HO1_ARFaMT_TL&SC - Traveling Waves Z S V + e - γz...

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Unformatted text preview: Traveling Waves Z S V + e - γz Ve γ = α + jβ γz Z z R e fe re n c e p la n e L • Voltage • Current V ( z ) = V + e −γz + V − eγz I ( z ) = I + e −γz − I − eγz Reflection Coefficient at Load V V Z S + - V + e - γz Ve γz Z z L • Load reflection coefficient • In polar form V− ΓL = + V ΓL = ρeθ L Load Impedance and ΓL • At the load VL V (0) V + +V − ZL = = = Z0 + I L I ( 0) I −I− L • Normalizing z Z L 1 + ΓL = = Z 0 1 − ΓL Reflection Coefficient Along Transmission Line Z S Z z L Γ(z ) • At distance z • In polar form V − eγz Γ( z ) = + −γz Ve Γ( z ) = ΓL e 2γz Reflection Coefficient Plane z • Positive phase change is the same so u rc e / lo a d / p o s itiv e as movement in the g e n e r a t o r z d ir e c tio n te r m in a tio n positive z direction • Positive phase Im a g in a ry change refers to a 1 movement away p o s itiv e p h a s e from the change Γ -P la n e generator/source • This is the same as R eal movement towards 1 the load/termination Terminated Loss Less Transmission Line l Z Γ IN Z IN 0 Z ΓL ZL L γ = jβ Z L + jZ 0 tan ( βl ) Input impedance Z IN = Z 0 Z 0 + jZ L tan ( βl ) 2π λ= Wavelength β ω v p = = λf Phase velocity β Terminated Lossy Transmission Line l Z Γ IN Z IN Input impedance Z IN Attenuation 0 Z ΓL ZL L γ = α + jβ Z L + jZ 0 tanh ( γl ) = Z0 Z 0 + jZ L tanh ( γl ) 10 log10 e − 2αl = −20αl log10 e1 = −8.686αl ⇒ 1neper = 8.686dB ( ) () Smith Chart • Impedance/admittance chart plotted directly onto the reflection coefficient plane ( r − 1) + jx z −1 Γ = U + jV = Γ= ( r + 1) + jx z +1 • Equating real an imaginary gives 2x r 2 −1+ x2 V= U= 2 2 2 ( r + 1) + x 2 ( r + 1) + x r 1 2 • Eliminating x, U − r + 1 + V = r + 1 • Eliminating r, 2 2 (U − 1) 2 + V 1 1 − = x x 2 2 Circles of Constant Resistance r 1 2 U − +V = r +1 r +1 Γ -P la n e r=0 2 2 V r = 0 .5 r= 1 r= 2 V (im a g in a ry ) |Γ | = 1 U (r e a l) U Circles of Constant Reactance (U − 1) 2 + V − 1 1 = x x Γ -P la n e x=0 |Γ | = 1 U (r e a l) x = - 0 .5 x = -1 U 2 2 V x=1 x=3 x = 0 .5 V (im a g in a ry ) x = -3 0 .4 8 0 .4 9 7 0 .4 ± 180 0 .4 0. 05 ­8 0 0 ­1 5 jB E (­ NC TA ) /Yo 0 .2 5 6 0 .4 4 0 .0 A D < – A R D L O TOW TH S ­1 7 0 EN G VEL W A 0 <– ­9 0 ­1 6 ­8 5 0 .1 0 .1 0 .4 0. 3 0 .2 05 5 0. 