Smith-Chart Smith-Chart Division of Electrical- and Communication Engineering Engineering and Information Technology Berne University of Applied Sciences Introduction to the School of Division of Electrical- and Communication Engineering Engineering and Information Technology School of Berne University of Applied Sciences From theory of lines: Z −1 Z − Z0 Z0 Γ= = Z Z + Z0 +1 Z0 Eq. 1 Γ : complex Reflection Coefficient = Γ ∠θ Z : complex Impedance = R + jX Z0 : Normalizing Impedance (usually real) Smith Chart Normalized Z = z Prof. F. Dellsperger 1 2 NT2 z= Z R X = +j = r + jx Z0 Z0 Z0 Eq. 2 Γ= z − 1 r + jx − 1 = z + 1 r + jx + 1 Eq. 3 NT2 =0 .8 45 Γ Γ =1 Engineering and Information Technology Γ = 0.8∠60o 135 Berne University of Applied Sciences 90 School of Γ = Γ ∠θ Division of Electrical- and Communication Engineering Smith-Chart Representation of Reflection Coefficient in Polar Diagram: Division of Electrical- and Communication Engineering According to Eq.1, each reflection coefficient must also represent a complex impedance Z, or according to Eq.3 a complex normalized impedance z. For Γ = 0.8∠60o and Z0 = 50Ω from Eq.1 and 3: 90 1+ Γ 1 + 0.8∠60o Z = Z0 = 50Ω 1− Γ 1 − 0.8∠60o = (21.429 + j82.479)Ω Γ = 0.8∠60o 135 0.6 0.8 1 + Γ 1 + 0.8∠60o = 1 − Γ 1 − 0.8∠60o = 0.429 + j1.65 z= 1 0 θ θ= 0.4 Γ o 60 0.2 o 60 ±180 45 .8 θ= θ Γ = 0.8∠60o Z = (21.429 + j82.479)Ω z = 0.429 + j1.65 =0 Engineering and Information Technology School of Berne University of Applied Sciences Smith-Chart 0.2 ±180 0.4 0.6 0.8 1 0 0≤ Γ ≤1 225 -135 3 315 -45 270 -90 NT2 225 -135 4 NT2 315 -45 270 -90 1 Smith-Chart Engineering and Information Technology School of This would help to visualize Eq. 1 and 3. It would also help to visualize the effect of changing elements in a network. Division of Electrical- and Communication Engineering Let‘s first consider a few special points. Why not overlay a grid with impedance values? Berne University of Applied Sciences Division of Electrical- and Communication Engineering Engineering and Information Technology School of Berne University of Applied Sciences Smith-Chart For example: In a given load impedance consisting of a serie R – L network the value of L changes by a certain value. How does it affect the reflection coefficient (or VSWR)? Short circuit: R=0, X=0, (r=0, x=0) Γ= −1 = −1 = 1∠180o 1 90 Open circuit: R=∞, X=0, (r=∞, x=0) 135 Γ = 1 = 1∠0o 45 Z=Z0: R=50, X=0, (r=1, x=0) Γ=0 0.2 ±180 Short 0.4 0.6 6 NT2 315 -45 270 -90 NT2 Smith-Chart Smith-Chart r= 1 ±180 1 0.2 0.4 0.6 0.8 Engineering and Information Technology 45 School of 135 7 r + jx − 1 −0.5 + jx = r + jx + 1 1.5 + jx 135 45 r = 0.5 ±180 0.5 1 0.2 0.4 0.6 0.8 1 0 -0.1 -0.5 -1 -2 -4 -10 +1 +0.1 +∞ +10 +4 +2 0 -∞ x = +0.5 225 -135 This gives a constant-r-circle for r=0.5 315 -45 270 -90 NT2 90 Γ= 1 0 -0.1 -0.5 -1 -2 -4 -10 +1 +0.1 +∞ +10 +4 +2 0 -∞ x = +0.5 This gives a constant-r-circle for r=1 Division of Electrical- and Communication Engineering Draw the locus for r=0.5 (R=0.5Z0) and -∞<x<+∞ 90 r + jx − 1 jx = r + jx + 1 2 + jx Berne University of Applied Sciences Division of Electrical- and Communication Engineering Engineering and Information Technology School of Berne University of Applied Sciences Draw the locus for r=1 (R=Z0) and -∞<x<+∞ Γ= 1 0 Open 225 -135 5 0.8 Z0 8 225 -135 315 -45 270 -90 NT2 2 Smith-Chart Smith-Chart 2 4 r = 0.2 ±180 0.2 0.5 1 0.2 0.4 2 0.6 4 0.8 Engineering and Information Technology 45 School of 135 Division of Electrical- and Communication Engineering Locus for r=0.2, 0.5, 1, 2, 4 and -∞<x<+∞ 90 r + jx − 1 Γ= r + jx + 1 Berne University of Applied Sciences Division of Electrical- and