C 546 Lecture 0 Review f lectrmagnetics Spring 018 Jse. Schutt-Aine lectrical & Cmputer ngineering University f Illinis jesa@illinis.edu C 546 Jse Schutt Aine 1
Printed Circuit Bard C 546 Jse Schutt Aine
High-Density Package C 546 Jse Schutt Aine 3
D H B lectrmagnetic Quantities lectric field (Vlts/m) lectric flux density (Culmbs/m ) Magnetic field (Amperes/m) Magnetic flux density (Webers/m ) J Current density (Amperes/m ) Charge density (Culmbs/m ) C 546 Jse Schutt Aine 4
Maxwell s quatins B t H J D t D Faraday s Law f Inductin Ampère s Law Gauss Law fr electric field B 0 Gauss Law fr magnetic field C 546 Jse Schutt Aine 5
Cnstitutive Relatins B H D Permittivity : Farads/m Permeability : Henries/m Free Space 1 8.8510 F / m 7 4 10 H / m C 546 Jse Schutt Aine 6
D H J t Cntinuity quatin D H J J D t t 0 D J t 0 C 546 Jse Schutt Aine 7
lectrstatics t Assume n time dependence 0 0 Since 0, such that where is the scalar ptential with we get we get Pissn s quatin if n charge is present 0 Laplace s quatin C 546 Jse Schutt Aine 8
Integral Frm f M C dl BdS t S H dl J ds D ds t C S S S DdS dv V S BdS 0 C 546 Jse Schutt Aine 9
Bundary Cnditins nˆ 0 1 nˆ H H J 1 nˆ D D 1 1 nˆ B B 0 S S, H, D, B 1 1 1 1, H, D, B ˆn C 546 Jse Schutt Aine 10
Free Space Slutin H t H t 0 Faraday s Law f Inductin Ampère s Law Gauss Law fr electric field B 0 Gauss Law fr magnetic field C 546 Jse Schutt Aine 11
C 546 Jse Schutt Aine 1 H t t t Wave quatin t Wave quatin can shw that H H t
Wave quatin x y z t separating the cmpnents x x x x x y z t y y y y x y z t x y z t z z z z C 546 Jse Schutt Aine 13
Wave quatin Plane Wave (a) Assume that nly x exists y = z =0 (b) Only z spatial dependence x y 0 x x z t This situatin leads t the plane wave slutin In additin, assume a time harmnic dependence x j t e t then j C 546 Jse Schutt Aine 14
Plane Wave Slutin z x x slutin j z x e frward wave j z x e backward wave where prpagatin cnstant In the time dmain slutin x() t cst z frward traveling wave x() t cst z backward traveling wave C 546 Jse Schutt Aine 15
Plane Wave Characteristics where prpagatin cnstant 1 v prpagatin velcity In free space 1 8 vc 310 m/ s C 546 Jse Schutt Aine 16
Slutin fr Magnetic Field H jh t xˆ yˆ zˆ 1 1 H j j x y z x 0 0 j z If we assume that x ˆ e j z then H yˆ e j z If we assume that x ˆ e j z then H yˆ e intrinsic impedance f medium C 546 Jse Schutt Aine 17
Time-Average Pynting Vectr P() t t H t Pynting vectr W/m time average Pynting vectr W/m 1 T 1 T P P() t dt t H t dt T 0 T 0 We can shw that 1 Re * P H where and H are the phasrs f ( t) and H( t) respectively C 546 Jse Schutt Aine 18
H t H J t Material Medium J jh H J j : cnductivity f material medium ( 1 m 1 ) since H j j j 1 j 1 j r then 1 j C 546 Jse Schutt Aine 19
Wave in Material Medium 1 j 1 j is cmplex prpagatin cnstant j 1 j j assciated with attenuatin f wave assciated with prpagatin f wave C 546 Jse Schutt Aine 0
Wave in Material Medium Slutin: x ˆ e x ˆ e e z z jz decaying expnential 1 1 1/ 1 1 Magnetic field H yˆ e e z j z 1/ Cmplex intrinsic impedance j j C 546 Jse Schutt Aine 1
Wave in Material Medium Phase Velcity: Wavelength: v p 1 1 1/ 1 1 f 1/ Special Cases 1. Perfect dielectric 0 air, free space 0 and C 546 Jse Schutt Aine
Wave in Material Medium. Lssy dielectric Lss tangent: 1 1 8 1 8 3 1 j 8 C 546 Jse Schutt Aine 3
Wave in Material Medium 3. Gd cnductrs Lss tangent: 1 j j j f f j f 1 j j C 546 Jse Schutt Aine 4
