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NUS/ECE EE2011 Plane Wave Propagation in Lossless Media See animation “Plane Wave Viewer” 1 Plane Waves in Lossless Media In a source free lossless medium, J = ρ = σ = 0. Maxwell’s equations: ∂H ∇ × E = -μ ∂t ∂E ∇×H = ε ∂t ε∇ ⋅ E = 0 J =current density ρ =charge density σ =conductivity μ∇ ⋅ H = 0 Hon Tat Hui 1 Plane Wave Propagation in Lossless Media NUS/ECE EE2011 Take the curl of the first equation and make use of the second and the third equations, we have: Note : 2 ∂ ∂ ∇ 2 E = μ ∇ × H = με 2 E ∂t ∂t ∇ × ∇ × E = ∇(∇ ⋅ E ) − ∇ 2 E This is called the wave equation: 2 ∂ ∇ 2E − με 2 E = 0 ∂t A similar equation for H can be obtained: ∂ ∇ H − με 2 H = 0 ∂t 2 2 Hon Tat Hui 2 Plane Wave Propagation in Lossless Media NUS/ECE EE2011 In free space, the wave equation for E is: ∂ ∇ E − μ 0ε 0 2 E = 0 ∂t 2 2 where 1 μ 0ε 0 = 2 c c being the speed of light in free space (~ 3 × 108 (m/s)). Hence the speed of light can be derived from Maxwell’s equation. Hon Tat Hui 3 Plane Wave Propagation in Lossless Media NUS/ECE EE2011 To simplify subsequent analyses, we consider a special case in which the field (and the source) variation with time takes the form of a sinusoidal function: sin(ωt + φ ) or cos(ωt + φ ) Using complex notation, the E field, for example, can be written as: • ⎧ ⎫ E( x, y , z , t ) = Re⎨E( x, y , z )e jωt ⎬ ⎩ ⎭ • where E( x, y, z ) is called the phasor form of E(x,y,z,t) and is in general a complex number depending on the spatial coordinates only. Note that the phasor form also includes the initial phase information and is a complex number. Hon Tat Hui 4 Plane Wave Propagation in Lossless Media NUS/ECE EE2011 The benefits of using the phasor form are that: ⎧ ∂n • ∂n j ωt ⎫ E( x, y, z , t ) = Re ⎨ n E( x, y, z )e ⎬ n ∂t ⎩ ∂t ⎭ • n ⎧ jωt ⎫ = Re ⎨( jω ) E( x, y, z )e ⎬ ⎩ ⎭ • ⎧ ⎫ jωt ∫ "∫ E(x, y, z, t )dt " dt = Re⎨⎩∫ "∫ E(x, y, z )e dt " dt ⎬⎭ ⎧ 1 • jωt ⎫ = Re⎨ E( x, y, z )e ⎬ n ⎩ ( jω ) ⎭ Hon Tat Hui 5 Plane Wave Propagation in Lossless Media NUS/ECE EE2011 Therefore differentiation or integration with respect to time can be replaced by multiplication or division of the phasor form with the factor jω. All other field functions and source functions can be expressed in the phasor form. As all time-harmonic functions involve the common factor ejωt in their phasor form expressions, we can eliminate this factor when dealing with the Maxwell’s equation. The wave equation can now be put in phasor form as (dropping the dot on the top, same as below): 2 • • ∂ 2 ∇ 2 E − μ0ε 0 2 E = 0 ⇒ ∇ 2 E− μ0ε 0 ( jω ) E = 0 ∂t (dropping the dot sign) ⇒ ∇ 2 E + μ0ε 0ω 2 E = 0 Hon Tat Hui 6 Plane Wave Propagation in Lossless Media NUS/ECE EE2011 In phasor form, Maxwell’s equations can be written as: ∇ × E = - jωB ∇ × H = jωD ∇⋅D = ρ ∇⋅B = 0 Using the phasor form expression, the wave equation for E field is also called the Helmholtz’s equation, which is: ∇ 2 E + μ0ε 0ω 2 E = ∇ 2 E + k 2 E = 0 where k = ω μ0ε 0 Hon Tat Hui 7 Plane Wave Propagation in Lossless Media NUS/ECE EE2011 k is called the wavenumber or the propagation constant. 