2 Parallel-Plate Transmission Line (Geometric Model) = c Assume it s a plane wave propagate in the z with polarization in y direction. d dz ~ ˆ.

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1 Tansmission ines 1 ntoution When the soue aiates in a ie aea, the eneg speas out. The aiate eneg is not guie an the tansmission of eneg though aiation is ineffiient. Dietive antenna oul have huge imensions in the oe of the ave length of the boaasting eletomagneti aves. We stu the tansvese eletomagneti (TM) aves guie b tansmission lines. The haateistis of TM aves guie b tansmission lines ae the same as those fo a unifom plane ave popagating in an unboune ieleti meium. The thee most ommon tpes of guiing stutues: 1. Paallel-plate tansmission line. At mioave fequenies, paallel-plate tansmission lines an be fabiate on a ieleti substate using pint-iuit tehnolog. The ae often alle stiplines.. To-ie tansmission line: poe an telephone lines. 3. oaial tansmission line: T ables an the input ables to high-fequen peision measuing instuments. Paallel-Plate Tansmission ine (eometi Moel) Phsial Moel 1 i t sine e ω ω + ( + k ) Assume it s a plane ave popagate in the ith polaiation in ietion. + k ik + e 1 1 v (+), ik e + ik +, ( ) e 1 ik+ ω e ( k ω ) v + ik + e (-) When the TM ave popagates in the paallel-plate tansmission line, hage an uent ma be inue in the plates. ossing the bouna fom ieleti meium to the pefet onution plates:, // D // D

2 ++++ at (loe plate) n, D ρ loe plate & n D ik e + e ik+ K J K D K 1 e Use an Ampee s loop to obtain the sufae uent. ik+ at (uppe plate) U ik+ e n & K U 1 e ik+ K n the ieleti meia, the eleti an magneti fiels satisf the Maell s eqs: B & B iω & iω,, (, ) ( basi iffeential equations i j k iω ω ω i & i Use Maell s equation to get the to basi iffeential equations. iω, sine KU D K U

3 K & u iω KU u iω ( K ) iω iω ' is the inutane pe unit length ( ) l l t Use integation to obtain the voltage an uent elation, an to alulate the inutane pe unit length fom ou phsial moel. iω l iω ( ) ( K ) ( ) iω iω iω hee 1 Q 1 l t l t U ( ' ) ' l t iω K U is the apaitane pe unit length Use integation to obtain the apaitane pe unit length. l Time hamoni tansmission line equations: iω & ik e ω & iω ω, ω ω v ω ik + k, ( e, ik ik e, (, e + iω Deive iffeential equation / ik iω iω Use the eive iffeential equations to obtain the uent an voltage vaiation as a funtion of an t. The impeane: ω / k / Z (not ivie b length) The veloit of popagation along the line is:

4 ω v k ( / )( / ) Obtain impeane an spee of popagation aves. Miostip ines The evelopment of soli-state mioave evies an sstems has le to the ie spea use of a fom of paallel-plate tansmission lines alle miostip lines o simpl stiplines. The miostip lines ae lose to the paallel-plate tansmission lines if >>. oss Paallel-Plate Tansmission ines Q D s s l l R l J s l s R et sufae impeane an esistane fom the onept of poe loss. When the apaitane beteen the to onutos, the pemittivit an the onutivit ae kno, e have ( ) 1 Aveage Poe Dissipation: pavg p sufae Re{ } (Ponting s theoem) Uppe plate: K Define a sufae impeane: U Z R + ix Sufae Sufae Sufae Z Sufae K Sufae K S

5 Obtain the sufae impeane fom the onept of attenuate aves in a pefet onuto. A goo onuto is a meium fo hih >> ω. i ( ) ( k +ωt t, e ), + k i ω ω i ω ω ω ω i ω 1 1 ω + i 1 1 k k i ω 1 ω Z R + ix ( 1 i) K ω Sufae Sufae Sufae + U Obtain esistane of a paallel-plate tansmission line. ( + i) RSufae psufae Re{ KU ZSufae} KU RSufeae, P psufae KU RSufae The effetive seies esistane pe unit length fo both plates of a paallel-plate tansmission line of ith is R R ω Sufae (to times fom the viepoint of poe issipation) (pe unit length) l R Z sufae (not ivie b length) K Kl l l

6 3 eneal Tansmission-ine quations (letoni iuit Moel) R Q D s s l l l J s l s R The apaito is the apabilit of stoing hages pe volt. The onuto is the uent flo pe volt. ( in paallel plate tansmission lines) i(, -Q +Q i(, v(, R i(+, v(+, i v R i v, v Ri i + ( + v i v i, i v v + { } ( + { } let ( ) ( ) i ω v, t Re e t an ( ) ( ) i ω i, t Re e t ( R + iω) an ( + iω) let k ( ( R + iω ) ) ( R + iω) ( R + iω)( + iω) e & k k ( R + iω )( + iω) k ( R + iω )( + iω)

