Revision of Plane Wave Propagation

Revision of Plane Wave Propagation - NUS/ECE EE4101...

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Hon Tat Hui Revision of Plane Wave Propagation NUS/ECE EE4101 1 Revision of Plane Wave Propagation 1 Plane Waves in Lossless Media In a source free lossless medium, . 0 = = = σ ρ J 0 0 = = = × = × H E E H H - E μ ε t t Maxwell’s equations: See animation “Plane Wave Viewer”
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Hon Tat Hui Revision of Plane Wave Propagation NUS/ECE EE4101 2 Take the curl of the first equation and make use of the second and the third equations, we have: E H E 2 2 2 t t = × = με μ () E E E 2 : Note = × × This is called the wave equation : 0 2 2 2 = E E t A similar equation for H can be obtained: 0 2 2 2 = H H t
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Hon Tat Hui Revision of Plane Wave Propagation NUS/ECE EE4101 3 In free space, the wave equation for E is: 0 2 2 0 0 2 = E E t ε μ where 2 0 0 1 c = c being the speed of light in free space (~ 3 × 10 8 (m/s)). Hence the speed of light can be derived from Maxwell’s equation.
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Hon Tat Hui Revision of Plane Wave Propagation NUS/ECE EE4101 4 In phasor form, Maxwell’s equations can be written as: 0 = = = × = × B D D H B - E ρ j ω j ω 0 0 2 2 2 0 0 2 where 0 ε μ ω = = + = + k k E E E E Using the phasor form expression, the wave equation for E field is also called the Helmholtz’s equation , which is:
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Hon Tat Hui Revision of Plane Wave Propagation NUS/ECE EE4101 5 k is called the wavenumber or the propagation constant . 0 0 0 0 2 2 λ π ε μ ω = = = = c f k k where 0 is the free space wavelength.
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Hon Tat Hui Revision of Plane Wave Propagation NUS/ECE EE4101 6 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 E x component is: 0 2 2 2 2 2 2 2 = + + + x E k z y x Consider a special case of the E x in which there is no variation of E x in the x and y directions, i.e., 0 2 2 2 2 = = x x E y E x
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Hon Tat Hui Revision of Plane Wave Propagation NUS/ECE EE4101 7 This is called the plane wave condition and E x ( z ) now varies with z only. The wave equation for E x becomes: ( ) () 0 2 2 2 = + z E k dz z E d x x Note that a plane wave is not physically realizable because it extends to an infinite extent in the x and y directions.
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This note was uploaded on 10/21/2011 for the course EE 4101 taught by Professor Yeotatsoon during the Spring '11 term at National University of Singapore.

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Revision of Plane Wave Propagation - NUS/ECE EE4101...

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