Ứng dụng đồ thị Smith

Ứng dụng đồ thị Smith để giải bài tập trường điện từ, siêu cao tần Điện tử viễn thông

Lecture 08 The Smith Chart and Basic Impedance-Matching Concepts Sections: 6.8 and 6.9 Homework: From Section 6.13 Exercises: 12, 13, 14, 15, 16, 17, 18, 19, 20 Nikolova 2012 2 The Smith Chart: Γ plot in the Complex Plane • Smith’s chart is a graphical representation in the complex Γ plane of the input impedance, the load impedance, and the reflection coefficient Γ of a loss-free TL • it contains two families of curves (circles) in the complex Γ plane • each circle corresponds to a fixed normalized resistance or reactance Nikolova 2012 Lecture 08: The Smith Chart 3 The Smith Chart: Normalized Impedance and Γ 0 00 1 where and 1 =| | = L LL LLL LL j ri Z Z z Z zrjx ZZ z Z ej             1 1 L z    relation #1: normalized load impedance z L and reflection Γ 22 22 22 1 (1 ) 2 (1 ) ri L ri i L ri r x           22 2 22 2 1 11 11 (1) L ri LL ri LL r rr x x              Nikolova 2012 Lecture 08: The Smith Chart 4 The Smith Chart: Resistance and Reactance Circles 22 2 1 11 L ri LL r rr       22 2 11 (1) ri LL x x       let the abscissa be Γ r and the ordinate be Γ i (the Γ complex plane) • resistance and reactance equations are circles in the Γ complex plane • resistance circles have centers lying on the Γ r axis (Γ i = 0 or ordinate = 0) • reactance circles have centers with abscissa coordinate = 1 • a complex normalized impedance z L = r L + jx L is a point on the Smith chart where the circle r L intersects the circle x L resistance circles reactance circles Nikolova 2012 Lecture 08: The Smith Chart 5 The Smith Chart: Resistance Circles r  i  1 || 1   1 L r  0 L r  0.5 0   1 0.2 0.25 L r  1 short open Nikolova 2012 Lecture 08: The Smith Chart 6 The Smith Chart: Reactance Circles inductive loads capacitive loads Nikolova 2012 Lecture 08: The Smith Chart 7 The Smith Chart: Nomographs at the bottom of Smith’s chart, a nomograph is added to determine • SWR and SWR in dB, • return loss in dB, • power reflection |Γ| 2 (P) • reflection coefficient |Γ| (E or I), etc. perfect match 10 20lo g||   10 20log SWR Nikolova 2012 Lecture 08: The Smith Chart 8 The Smith Chart: SWR Circles  a circle of radius Γ m centered at Γ = 0 is the geometrical place for load impedances producing reflection of the same magnitude, | Γ| = Γ m  such a circle also corresponds to constant SWR 1| | 1| | SWR    SWR circle 0.4 0.7 L z j   3.87SWR  || 0.59   Nikolova 2012 Lecture 08: The Smith Chart 9 The Smith Chart: Plotting Impedance and Reading Out Γ 0.5 1.0 L zj   0.5 L r  1 L x  ||  (1 0.135 / 0.25) 0.46 83         || 0.62 What is Z L if Z 0 = 50 Ω? 0.135  R  getting |Γ| with a ruler: 1) measure 2) measure 3) | | / R R    83  Nikolova 2012 Lecture 08: The Smith Chart 10 The Smith Chart: Tracking Impedance Changes with L () 0 () 0 () 0 () () j LjL zL in z L j LjL zL V Ve e ZZ Z I Ve e               2 0 2 1 1 j L in j L e ZZ e              relation #2: input impedance versus the TL length L compare with 1 1 L z      2 2 1 1 j L in j L e z e          on the Smith chart, the point corresponding to z in is rotated by −2βL (decreasing angle, clockwise rotation) with respect to the point corresponding to z L along an SWR circle  one full circle on the Smith chart is 2βL max = 2π, i.e., L max = λ/2; this reflects the periodicity of z in  [...]