Notes 3 - Smith chart examples.pptx for electrical engineers

1. 1
Prof. David R. Jackson
Dept. of ECE
Notes 3
ECE 5317-6351
Microwave Engineering
Fall 2011
Smith Chart
Examples

2. 2
0 50
100 50
L
Z
Z j
 
  
I(-d)
V(-d)
+
d
ZL
-
0,
Z 
Find Z(-d)
/ 1/ 4, 3/ 8, 1/ 2
g
d  
at
,
0
2 1
L
L n
Z
Z j
Z
  
 
/ 4
0.4 0.2
/ 4 20 10
g
n
g
d
Z j
Z j



 
    
n
L
Z
3/ 8 g
d 

/ 4
g
d 

a
b
0
1/ 2 g
d 

or
Example 1
a
Impedance chart

3. 3
 
3/ 8
0.5 0.5
3/8 25 25
g
n
g
d
Z j
Z j



 
    
b
 
/ 2
2 1
/ 2 100 50
g
n
g
d
Z j
Z j



 
    
c
n
L
Z
3/ 8 g
d 

/ 4
g
d 

a
b
0
/ 2
g
d



0.087lg
c
0.5lg
0.462lg
0.212lg
Example 1 (cont.)
I(-d)
V(-d)
+
d
ZL
-
0,
Z 

4. 4
 
0 0
50 20mS
8mS 4mS
L
Z Y
Y j
  
 
Find Y(-d)
/ 1/ 4, 3/ 8, 1/ 2
g
d  
at
,
0
0.4 0.2
L
L n
Y
Y j
Y
  
 
/ 4
2 1
/ 4 40mS 20mS
g
n
g
d
Y j
Y j



 
   
n
L
Y
3/ 8 g
d 

/ 4
g
d 

a
b
0
1/ 2 g
d 

or
c
Example 2
a
I(-d)
V(-d)
+
d
ZL
-
0,
Z 
Admittance chart

5. 5
 
3/ 8
1 1
3/8 20mS 20mS
g
n
g
d
Y j
Y j



 
   
 
/ 2
0.4 0.2
/ 2 8mS 4mS
g
n
g
d
Y j
Y j



 
   
Example 2 (cont.)
b
c
I(-d)
V(-d)
+
d
ZL
-
0,
Z 
n
L
Y
3/ 8 g
d 

/ 4
g
d 

a
b
0
1/ 2 g
d 


c
Admittance chart

6. 6
Simple answer:
* When adding elements in series use Z-chart
* When adding elements in parallel use Y-chart
A B C
Z Z Z
 
A B C
Y Y Y
 
ZA ZB ZC

YA YB YC

Which Chart to Use?

7. 7
 Use reactively loaded section of
transmission line.
most common to use open or short load
d
ZL
0 ,
Z 
 
Z d

L L
Z jX

 
L
X    
0
L
X 
SC OC
d
d = 0
n
L
X
 
n
X d

Non-absorbing load
(RL
= 0)
Using Reactive Loads
Impedance chart
Reactance

8. 8
d
YL
0 ,
Z 
 
Y d

L L
Y j

SC OC
d
d = 0
n
L
Y
 
n
Y d

Susceptance
   
 
0
0
/
n
n
Y d Y d Y
Y d Z
  
 
Using Reactive Loads (cont.)
Admittance chart

9. 9
Use a short-circuited section of air-filled TEM, 50  transmission line
( = k0, g =0) to create an impedance of Zin = -j25  at f = 10 GHz.
SC
-1/2
0.426 g

50W
0 g
 L
25
in
Z j
 
SC
50 Ω , k0
 
,
25
1/ 2
50
in n
Z j j
   
0.426 0 0.426
g g g
L   
  
Example 3
L = 1.28 cm
0
0 0 0
2 2
c
f k
 

  
   0 3.0 cm
 
Impedance chart

10. 10
Use an open circuited section of 75  (Y0 = 1/75 S) air-filled
transmission line at f = 10 GHz to create an admittance of
j1
1/75 S
OC
L
L
 
1/ 75
in
Y j
 
OC
75 Ω , k0
 
, 1
in n
Y j

0
1
75
Y  
1
S 13.3mS
75
in
Y j j
  
0.375cm
L
 
Example 4
0
0.125
L 
 0 3.0 cm
 
Admittance chart

11. 11
Y0
d
1
n
jb
 ,
A n
Y
,
L n
Y
1
n
G 
,
in in
Z Y d
ZL
YS
A
Y
0
Z
 
0
in
Z Z

,
,
0 0 ,
1
L
L n
L
L n
L n
Y
Y
Y Z
Z
Z
Z
 
 
, , ,
in n A n S n
Y Y Y
 
Note: At d we have
Want to pick d and Ys such that Yin = Y0.
0
0
1
Y
Z

Matching Circuit
Admittance chart

12. 12
,
S n n
Y jb
 
, ,
1 1 n S n
in n Y
Y jb
  

We want
Choose
 
0
,
0
1
1 1
in n n n
in
in
in
Y jb b
Y
j
Y Y
Z Z
    

  

,
in in
Z Y d
ZL
YS
A
Y
0
Z
Y0
d
1
n
jb
 ,
A n
Y
,
L n
Y
1
n
G 
Matching Circuit (cont.)
Admittance chart

13. 13
Example 5
ZL
Z0
ls
Z0s
d
0 50[ ]
Z  
100 100 [ ]
L
Z j
  
, 2 2
L n
Z j
 
 
,
1
0.25 .25
2 2
L n
Y j
j
  

/6 o
0.62 0.62 30
j
L e 
   
0
0
1
1
L Ln
L
L Ln
Z Z Z
Z Z Z
 
  
 
In this example we will
use the “usual” Smith
chart, but as an
admittance calculator.

14. 14
Example 5 (cont.)
X
X
0.178 g

,
Y 0.25 0. 5
2
L n j
 
1 1.57
j

1 1.57
j

0.322 g

0.363 g

0.219 g

0.041 g

X
0.219
-
0.36
1.57
1.57 3
g
g
d
d
j
j





Solution:
Add at
or at
0.17
0 0.2
.0 9
8
1 1
4 g
 
 

0.32
0 0.3
.0 3
2
1 6
4 g
 
 

wavelengths toward load
wavelengths toward generator
Smith chart scale:
Admittance calculator
 plane
SC
OC
(We’ll use the first choice.)

15. 15
Example 5 (cont.)
S / C
X
0 1.57
j

0.09
O / C
Admittance calculator
 
, cot
1.57 cot
1
cot 1.57; tan
1.57
2
0.567 [radians]
s n
B l
l
l l
l l


 



 
  
 
 
0.09
s g
l 

From the Smith chart:
0.0903
s g
l 

Analytically:

16. 16
0
0
0
tan
tan
L T
in T
T L
Z jZ
Z Z
Z jZ


 

  

 


  2
/ 4
4 2
g
g
g

 
 

   

 
2
0
1
2
0
T
in
L
T in L
Z
Z
Z
Z Z Z
 

 
0
1
2
0 0
in
T L
Z Z
Z Z Z

 
ZL
Z0 Z0T
Zin, Gin
ZL is real
/ 4
g

Quarter-Wave Transformer
At f0 :
For in = 0

17. 17
Match 100  load to 50  transmission Line at f0.
0 100 50
70.7
T
Z  

0
0
0
0
2 2 2
g f
r r
f
k k

  

  
   
Example 6
0 70.7
T
Z  
 
1
r
 
b, 50 Ω Z0T
/ 4
g

100 Ω