E.E. Essential Knowledge Sereies My Online Courses: https://www.byparams.com/courses Smith Chart

1. Chapter 2-2
The Smith Chart
Chien-Jung Li
Department of Electronics Engineering
National Taipei University of Technology

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.
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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
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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
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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  
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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 
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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  
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8. Department of Electronic Engineering, NTUT
Complete the Smith Chart
Short OpenLoad
+jx
-jx
Inductive
Capacitive
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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 
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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
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11. Department of Electronic Engineering, NTUT
Example – Find from Impedance
19.44 

1 3 3z j 
1z
0.721 19.44   
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12. Department of Electronic Engineering, NTUT
Example – Find Impedance from
0.447 26.56  
2 1z j 
26.56

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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
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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
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15. Department of Electronic Engineering, NTUT
The ZY Chart
U
V
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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


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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


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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


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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

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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
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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
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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)
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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
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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
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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 



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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  



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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 
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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 
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29. Department of Electronic Engineering, NTUT
Constant Q Contour (I)
n
X x
Q
R r
 
1nQ 
2nQ 
Short Open
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30. Department of Electronic Engineering, NTUT
Constant Q Contours (II)
Short Open
very intensive
very intensive
intensive
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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:
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32. Department of Electronic Engineering, NTUT
Matching with Particular Q Requirement (II)
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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 
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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.
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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)
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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?
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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.
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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
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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 
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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 
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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 
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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.
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