7 let k α + iβ ( R + iω)( + iω) i (α : attenuation onstant, β : phase onstan + k + k e + e k + k+ iω t + k+ ( ) ( e + e + k + an e + e ( v + k+ iω t + k+ ( e + e, ), an + k k + k k ( R + iω) k e k e ( R + i )( e + e ) + R + iω k ω haateisti impetane: Z R + iω k R + iω + iω ( / l / l ) (not ivie b length) ample: Demonstate the analog beteen the ave haateistis on a tansmission line an unifom plane aves in a loss meium. n a loss meium, e appl the moel use to esibe fequen epenent pemittivit. The pemittivit is theefoe a omple numbe, ' i ''. The same as pemittivit, e ma have omple pemeabilit, ' i' '. The Maell s equations to elative magneti an eleti fiels ae: iω( ' i '') & iω( ' i '') Assume a unifom plane ave haateie b an that vaies ith onl ith. (Keep in min that a ave funtion popagating in the ietion ith a polaiation in the ietion. The magneti fiel is in the ietion.) ( ' i '') ( ω '' iω ') iω + i j k i i ( ' i '') i ω ( ω '' + iω ') ( ω '' + iω ') ( ω '' + iω ')( ω '' + iω ') k let e & k ( ω '' + iω ')( ω '' + iω ') k & k

8 Deive fom geneal tansmission line equations, e have ( R + iω )( + iω) k. Thee limiting ases: 1. ossless ine ( R,. Thee is no eal pat in k.) (a) Popagation onstant: k iω, α, β ω ω 1 (b) Phase veloit: v phase β () haateisti impeane: X R + iω Z R + ix + iω, R,. o-oss ine ( R << ω, << ω ) (a) Popagation onstant: k iω R + R 1 i 1 i ω ω + iω iω R 1 i i ω ω R α +, β ω ω 1 (b) Phase veloit: v phase β R + iω () haateisti impeane: Z R + ix + iω Z 1/ 1/, R R 1 i 1 i 1 i + i ω ω ω ω R, X ω R ω 1 3. Distotionless ine ( R / / ) R (a) Popagation onstant: k iω 1 i R + iω ω α R, β ω

9 ω 1 (b) Phase veloit: v phase β R + iω R () haateisti impeane: Z R + ix, + iω R, X The iffeent fequen omponents tavel along a tansmission line at the same veloit in oe to avoi istotion. Fo a loss line, ave amplitue ill be attenuate, an istotion ill esult hen iffeent fequen omponents attenuate iffeentl, even hen the tavel ith the same veloit. ample: t is foun that the attenuation on a 5 (Ω) istotionless tansmission line is.1 (B/m). The line has a apaitane of.1 (nf/m). a) Fin the esistane, inutane, an onutane pe mete of the line. b) Fin the veloit of ave popagation. ) Detemine the peentage to hih the amplitue of a voltage taveling ave eease in 1 (km) an in 5 (km). 5 (Ω) eal pat of haateisti impeane 1 nepe eibels R / /.1 3 a) R 5Ω, α R.1( B / m) ( Np / m) ( Np / m) ( F / m) R.5 1 ( / m).5( / m) ( m) 3 R α R / Ω R.8 / ( S m) 1 8 b) v phase 1 m / s α ) Afte 1 (km), / e e α Afte 5 (km), / e e Tansmission-ine Paametes The eletial popeties of a tansmission line ae ompletel haateie b its fou paametes R,,, an.

10 et R k iω ( + iω) iω 1 + iω ompae ith k iω ω k iω 1 + iω 1. To-ie tansmission line. The ies have a aius a an ae sepaate b a istane D. (Obtain the apaitane fom the metho of images.) P Sufae osh 1 π πap ( D / a) Sufae D ( ) osh 1, π a 1 1 RSufae πa K RSufae πa osh 1 π ( D / a) ω RS R (The fato of is not fom seies esistane but fom poe πa πa issipation.). oaial tansmission line. b ln ( ) π a π an ln ( b / a) πa ln ( b / a) 1 RS R πak Si πbk So, PS, i πaps, i & P 1 S S, o πa πb ω RS R + + π a b π a b Attenuation onstant fom Poe Relations The attenuation onstant of a taveling ave on a tansmission line: { } {} k ( R + iω)( iω) α Re Re + Fin the attenuation onstant fom poe elation: k k k e & e e Z The time aveage poe popagate along the tansmission line is: P 1 α Re{ } e Z Time-aveage poe loss pe unit length: P P P

11 P P αp α P P eise: The folloing haateistis have been measue on a loss tansmission line at 1 M: Z 5 + (Ω), α.1 (B/m), β.8p (a/m) i Detemine R,,, an fo the line.

2. Electrostatics. 2.1 What is electrostatics and electrostatic induction? 2.2 Explain coulomb s law of electrostatics

2. Electrostatics. 2.1 What is electrostatics and electrostatic induction? 2.2 Explain coulomb s law of electrostatics . Eletostatis. What is eletostatis an eletostati inution? Eletostatis is a banh of siene ealing with eletiity at est i.e. stati eletiity. Nomally evey atom has an ual numbe of potons (ve hage) an eletons