ecture 9 The Smith Chart and Basic Impedance-Matching Concepts
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 T it contains two families of curves (circles) in the complex Γ plane each circle corresponds to a fixed normalized resistance or reactance Im j1 i 1 0 1 Re r ElecEng4FJ4 2
The Smith Chart: Normalized Impedance and Γ relation #1: normalized load impedance z and reflection Γ Z Z z 1 Z where z r jx and Z Z z Z 0 0 1 0 j = e = r j i z 1 1 r x 2 2 r i 2 2 r i 1 (1 ) 2 (1 ) r i 2 2 i 2 2 2 1 i r 2 2 r r 1r 1 2 1 1 ( r 1) i x x ElecEng4FJ4 3
The Smith Chart: Resistance and Reactance Circles 2 2 2 1 i r r r 1r 1 resistance circles 2 2 2 1 1 ( r 1) i x x reactance circles let the abscissa be Γ r and the ordinate be Γ i (the complex Γ plane) resistance and reactance equations describe circles in the Γ complex plane resistance circles have centers lying on the Γ r axis (with Γ i = 0, i.e., ordinate = 0) reactance circles have centers with abscissa coordinate = 1 a complex normalized impedance z = r + jx is a point on the Smith chart where the circle r intersects the circle x ElecEng4FJ4 4
The Smith Chart: Resistance Circles ElecEng4FJ4 5
The Smith Chart: Reactance Circles ElecEng4FJ4 6
The Smith Chart: Nomographs at the bottom of Smith s chart (left side), nomograph is added to read out with a ruler the following (1 st ruler) above: SWR, below: SWR in db, 20log (2 nd ruler) above: return loss in db, 20log 10 10 SWR below: power reflection Γ 2 (P) (3 rd ruler) above: reflection coefficient Γ (E or I) 2 10log 10(1 ) T 2 perfect match 1 1 ElecEng4FJ4 7
a circle of radius Γ centered at Γ = 0 is the geometrical place for load impedances producing reflection of the same magnitude Γ such a circle also corresponds to constant SWR 1 SWR 1 The Smith Chart: SWR Circles z SWR 3.87 0.4 j0.7 0.59 SWR circle ElecEng4FJ4 8
The Smith Chart: Plotting Impedance and Reading Out Γ x 1 0.135 What is Z if Z 0 = 50 Ω? 83 R r 0.5 z 0.5 j1.0 83 0.62 getting Γ with a ruler: 1) measure R 2) measure 3) / R ElecEng4FJ4 9
The Smith Chart: Tracking Impedance Changes with relation #2: input impedance versus the T length 1 g j2 at generator: Zin Z0, g e 1 (see 08, sl. 4) g j2 1e j2 Zin Z 1e 0 j2 zin 1e j2 1 e compare with at load: z 1 1 on the Smith chart, the point corresponding to z in is rotated by 2β (decreasing angle, clockwise rotation) with respect to the point corresponding to z along an SWR circle (toward generator) one full circle on the Smith chart is 2β max = 2π, i.e., max = λ/2; this reflects the π-periodicity of z in ElecEng4FJ4 10
The Smith Chart: Tracking Impedance Changes with 2 toward generator /4 z toward load ElecEng4FJ4 SWR circle i j1 zin 0 1 0.5 j0.5 1 j1 r for Z 0 = 50 Ω, the quarter-wavelength T transforms a load of Z 25 j25 to an input impedance of Zin 50 j50 check and see whether Z Z Z 0 in For a frequency-independent load Z, what would be the direction of the locus of Z in as frequency increases? 11
The Smith Chart: Read Out Distance to oad unknown distance to load in terms of λ D D/ n toward generator 0.194 A known load Z Z 75 j75 known Z 0 Z0 50 A z 1.5 j1.5 measured Z in 23 j34 Zin Dn B A 0.2 n 2 ElecEng4FJ4 12 B zin B 0.46 j0.68 0.394
The Smith Chart: Reading Out SWR SWR B A r r B, B, SWR 1 1 B 1 B SWRB 1 B SWR r B B, A z, 1 j1 A r, 2.6 B B SWR r B, 2.6 SWR circle ElecEng4FJ4 13
normalized load admittance y The Admittance Smith Chart 1 1 1 1 1e z 1 1 1e normalized input admittance (at generator) y j2 1 1e in zin j2 1 e j j () the relation between y in and y is the same as that between z in and z one can get from load to generator (and vice versa) by following a circle clockwise (counter-clockwise) standard Smith chart gives resistance and reactance values admittance Smith chart is exactly the same as the impedance Smith chart but rotated by 180 [see eq. (*) ] in the complex Γ plane ElecEng4FJ4 14
