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Prof. David R. JacksonDept. of ECE
Notes 2
ECE 5317-6351 Microwave Engineering
Fall 2011
Smith Charts
1
Recall,
0
0
20
20
2
2
00
0 0
1
1
1
1
1
1
1
1
L
L
L
L
V e
Ve
V V e e
VI e e
Z
V eZ
Z
ZZI e
Generalized reflection Coefficient: 2L e
Generalized Reflection Coefficient I(-l )
V(-l )+
l
ZL-
0,Z
2
2
2LL
I
L
j
R
e e
j
e
Lossless transmission line ( = 0)
2LjL e
0
0
LL
L
Z Z
Z Z
Generalized Reflection Coefficient (cont.)
Re 0
1
L
L
Z
For
0
0
0
0
L LL
L L
L L
L L
R jX Z
R jX Z
R Z jX
R Z jX
2 22 0
2 20
L LL
L L
R Z X
R Z X
Proof:
3
Complex Plane
2
2L
R I
jL
jL
j
e
e
Increasing l (toward generator)
l
Re
Im Decreasing l (toward load)
L
L
L
2L
Lossless line
4
0
1
1Z Z
0
1
1n
ZZ
Z
Define;n n n R IZ R jX j
Substitute into above expression for Zn(-l ):
Impedance (Z) Chart
1
1R I
Rn n
I
j
jR jX
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
1) Equation #1:2
2 1
1 1n
R In n
R
R R
Equation for a circle in the plane
,01
1
1
n
n
n
R
R
R
center
radius
Impedance (Z) Chart (cont.)
6
1
0nR 1nR
1nR 1 nR
R
I
2 2
2 1 11R I
n nX X
Equation for a circle in the plane:
11,
1n
n
X
X
center
radius
Impedance (Z) Chart (cont.)2) Equation #2:
7
1
0nX
1 nX
1nX 0 1nX
0 1nX
1nX
1nX
R
I
Impedance (Z) Chart (cont.)
Short-hand version
Rn = 1
Xn = 1
Xn = -1
8
Open ckt. (1
Imag. (reactive)impedance
Match pt. (0
Realimpedance
Short ckt. ( 1
Inductive (Xn > 0)
Capacitive (Xn < 0)
Impedance (Z) Chart (cont.)
planeRn = 1
Xn = 1
Xn = -1
9
Note:
0
11 1
1Y
Z Z
0
1
1Y
0
1
1n n nG jBY
YY
00
1Y
Z
Admittance (Y) Calculations
Define: 1
1nY
Same mathematical form as for Zn:
Conclusion: The same Smith chart can be used as an admittance calculator.
10 1
1nZ
Short ckt. ( 1
Imag. (reactive)admittance
Match pt. ( 0
Realadmittance
Open ckt. ( 1
Capacitive (Bn > 0)
Inductive (Bn < 0)
Admittance (Y) Calculations (cont.)
planeBn = 1
Bn = -1
Gn = 1
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
As an alternative, we can continue to use the original plane, and add admittance curves to the chart.
1
1n n nG jBY
Admittance (Y) Chart
1
1n n nR jXZ
Compare with previous Smith chart derivation, which started with this equation:
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).
Side note: A 180o rotation on a Smith chart makes a normalized impedance become its reciprocal. 13
Þ Rn = a circle, rotated 180o, becomes Gn = a circle. and Xn = b circle, rotated 180o, becomes Bn = b circle.
Open ckt.
Match pt.
Gn = 0
Short ckt.
Inductive (Bn < 0)
Capacitive (Bn > 0)
Gn = 1
Bn = +1
Bn = -1
Bn = 0
Admittance (Y) Chart (cont.)
plane
14
Short-hand version
Gn = 1
Bn = -1
Bn = 1
plane
15
Admittance (Y) Chart (cont.)
Impedance and Admittance (ZY) Chart
Short-hand version
Gn = 1 Rn = 1
Xn = 1
Xn = -1
Bn = -1
Bn = 1
plane
16
The SWR is given by the value of Rn on the positive real axis of the Smith chart.
Standing Wave Ratio
L
Proof:1
1L
L
SWR
2
2
2
2
1
1
1
1
1
1
L
L
n
jL
jL
j jL
j jL
Z
e
e
e e
e e
17
max 1
1L
nL
R
maxn nR R
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
Electronic Smith Chart
18