2πf 2π k = k 0 = ω μ 0ε 0 = = λ0 c where λ0 is the free space wavelength. In an arbitrary medium with ε =ε0εr and μ =μ0μ r, 2πf k = ω μ 0ε 0 μ r ε r = c We call, 2π λ= = k Hon Tat Hui μ rε r = 2π λ0 μrε r λ0 = wavelength in the medium μrε r 8 Plane Wave Propagation in Lossless Media NUS/ECE EE2011 In Cartesian coordinates, the Helmholtz’s equation can be written as three scalar equations in terms of the respective x, y, and z components of the E field. For example, the scalar equation for the Ex component is: ⎛ ∂2 ∂2 ∂2 2⎞ ⎜⎜ 2 + 2 + 2 + k ⎟⎟ Ex = 0 ∂y ∂z ⎠ ⎝ ∂x Consider a special case of the Ex in which there is no variation of Ex in the x and y directions, i.e., ∂2 ∂2 Ex = 2 Ex = 0 2 ∂y ∂x Hon Tat Hui 9 Plane Wave Propagation in Lossless Media NUS/ECE EE2011 This is called the plane wave condition and Ex(z) now varies with z only. The wave equation for Ex becomes: d 2 Ex (z ) 2 + k Ex (z ) = 0 2 dz Note that a plane wave is not physically realizable because it extends to an infinite extent in the x and y directions. However, when considered over a small plane area, its propagation characteristic is very close to a spherical wave, which is a real and common form of electromagnetic wave propagating. Hon Tat Hui 10 Plane Wave Propagation in Lossless Media NUS/ECE EE2011 Solutions to the plane wave equation take one form of the following functions, depending on the boundary conditions: 1. E x ( z ) = E0+ e − jkz 2. E x ( z ) = E0− e + jkz 3. E x (z ) = E0+ e − jkz + E0− e + jkz E0+ and E0- are constants to be determined by boundary conditions. + − jkz 0 − + jkz 0 are plane waves propagating along the E e and E e +z direction and –z direction. Hon Tat Hui 11 Plane Wave Propagation in Lossless Media NUS/ECE EE2011 1.1 Solutions to plane wave equation In time domain, (1) E x ( z ) = E0+ e − jkz → E x ( z , t ) = Re{E0+ e − jkz e jωt } = E0+ cos(ωt − kz ) Assume E0+ = ω = k = 1, then E x (z , t ) = cos(t − z ). We can plot this solution for several seconds to see its motion in space. We focus on one period of the sine function only while keeping in mind that this period repeats itself continuously in both left and right directions. Hon Tat Hui 12 Plane Wave Propagation in Lossless Media NUS/ECE EE2011 At t = 0, E x (z ,0 ) = cos(− z ) -2π -3π/2 -π Ex 1 0 -π/2 π/2 π 3π/2 2π π 3π/2 2π π 3π/2 2π z -1 At t = 1s, E x ( z ,1) = cos(1 − z ) -2π -3π/2 -π Ex 1 0 -π/2 π/2 z -1 At t = 2 s, E x (z ,2 ) = cos(2 − z ) -2π -3π/2 -π Ex 1 0 -π/2 π/2 z -1 Hon Tat Hui 13 Plane Wave Propagation in Lossless Media NUS/ECE EE2011 ( 2) − + jkz − + jkz jωt ( ) ( ) { E x z = E0 e → E x z , t = Re E0 e e } = E0− cos(ωt + kz ) Assume E0− = ω = k = 1, then E x ( z , t ) = cos(t + z ). The solution is plotted on the next page for the first several seconds. Hon Tat Hui 14 Plane Wave Propagation in Lossless Media NUS/ECE EE2011 At t = 0, E x (z ,0 ) = cos( z ) -2π -3π/2 Ex 1 -π 0 -π/2 π/2 π 3π/2 2π π 3π/2 2π π 3π/2 2π z -1 