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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