Notes 3 - Smith chart examples
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Transcript Notes 3 - Smith chart examples
ECE 5317-6351
Microwave Engineering
Fall 2011
Prof. David R. Jackson
Dept. of ECE
Notes 3
Smith Chart
Examples
1
Example 1
I(-d)
Z0 50
+
Z0 ,
Z L 100 j50
-
ZL
V(-d)
d
Find Z(-d)
d 3 / 8g
at d / g 1/ 4, 3/ 8, 1/ 2
Z L ,n
a
d
ZL
2 j1
Z0
0
o
/
r1
2 g
Impedance chart
b
Z Ln
d g / 4
Z n 0.4 j 0.2
a
Z g / 4 20 j10
d g / 4
2
Example 1 (cont.)
b
I(-d)
d 3 / 8g
Z n 0.5 j 0.5
Z0 ,
Z 3 / 8g 25 j 25
+
-
ZL
V(-d)
d
d 3 / 8g
c
d g / 2
d
0.087g
0.212g
Z n 2 j1
Z g / 2 100 j50
b
0
g
Z Ln
c
0.5g
a
d g / 4
0.462g
3
/2
Example 2
Z 0 50 Y0 20 mS
I(-d)
YL 8mS j 4 mS
+
Z0 ,
-
d
Find Y(-d)
at d / g 1/ 4, 3/ 8, 1/ 2
YL,n
ZL
V(-d)
d 3 / 8g
0
d 1 / 2 g
r
YL
0.4 j 0.2
Y0
o
Admittance chart
b
a
YLn
c
d g / 4
a
Yn 2 j1
Y g / 4 40 mS j 20 mS
d g / 4
4
Example 2 (cont.)
I(-d)
b
d 3 / 8g
+
Z0 ,
Yn 1 j1
Y 3 / 8g 20 mS j 20 mS
-
ZL
V(-d)
d
d 3 / 8g
c
d
d g / 2
0
/
1
2 g
Admittance chart
Yn 0.4 j 0.2
Y g / 2 8mS j 4 mS
b
YLn
c
a
d g / 4
5
Which Chart to Use?
Simple answer:
* When adding elements in series use Z-chart
ZA
ZB
ZC
Z A Z B ZC
* When adding elements in parallel use Y-chart
YA
YB
YC
YA YB YC
6
Using Reactive Loads
Use reactively loaded section of
transmission line.
most common to use open or short load
X L
Impedance chart
X L 0
Z d
SC
OC
X Ln
d
X n d
Z0 ,
ZL
d
Z L jX L
d=0
7
Using Reactive Loads (cont.)
Y d
Z0 ,
SC
YL
d
OC
YL j L
Y Ln
d
Yn d
Admittance chart
d=0
Y d Yn d Y0
Yn d / Z 0
8
Example 3
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.
Z in ,n j
Impedance chart
50
25
j 1/ 2
50
Zin j25
50 Ω , k0
SC
0g
L
-1/2
0 .4
SC
L 0.426g 0g 0.426g
L = 1.28 cm
g
26
0
c 2
2
f k0 0 0
0 3.0 cm
9
Example 4
Use an open circuited section of 75 (Y0 = 1/75 S) air-filled
transmission line at f = 10 GHz to create an admittance of
Yin j
1
S j13.3mS
75
Admittance chart
Yin,n j 1
1/75 S
Yin j 1/ 75
75 Ω , k0
OC
L
OC
Y0
1
75
j1
L 0.125 0
0 3.0 cm
L
L 0.375cm
10
Matching Circuit
Z in , Yin
Y0
1
Z0
d
Z0
ZL
YS
Admittance chart
YA
Y0
Want to pick d and Ys such that Yin = Y0.
Gn 1
Zin Z0
YL ,n
Z L,n
1
jb
n
YA,n
ZL
Z0
YL,n
YL
1
Y0 Z L,n
Note: At d we have
d
Yin,n YA,n YS ,n
11
Matching Circuit (cont.)
Z in , Yin
d
We want
Yin,n 1 1 jbn YS ,n
Z0
ZL
YS
YA
Choose
YS ,n jbn
Y0
Gn 1
Yin ,n 1 jbn jbn 1
YL ,n
Yin Y0
1
Z0
Yin
YA,n
jb
n
d
1
Z in
Admittance chart
12
Example 5
Z0 50[]
ZL 100 j100 []
d
Z0
In this example we will
use the “usual” Smith
chart, but as an
admittance calculator.
L
Z L Z0 Z Ln 1
Z L Z0 Z Ln 1
ZL
Z L, n 2 j 2
Z0s
ls
YL ,n
1
0.25 j .25
2 j2
L 0.62 e j /6 0.62 30o
13
Example 5 (cont.)
0.041 0.178 0.219g
0.041 0.322 0.363g
Solution :
0.178g
Add - j1.57 at d 0.219g
or
j1.57 at d 0.363g
Admittance calculator
1 j1.57
(We’ll use the first choice.)
plane
0.219g
SC
OC
0.363g
0.041g
1 j1.57
YL, n 0.25 j 0.25
Smith chart scale:
wavelengths toward load
wavelengths toward generator
0.322g
14
Example 5 (cont.)
From the Smith chart:
ls 0.09g
Admittance calculator
Analytically:
Bs , n cot l
1.57 cot l
cot l 1.57; tan l
l
2
S/ C
O/ C
1
1.57
l 0.567 [radians]
0.09
0 j1.57
ls 0.0903g
15
Quarter-Wave Transformer
g / 4
Z jZ 0T tan
Z in Z 0T L
Z
jZ
tan
L
0T
At f0 :
2 g
g / 4
g 4 2
Z0T
Zin, in
ZL
ZL is real
For in = 0
Z 02T
Zin
ZL
Z 0T Zin Z L
Z0
Z in Z 0
1
2
Z 0T Z 0 Z L
1
2
16
Example 6
Match 100 load to 50 transmission Line at f0.
g / 4
, 50 Ω
Z0T
Z 0T 100 50
70.7
g
f0
2
f0
0
2
2
k k0 r
r
100 Ω
r 1
Z0T 70.7
17