Elctomagntcs: Th Smth Chat (9-6 Yoonchan Jong School of Elctcal Engnng, Soul Natonal Unvsty Tl: 8 (0 880 63, Fax: 8 (0 873 9953 Emal: yoonchan@snu.ac.k
A Confomal Mappng ( Mappng btwn complx-valud vaabls: f ( f ( f Suppos a mappng functon: x Catsan coodnats Consd: f ( : ( Mappng functon f ( Pola coodnats x ( & ( x plan x x f ( f ( plan -ccl x-ccl
A Confomal Mappng ( -ccl: plan x f ( plan x-ccl: ( Tansfomaton:.g. x 0.5 x x f ( x x 0.5 f ( Invs tansfom s possbl, too! plan -ccl: ( 0.5 0.5 0.5 call: x-ccl: ( ( 0.5 0.5-0.5 4.5-0.5-0.5-0.5 ( (, x Fo all ccls: Sngulaty 3
Th Smth Chat: A Confomal Mappng ( Voltag flcton coffcnt at th load fo a losslss tansmsson ln: Nomald load mpdanc: 0 0 0 X x 0 0 θ θ x ( ( ( ( x ( x x Pola coodnats θ 0 0 -ccl: x-ccl: 4 Catsan coodnats
Th Smth Chat: A Confomal Mappng ( -ccl: x-ccl:.g. (. t us assum that th voltag flcton coffcnt s gvn by: x x flcton coffcnt In th Smth chat, th a many ccls alady dawn fo vaous valus of and x! n sold lns plan n dashd lns. Fnd - and x-ccls that pass though 3. ad th - and x-valus fo th ccls:, x 0.5 In sult, whn th voltag flcton coffcnt s, th nomald load mpdanc s gvn by: x 0. 5 Ths can also b vfd: 0.5 0.5 0.5 4.5 D. K. Chng, Fld and Wav Elctomagntcs, nd d., Addson-Wsly, 989. 5
Th Smth Chat: A Confomal Mappng (3 ( x x Th -ccls:. Cntd at ( /(, 0: On th -axs. 0: Cntd at th ogn wth a unty adus 3. : Cntd at (, 0 4. All -ccls: Passng though (, 0. SC pont OC pont Th x-ccls:. Cntd at (, /x: On th ln. x 0: 0 ln,.., -axs 3. x : Cntd at (, 0 0, x 0 4. All x-ccls: Passng though (, 0., x Th - and x-ccls a vywh othogonal to on anoth! D. K. Chng, Fld and Wav Elctomagntcs, nd d., Addson-Wsly, 989. 6
Smth Chat wth Pola Coodnats ( Voltag flcton coffcnt at th load fo a losslss tansmsson ln: 0 0 θ Catsan coodnats Pola coodnats plan 0.5 Popts to not:. All -ccls: ad fom 0 to - -0.5 θ 0.5. θ : Masud fom th postv al axs -0.5-7
Smth Chat wth Pola Coodnats ( Input mpdanc: ( Nomald nput mpdanc: β β 0 φ θ β Nomald load mpdanc: Tansfomaton: ( V ( I( ( 0 β φ θ β β θ θ V (, I( Effctv flcton coffcnt at : φ 4 Towad gnato Towad load V ( ( I( Of xactly sam fom!! - λ -0.5 o 0.5-0.5 plan θ V, Towad load 0.5 I β π 8 - Towad gnato β
Smth Chat wth Pola Coodnats (3 o plan Nomald load mpdanc: θ Nomald nput mpdanc: Addtonal popts to not:. P M : Th -ccl valu S (SW. P m : Th -ccl valu /S θ φ θ β D. K. Chng, Fld and Wav Elctomagntcs, nd d., Addson-Wsly, 989. call: Vmax I S Vmn I max mn M S S m 9
Exampl 9-3: Gvn: 0 50 ( Ω 0 0.λ Soluton:. Pont P sc : x 0. Amuthal shft n th CWD by 0. fom P sc to P 3. At P, ad & x : In sult: Towad gnato 0, x 0. 75 0 50 0.75 36.3 ( Ω Smth chat: D. K. Chng, Fld and Wav Elctomagntcs, nd d., Addson-Wsly, 989. 0
Exampl 9-5: call: Smth chat: S V V max mn I I max mn S S. Daw a ccl fo 0.5 3. Voltag mnmum (/S /3. S 3 (.., 3, x 0 D. K. Chng, Fld and Wav Elctomagntcs, nd d., Addson-Wsly, 989. Gvn: 0 50 ( Ω S λ 3.0 0. 0.4 ( m m 0.05 ( m (Fst voltag mnmum Not: 4. Amuthal shft n CCWD ( Towads load by 0.5 to P D. K. Chng, Fld and Wav Elctomagntcs, nd d., Addson-Wsly, 989. / λ 0.05/ 0.4 0.5 m 5. At P, ad & x: 50(0.60 0.80 30 40 ( Ω / π 0. 5 x 0.60 0.80
D. K. Chng, Fld and Wav Elctomagntcs, nd d., Addson-Wsly, 989. Smth chat: Smth-Chat Calculatons fo ossy ns β α β α φ α φ α Nomald nput mpdanc fo a lossy ln: β θ φ Consd shnkag du to attnuaton: α α