Lecue 5 Plane Wave Reflecon and Tansmsson
Incden wave: 1z E ( z) xˆ E (0) e 1 H ( z) yˆ E (0) e 1 Nomal Incdence (Revew) z 1 (,, ) E H S y (,, ) 1 1 1 Refleced wave: 1z E ( z) xˆ E E (0) e S H 1 1z H ( z) yˆ E (0) e 1 z E (0) Tansmed wave: E ( z) xˆ E (0) e noe mnus sgn! E (0) E (0) 1 T H ( z) yˆ E (0) e eflecon coeffcen E (0) ansmsson coeffcen x E H Nkolova 01 LECTURE 05: PLANE WAVE REFLECTION AND TRANSMISSION S z z
Oblque Incdence Paallel (o p) Polazaon (Revew) plane of ncdence he plane defned by he neface nomal ẑ) and he decon of ncdence xˆ xˆsn zˆcos yˆ xˆcos zˆsn xˆ xˆsn zˆcos yˆ xˆcos zˆsn Nkolova 01 3 x 1 1 PLANE OF INCIDENCE xˆ yˆ (, ) angle of eflecon angle of ncdence H E y ˆ xˆ H E y yˆ E H x ˆ (, ) angle of ansmsson xˆ xˆsn zˆcos yˆ xˆcos zˆsn p-polazaon E-feld s paallel o he plane of ncdence z
Oblque Incdence p Polazaon (Revew) () Incden wave: E ( xˆcos zˆsn ) E e E H 1( xsnzcos ) 0 0 1( sn cos ) yˆ e x z 1 Refleced wave: E ( xˆcos zˆsn ) E e x z 0 1 E ( sn cos ) E e Tansmed wave: E ( xˆcos zˆsn ) TE e x z 0 E ( sn cos ) 0 ˆ 1( xsn zcos ) ˆ x z H y noe mnus sgn! 1 eflecon coeffcen H TE y e 0 ( sn cos ) ansmsson coeffcen Nkolova 01 LECTURE 05: PLANE WAVE REFLECTION AND TRANSMISSION 4
Oblque Incdence Pependcula (o s) Polazaon (Revew) E-veco s pependcula o he plane of ncdence (angenal o he neface) x xˆ yˆ E xˆ xˆsn zˆcos yˆ xˆcos zˆsn xˆ xˆsn zˆcos yˆ xˆcos zˆsn E H xˆ y E H yˆ xˆ xˆ xˆsn zˆcos yˆ xˆcos zˆsn z 1 1 (, ) H (, ) Nkolova 01 LECTURE 05: PLANE WAVE REFLECTION AND TRANSMISSION 5 yˆ
Oblque Incdence s Polazaon (Revew) () Incden wave: E yˆ E e 1( xsnzcos ) 0 0 ( ˆcos ˆsn ) 1( xsn zcos ) e 1 H x z E Refleced wave: E ˆ E 1 x z 0 e y E 0 H ( xˆcos zˆsn ) 1 E 0 e ( xsn zcos ) 1 ( sn cos ) eflecon coeffcen Tansmed wave: E ˆ T E x z 0 e y E 0 H ( xˆcos zˆsn ) T E e 0 ( xsn zcos ) ( sn cos ) ansmsson coeffcen Nkolova 01 LECTURE 05: PLANE WAVE REFLECTION AND TRANSMISSION 6
Snell s Law (Paal Revew) vald fo all polazaons follows fom he connuy of he angenal feld componens and he machng of he phases sn sn n n 1 1 1 1 ansmsson no vey good conducos If medum 1 s a loss-fee delecc and medum s a vey good conduco (σ >> ωε ), show ha θ s complex and ends o zeo egadless of he angle of ncdence. Deve he expesson fo he popagaon faco e ( xsn zcos ) n medum when σ >> ωε. Nkolova 01 LECTURE 05: PLANE WAVE REFLECTION AND TRANSMISSION 7
Reflecon and Tansmsson (Fesnel) Coeffcens (Revew) by defnon he eflecon coeffcen Γ gves he ao of he efleced o ncden angenal E-feld componen a he neface he ao of he efleced o ncden angenal H-feld componen a he neface s Γ by defnon he ansmsson coeffcen T gves he ao of he ansmed o ncden angenal E-feld componen a he neface Nkolova 01 LECTURE 05: PLANE WAVE REFLECTION AND TRANSMISSION 8
Fesnel Coeffcens (Revew) fo p-polazaon: cos 1cos cos cos T 1 cos cos cos 1 T 1 cos cos fo s-polazaon: T cos 1cos cos 1cos cos cos 1cos 1 a nomal ncdence (θ = θ = 0), he above cases educe o 1, T 1 1 1 Nkolova 01 LECTURE 05: PLANE WAVE REFLECTION AND TRANSMISSION 9 T
