of Moratuwa) Antennas and Propagation