... standard Smith chart gives resistance and reactance values • admittance Smith chart is exactly the same as the “impedance” (or standard) Smith chart but rotated by 180° [see eq (*) and sl 17] Nikolova 2012 Lecture 08: The Smith Chart 14 The Smith Chart: Admittance Interpretation – 2 normalized reactance normalized resistance Nikolova 2012 impedance Smith Chart Lecture 08: The Smith Chart 15 The Smith. .. Interpretation – 3 combined impedance and conductance Smith Chart Nikolova 2012 Lecture 08: The Smith Chart 16 The Smith Chart: Admittance Interpretation – 4 • impedance values from a standard Smith chart can be easily converted to admittance (conductance + susceptance) values by rotation along a circle of exactly 180° • rotation by 180° on the impedance Smith chart corresponds to  impedance transformation... opposite on the Smith chart from an impedance value is the respective “admittance” value Nikolova 2012 Lecture 08: The Smith Chart 17 The Smith Chart: Admittance Interpretation – 5 r ato ner e dg ar tow i Check whether in this example the yL found from the Smith chart satisfies 1 yL  zL 1 zin  1  j1 yL  1  j1 L/4 0 1 r z L  0.5  j 0.5 tow ard loa Nikolova 2012 d Lecture 08: The Smith Chart... increases? Lecture 08: The Smith Chart 11 The Smith Chart: Read Out Distance to Load • unknown distance to load in terms of λ Dn  D /  toward generator • known load ZL Z L  75  j 75  A • known Z0 Z 0  50  LA  0.194 z L  1.5  j1.5 • measured Zin Z in  23  j 34  B zin  0.46  j 0.68 Dn  LB  LA  0.2 Nikolova 2012 Lecture 08: The Smith Chart LB  0.394 12 The Smith Chart: Reading Out... Reading Out SWR A z L , A  1  j1 SWRA  SWRB rL , B  1 B  rL , B  1 rL , B  2.6 1 |  B | SWRB  1 |  B |  SWRB  rL , B B SWR  rL,B  2.6 SWR circle Nikolova 2012 Lecture 08: The Smith Chart 13 The Smith Chart: Admittance Interpretation • normalized load admittance 1 1  1    () yL  zL1     1  1   • normalized input admittance (at generator) yin   zin1 1  e  j 2... The Smith Chart 19 Quarter-wave Transformer Revisited – 2  the impedance match with the λ/4 transformer holds perfectly at one frequency only, f0, where L = λ0/4  this impedance-match device is narrow-band  Z L  jZ 0 tan(  L)  2 0  f Z in ( f )  Z 0    , where  L    4  2  f 0  Z 0  jZ L tan(  L)  Z L  100  Z G  50  Z 0  70.71  | ( f ) | Nikolova 2012 Lecture 08: The Smith. ..The Smith Chart: Tracking Impedance Changes with L – 2 r i ato for Z0 = 50 Ω, the r ne ge quarter-wavelength TL 1 rd a ow transforms a load of t SWR circle Z L  25  j 25  to an input impedance of zin ... I  }  1 | V |2 Re Y  1 | V |2    ( Pin ) av  Re{Vin in Re   in in G Z G  Z in 2 2 2  Z in  Rin 1  2  ( Pin ) av  | VG | 2 ( Rin  RG ) 2  ( X in  X G ) 2 Nikolova 2012 Lecture 08: The Smith Chart 21 Optimal Power Delivery: Review (Homework)  assume generator’s impedance ZG = RG + jXG is known and fixed  optimal matching is achieved when maximum active power is delivered to the load... Rin Pin  0  X in ( X in  X G )  0 X in maximum power is delivered to the load under conditions of conjugate match opt opt opt  Rin  RG and X in   X G  Z in  Z G Nikolova 2012 Lecture 08: The Smith Chart 22