Electromagnetic Waves and Antenna Sources: Lectures on Superposition and Scattered Radiation
Explore the concepts of superposition and scattered radiation in electromagnetic waves, along with discussions on scalar and vector potentials, antenna sources, and vector potential generation. Delve into the intricate details of electromagnetic wave behavior from time-harmonic sources and the implications for radiation in the zone.
Download Presentation
Please find below an Image/Link to download the presentation.
The content on the website is provided AS IS for your information and personal use only. It may not be sold, licensed, or shared on other websites without obtaining consent from the author. If you encounter any issues during the download, it is possible that the publisher has removed the file from their server.
You are allowed to download the files provided on this website for personal or commercial use, subject to the condition that they are used lawfully. All files are the property of their respective owners.
The content on the website is provided AS IS for your information and personal use only. It may not be sold, licensed, or shared on other websites without obtaining consent from the author.
E N D
Presentation Transcript
PHY 712 Electrodynamics 9-9:50 AM MWF Olin 105 Plan for Lecture 24: Complete reading of Chap. 9 & 10 A. Superposition of radiation B. Scattered radiation 03/19/2018 PHY 712 Spring 2018 -- Lecture 24 1
03/19/2018 PHY 712 Spring 2018 -- Lecture 24 2
Electromagnetic waves from time harmonic sources review: c For scalar potential (Lorentz gauge, ) k r r ' ik 1 e ~ ~ ( ) ( ) ( ) ~ = + 3 r r , ' r , , ' d r 0 r r 4 ' 0 For vector potential (Lorentz c gauge, ) k r r ' ik ~ A ~ A e ~ J ( ) ( ) ( ) = + 3 r r , ' r 0 , , ' d r 0 r r 4 ' 03/19/2018 PHY 712 Spring 2018 -- Lecture 24 3
Consider antenna source (center-fed) Note these notes differ from previous formulation d/2 d r z d y d x ~ J ( ( ) ) ( ) ( ) x ( ) , = r z sin for I k d z y d z d k c 03/19/2018 PHY 712 Spring 2018 -- Lecture 24 4 4
Consider antenna source -- continued ~ J ( ( ) ) ( ) ( ) x ( ) = r z , sin for I k d z y d z d n = 3 , 2 , 1 = ; .... k n c d n=1 n=2 n=3 03/19/2018 PHY 712 Spring 2018 -- Lecture 24 5
Consider antenna source -- continued ( ) ( , sin I k d ) ( ) ( ) x ( ) = J r z for z y d z d k c Vector potential from source: r r ' ik e r ( ) ( ) = 3 A r J r , ' ', 0 d r r 4 ' ikr e ( ) ( ) r r' 3 ik A r J r For , ' ', r d d r e 0 4 r d ikr e ( ) ( ) ( ) 'cos ikz A r z , ' sin ' 0 I dz e k d z 4 r d 03/19/2018 PHY 712 Spring 2018 -- Lecture 24 6
Consider antenna source -- continued ( ) 4 d ikr ~ A e ( ( ) ) , cos ikz r z 0 sin I dz e k d z r d ( ) ( ) 2 ikr cos cos cos e kd kd = z 0 2 I 4 : sin kr In the radiation = zone ~ A ~ A ~ B ( ( ) ) ( ) ( ) r r r r , , , ik ( ( ) ~ A ~ E ( ) ~ B r r r r , , ikc ) 2 1 ~ A ~ A dP k c ~ E ( ) ( ) ( ) ( ) 2 2 = = 2 * 2 r r r r r r , , , , r r 2 2 d 0 0 ( ) ( ) 2 cos cos cos c dP kd kd = 2 0 I 2 8 sin d 03/19/2018 PHY 712 Spring 2018 -- Lecture 24 7
Consider antenna source -- continued ( ) ( ) 2 cos cos cos c dP kd kd = 2 0 I 2 8 sin d n=1 n=2 n=3 03/19/2018 PHY 712 Spring 2018 -- Lecture 24 8
Consider antenna source -- continued ( ) ( ) 2 cos cos cos c dP kd kd = 2 0 I 2 8 sin n d = For kd : n=1 n=2 n=3 03/19/2018 PHY 712 Spring 2018 -- Lecture 24 9
Radiation from antenna arrays z y x 2d a x + 2 1 N ~ J ( ( ) ) ( ) ( ( ) ) ( ) = j = + r z , sin 1 for I k d z N j a y d z d 1 n = 3 , 2 , 1 = ; .... k n c d 03/19/2018 PHY 712 Spring 2018 -- Lecture 24 10
