Ứng dụng đồ thị Smith

Lecture 08
The Smith Chart and Basic
Impedance-Matching Concepts
Sections: 6.8 and 6.9
Homework: From Section 6.13 Exercises: 12, 13, 14, 15,
16, 17, 18, 19, 20
Nikolova 2012 2
The Smith Chart: Γ plot in the Complex Plane
• Smith’s chart is a graphical representation in the complex Γ plane of
the input impedance, the load impedance, and the reflection
coefficient Γ of a loss-free TL
• it contains two families
of curves (circles) in
the complex Γ plane
• each circle corresponds
to a fixed normalized
resistance or reactance
Nikolova 2012 Lecture 08: The Smith Chart 3
The Smith Chart: Normalized Impedance and Γ
0
00
1
where and
1
=| | =
L
LL
LLL
LL
j
ri

Z
Z
z
Z
zrjx
ZZ z Z
ej




    

 
1
1
L
z



relation #1: normalized load impedance z
L
and reflection Γ
22
22
22
1
(1 )
2

(1 )
ri
L
ri
i
L
ri
r
x
 

 



 
22
2
22
2
1
11
11
(1)
L
ri
LL
ri
LL
r

rr
x
x

 




  



Nikolova 2012 Lecture 08: The Smith Chart 4
The Smith Chart: Resistance and Reactance Circles
22
2
1
11
L
ri
LL
r
rr

 



22

2
11
(1)
ri
LL
x
x

  


let the abscissa be Γ
r
and the ordinate be Γ
i
(the Γ complex plane)
• resistance and reactance equations are circles in the Γ
complex plane
• resistance circles have centers lying on the Γ
r
axis (Γ
i
= 0 or
ordinate = 0)
• reactance circles have centers with abscissa coordinate = 1
• a complex normalized impedance z
L
= r
L
+ jx

L
is a point on
the Smith chart where the circle r
L
intersects the circle x
L
resistance circles
reactance circles
Nikolova 2012 Lecture 08: The Smith Chart 5
The Smith Chart: Resistance Circles
r

i

1
||
1


1
L
r 
0
L
r

0.5
0



1
0.2
0.25
L
r

1
short
open
Nikolova 2012 Lecture 08: The Smith Chart 6
The Smith Chart: Reactance Circles
inductive
loads
capacitive
loads
Nikolova 2012 Lecture 08: The Smith Chart 7
The Smith Chart: Nomographs
at the bottom of Smith’s chart, a nomograph is added to determine
• SWR and SWR in dB,
• return loss in dB,
• power reflection |Γ|
2
(P)
• reflection coefficient |Γ| (E or I), etc.
perfect match
10
20lo
g||



10
20log SWR
Nikolova 2012 Lecture 08: The Smith Chart 8
The Smith Chart: SWR Circles
 a circle of radius Γ
m
centered
at Γ = 0 is the geometrical
place for load impedances
producing reflection of the
same magnitude, | Γ| = Γ
m
 such a circle also corresponds
to constant SWR
1| |
1| |
SWR



SWR circle
0.4 0.7
L
z
j


3.87SWR

||

0.59


Nikolova 2012 Lecture 08: The Smith Chart 9
The Smith Chart: Plotting Impedance and Reading Out Γ
0.5 1.0
L
zj


0.5
L
r

1
L
x

||

(1 0.135 / 0.25) 0.46 83



   

||
0.62
What is Z
L

if Z
0
= 50 Ω?
0.135

R

getting |Γ| with a ruler:
1) measure
2) measure
3) | | /
R
R



83

Nikolova 2012 Lecture 08: The Smith Chart 10
The Smith Chart: Tracking Impedance Changes with L
()
0
() 0
()
0
()
()
j
LjL
zL

in z L
j
LjL
zL
V
Ve e
ZZ Z
I
Ve e














2
0
2
1
1
j
L

in
j
L
e
ZZ
e













relation #2: input impedance versus the TL length L
compare with
1
1
L
z






2
2
1
1
j
L
in
j
L
e
z
e








 on the Smith chart, the point corresponding to z
in
is rotated by
−2βL (decreasing angle, clockwise rotation) with respect to the
point corresponding to z
L
along an SWR circle
 one full circle on the Smith chart is 2βL
max
= 2π, i.e., L

max
= λ/2;
this reflects the periodicity of z
in

Nikolova 2012 Lecture 08: The Smith Chart 11
The Smith Chart: Tracking Impedance Changes with L –2
for Z
0
= 50 Ω, the
quarter-wavelength TL
transforms a load of
25 25
L
Z
j


to an input impedance of
r

i

1
0
1
0.5 0.5
L
z
j

11
in
zj


to
w
a
r
d

g
e
n
e
r
a
to
r
t
o
w
a
r
d

l
o
a
d

/4L


50 50
in
Z
j


check and see whether
0 Lin
Z
ZZ
For a frequency-independent
load Z
L
, what would be the
direction of the locus of Z
in
as
frequency increases?
SWR circle
Nikolova 2012 Lecture 08: The Smith Chart 12
The Smith Chart: Read Out Distance to Load
• unknown distance to load
in terms of λ
/
n
DD