At t = 1s, E x (z ,1) = cos(1 + z ) -2π -3π/2 Ex 1 -π 0 -π/2 π/2 z -1 At t = 2 s, E x ( z ,2 ) = cos(2 + z ) -2π -3π/2 -π Ex 1 0 -π/2 π/2 z -1 Hon Tat Hui 15 Plane Wave Propagation in Lossless Media NUS/ECE EE2011 E0+ e − jkz is a wave propagating in the + z direction. E0− e + jkz is a wave propagating in the − z direction. For the wave moving in +z direction, in a time of 1 second, the wave moves in 1 unit of distance (for example, meter). Then the speed of propagation is (1 m/1 s = 1ms-1). A similar result can be obtained for the wave moving in –z direction. Hon Tat Hui 16 Plane Wave Propagation in Lossless Media NUS/ECE EE2011 1.2 Propagation speed of a general plane wave If ω ≠ 1, k ≠ 1, and E+0 ≠ 1, then for the wave moving to the right: E x ( z , t ) = E0+ cos(ωt − kz ). + ( ) E z , 0 = E (1) At t = 0, x 0 cos(− kz ). Consider a zero point (z coordinate = z0) of the wave (for example the first one to the right of the origin), then E x (z0 ,0 ) = 0 ⇒ cos(− kz0 ) = 0 ⇒ − kz0 = − Hon Tat Hui 17 π 2 ⇒ z0 = π 2k Plane Wave Propagation in Lossless Media NUS/ECE EE2011 (2) At t = 1s, E x ( z ,0 ) = E0+ cos(ω − kz ). Consider the same zero point of the wave as in (1) but now its z coordinate has move from z0 to z1, then E x (z1 ,1) = 0 ⇒ cos(ω − kz1 ) = 0 ⇒ ω − kz1 = − π 2 ⇒ z1 = Distance change in 1s = z1 − z0 = Thus propagation speed = ω ω k + π 2k / 1s = ω ω k − + π π 2k 2k ms = ω k −1 k k A same result can be obtained for the wave moving to the left. Hon Tat Hui 18 Plane Wave Propagation in Lossless Media NUS/ECE EE2011 1.3 Solution for the magnetic field Once the electric field is known, the accompanying magnetic field H can be found from the Maxwell’s equation ∇ × E = - jωμH For example, if the solution for E is, E( z ) = xˆ E x = xˆ E0+ e − jkz , then the solution for H is: E0+ ∂e − jkz k + − jkz H (z ) = yˆ = yˆ E0 e = yˆ H y ωμ − jωμ ∂z Note that H is ⊥ to E and they are shown on next page. Hon Tat Hui 19 Plane Wave Propagation in Lossless Media NUS/ECE EE2011 H and E propagate in free space See animation “Plane Wave E and H Vector Motions ” Hon Tat Hui 20 Plane Wave Propagation in Lossless Media NUS/ECE EE2011 The ratio of Ex to Hy is called the intrinsic impedance of the medium, η. ωμ E x ( z ) ωμ η= = = = H y (z ) k ω με μ (Ω) ε Note that η is independent of z. In free space, μ0 η0 = = 120π ≈ 377 Ω ε0 and Ex and Hy are in phase (as η0 is a real number). Hon Tat Hui 21 Plane Wave Propagation in Lossless Media NUS/ECE EE2011 The phase velocity (propagation speed of a constantphase point) of the wave up is given by: 1 ω ω up = = = (m/s) k ω με με See animation “Plane Wave in 3D” The plane wave is also called the TEM wave (TEM = Transverse ElectroMagnetic) in which Ez = Hz = 0 where z is the direction of propagation. Hon Tat Hui 22 Plane Wave Propagation in Lossless Media NUS/ECE EE2011 1.4 Expressions for a general plane wave The general expressions of a plane wave are: E = E0 e − jk ⋅r H = H 0e − jk ⋅r E0 and H0 are vectors in arbitrary directions. k is the vector