Communication Engineering Engineering and Information Technology School of Berne University of Applied Sciences Draw the locus for r=0.2, 2, 4 and -∞<x<+∞ 90 Γ= r + jx − 1 r + jx + 1 45 r = 0.2, 0.5, 1, 2, 4 -∞<x<+∞ 1 0 135 180 0.2 0.5 1 0.2 0.4 2 0.6 4 0.8 1 0 -0.1 -0.5 -1 -2 -4 -10 +1 +0.1 +∞ +10 +4 +2 0 -∞ x = +0.5 Locus for constant real part of impedance More 225 -135 315 -45 225 -135 315 -45 constant-r-circles 270 -90 9 270 -90 10 NT2 NT2 Smith-Chart Smith-Chart 45 0.2 2 x= 1 ±180 0.2 0.5 1 0.2 0.4 2 0.6 4 0.8 Engineering and Information Technology 135 School of 1 Division of Electrical- and Communication Engineering Draw the locus for x=-1 (X=-Z0) and 0<r<+∞ 90 r + jx − 1 r + j1 − 1 Γ= = r + jx + 1 r + j1 + 1 Berne University of Applied Sciences Division of Electrical- and Communication Engineering Engineering and Information Technology School of Berne University of Applied Sciences Draw the locus for x=1 (X=Z0) and 0<r<+∞ 90 Γ= r + jx − 1 r − j1 − 1 = r + jx + 1 r − j1 + 1 1 135 0.2 2 x = -1 1 0 ±180 1 4 +∞ 0 0 20.2 r = 0.5 45 0.2 0.5 1 0.2 0.4 2 0.6 4 0.8 1 0 1 4 +∞ 0 0 20.2 r = 0.5 225 -135 315 -45 225 -135 315 -45 -1 11 270 -90 NT2 12 270 -90 NT2 3 Smith-Chart Smith-Chart Impedance plane 0.2 4 -10 -4 -2 +0.2 -0.5 +2 +10 +0.5 +4 x = -0.2 0.5 1 0.4 0.6 2 0.8 4 1 0 0 r = 0<r<+∞ 4 0.2 0.5 2 1 -4 225 -135 -2 -0.5 315 -45 -2 -0.5 -1 270 -90 -1 270 -90 14 NT2 NT2 Smith-Chart Impedance plane ±180 0 Zin 0.2 0.5 2 1 Zin L/Z 0 225 -135 -0.5 Z1 jx=-jZ0/ωC Zin -2 1 135 Zin R 0.5 2 4 r=R /Z NT2 0 Zin Z1 ±180 0 0.2 0.5 2 1 0 4 ∞ Zin r= Z1 = R1 + jX1 Zin = (R1 + R ) + jX1 315 -45 -0.2 225 -135 0 R/Z -4 Z1 -2 -0.5 -1 Real part of impedance = constant 45 0.2 Z1 ∞ -4 90 Serie R 0 4 -0.2 jx= jω School of 4 Berne University of Applied Sciences Z1 Zin = R + j(X1 ± X) +jx (inductive) Z1 0.2 Z1 = R1 + jX1 45 -jx (capacitive) jx= jω L/Z jX 2 0 0.5 135 Engineering and Information Technology 1 Zin Division of Electrical- and Communication Engineering 90 Serie L or C jx=-jZ /ωC 0 Division of Electrical- and Communication Engineering Engineering and Information Technology School of Berne University of Applied Sciences Impedance plane 315 -45 +jx (inductive) 225 -135 Smith-Chart 15 ∞ -0.2 Locus for constant imaginary part of impedance Zin 0 4 -4 -0.2 13 45 -jx (capacitive) 0.2 0.2 2 0.2 ±180 ±180 0.5 +jx (inductive) 45 1 135 -jx (capacitive) 2 Engineering and Information Technology 0.5 135 School of 1 Division of Electrical- and Communication Engineering 90 r + jx − 1 Γ= r + jx + 1 Berne University of Applied Sciences Division of Electrical- and Communication Engineering Engineering and Information Technology School of Berne University of Applied Sciences Draw the locus for x=±0.2, ±0.5, ±2, ±4, ±10 and 0<r<+∞ 90 315 -45 -1 270 -90 Imaginary part of impedance = constant 16 270 -90 NT2 4 Smith-Chart l Z1 1− Γ |Γ| = = 2 0.6 The center of constant VSWR-circle is the center of Smith-Chart if SmithChart is normalized to the line impedance. 18 NT2 -0.2 225 -135 0.5 VS W R = Division of Electrical- and Communication Engineering Engineering and Information Technology School of Y : complex Admittance = G + jB -4 -2 -0.5 270 -90 .8 =0 Γ 1 − Γ 1 − 0.8∠60o = 1 + Γ 1 + 0.8∠60o = 0.163 − j0.675 y= 0.2 ±180 225 -135 20 NT2 θ o Eq. 6 45 60 NT2 1 − y 1 − g − jb = 1 + y 1 + g + jb Γ = 0.8∠60o 135 Γ = 0.8∠60o Y = (3.253 − j13.5)mS y = 0.163 − j0.675 θ= 19 Γ= 315 -45 -1 90 1− Γ 1 − 0.8∠60o = 20mS Y = Y0 1+ Γ 1 + 0.8∠60o = (3.253 − j13.5)mS Normalized Y = y Eq. 5 0 8 4 For Γ = 0.8∠60o and Z0 = 50Ω from Eq.1 and 3: Y0 = 1/ Z0 Y G B = +j = g + jb Y0 Y0 Y0 1 According to Eq.4, each reflection coefficient must also represent a