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# RF Circuit Design - [Ch2-2] Smith Chart

Smith Chart

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### RF Circuit Design - [Ch2-2] Smith Chart

1. 1. Chapter 2-2 The Smith Chart Chien-Jung Li Department of Electronics Engineering National Taipei University of Technology
2. 2. Department of Electronic Engineering, NTUT The Smith Chart • The analysis of transmission-line problems and of matching circuits at microwave frequencies can be cumbersome in analytical form. The smith chart provides a very useful graphical aid to the analysis of these problems. • Matching circuits can be easily and quickly designed using the normalized impedance and admittance Smith chart (Z and Y charts). • The Smith chart is also used to present the frequency dependence of scattering parameters and other amplifier characteristics. 2/42
3. 3. Department of Electronic Engineering, NTUT Development of the Smith Chart (I)       o o Z Z x Z Z • The Smith chart is the representation in the reflection coefficient plane, called the plane, of the relation for all values of Z, such that Re{Z}≥0. Zo is the characteristic impedance of the transmission line or a reference impedance value. • Defining the normalized impedance z as      o o Z R jX z r jx Z Z               11 1 1 r jxz U jV z r jx        2 2 2 2 1 1 r x U r x      2 2 2 1 x V r x where and • Reflection Coefficient 3/42
4. 4. Department of Electronic Engineering, NTUT Development of the Smith Chart (II) r x  U jV  Γ-plane U V  1z j  1z  0z 1 1 z z     1 1 1 90z j j    0 1 1 180z        1 0z     1 90   0  1     z r jxz-plane 1 1 1 90z j j         1z j  Short Load Open 1z      1  Pure Imaginary: inductive 1 90    Pure Imaginary: capacitive 4/42
5. 5. Department of Electronic Engineering, NTUT Constant Resistance Circles (I) r x  U jV  Γ-plane U V  1 1z j  1 1z j  0z 0.447 63.4   0.447 63.4      z r jxz-plane 1 1z j  1 1z j  0.447 63.43   0.447 63.43    1 2z j  1 2z j  1 2z j  1 2z j  0.707 45   0.707 45    1j 2j 1j 2j 0.707 45   0.707 45   5/42
6. 6. Department of Electronic Engineering, NTUT Constant Resistance Circles (II) r x   z r jxz-plane U V 0z jx  0z r  0.5r  1r  3r  0.5z jx  1z jx  3z jx  0r  3r 1r  0.5r  6/42
7. 7. Department of Electronic Engineering, NTUT Constant Reactance Loci r x   z r jxz-plane U V 0.5z j 0.5z j 1z j 3z j 0.5z j  1z j  3z j  0j 0.5j 1j 3j 0.5j 1j 3j 0.5 0.5z j  1 0.5z j  1.5 0.5z j  1 126.87   0.447 116.56   0.243 75.97   0.2773 33.69   7/42
8. 8. Department of Electronic Engineering, NTUT Complete the Smith Chart Short OpenLoad +jx -jx Inductive Capacitive 8/42
9. 9. Department of Electronic Engineering, NTUT Reactance in the Smith Chart Short OpenLoad +jx -jx Inductive Capacitive +j0.1 +j0.2 +j0.3 +j0.4 +j0.5 +j0.6 +j1.6 +j1.7 +j1.8 +j2.0 +j3.0 +j4.0 +j5.0 +j6.0 0.4x  0.4x  0.4x  9/42
10. 10. Department of Electronic Engineering, NTUT Example – Impedance in the Smith Chart 1 1 1z j  2 0.4 0.5z j  3 3 3z j  4 0.2 0.6z j  5 0z  1z2z 3z 4z 5z 10/42
11. 11. Department of Electronic Engineering, NTUT Example – Find from Impedance 19.44   1 3 3z j  1z 0.721 19.44    11/42
12. 12. Department of Electronic Engineering, NTUT Example – Find Impedance from 0.447 26.56   2 1z j  26.56  12/42
13. 13. Department of Electronic Engineering, NTUT Use Smith Chart as an Admittance (Y) Chart y g jb 1 1 1y j  2 0.4 0.5y j  3 2 1.4y j  4 0.5 0.2y j  5y   1y2y 3y4y 5y 13/42
14. 14. Department of Electronic Engineering, NTUT Show Z and Y in One Chart y g jb  U V U Vz r jx  1 1 1 y g jb z         1 1 z      Impedance Chart (Z-Chart) Admittance Chart (Y-Chart) jx jx jb jb Short Load Open Short Load Open 14/42
15. 15. Department of Electronic Engineering, NTUT The ZY Chart U V 15/42
16. 16. Department of Electronic Engineering, NTUT Adding a Series Inductor 0.8Lz j 0.3 0.3z j  0.3 0.5inz j  0.3 0.3z j  0.3 0.5inz j  0.8x  -j0.3 +j0.5   16/42