Material Medium attenuatin prpagatin d p H, xamples PC 0 0 0 supercnd Gd cnductr f 1 j 1 f finite cpper Pr cnductr / 1 j finite Ice Perfect dielectric 0 / finite air C 546 Jse Schutt Aine 5
Radiatin - Vectr Ptential Assume time harmnicity ~ j H (1) H J j () D / (3) B 0 (4) j t e C 546 Jse Schutt Aine 6
Radiatin - Vectr Ptential Using the prperty: 0 anyvectr B 0 Asuch that AB A 0 A : vectr ptential ja j A 0 C 546 Jse Schutt Aine 7
vectr 0 vectr Vectr Ptential ja where is the scalar ptential Since a vectr is uniquely defined by its curl and its divergence, we can chse the divergence f A chse A such that A j 0 Lrentz cnditin C 546 Jse Schutt Aine 8
B J j AJ j j A A A J A j A j J A j Vectr Ptential A A J D Alembert s equatin C 546 Jse Schutt Aine 9
G r, r' Vectr Ptential Three-dimensinal free-space Green s functin j rr' e 4 r r' Vectr ptential A r V ' Jr' e 4 r r' j rr' dv' Frm A, get and H using Maxwell s equatins C 546 Jse Schutt Aine 30
Fr infinitesimal antenna, the current density is: J r zi dl ( x') ( y ') ( z ') ' ˆ A r z e ˆ 4 Idl r Vectr Ptential Calculating the vectr ptential, jr In spherical crdinates, zˆ rˆ cs ˆ sin C 546 Jse Schutt Aine 31
Vectr Ptential Idl ˆ ˆcs ˆ sin A r z r e 4 r Reslving int cmpnents, jr Idl cs A ˆ r z e 4 r jr A Idl 4 sin e r jr C 546 Jse Schutt Aine 3
and H Fields Calculate and H fields 1 H A 1 A H sin r A 0 sin H 1 1 A r ra r sin r 0 C 546 Jse Schutt Aine 33
and H Fields H 1 A ra r r r Idl 1 jr H j 1 e sin 4r jr 1 H j r r 1 H sin 1 sin j C 546 Jse Schutt Aine 34
and H Fields csidl 1 jr 1 r j 1 e 4 r jr j 1 rh 1 r r j j Idl 1 1 1 sin 1 jr j e 4 r jr r j C 546 Jse Schutt Aine 35
and H Fields csidl 1 jr 1 r j 1 e 4 r jr j j Idl j r 1 1 e sin 1 4 r jr jr Idl 1 jr H j 1 e sin 4r jr C 546 Jse Schutt Aine 36
Far field : r r Far Field Apprximatin 0 1 1 r r then, 0 r j Idl 4 r e jr sin H j Idl 4 r Nte that: e jr H sin,where C 546 Jse Schutt Aine 37
Far Field Apprximatin Characteristics f plane waves Unifrm cnstant phase lcus is a plane Cnstant magnitude Independent f Des nt decay Similarities between infinitesimal antenna far field radiated and plane wave (a) and H are perpendicular (b) and H are related by (c) is perpendicular t H C 546 Jse Schutt Aine 38
Pynting Vectr Pt t Ht Time-average Pynting vectr r TA pwer density 1 P Re H * and H here are PHASORS I dl ˆ rˆ P Re H r 4r sin C 546 Jse Schutt Aine 39
Time-Average Pwer Ttal pwer radiated (time-average) P= P ds with ds rr ˆ sindd 0 0 P= 0 0 I dl 4 r r sin sindd Idl P= 4 3 d sin 0 C 546 Jse Schutt Aine 40
Time-Average Pwer P= 4 Idl 3 4 Pynting Pwer Density Directive Gain = AveragePynting Pwer Density ver area f sphere with radius r Directive Gain = P P /4 r C 546 Jse Schutt Aine 41
Fr infinitesimal antenna, Directive Gain 3 Directive Gain sin Directivity I dl sin 4 r 4 I dl /4 r 3 4 C 546 Jse Schutt Aine 4
Directivity: gain in directin f maximum value Radiatin resistance: Frm 1 P P RI we have: rad I Fr infinitesimal antenna: Directivity R R rad 4 Idl dl I 3 4 3 C 546 Jse Schutt Aine 43
Radiatin Resistance Fr free space, 10 R rad 80 dl (fr Hertzian diple) The radiatin resistance f an antenna is the value f a fictitius resistance that wuld dissipate an amunt f pwer equal t the radiated pwer P r when the current in the resistance is equal t the maximum current alng the antenna A high radiatin resistance is a desirable prperty fr an antenna C 546 Jse Schutt Aine 44