complex admittance Y, or according to Eq.6 a complex normalized admittance y. Y0 : Normalizing Admittance (usually real) y= 0.8 4 Smith-Chart Berne University of Applied Sciences Division of Electrical- and Communication Engineering Engineering and Information Technology School of Berne University of Applied Sciences Γ : complex Reflection Coefficient = Γ ∠θ 0.6 2 NT2 Admittance Chart: Eq. 4 4 0.4 1 Smith-Chart Y 1− Y0 − Y Y0 Γ= = Y0 + Y 1 + Y Y0 45 Zin Z1 0.2 Load R 2 0.2 0 d towar Load 17 d towar VSWR = 1+ Γ -jx (capacitive) ± 180 VS W 0.2 Gener l Z = Z0 0.5 +jx (inductive) ator Zin 1 135 -jx (capacitive) with ZL = Z0 90 towar d Division of Electrical- and Communication Engineering Gener ator +jx (inductive) constant |Γ| and VSWR Engineering and Information Technology Serie Transmission line School of Circles of Berne University of Applied Sciences Impedance plane towar d Division of Electrical- and Communication Engineering Engineering and Information Technology School of Berne University of Applied Sciences Smith-Chart Impedance plane 0.4 0.6 0.8 1 0 315 -45 270 -90 5 -0.2 Division of Electrical- and Communication Engineering -4 Engineering and Information Technology -0.4 -2 School of -1 Berne University of Applied Sciences Division of Electrical- and Communication Engineering Engineering and Information Technology Locus for constant imaginary part of admittance Admittance plane Admittance plane -1 Parallel L or C ∞ 4 0.4 1 2 0.2 21 /ω L Y1 -4 jB Y1 4 2 1 0.4 -0.2 0.2 0 Yin 4 Yin Yin = G + j(B1 ± B) 0.2 jb 2 Real part of admittance = constant 0.4 22 1 NT2 jY 0 4 2 -0.4 jb =- 0 Yin Y1 = G1 + jB1 Locus for constant real part of admittance Yin -2 ∞ School of Using the similar procedure as with the impedance plane we get circles for constant conductance g and arcs for constant susceptance b. Smith-Chart /Y jb=jωC 0 Berne University of Applied Sciences Smith-Chart Yin jb=jωC/Y 0 Y1 jY =- /ω 0.2 L 0 0.4 1 NT2 Smith-Chart Smith-Chart 90 ±180 Constant Q circles -0.24 -40.2 4 0.2 0 0.4 1 2 1 0.4 0.2 4 2 4-0.2 225 -135 NT2 0 0.2-4 2 0.4 -0.4 1 23 45 -2 315 -45 Division of Electrical- and Communication Engineering 2 Engineering and Information Technology -0.4 School of 1 Berne University of Applied Sciences -2 +jx, -jb (inductive) Division of Electrical- and Communication Engineering Engineering and Information Technology School of Overlay of impedance and admittance plane ∞ Berne University of Applied Sciences 135 0.4 Q= X G = = cons tan t R B 0 ∞ -jx, +jb (capacitive) -1 Imittance Chart -1 270 -90 24 NT2 6 27 NT2 28 Division of Electrical- and Communication Engineering Engineering and Information Technology School of Berne University of Applied Sciences Division of Electrical- and Communication Engineering Engineering and Information Technology School of Berne University of Applied Sciences www.hta-be.bfh.ch/~dellsper Division of Electrical- and Communication Engineering NT2 Engineering and Information Technology DP-Nr. 1 DP-Nr. 2 DP-Nr. 3 DP-Nr. 4 DP-Nr. 5 DP-Nr. 6 School of 25 Berne University of Applied Sciences Division of Electrical- and Communication Engineering Engineering and Information Technology School of Berne University of Applied Sciences Smith-Chart Smith-Chart Example using „Smith“-Software L-networks for maching Z2 to Z0 (7.7 - j7.4)Ohm Q = 1.0 470.000 MHz (14.8 + j0.0)Ohm Q = 0.0 470.000 MHz (14.8 - j14.3)Ohm Q = 1.0 470.000 MHz (28.6 + j0.2)Ohm Q = 0.0 470.000 MHz (28.6 + j24.4)Ohm Q = 0.9 470.000 MHz (49.5 + j0.0)Ohm Q = 0.0 470.000 MHz 26 NT2 Smith-Chart Smith-Chart L-networks for maching Z2 to Z0 „Smith“-Software www.fritz.dellsperger.net NT2 7
© Copyright 2024 ExpyDoc