propagation constant whose magnitude is k and whose direction is the direction of propagation of the wave. r is the observation position vector. k = k x xˆ + k y yˆ + k z zˆ , k = k x2 + k y2 + k z2 r = xxˆ + yyˆ + zzˆ Hon Tat Hui 23 Plane Wave Propagation in Lossless Media NUS/ECE EE2011 Right-hand rule: k (index finger) (thumb) r E E k H H (middle finger) Hon Tat Hui 24 Plane Wave Propagation in Lossless Media NUS/ECE EE2011 Using Maxwell’s equations ∇ × E = - jωμH ∇ × H = jωεE it can be shown that we have the following relations for the field vectors and the propagation direction. E⊥H⊥k 1ˆ H = k × E, E = ηH × kˆ η Hon Tat Hui 25 Plane Wave Propagation in Lossless Media NUS/ECE EE2011 Example 1 A uniform plane wave with E = xˆ E x propagates in the +zdirection in a lossless medium with εr = 4 and μr = 1. Assume that Ex is sinusoidal with a frequency of 100 MHz and that it has a positive maximum value of 10-4 V/m at t = 0 and z = 1/8 m. (a) Calculate the wavelength λ and the phase velocity up, and find expressions for the instantaneous electric and magnetic field intensities. (b) Determine the positions where Ex is a positive maximum at the time instant t = 10-8s. Hon Tat Hui 26 Plane Wave Propagation in Lossless Media NUS/ECE EE2011 eˆ = unit vector of E Solutions ˆ ⋅r = kz, , kkk̂ (a) k̂ k up = ω k = 1 c = = 1.5 × 108 (m/s) 4 με (phasor form) k̂ (instantaneous form) Hon Tat Hui 27 Plane Wave Propagation in Lossless Media NUS/ECE EE2011 The cosine function has a positive maximum when its argument equals zero (ignoring the 2nπ ambiguity). Thus at t = 0 and z = 1/8, k̂ k̂ Hon Tat Hui 28 Plane Wave Propagation in Lossless Media NUS/ECE EE2011 kˆ = zˆ (m) See animation “Plane Wave Simulator” Hon Tat Hui 29 Plane Wave Propagation in Lossless Media NUS/ECE EE2011 2 Polarization of Plane Waves The polarization of a plane wave is the figure the tip of the instantaneous electric-field vector E traces out with time at a fixed observation point. There are three types of polarizations: the linear, circular, and elliptical polarizations. Ey Ey Ey Ex Eectric-field vector Linearly polarized Ex Ex Eectric-field vector Eectric-field vector Circularly polarized Elliptically polarized See animation “Polarization of a Plane Wave - 2D View” Hon Tat Hui 30 Plane Wave Propagation in Lossless Media NUS/ECE EE2011 See animation “Polarization of a Plane Wave - 3D View” Polarization of a Plane Wave Hon Tat Hui 31 Plane Wave Propagation in Lossless Media NUS/ECE EE2011 (a) Linear polarization A plane wave is linearly polarized at a fixed observation point if the tip of the electric-field vector at that point moves along the same straight line at every instant of time. (b) Circular Polarization A plane wave is circularly polarized at a a fixed observation point if the tip of the electric-field vector at that point traces out a circle as a function of time. Circular polarization can be either right-handed or lefthanded corresponding to the electric-field vector rotating clockwise (right-handed) or