0. I CT DU 06 ­1 40 ­7 5 0. 44 S VE U EP SC 75 0 .0 – > W A V E L E N GTH S T O W AR 0 .0 D G 0 .4 9 EN E 0 .4 8 RA TO 170 R – 0 .4 > 7 0 .0 160 90 4 0 .4 85 6 15 0 IN D 80 UC T IV E R EA CT AN 0 .1 CE C OM PO 14 N 0 0. 0. 0 .3 44 EN 06 3 0. 0. 4 07 , O T 0. ­7 0 0. 4 30 IN R 4 (+ 70 07 0. 0. o) /Z 0 .2 jX 43 1 ­1 T (­ jX /Z o) , O C R 30 2 N EN 0 .4 PO 0 ­6 5 .5 E C OM 0. A AP 5 65 T CI E IV 8 0 .0 4 0. 2 0 .0 8 0 .3 0 12 ­1 2 0 NC SC S U 9 0 .0 0 .4 1 CT A 0 .0 9 EA ­6 0 0 .6 0 .4 0 .6 60 EP TA 1 0 .4 E R E NC 110 ­1 1 0 0 .1 0 .4 0 .1 0 .7 0 .5 0 .6 0 .7 ) /Yo ( + jB 0 .4 ­5 0 .8 CAP AC IT I V 5 55 0 .8 0 .1 1 0 .3 9 0 .3 9 100 0 .1 1 ­1 0 0 0 .7 0 .8 0 .9 1 .0 0 .2 0 .2 ­ 0 .9 50 0 .9 50 0 .1 2 0 .3 8 0 .3 8 0 .1 2 R E S IS T A N C E C O M P O N E N T ( R / Z o ), O R C O N D U C T A N C E C O M P O N E N T (G / Y o ) ­9 0 1 .0 ­4 1 .0 0 .2 90 0 .2 5 0 .4 0 .4 45 0 .4 0 .3 7 0 .4 0 .1 3 0 .1 3 0 .3 7 0 .6 1 .2 0 .6 0. 0 .6 0 .6 8 0. 1 .2 0 ­4 1 .4 8 0. 8 0 .8 1 .2 80 0 0 .1 4 ­8 0 1. 0 .3 6 1. 0 0 .1 4 1 .6 1 .8 1. 0 40 0 .3 6 0 1. 1 .4 0 .1 5 1 .4 2 .0 0 .3 5 0 .1 5 0 .3 5 70 ­7 0 5 35 ­3 1 .6 6 0 .1 4 0 .3 1 .6 0 .3 4 0 .1 6 ­6 0 ­3 0 1 .8 3 .0 1. 8 60 30 0 3 7 0 .1 2. 3 0. 4 .0 5 .0 0 ­5 ­2 5 2. 0 0 .3 3 0 .1 7 25 50 0. 18 32 2 0. 0. 1 8 3 0. 0 3. 20 0 ­2 0 3. 4 .0 ­1 5 5 .0 Impedance Smith Chart ­1 0 10 50 20 20 50 2 0. 3 0. 1 0 .2 9 0 .2 30 20 0. 3 10 0. 19 31 40 2 0 .2 8 0 .2 10 ­4 0 0. 0. 20 0 .2 4 0 .2 3 0 .2 6 0 .2 7 R E F L E C T IO N L E O F ANG M IS S IO T R A N S L E O F ANG 0 .2 5 0 .2 6 0 .2 7 0 .2 5 0 .2 4 0 .2 3 C O E F F IC I E N T I N D E G R EES N C O E F F I C I E N T I N D E GRE ES ­2 0 0 .2 2 0 .2 8 0 .2 9 0 .2 1 ­3 0 0. 2 19 0. 31 4 .0 15 5 .0 10 50 0 .4 8 0 .4 9 0 0 50 50 20 20 10 10 4 .0 0 .0 TA o) B/ Y E (+ j NC 0 .2 06 3 .0 0. P CA I AC TI VE 43 0. 0. ­7 0 0. 4 0. 30 07 X ­1 o /Z ), O R 4 (­j T – > W A VELE N GTH S TO W A RD 0 .4 9 G E N ERA 0 .4 8 ± 180 TO R 170 –> 0 .4 7 0 .0 160 4 90 0 .4 85 6 0. 15 05 0 IN D 80 UCT 0. 45 IV E R E AC TA 0. 0 .1 75 NC 06 14 E C 0. 0 44 OM PO NE NT 70 (+ jX /Z 0 .2 o) , .4 2 N EN 0 PO 0 .0 8 OM 0 .3 20 ­1 NC E C 0 .4 1 CT A 0 .0 9 ­6 0 0 .6 ­1 1 0 0 .4 0 .1 0 .7 1 .4 0 .5 2 .0 1 .8 0 .6 1 .0 0 .1 1 0 .3 9 0 .3 9 100 0 .1 1 ­1 0 0 0 .8 0 .6 ­5 0 0. 