Reading Out Normalized Conductance and Susceptance Values normalized admittance y Y YZ0 Y0 1 z Y load admittance Y 0 characteristic admittance Y G jb y g jb conductance susceptance ElecEng4FJ4 15
Conductance and Susceptance Circles in Admittance Smith Chart combined impedance and conductance Smith Charts conductance circles resistance circles short circuit Y ( Z 0) 1 positive (capacitive) susceptance negative (inductive) susceptance open circuit Y 0 ( Z ) 1 susceptance circles reactance circles ElecEng4FJ4 16
Switching Between Impedance and Admittance on Smith Chart impedance values from a standard Smith chart can be easily converted to admittance by rotation along a circle by exactly 180 rotation by 180 on the impedance Smith chart corresponds to impedance transformation by a quarter-wavelength T j 2 1 e 1 4 z ( /4) in j 1e 1 1 zin( /4) y 1 z z 1 in impedance Smith chart, the point diametrically opposite from an impedance point shows its respective admittance value ElecEng4FJ4 17
Switching Between Impedance and Admittance: Example toward generator /4 i j1 zin 1 j1 same as y 1 j1 0 1 Check whether in this example the y found from the Smith chart satisfies 1 y z r z toward load 0.5 j0.5 Example: -network matching ElecEng4FJ4 18
Quarter-wave Transformer Revisited Z G 0 /4 V G (, Z ) 0 loss-free line Z from 08, sl. 18: Z in /4 Z Z 2 0 Z in for impedance match at the input terminals of the λ/4 T, Z in =Z G * Z 0 Z GZ z G z 1 z y G T must be designed to have this specific Z 0 ElecEng4FJ4 19
Quarter-wave Transformer Revisited 2 the impedance match with the λ/4 transformer holds perfectly at one frequency only, f 0, where = λ 0 /4 this impedance-match device is narrow-band Z jz0 tan( ) 2 0 f Zin ( f ) Z0, where Z jz tan( ) 4 2 f 0 0 perfect match Z Z Z G 0 100 50 70.71 ( f ) Zin ( f) Z Z ( f) Z in 0 0 ElecEng4FJ4 20
Optimal Power Delivery: Review (Homework) at the generator s terminals, a loaded T is equivalently represented by its input impedance Z in Z G I in V G V in Z in active (or average) power delivered to the loaded T (this is also the power delivered to the load Z if the line is loss-free) 2 1 1 2 1 2 Zin 1 ( Pin) av Re{ V ini in} V in Re Yin V G Re 2 2 2 ZG Zin Zin 1 2 Rin ( Pin) av V G 2 2 2 ( R R ) ( X X ) in G in G ElecEng4FJ4 21
Optimal Power Delivery: Review (Homework) assume generator s impedance Z G = R G + jx G is known and fixed optimal matching is achieved when maximum active power is delivered to the load Z in what is this optimal value of Z in? opt in in in Z Z max P ( Z ) in find the optimal R in and X in by obtaining the respective derivatives Pin 2 2 2 0 RG Rin ( Xin XG) 0 Rin Pin 0 Xin( Xin XG ) 0 X in maximum power is delivered to the load under conditions of conjugate match opt opt opt in G and in G in G R R X X Z Z ElecEng4FJ4 22
Summary the impedance Smith chart depicts a normalized load impedance as a point in the complex Γ plane the load impedance is normalized with respect to the characteristic impedance Z 0 of the T the admittance Smith chart depicts a normalized load admittance y as a point in the complex Γ plane the admittance Smith chart is rarely used because the impedance Smith chart can be readily used as an admittance chart as well the sense of rotation with increasing T length is the same resistance/reactance impedance values are determined from the resistance/reactance circles the input impedance z in of a T loaded with a known z is found by following the SWR circle, starting from z, and completing an angle of 2β in a clockwise direction (on chart: toward generator ) ElecEng4FJ4 23
Summary 2 if the input impedance z in of a T is known but the load z is not, z is determined by starting from z in, following the SWR circle and completing an angle of 2β in a counter-clockwise direction (on chart: toward load ) the normalized admittance y of a given impedance z is found by reading out the value of the point diametrically opposite to z on the Smith chart many more applications of the Smith chart will be shown during the tutorial ElecEng4FJ4 24