8/7/018 Cours Instructor Dr. Raymond C. Rumpf Offic: A 337 Phon: (915) 747 6958 E Mail: rcrumpf@utp.du EE 4347 Applid Elctromagntics Topic 3 Skin Dpth & Powr Flow Skin Dpth Ths & Powr nots Flow may contain copyrightd matrial obtaind undr fair us ruls. Distribution of ths matrials is strictly prohibitd Slid 1 Lctur Outlin Skin Dpth Powr Flow Skin Dpth & Powr Flow Slid 1
8/7/018 Skin Dpth Skin Dpth & Powr Flow Slid 3 Skin Dpth, Wavs in good conductors attnuat vry quickly. Th distanc ovr which thy dcay by a factor of 1/ is calld th skin dpth. E 0 Dfinition of Skin Dpth 1 m In Trms of Fundamntal Paramtrs 1 f Rlation to Impdanc 1 j 45 Skin Dpth & Powr Flow Slid 4
8/7/018 DC Rsistanc, RDC Currnt dnsity is uniform throughout th conductor so th ntir conductor contributs to currnt flow. J r Conductor Ara ADC r RDC ADC r Skin Dpth & Powr Flow Slid 5 AC Rsistanc, RAC Currnt dnsity is NOT uniform throughout th conductor so only part of th conductor contributs to currnt flow. Conductor Ara AAC r RAC J r AAC r Skin Dpth & Powr Flow Slid 6 3
8/7/018 Nots on Skin Dpth and AC Rsistanc High frquncis xprinc so much loss that thy do pntrat vry far into a conductor. Th dpth of pntration is calld skin dpth. Du to th skin dpth at high frquncis, only part of th conductor contributs to currnt flow. This maks rsistanc incras as a function of frquncy. Drawbacks High frquncis xprinc mor loss. Signals gt distortd Bnfits Conductors can b mad hollow chapr, lightr, tc. Can mak innr part of conductor out of a diffrnt matrial. Skin Dpth & Powr Flow Slid 7 Powr Flow Skin Dpth & Powr Flow Slid 8 4
8/7/018 Poynting s Thorm Poynting s thorm is a consrvation of powr quation. Th total powr laving a volum must b qual to th rat of dcras of th total nrgy stord in th fild plus th nrgy lost du to hat (or somthing ls). Powr laving Powr ntring Ohmic Loss Stord magntic nrgy Stord lctric nrgy 1 EHds H E dv E t S V V Total powr Rat of dcras of stord Ohmic powr laving volum lctric and magntic nrgy dissipatd Skin Dpth & Powr Flow Slid 9 dv Poynting Vctor, W rad off of Poynting s thorm th trm rsponsibl for powr laving th volum. 1 E H ds H E dv E dv t S V V t E t H t Hr w ar intgrating a flux to gt total powr. Th argumnt must b powr dnsity (W/m ). W call this argumnt th instantanous Poynting vctor. Du to th cross product, th Poynting vctor is prpndicular to both E and H. For LHI matrials, th Poynting vctor is in th sam dirction as th wav vctor. E k H E H k Skin Dpth & Powr Flow Slid 10 5
8/7/018 Instantanous Poynting Vctor Rcall th lctric and magntic fild componnts of a plan wav travlling in th + dirction can b writtn as E, t cost a x E H t, costa 0 y Substituting ths xprssions into th dfinition of th instantanous Poynting vctor givs t Et Ht E 0 costa cos x t ay t cosa cos t a Constant powr flow Rapidly oscillating fluctuation in powr flow Skin Dpth & Powr Flow Slid 11 Animation of Instantanous Powr Flow t cosa cos t a Constant powr flow Rapidly oscillating fluctuation in powr flow Skin Dpth & Powr Flow Slid 1 6
8/7/018 Avrag Poynting Vctor Th instantanous powr flow is rarly of intrst bcaus th rapidly fluctuating trm dos not transport any nt powr. Th mor practical and usful quantity is th tim avrag Poynting Vctor. To obtain th tim avrag Poynting vctor, w intgrat ovr on wav cycl. 1 avg t dt avg 0 1 E 0 1 E 0 cos a cos dt t a dt 0 0 a cos Skin Dpth & Powr Flow Slid 13 Complx Poynting Vctor For tim harmonic signals, th frquncy domain Poynting vctor is complx. * EH Th fild xprssions for our plan wav ar E E 0 a x j H a y Substituting ths into th dfinition of complx Poynting vctor givs * E 0 j a x ay * j E 0 j j a x ay E E 0 j 0 j a x a y a Skin Dpth & Powr Flow Slid 14 7
8/7/018 RMS Poynting Vctor Th complx Poynting vctor is lik th instantanous Poynting vctor and contains th rapidly varying fluctuations in powr flow. A mor maningful quantity is th root man squar (RMS) powr flow that is asily calculatd from th complx Poynting vctor. 1 1 avg R * R E H 1 avg R 1 E 0 j R a cos a avg a cos Skin Dpth & Powr Flow Slid 15 Total Powr Th Poynting vctor is a powr dnsity with units of W/m. To calculat total powr flow through som ara, w must intgrat th Poynting vctor ovr that ara. P t t ds S Th avrag powr flow is simply calculatd from th avrag Poynting vctor. P avg ds S avg Skin Dpth & Powr Flow Slid 16 8
8/7/018 Indx Ellipsoids and Powr Flow Isotropic Matrials y P k x Anisotropic Matrials y k P x Phas propagats in th dirction of k. Thrfor, th rfractiv indx drivd from k is bst dscribd as th phas rfractiv indx. Vlocity hr is th phas vlocity. Powr propagats in th dirction of th Poynting vctor which is always normal to th surfac of th indx llipsoid. From this, w can dfin a group vlocity and a group rfractiv indx. Skin Dpth & Powr Flow Slid 17 9