3 5 Sinusoidally Time Varying Uniform Plane Waves in Free Space Slide Presentations for ECE 329, Introduction to Electromagnetic Fields, to supplement “Elements of Engineering Electromagnetics, Sixth[.]
Slide Presentations for ECE 329, Introduction to Electromagnetic Fields, to supplement “Elements of Engineering Electromagnetics, Sixth Edition” by Nannapaneni Narayana Rao Edward C Jordan Professor of Electrical and Computer Engineering University of Illinois at Urbana-Champaign, Urbana, Illinois, USA Distinguished Amrita Professor of Engineering Amrita Vishwa Vidyapeetham, Coimbatore, Tamil Nadu, India 3.5 Sinusoidally Time-Varying Uniform Plane Waves in Free Space 3.5-3 Sinusoidal Traveling Waves f t z vp cos t z vp cos t z g t z v p cos t z vp cos t z where vp 0 3.5-4 f z , t cos t z t 4 f t 2 t 0 -1 2 z 3.5-5 g z , t cos t z t 2 t 4 g t 0 -z 2 -1 3.5-6 For J S t J S cos t ax for z 0, The solution for the electromagnetic field is 0 J S E cos w t z vp ax for z 0 J S = cos t z ax for z JS H cos t z vp a y for z JS cos t z a y for z where w vp w 0 3.5-7 Parameters and Properties t z Phase, radian frequency = t rate of change of phase with time for a fixed value of z (movie) f frequency 2 = number of 2 radians of phase change per sec 3.5-8 phase constant = z = magnitude of rate of change of phase with distance z for a fixed value of t (still photograph) vp phase velocity = velocity with which a constant phase progresses along the direction of propagation follows from d t z 3.5-9 2 = wavelength = distance in which the phase changes by 2 for a fixed t Note that 2 f vp f 2 in m f in MHz = 300 Ex Ex 0 Hy Hy = Ratio of the amplitude of E to the amplitude of H for either wave 3.5-10 E × H (Poynting Vector, P) ax × a y az for (+) wave ax × a y az for ( ) wave is in the direction of propagation x x E E H y P z P y H z 3.5-11 Example: Consider E 37.7 cos 6 108 t 2 z a y V m Then 6 10 , f 108 Hz 2 2 2 , 1 m 6 108 vp 108 m s 2 Direction of propagation is –z H 0.1 cos 6 108 t 2 z ax A m 3.5-12 Array of Two Infinite Plane Current Sheets JS1 z 0 JS z J S J S cos t ax for z 0 J S J S sin t ax for z For J S , 0 J S cos t z ax for z E1 0 J S cos t z a for z x 3.5-13 For J S , 0 J S sin t z ax for z E2 0 J S sin t z a for z x 4 0 J S sin t z ax for z 0 J S sin t z a for z x 2 0 J S cos t z ax for z 0 J S cos t z a for z x 3.5-14 For both sheets, E = E1 E2 E1 E2 for z z 0 z E1 z 0 E2 z for z E1 z 0 E2 z for z 0 J S cos t z ax for z 0 J S sin t sin z ax for z 0 for z No radiation to one side of the array “Endfire” radiation pattern