17. 17. Department of Electronic Engineering, NTUT Adding a Series Capacitor 0.8Cz j  0.3 0.3z j  0.3 1.1inz j  0.3 0.3z j  0.3 1.1inz j 0.8x   -j0.3 -j1.1   17/42
18. 18. Department of Electronic Engineering, NTUT Adding a Shunt Inductor 1.6 1.6y j  1.6 0.8iny j  2.4Ly j  1.6 1.6y j  1.6 0.8iny j  2.4y   +j1.6 -j0.8   18/42
19. 19. Department of Electronic Engineering, NTUT Adding a Shunt Capacitor 1.6 1.6y j  1.6 5iny j  3.4Cy j 1.6 1.6y j  1.6 5iny j  3.4y  +j1.6 +j5  19/42
20. 20. Department of Electronic Engineering, NTUT Series/Shunt Inductor or Capacitor Higher impedanceLower impedance Series L Series C Shunt L Shunt C +jx -jx Inductive Capacitive Short Open Lower admittanceHigher admittance -jb +jb 20/42
21. 21. Department of Electronic Engineering, NTUT Matching Networks (Two-Element L-Shape) LZ1C 2C LZL C LZ1L 2L LZC L LZC L LZ2C 1C LZL C LZ2L 1L 21/42
22. 22. Department of Electronic Engineering, NTUT Match to the Reference Impedance • Usually the goal is to transform a particular impedance to the reference impedance (center of the Smith chart). In practical systems, the reference impedance . 50refZ   1z2z 3z 4z 5z Goal Goal circle (r=1) Goal circle (g=1) 22/42
23. 23. Department of Electronic Engineering, NTUT Matching from Load to the Reference Impedance (I)  10 10LZ j   0.2 0.2Lz j  Goal 0.2j 0.4j 0.2x j  2j 0j 2y j  0.2 0.4z j   50refZ          C L 01@ 500 MHzinz f  0.2 0.2j  0.2j  0.5j 02 0.2 50 10f L     0 1 2 2 0.04 50 f C     3.18 nHL  12.74 pFC  C L 10  3.18 nH 3.18 nH 12.74 pF 23/42
24. 24. Department of Electronic Engineering, NTUT Matching from Load to the Reference Impedance (II)  10 10LZ j   0.2 0.2Lz j  Goal 0.2j 0.4j 0.6x j   2j 0j 2y j   0.2 0.4z j  L C 0.2 0.2j 01@ 500 MHzinz f          0.6j   1 02 0.6 50 30f C        1 0 1 2 2 0.04 50 f L      10.6 pFC  7.95 nHL  L C 10.6 pF 7.95 nH 10  3.18 nH 24/42
25. 25. Department of Electronic Engineering, NTUT Matching from the Reference Impedance 1 L C  8 12 mSoutY j  Goal  50  0.4 0.6outy j     25/42
26. 26. Department of Electronic Engineering, NTUT Matching from Load to an Arbitrary Impedance LZC L 50 20inZ j   100 100LZ j   Goal 100refZ   LZC L 0.5 0.2inZ j   1 1Lz j      26/42
27. 27. Department of Electronic Engineering, NTUT Impedance with Frequency Increasing L R C R L R C L R C  1inZ R j L      1 1 50 in in Z z r jx        1in aZ   1in bZ   2inZ   2in aZ   2in bZ   3inZ   3in aZ   1 1 inZ R j C      3in bZ   4inZ   4in bZ   4in aZ  27/42
28. 28. Department of Electronic Engineering, NTUT Impedance with Frequency Increasing L R C R L R C C R L  2inZ   1inZ   4inZ   3inZ   1in aZ   1in bZ   2in aZ   2in bZ   3in aZ   3in bZ   4in aZ   4in bZ  28/42
29. 29. Department of Electronic Engineering, NTUT Constant Q Contour (I) n X x Q R r   1nQ  2nQ  Short Open 29/42
30. 30. Department of Electronic Engineering, NTUT Constant Q Contours (II) Short Open very intensive very intensive intensive 30/42
31. 31. Department of Electronic Engineering, NTUT Matching with Particular Q Requirement (I) • At matched condition: 2 n L Q Q  • For certain BW spec., the designed QL meets 0 1 L f BW Q  • Design a T-shape matching networks to transform to . The matching should meet relative bandwidth requirement of 40%. 50LZ   10 15inZ j   1 0.4 LQ  1 2.5 0.4 LQ   At matched condition: 2.5 2 n L Q Q   5nQ Thus in the design stage, the network should have a node Q: 31/42
32. 32. Department of Electronic Engineering, NTUT Matching with Particular Q Requirement (II) 32/42
33. 33. Department of Electronic Engineering, NTUT Low Q Matching with 5% LC Variations 1.2 nHL  1.8 pFC  1.1nQ 1nQ  1nQ  1.06nQ  LZ1.8 pFC  1.2 nHL   24.26 11.62LZ j   50inZ   • Application example: Match a certain impedance to 50-Ohm in a 1800 MHz GSM handset front-end with node Q = 1. 1.26 nHL  1.8 pFC  5% L variation 1.2 nHL  1.89 pFC  5% C variation 1.26 nHL  1.89 pFC  5% L+C variation 50.4 0.61inZ j  51.8 0.57inZ j  50.34 1.97inZ j  51.75 2.16inZ j  33/42