Oblque Incdence: Suface Impedance defnon: he ao of he values (complex n geneal) of he angenal E and H feld vecos a he neface E H z an Z s ( an ˆ) z0 noe: n medum 1, hese ae oal (ncden + efleced) feld values suface mpedances (n he chosen coodnae sysem) fo p and s polazaons ae hen obaned as Z s E H x y z 0 Z s E H y x z 0 Nkolova 01 LECTURE 05: PLANE WAVE REFLECTION AND TRANSMISSION 10
Z Oblque Incdence: Suface Impedance (1) E x0 oal () () E x0 / H y0 E 0 (1 ) cos 1 1 s 1 cos cs o 1 sn ( E 0 / 1 )(1 ) 1 (1) H y0 oal (1) E oal y0 E () () y0/ Hx0 Z s ( E / )(1 )cos co E0(1 ) 1 1 = 0 1 cos 1 s 1 (1) H oal x0 1 sn Nkolova 01 LECTURE 05: PLANE WAVE REFLECTION AND TRANSMISSION 11
Oblque Incdence: Suface Impedance of a Vey Good Conduco we have aleady poven (sl. 7) ha f medum s a vey good conduco, egadless of he angle of ncdence, cosθ 1 Z Z (1 j) s s he powe-flux densy peneang no medum (along z) s 1 1 S z,,, ˆ Z 0 s an z 0 E H z H z hs powe-flux densy s dsspaed powe pe un aea J, J zˆ H 1 p Re S z, Rs H an (0), W/m 1 Rs Re Nkolova 01 LECTURE 05: PLANE WAVE REFLECTION AND TRANSMISSION 1 Z s s s an z0
Exeme Cases: Full Tansmsson, Full Reflecon full ansmsson occus when he wo meda have he same consuve paamees (no efleced wave) 1 and 0, T 1 hee s no efleced wave he feld n medum 1 s smply he avelng ncden wave he feld n medum s he same avelng wave Can oal ansmsson happen f he meda ae dffeen? Explan. full eflecon occus when medum s a PEC (sho ccu) medum s a PMC (open ccu) oal nenal eflecon a a delecc neface he efleced wave s as song as he ncden one hee s no feld n medum Nkolova 01 LECTURE 05: PLANE WAVE REFLECTION AND TRANSMISSION 13
Full Reflecon full eflecon a PEC he conducng medum shos ou medum 1 by focng a zeo angenal E-feld componen (volage beween any pons on neface s 0); angenal H-feld s doubled Pove ha Γ = 1 and T = 0 f σ. full eflecon a PMC PMC acs as an open ccu focng a zeo angenal H-feld componen and doublng he angenal E-feld Pove ha Γ = 1 and T = f µꞌꞌ. oal feld n medum 1 when Γ = ±1 fo nomal ncdence z z E 1( z) xˆ E (0) e e z z H 1( z) yˆ H (0) e e 1 1 1 1 Nkolova 01 LECTURE 05: PLANE WAVE REFLECTION AND TRANSMISSION 14
Suface Cuen Densy a he Suface of PEC 1, T 0 efleced: H (0) H (0) ansmed: H (0) 0 H 1(0) H (0) H (0) H (0) H (0) H (0) 0 accodng o he bounday condons: J s xˆ H an (0) xˆ E an (0) / 1 powe loss n a vey good conduco [see sl. 1] p 0.5 R (0) R (0), W/m s Han s Han zˆ ( H H 1) J s, A/m zˆ yˆ H (0) J, A/m he ncden wave s shoed and hus nduces cuen a he conduco s suface hs cuen s he souce of he efleced wave Nkolova 01 Han (0) 15 1 s
Full Reflecon a PEC Anmaons E eveses phase a a pefecly conducng wall 1 E(0) 0 H does no evese phase a a pefecly conducng wall 1 o H 1 H(0) H (0) Nkolova 01 LECTURE 05: PLANE WAVE REFLECTION AND TRANSMISSION 16
Sandng Waves Due o Reflecon fom PEC (Γ = 1) assume medum 1 s loss-fee (γ 1 = jk 1 ) E z xe e e xe j k z H ( z) yh (0)( e e ) y cos( k z) 1 ˆ jk z jk z ˆ 1 jk1z jk (0) 1z E 1 ˆ ˆ 1 1 1 1 ( ) (0)( ) (0) sn( ) feld foms a sandng wave wh clealy defned nulls and maxma along z (a PEC wall, z = 0, E has a null, H has a maxmum) whee E has a null, H has a maxmum, and vce vesa fo any z, E and H ae n phase quadaue Poynng s veco s puely magnay no powe ansfeed along z S E H zˆ sn( ) cos( kz) 1 j E (0) 1 1 1 kz 1 1 1 0.5sn( kz) 1 Nkolova 01 LECTURE 05: PLANE WAVE REFLECTION AND TRANSMISSION 17