0 .9 0 .4 0 .8 0 .6 0 .4 0 .1 2 0 .3 8 0 .3 8 0 .1 2 0 .9 0 .4 0 .2 ­9 0 1 .0 ­4 1 .0 1 .0 0 .2 0 .2 R E S IS T A N C E C O M P O N E N T (R / Z o ), O R C O N D U C T A N C E C O M P O N E N T (G / Y o ) 0 .2 0 .2 5 0 .4 0 .2 0 .4 0 .1 3 0 .1 3 0 .3 7 0 .6 1 .2 0 .6 0 .6 0 .8 ­8 0 1 .2 1 .4 0. 8 0. 8 1 .2 0 .8 0 0 ­4 0 .1 4 1. 0 .3 6 80 0 .3 6 1 .0 0 .1 4 1 .8 2 .0 0 .5 0 .1 5 1 .4 1 .4 0 .3 5 ­7 0 ­3 5 4 0 .3 6 0 .1 ­6 0 ­3 0 1 .8 1 .8 7 3 0 .1 0 .3 5 .0 0 ­5 ­2 5 18 0. 4 32 0. 0. 0 ­2 0 3. 4 .0 ­1 5 5 .0 0 .2 ­1 0 20 50 0 .1 50 ­2 0 0 .2 8 0 .2 2 0 .2 0 .2 9 Admittance on Impedance Smith Chart 0 .2 0 .3 20 0 .2 ­3 0 1 0. 19 31 40 10 0 .1 ­4 0 0. 5 0 2. 4 .0 0 .5 2. 0 0 .6 0. 1 0 .2 9 0 .2 30 2 0 .2 8 0 .2 20 0 .2 5 0 .2 6 0 .2 4 0 .2 7 0 .2 3 0 .2 5 0 .2 4 0 .2 6 0 .2 3 0 .2 7 E F L E C T I O N C O E F F C IE N T IN D E G L E O F R R EE S ANG S O N C O E F F I C I E N T I N T R A N SM IS D E G LE O F REES ANG 0 .3 0 .4 1 .6 1 .6 0 .3 4 0 .1 6 3 .0 0 .8 1 .6 0 .6 1. 0 1 .0 0 .9 1 .2 0. 8 0 .7 1 .0 1 .6 1 .4 1 .2 0 .3 7 0 .9 0 .8 0 .7 0 .2 1. 0 .8 C AP AC IT I VE RE A 0 .7 1 .4 ­5 5 1 .6 0 .4 1 .8 2 .0 4 .0 ­ 6 5 0 .5 10 0. ­1 40 ­7 5 S U 0 .3 44 EP SC 20 5 6 0 .4 4 0 .0 0 ­1 5 ­8 0 5 .0 0 .4 5 50 A D <– A R D LO TOW 7 T H S 0 .4 ­1 7 0 EN G VEL W A <– ­9 0 ­8 5 0 .1 0 .1 0 .2 5 .0 4 .0 5 .0 0. 3 3. 0 0. 07 0. 43 0 13 OR IN 65 8 0 .0 0 .4 D 0 .5 2 3 .0 0 0. 2 0. 3 0. 19 0. 31 2. 8 0 0 .6 T UC I 12 0 SU VE 1 .8 0 .0 9 0 .6 60 EP SC TA 1 0 .4 0 .6 1 .6 E NC 110 (­ 0 .1 0 .7 0 .4 55 o) jB / Y 0 .8 1. 0 0 .8 0 .7 0 .8 1 .2 0 .9 0 .9 0 .6 50 0 .4 0 .2 1 .0 45 90 1 .0 0. 4 0 .2 0 .4 0 .4 0 .6 40 1 .0 60 30 25 0 .3 5 0 .1 5 70 35 0 .3 3 7 0 .1 50 0. 32 0. 18 3 .0 20 0 .3 4 .0 15 5 .0 10 10 20 50 Smith Chart & Reflection Coefficient • Conversion between impedance, admittance and reflection coefficient using the smith chart. • Rotation forwards towards load/termination and forwards towards source/generator. • Rotation backwards towards load/termination and backwards towards source/generator. • Change of normalizing characteristic impedance. • Relationship between wavelength and phase angle. • Differences between reflection diagram and transmission diagram ...
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This note was uploaded on 01/11/2011 for the course EE 5602 taught by Professor Wingshingchan during the Spring '10 term at City University of Hong Kong.

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