34. 34. Department of Electronic Engineering, NTUT High Q Matching with 5% LC Variations 50inZ  44 5inZ j  40 8inZ j  35.4 13.5inZ j  LZ1.8 pFC  5.5 nHL   24.26 11.62LZ j   50inZ   8.8 nHL  5% L variation 5% C variation 5% L+C variation • Application example: Match a certain impedance to 50-Ohm in a 1800 MHz GSM handset front-end with node Q = 3. 34/42
35. 35. Department of Electronic Engineering, NTUT Small Impedance Matched to 50 Ohm (I) 4.9nQ  5.1nQ  4.9nQ  5.1nQ  LZ8.6 pFC  0.78 nHL   2 1LZ j   50inZ   50.2 1.26inZ j  52.8 10inZ j  48.2 9.76inZ j  45.94 0.44inZ j  0.78 nHL  8.6 pFC  0.82 nHL  8.6 pFC  5% L variation 0.78 nHL  9 pFC  5% C variation 0.82 nHL  9 pFC  5% L+C variation • Application example: Match a certain small impedance to 50-Ohm in a 1800 MHz GSM handset front-end. (node Q = 4.9)  In this case, the major problem is not easy to find a small inductor for matching.  Practically, a higher value of inductor would be used. (see next page) 35/42
36. 36. Department of Electronic Engineering, NTUT Small Impedance Matched to 50 Ohm (II) 11.2nQ  2.2nQ  11.8nQ  1.15nQ 11.8nQ  1.62nQ  LZ3.2 pFC  1.9 nHL   2 1LZ j   50inZ   50.15inZ  74.3 22inZ j  119.6 41.8inZ j   To avoid a small inductor, use a higher value of L with increasing the node Q. 5% L variation 5% L+C variation  Problems arise: (1) Fail to meet broadband spec. (not a case for GSM in this example) (2) Sensitive to component variations (3) Use parallel-connected Ls to maintain a low-Q matching (area consuming)  How about using a series-C and shunt-L? 36/42
37. 37. Department of Electronic Engineering, NTUT Small Impedance Matched to 50 Ohm (III) 5.1nQ  5.8nQ  LZ1 pFC  1.3 nHL   2 1LZ j    50inZ   12 pFC  9.3 pFC   Use more components to trade the matching bandwidth. (area consuming)  Variations affect node Q easily in low-impedance region. 37/42
38. 38. Department of Electronic Engineering, NTUT High Impedance Matched to 50 Ohm 9.94nQ  10.44nQ  LZ0.176 pFC  44 nHL   5000 60LZ j   50inZ   50.08 0.32inZ j  45.47 23inZ j  • Application example: Match a certain small impedance to 50-Ohm in a 1800 MHz GSM handset front-end. (node Q = 9.9)  In this case, the major problem is not easy to find a small capacitor for matching. (In ICs, it is possible)  The components variation affects. 5% L+C variation 38/42
39. 39. Department of Electronic Engineering, NTUT Frequency Sweeping (Low Q v.s. High Q) 5.7nQ  2.2nQ  LZ0.94 pFC  9.6 nHL   160.5 44LZ j   50inZ   LZ3.9 pFC  6.1 nHL   160.5 44LZ j   50inZ   2.0 pFC   47.8 2 @1.8 GHzinZ j   44.2 10 @1.9 GHzinZ j   57.3 13 @1.7 GHzinZ j  50@1.8 GHzinZ   23.6 4.7 @1.7 GHzinZ j   47.8 46 @1.9 GHzinZ j  39/42
40. 40. Department of Electronic Engineering, NTUT Frequency Sweeping Low Q with 5% LC Variations LZ0.94 pFC  9.6 nHL   160.5 44LZ j   50inZ   LZ0.99 pFC  10.1 nHL   160.5 44LZ j   50inZ    47.8 2 @1.8 GHzinZ j   44.2 10 @1.9 GHzinZ j   57.3 13 @1.7 GHzinZ j   44.7 9 @1.8 GHzinZ j   39.4 20.7 @1.9 GHzinZ j   51.1 3 @1.7 GHzinZ j  40/42
41. 41. Department of Electronic Engineering, NTUT Frequency Sweeping High Q with 5% LC Variations LZ3.9 pFC  6.1 nHL   160.5 44LZ j   50inZ   2.0 pFC  LZ4 pFC  6.4 nHL   160.5 44LZ j   50inZ   2.1 pFC  50@1.8 GHzinZ   23.6 4.7 @1.7 GHzinZ j   47.8 46 @1.9 GHzinZ j   53 44 @1.8 GHzinZ j   46.4 3 @1.7 GHzinZ j   17.6 49 @1.9 GHzinZ j  41/42
42. 42. Department of Electronic Engineering, NTUT Summary • Although the Smith chart is seldom used nowadays for the computation of reflection coefficients. It is very useful and helpful for the engineers on the high-frequency circuit designs. • Just remember that a higher-Q circuit corresponds to a narrower bandwidth, and a lower-Q circuit corresponds to a wider bandwidth. Thus a higher-Q circuit is more sensitive to the frequency and components variations. 42/42