Module 3.1 Time Varying Uniform Plane Waves

Introduction:

     This module discusses the time-varying behavior of the electric field of a uniform plane wave in free space.
     Ex: There are two infinite current sheets \(J_{s1}\) and \(J_{s2}\) in the space which have the same amplitude of current density. The spacing between \(J_{s1}\) and \(J_{s2}\) is 0.25 \(\lambda\) ( z = 0.25 \(\lambda\) ) in the xy-plane, and the initial phase of \(J_{s1}\) leads the phase of \(J_{s2}\) by 0.5 \(\pi\). It will have an endfire radiation in the +z direction and null field in the –z direction.

Formula:

1. Etotal (z,t) (V/m) : Total eletrical field due to three infinite plane parallel current sheets, defined by
     total (z,t) \( = \frac{\eta_{0} J_{s1} }{ 2 }cos[\omega (t-\frac{z -d_{1} }{ c })] x̂ + \frac{\eta_{0} J_{s2} }{ 2 }cos[\omega (t-\frac{z -d_{2} }{ c })] x̂ + \frac{\eta_{0} J_{s3} }{ 2 }cos[\omega (t-\frac{z -d_{3} }{ c })] x̂ \)
2. E (z,t) (V/m) : Eletrical field due to a infinite plane current sheet, defined by
     n (z,t) \( = \frac{\eta_{0} J_{sn} }{ 2 }cos[\omega (t-\frac{z -d_{n} }{ c })] x̂, \) (n = 1, 2, 3)
3. J (t) (A/m) : A infinite plane current sheet, defined by
     n (z,t) \( = \ -J_{sn} cos[\omega (t-\phi_{n})] x̂, z = a_{n}, \) (n = 1, 2, 3)

Parameters:

1.   \(J_s\) : Amplitude of the surface current density, (A/m)
2.   \(\phi\)   : Phase angle of the surface current density, (rad),
3.   \(f\)   : Operating frequency, (GHz),
4.   \(d\)   : Location of the surface current density from zero, (\(\lambda\)),
5.   \(\eta_{0}\) : Instrnsic impdance of the free space, (\(\Omega\))
6.   \(c\)    : Speed of light, (m/s)