Snusodal Sandng Wave Anmaon supeposon of wo sne waves avelng n oppose decons poson Nkolova 01 LECTURE 05: PLANE WAVE REFLECTION AND TRANSMISSION 18
Sandng Wave Anmaon sandng H-feld plane wave has a maxmum a he shoed end (he E-feld wave looks exacly he same only shfed by a quae wavelengh) / Nkolova 01 LECTURE 05: PLANE WAVE REFLECTION AND TRANSMISSION 19
Envelopes of Tavelng, Sandng, and Mxed Waves he envelope of a sandng wave has nulls and maxma spaced a λ/4 [he dsance beween any wo neghbong nulls (o maxma) s λ/] he envelope of a avelng wave s a consan lne wh no mnma and maxma beween he exemes of a avelng and a sandng wave le nfne combnaons of ncden and efleced waves mxed waves mxed waves can be vewed as a supeposon of a avelng wave and a sandng wave whee Γ can be any complex numbe 0 1 hp://wn.com/sandng_wave_envelope Nkolova 01 LECTURE 05: PLANE WAVE REFLECTION AND TRANSMISSION 0
Locaons of he Envelope Mnma/Maxma maxma occu whee he sandng wave and he avelng wave nefee consucvely mnma occu whee he sandng wave and he avelng wave nefee desucvely maxma ae (1 + Γ ) mes he magnude of he ncden wave mnma ae (1 Γ ) mes he magnude of he ncden wave jk z jkz j 1 1 ( ) (0) (0), E z E e E e e Ez ( ) EzE ( ) ( z) E (0) 1 cos( kz ) envelope max 1 mn 1 Ez ( ) E (0) 1 fo cos( kz ) 1 Ez ( ) E(0) 1 fo cos( kz ) 1 1 Nkolova 01 LECTURE 05: PLANE WAVE REFLECTION AND TRANSMISSION 1
Locaons of he Envelope Mnma/Maxma consde l = z (he dsance fom he neface back no medum 1) he posons of he E-wave mnma l mn,n (n = 0,1,, ) ae found fom cos( kl ) 1 1mn, n Pove ha he mnma posons ae gven by lmn, n (n 1), n 0,1, whee 1 4 4 Ls he fs 3 mnma posons (n = 0,1,) f medum s PEC and he wavelengh s 10 cm. Hn: Noe ha l mn,n 0 mus hold. To ge he coec sgn consde ha cos( kl ) cos( kl ) 1 1 Nkolova 01 LECTURE 05: PLANE WAVE REFLECTION AND TRANSMISSION
Sandng Wave Rao he SWR (Sandng Wave Rao) s he ao of he maxma and he mnma of he oal wave n medum 1 SWR Ez ( ) Ez ( ) max mn SWR 1 1 1 SWR SWR 1 1 SWR has a mnmum value of 1 n he case of a avelng wave SWR s nfny n he case of a sandng wave n mcowave engneeng, a SWR s consdeed sasfacoy fo he puposes of machng and good powe ansfe Wha s Γ f SWR =? Wha s he ao of efleced o ncden powe densy S /S n hs case? Wha s he ao of ansmed o ncden powe densy S /S? Nkolova 01 LECTURE 05: PLANE WAVE REFLECTION AND TRANSMISSION 3
Sandng Wave Rao () E max E mn 0.5 SWR E E max mn 3 Nkolova 01 LECTURE 05: PLANE WAVE REFLECTION AND TRANSMISSION 4
Summay he eflecon and ansmsson coeffcens elae he especve angenal E-feld componens a he neface a nomal ncdence full eflecon, Γ = 1, s due o PEC o PMC emnaons losses pe un aea on vey good conducos ae calculaed usng 1 1 p Rs J s, W/m whee Rs and s an J H he SWR gves he ao of he wave envelope maxmum and mnmum (SWR 1) he dsance beween wo neghboung mnma (o maxma) s λ/ a a PEC wall (sho), he E-feld has a null whle he H-feld has a maxmum, whch s double he value of he ncden feld magnude a a PMC wall (open), he suaon s evesed