Radiation from antenna arrays -- continued Vector potential from array source : r r ' ik ikr ~ A e r e ~ J ~ J ( ) ( ) ( ) , = r r' 3 3 ik r , ' r , ' r 0 0 ' ' e d r d r 4 4 r ' r + 2 1 N ~ J ( ( ) ) ( ) ( ( ) ) ( ) = , = + r z sin 1 for I k d z x N j a y d z d 1 j d ikr N ~ A e ( ( ) ) ( ) = , sin cos cos ikaj ikz r z 0 sin e I dz e k d z 4 r j N d ( ) ( 1 ) + sin 2 1 sin cos ka N N 1 = = sin cos ikaj 2 e ( ) sin sin cos ka j N 2 03/19/2018 PHY 712 Spring 2018 -- Lecture 24 11
Radiation from antenna arrays -- continued In the radiation = zone r : ~ A ~ A ~ B ( ( ) ) ( ) ( ) r r r , , , ik ( ( ) ~ A ~ E ( ) ~ B r r r r , , ikc ) 2 2 1 ~ A ~ A dP k cr ~ E ( ) ( ) ( ) ( ) 2 2 = = 2 * r r r r r r , , , , r 2 2 d 0 0 ( ) ( 1 ) ( ) ( ) 2 2 + sin 2 1 sin cos ka N cos cos cos c dP kd kd 1 = 2 0 2 sin I ( ) 2 8 sin sin cos d ka 2 N=2;a=d/2; = N=2;a=d; = N=0 03/19/2018 PHY 712 Spring 2018 -- Lecture 24 12
( ) 2 ( ) ( ) ( ) 2 + = sin 2 1 sin cos sin cos ka N ka 1 2 sin cos cos cos kd kd c dP d = 2 0 I ( ) 2 8 sin 1 2 = = = Example for 0, 10, 2 N kd ka Additional amplitude patterns can be obtained by controlling relative phases of antennas. 03/19/2018 PHY 712 Spring 2018 -- Lecture 24 13
Brief introduction to multipole expansion of electromagnetic fields (Chap. 9.7) Sourceless Maxwell's equations in terms of and fields with time depende 0 i t E H nce : e = = = E E H H E / ikZ ik Z 0 0 = H 0 where / and c / k Z 0 0 0 Decoupled equations: + E ( ) ( ) = + = 2 2 2 2 H 0 0 k k iZ k i H E E H = = 0 kZ 0 03/19/2018 PHY 712 Spring 2018 -- Lecture 24 14
Multipole expansion of electromagnetic fields -- continued Note that: + ( )( ( )( ) ) = + = 2 2 2 2 r E r H k 0 0 k Convenient operators for angular momentum analysis 1 Defin e: i = r L ( ) L r Note that 0 2 2 1 r r r L r = 2 2 2 Eigenfunctions: 2 1 1 n ( , ) = + ( , ) = + ( , ) 2 sin ( 1) LY Y l l Y lm lm lm 2 2 sin si 03/19/2018 PHY 712 Spring 2018 -- Lecture 24 15
Multipole expansion of electromagnetic fields -- continued Magnetic multipole field: l l k = = + L E ( ) + 1 ( , ) M lm r H ( ) g kr Y l lm spherical Bessel function M lm r E 0 ( ) ( , ) M lm 1 ( ) l l Z g kr Y 0 l lm Electric multipole fie ld: ( ) + 1 l l ( , ) E lm r E ( ) Z f kr Y 0 l lm k spherical Bessel function = = E lm r L H H 0 ( ) + ( , E lm 1 ( ) ) l l f kr Y l lm 03/19/2018 PHY 712 Spring 2018 -- Lecture 24 16
Multipole expansion of electromagnetic fields -- continued Vector spherical harmonics: (fo 1 ( ( l l r 0) l = X L , ) ( , ) Y lm lm + 1) Orthogonality condition s: = * ' X X ( , ) ( , ) d ' l m lm ll mm ( ) = * ' X r X ( , ) ( , ) 0 d ' l m lm General expansion of fields: i k ( ) = E lm M lm H X X ( ) ( , ) ( ) ( , ) a f kr a g kr l lm l l m lm i k ( ) = + E lm M lm E X X ( ) ( , ) ( ) ( , ) a f kr a g kr l lm l lm lm 03/19/2018 PHY 712 Spring 2018 -- Lecture 24 17
Multipole expansion of electromagnetic fields -- continued Time averaged power distribution of radiation far from source: 2 Z k dP d + = ( ) + 1 l E lm M lm r X X ( , ) ( , ) 0 i a a lm lm 2 2 lm E lm M For a pure multipole radiation with either Z d P a d k 1 2 ( 1) l l + or : a a lm 2 2 = X ( , ) 0 lm lm 2 2 2 2 ( )( ) ( )( ) 2 2 , ) = + + + + + + 2 X ( 2 1 1 m Y l m l m Y l m l m Y ( ) ( ) + lm lm 1 1 l m l m 03/19/2018 PHY 712 Spring 2018 -- Lecture 24 18
= For exa mple: 1 l 3 3 ( ) 2 2 2 = = = 1 cos + 2 2 X X X ( , ) sin ( , ) ( , ) 1 1 10 11 8 16 03/19/2018 PHY 712 Spring 2018 -- Lecture 24 19
= For example: 2 l 15 8 5 5 6 ( ) ( ) 2 2 2 = cos = 1 3cos + = 2 2 2 4 4 X X X ( , ) sin ( , ) 4cos ( , ) 1 cos 20 21 22 16 1 03/19/2018 PHY 712 Spring 2018 -- Lecture 24 20