• known load Z
L
75 75
L
Zj
 
• known Z
0
0
50
Z

1.5 1.5
L
zj


A
• measured Z
in
23 34
in
Zj
 
B
0.46 0.68
in
zj


toward generator
0.194
A
L


0.394
B
L


0.2
nBA
DLL


Nikolova 2012 Lecture 08: The Smith Chart 13
The Smith Chart: Reading Out SWR
,
11
LA
z
j


,
2.6
LB
r


,
2.6
LB
SWR r


SWR circle
A
B
A
B
SWR SWR
,
,
1
1
LB
B
LB
r
r



,
1| |
1| |
B
B
B

B
LB
SWR
SWR r




Nikolova 2012 Lecture 08: The Smith Chart 14
The Smith Chart: Admittance Interpretation
• normalized load admittance
1
1
11
11
LL
yz





 




• normalized input admittance (at generator)
2
1

2
1
1
j
L
in in
j
L
e
yz
e









• the relation between y
in
and y
L
is the same as that between z
in
and z
L
– one can get from load to input terminals (and vice versa) by
following a circle clockwise (counter-clockwise)

• standard Smith chart gives resistance and reactance values
• admittance Smith chart is exactly the same as the “impedance” (or
standard) Smith chart but rotated by 180° [see eq. (*) and sl. 17]
()

Nikolova 2012 Lecture 08: The Smith Chart 15
The Smith Chart: Admittance Interpretation – 2
normalized resistance
normalized reactance
impedance Smith Chart
Nikolova 2012 Lecture 08: The Smith Chart 16
The Smith Chart: Admittance Interpretation – 3
combined impedance and conductance Smith Chart
Nikolova 2012 Lecture 08: The Smith Chart 17
The Smith Chart: Admittance Interpretation – 4
• impedance values from a standard Smith chart can be easily
converted to admittance (conductance + susceptance) values by
rotation along a circle of exactly 180°
• rotation by 180° on the impedance Smith chart corresponds to
impedance transformation by a quarter-wavelength TL
2
4












11
(/4)
1
1
1
1
j
in
j
L
e
zL
e
z







 







1
(/4)
in L
L
z
Ly
z



• the value diametrically opposite on the Smith chart from
an impedance value is the respective “admittance” value
Nikolova 2012 Lecture 08: The Smith Chart 18
The Smith Chart: Admittance Interpretation – 5
r

i

1
0
1
0.5 0.5
L
z
j
11
in
zj



t
o
w
a
r
d

g
e
n
e
r
a
t
o
r
t
o
w
a
r
d

l
o
a
d
/4L



11
L
yj


Check whether in this
example the y
L
found
from the Smith chart
satisfies
1
L
L
y
z

Nikolova 2012 Lecture 08: The Smith Chart 19
Quarter-wave Transformer Revisited
from L07, sl. 18:
2
0
/4
in
L
L
Z
Z
Z




 for impedance match at the input terminals of the λ/4 TL, Z
in
=Z
G
*
0 GL
Z
ZZ


⇒
⇒
in
Z
0
/4
L


0
(, )Z

L
Z
G
Z
G

V
loss-free line
1
GL
zz



⇒
GL
z
y


 in this case the TL must be designed to have this specific Z
0
Nikolova 2012 Lecture 08: The Smith Chart 20
Quarter-wave Transformer Revisited – 2
 the impedance match with the λ/4 transformer holds perfectly at
one frequency only, f
0
, where L = λ
0
/4
 this impedance-match device is narrow-band
00
0
00
tan( )
2

( ) , where
tan( ) 4 2
L
in
L
ZjZ L
f
Zf Z L
Zj
ZL
f






  



0
0
()
|()|
()
in
in
Z
fZ

f
Z
fZ



0
100
50
70.71
L
G
Z
Z
Z






Nikolova 2012 Lecture 08: The Smith Chart 21
Optimal Power Delivery: Review (Homework)
 at the generator’s terminals, a loaded TL is equivalently represented
by its input impedance Z
in
G
Z
in
Z

in
V

in
I

G
V

 active (or average) power delivered to the loaded TL (this is also
the power delivered to the load Z
L
if the line is loss-free)
2
22
11 1 1
( ) Re{ } | | Re | | Re
22 2
in
in av in in in in G
Gin in
Z
PVIVYV
Z
ZZ



 





  
2
22
1
() | |
2
()( )
in
in av G
in G in G
R
PV
RR X X



Nikolova 2012 Lecture 08: The Smith Chart 22
 optimal matching is achieved when maximum active power is
delivered to the load Z
in
– what is this optimal value of Z
in
?
 assume generator’s impedance Z
G
= R
G

+ jX
G
is known and fixed
opt
max ( )
in
in in in
Z
Z
PZ
find the optimal R
in
and X
in
by obtaining the respective derivatives
22 2
0 ( ) 0
in
Gin in G
in
P
RR X X
R


 

0 ( ) 0
in
in in G

in
P
XX X
X




maximum power is delivered to the load under conditions of
conjugate match
opt opt opt
and
in G in G in G
R
RX X ZZ


Optimal Power Delivery: Review (Homework)