Notes 2 - Smith charts

Report
ECE 5317-6351
Microwave Engineering
Fall 2011
Prof. David R. Jackson
Dept. of ECE
Notes 2
Smith Charts
1
Generalized Reflection Coefficient
I(-l )
+
Z0 , 
V(-l )
ZL
l
Recall,
V 
  V 0 e 
I 
V0
Z 



e

Z0
V 
I 
1  
1  


e
L
2
e
L
2 
V


0
e
1      


V0
e

Z0
1      
1  
 1   L e 2 
 Z0 
 Z0 
2 
1  
 1   Le


Generalized reflection Coefficient:
 




 L e
2
2
Generalized Reflection Coefficient (cont.)
 
  L e
 L e
 R 
j L
L 
2 
e
2 

For
j I  
ZL  Z0
ZL  Z0
Re Z L  0

 L  1
Proof:
Lossless transmission line ( = 0)
 

 L e
j  L  2 
L 


 L
2

 RL 
 RL 
jX L   Z 0
jX L   Z 0
 RL  Z 0  
 RL  Z 0  
jX L
jX L
 RL  Z 0   X L
 RL  Z 0 
2
2
2
 XL
2
3
Complex  Plane
   

Im 
  R  j I
  Le
j 2 
 L e
Decreasing l (toward load)

j  L  2 

l
L
L
L
L  2

Increasing l (toward
generator)
Re 
Lossless line
4
Impedance (Z) Chart
Z 

1  
 Z0 

1




Zn 
Define
   

Z 
Z0


1  


1




Z n  R n  jX n ;    R  j  I
Substitute into above expression for Zn(-l ):
 1    R  j I  
R n  jX n  
 1     j   
R
I


Next, multiply both sides by the RHS denominator term and equate real and imaginary
parts. Then solve the resulting equations for R and I in terms of Rn and Xn. This gives
two equations.
5
Impedance (Z) Chart (cont.)
1) Equation #1:

 1 
Rn 
2
R 
  I  

1

R
1

R
n 
n 


I
2
Equation for a circle in the  plane
Rn  0
Rn  1
Rn 1
1  Rn  
1
R
 Rn

cen ter  
,0
 1  Rn

rad iu s 
1
1  Rn
6
Impedance (Z) Chart (cont.)
2) Equation #2:
2
  R  1
2

 1 
1 
 I 
 

X
X
n 

 n
I
0  Xn 1
Xn 1
2
Equation for a circle in the  plane:
1 Xn  

1 
cen ter   1,

X
n 

1
Xn  0
R
rad iu s 
1
Xn
  X n  1
0  X n  1
X n  1
7
Impedance (Z) Chart (cont.)
Short-hand version
Xn = 1
Rn = 1
Xn = -1
8
Impedance (Z) Chart (cont.)
Imag. (reactive)
impedance
Inductive (Xn > 0)
 plane
Xn = 1
Rn = 1
Short ckt.
( 1
Match pt.
(0
Open ckt.
(1
Real
impedance
Xn = -1
Capacitive (Xn < 0)
9
Admittance (Y) Calculations
Note:
Y 

1
Z 

1 1  


Z0 1  
 1     
 Y0 
 1     

 Yn  
Define:     

Y 
Y0

 

  
 1     

 1     

Yn  




Y0 
1
Z0
 
  Gn    
  
 1   



1




Same mathematical form as for Zn:
Zn 
jB n  

Conclusion: The same
Smith chart can be used as
an admittance calculator.

1  


1  
10
Admittance (Y) Calculations (cont.)
Imag. (reactive)
admittance
Capacitive (Bn > 0)
 plane
Bn = 1
Gn = 1
Open ckt.
(  1
Match pt.
( 0
Short ckt.
( 1
Real
admittance
Bn = -1
Inductive (Bn < 0)
11
Impedance or Admittance (Z or Y) Calculations
The Smith chart can be used
for either impedance or
admittance calculations, as
long as we are consistent.
12
Admittance (Y) Chart
As an alternative, we can continue to use the original  plane, and
add admittance curves to the chart.
Yn  

 1     

 1     

 
  Gn    
  
jB n  

Compare with previous Smith chart derivation, which started
with this equation:
Zn 

 1   

 1   

 
  Rn    
  
jX n  

If (Rn Xn) = (a, b) is some point on the Smith chart corresponding to  = 0,
Then (Gn Bn) = (a, b) corresponds to a point located at  = - 0 (180o rotation).
 Rn = a circle, rotated 180o, becomes Gn = a circle.
and Xn = b circle, rotated 180o, becomes Bn = b circle.
Side note: A 180o rotation on a Smith chart makes a normalized impedance become its reciprocal.
13
Admittance (Y) Chart (cont.)
Gn = 0
Inductive (Bn < 0)
 plane
Bn = -1
Gn = 1
Short ckt.
Match pt.
Bn = 0
Open ckt.
Bn = +1
Capacitive (Bn > 0)
14
Admittance (Y) Chart (cont.)
Short-hand version
Bn = -1
Gn = 1
Bn = 1
 plane
15
Impedance and Admittance (ZY) Chart
Short-hand version
Bn = -1
Xn = 1
Gn = 1
Rn = 1
Bn = 1
Xn = -1
 plane
16
Standing Wave Ratio
The SWR is given by the
value of Rn on the positive
real axis of the Smith chart.
Proof:
SW R 
L
Zn 
Rn  Rn
m ax

 R
m ax
n

1 L
1 L

1  
1 L
1 L


1  
1 L e
2 j
1 L e
2 j
1 L e
j L
1 L e
j L
e
e
2 j
2 j
17
Electronic Smith Chart
At this link
http://www.sss-mag.com/topten5.html
Download the following .zip file
smith_v191.zip
Extract the following files
smith.exe
smith.hlp
smith.pdf
This is the application file
18

similar documents