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: J =current density ρ =charge density ∂H ∇× E = -μ σ =conductivity ∂t ∂E ∇× H = ε ∂t ε∇ ⋅E = 0 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 : ∇×∇×E = ∇()∇ ⋅E − ∇2E ∂ ∂ 2 ∇ 2E = μ ∇× H = με E ∂t ∂t 2 This is called the wave equation: ∂2 ∇2E − με E = 0 ∂t 2 A similar equation for H can be obtained: ∂2 ∇2H − με H = 0 ∂t 2 Hon Tat Hui 2 Plane Wave Propagation in Lossless Media NUS/ECE EE2011 In free space, the wave equation for E is: ∂ 2 ∇ 2E − μ ε E = 0 0 0 ∂t 2 where 1 μ ε = 0 0 c2 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 Signal Propagation - 시간 지연 - 시간영역 파형 유지 Ef(0)= (t ) Ef()r= ( t − rc / ) 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: • ⎧ jωt ⎫ E()x, y, z,t = Re⎨E ()x, y, z e ⎬ ⎩ ⎭ • 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 • ∂ ⎧ ∂ jωt ⎫ n E()x, y, z,t = Re⎨ n E()x, y, z e ⎬ ∂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⎨ n E()x, y, z e ⎬ ⎩()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 ∇−22EEμε =0 ⇒∇− E με()j ω E =0 00∂t 2 00 22 (dropping the dot sign) ⇒∇+ EEμεω00 =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 2 2 2 ∇ E + μ0ε0ω E = ∇ E + k E = 0 where k = ω μ0ε0 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 = k0 = ω μ0ε0 = = c λ0 where λ0 is the free space wavelength. In an arbitrary medium with ε =ε0εr and μ =μ0μ r, 2πf 2π k = ω μ0ε0μrε r = μrε r = μrε r c λ0 We call, 2π λ λ = = 0 = wavelength in the medium k μrε r Hon Tat Hui 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 ⎞ ⎜ + + + k 2 ⎟E = 0 ⎜ 2 2 2 ⎟ x ⎝ ∂x ∂y ∂z ⎠ 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 E = E = 0 ∂x2 x ∂y2 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 2E (z) x + k 2E ()z = 0 dz2 x 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: + − jkz 1. Ex (z) = E0 e − + jkz 2. Ex (z) = E0 e + − jkz − + jkz 3. Ex (z) = E0 e + E0 e + - E0 and E0 are constants to be determined by boundary conditions. + − jkz − + jkz E0 e and E0 e are plane waves propagating along the +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, + − jkz + − jkz jωt (1) Ex ()z = E0 e → Ex (z,t) = Re{E0 e e } + = E0 cos()ωt − kz + Assume E0 = ω = k = 1, then Ex (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 ()z,0 = cos(− z) Ex x 1 z -2π -3π/2 -π -π/2 0 π/2 π 3π/2 2π -1 Ex At t = 1s, Ex ()z,1 = cos(1− z) 1 z -2π -3π/2 -π -π/2 0 π/2 π 3π/2 2π -1 Ex At t = 2s, Ex ()z,2 = cos(2 − z) 1 z -2π -3π/2 -π -π/2 0 π/2 π 3π/2 2π -1 Hon Tat Hui 13 Plane Wave Propagation in Lossless Media NUS/ECE EE2011 − + jkz − + jkz jωt (2) Ex ()z = E0 e → Ex (z,t) = Re{E0 e e } − = E0 cos()ωt + kz − Assume E0 = ω = k = 1, then Ex (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 Ex At t = 0, Ex ()z,0 = cos(z) 1 z -2π -3π/2 -π -π/2 0 π/2 π 3π/2 2π -1 Ex At t = 1s, Ex ()z,1 = cos(1+ z) 1 z -2π -3π/2 -π -π/2 0 π/2 π 3π/2 2π -1 Ex At t = 2s, Ex ()z,2 = cos(2 + z) 1 z -2π -3π/2 -π -π/2 0 π/2 π 3π/2 2π -1 Hon Tat Hui 15 Plane Wave Propagation in Lossless Media NUS/ECE EE2011 + − jkz E0 e is a wave propagating in the + z direction. − + jkz E0 e 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: + Ex (z,t)()= E0 cos ωt − kz . + (1) At t = 0, Ex (z,0)()= E0 cos − kz . Consider a zero point (z coordinate = z0) of the wave (for example the first one to the right of the origin), then Ex ()z0 ,0 = 0 ⇒ cos(− kz0 ) = 0 π π ⇒ −kz = − ⇒ z = 0 2 0 2k Hon Tat Hui 17 Plane Wave Propagation in Lossless Media NUS/ECE EE2011 + (2) At t = 1s, Ex (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 Ex ()z1,1 = 0 ⇒ cos(ω − kz1 ) = 0 π ω π ⇒ ω − kz = − ⇒ z = + 1 2 1 k 2k ω π π ω Distance change in 1s = z − z = + − = 1 0 k 2k 2k k ω ω Thus propagation speed = /1s = ms−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, + − jkz E(z) = xˆEx = xˆE0 e , then the solution for H is: E + ∂e− jkz k H()z = yˆ 0 = yˆ E +e− jkz = yˆH − jωμ ∂z ωμ 0 y Note that H is ⊥ to E and they are shown on next page.