anti-clockwise (left-handed). Hon Tat Hui 32 Plane Wave Propagation in Lossless Media NUS/ECE EE2011 (c) Elliptical Polarization A plane wave is elliptically polarized at a a fixed observation point if the tip of the electric-field vector at that point traces out an ellipse as a function of time. Elliptically polarization can be either right-handed or left-handed corresponding to the electric-field vector rotating clockwise (right-handed) or anti-clockwise (left-handed). Hon Tat Hui 33 Plane Wave Propagation in Lossless Media NUS/ECE EE2011 For example, consider a plane wave: E x = E x 0 e − jkz E = xˆ E x + yˆ E y = xˆ E x 0 e − jkz Ex0 and Ey0 are both real numbers − yˆ jE y 0 e − jkz E y = − jE y 0 e − jkz Note that the phase difference between Ex and Ey is 90º. The instantaneous expression for E is: { E( z , t ) = Re xˆ E x 0 e jωt − jkz − yˆ jE y 0 e jωt − jkz } = xˆ E x 0 cos(ωt − kz ) + yˆ E y 0 sin (ωt − kz ) Let: X = Ex =Ex 0 cos (ωt − kz ) , Y = E y = E y 0 sin (ωt − kz ) Hon Tat Hui 34 Plane Wave Propagation in Lossless Media NUS/ECE EE2011 Case 1: Exo = 0 or E yo = 0, then X = 0 or Y = 0 Both are straight lines. Hence the wave is linearly polarized. Case 2: Exo = E yo = C , then X 2 + Y 2 = C 2 ⎡⎣cos 2 (ωt − kz ) + sin 2 (ωt − kz ) ⎤⎦ = C 2 X and Y describe a circle. Hence the wave is circularly polarized. Case 3: Exo ≠ E yo , then X2 Y2 2 2 t kz + = cos ω − + sin ( ) (ωt − kz ) = 1 2 2 Ex 0 E y 0 X and Y describe an ellipse. Hence the wave is elliptically polarized. Hon Tat Hui 35 Plane Wave Propagation in Lossless Media NUS/ECE EE2011 Example 2 Two circularly polarized plane waves watched at z = 0 are given by: E1 (t ) = xˆ 5 cos(ωt + 53.1°) + yˆ 5 sin (ωt + 53.1°) E 2 (t ) = xˆ 5 cos(ωt − 53.1°) − yˆ 5 sin (ωt − 53.1°) Show that they combine together to form a linearly polarized wave. Solutions: E = E1 + E2 = xˆ [5 cos(ωt + 53.1°) + 5 cos(ωt − 53.1°)] + yˆ [5 sin (ωt + 53.1°) − 5 sin (ωt − 53.1°)] = xˆ 10 cos(ωt )cos(53.1°) + yˆ 10 cos(ωt )sin (53.1°) = xˆ 6 cos(ωt ) + yˆ 8 cos(ωt ) Hon Tat Hui 36 Plane Wave Propagation in Lossless Media NUS/ECE EE2011 Now, E x = 6 cos(ωt ), E y = 8 cos(ωt ) Let X = Ex , Y = E y Y 8 cos(ωt ) 4 = = Then X 6 cos(ωt ) 3 4 4 Y = X ⇒ equation of a straight line with slope = 3 3 Hence the locus of the combined electric field falls on a straight line and the polarization of the combined wave is thus linear. Hon Tat Hui 37 Plane Wave Propagation in Lossless Media NUS/ECE EE2011 Generic Polarization Description Method In general, the polarization state of an EM wave is characterized by two parameters. E x = E x 0 e − jkz , E y = e jδ E y 0 e − jkz 1. Ratio of E y 0 to E x 0 ⎛ Ey0 ⎞ ⎟⎟, ⇒ γ = tan ⎜⎜ ⎝ Ex0 ⎠ −1 0 ≤ γ ≤ 90° 2. Phase difference between E x and E y , i.e., δ , Hon Tat Hui - 180° ≤ δ ≤ 180° 38 Plane Wave Propagation in Lossless Media NUS/ECE EE2011 For example: γ = 0 or 90º and for any value of δ ⇒ linearly polarized γ = 45° and δ = 90° ⇒ Hon Tat Hui (right - hand) circularly polarized 39 Plane Wave Propagation in Lossless Media