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Lecture 9
The Smith Chart and Basic Impedance-Matching Concepts
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ElecEng4FJ4 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
Re r
Im i 1j
| | 1
0 1
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ElecEng4FJ4 3
The Smith Chart: Normalized Impedance and Γ
0
0 0
1 where and1
=| | =
L L LL L L
L Lj
r i
Z Z z Zz r jxZ Z z Z
e j
11Lz
relation #1: normalized load impedance zL and reflection Γ
2 2
2 2
2 2
1(1 )
2(1 )
r iL
r i
iL
r i
r
x
2 22
2 22
11 1
1 1( 1)
Lr i
L L
r iL L
rr r
x x
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ElecEng4FJ4 4
The Smith Chart: Resistance and Reactance Circles
2 22 1
1 1L
r iL L
rr r
2 22 1 1( 1)r i
L Lx x
let the abscissa be Γr and the ordinate be Γi (the complex Γ plane)
• resistance and reactance equations describe circles in the Γcomplex plane
• resistance circles have centers lying on the Γr axis (with Γi= 0, i.e., ordinate = 0)
• reactance circles have centers with abscissa coordinate = 1• a complex normalized impedance zL = rL + jxL is a point on
the Smith chart where the circle rL intersects the circle xL
resistance circles reactance circles
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ElecEng4FJ4 5
The Smith Chart: Resistance Circles
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ElecEng4FJ4 6
The Smith Chart: Reactance Circles
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ElecEng4FJ4 7
The Smith Chart: Nomographs at the bottom of Smith’s chart (left side), nomograph is added to
read out with a ruler the following • (1st ruler) above: SWR, below: SWR in dB,• (2nd ruler) above: return loss in dB,
below: power reflection |Γ|2 (P)• (3rd ruler) above: reflection coefficient |Γ| (E or I)
perfect match
1020log | | 1020log SWR
21 | | 1T
21010log (1 | | )
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ElecEng4FJ4 8
The Smith Chart: SWR Circles
a circle of radius |Γ| centered at Γ = 0 is the geometrical place for load impedances producing reflection of the same magnitude |Γ|
such a circle also corresponds to constant SWR
1 | |1 | |
SWR
SWR circle
0.4 0.7Lz j
3.87SWR
| | 0.59
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The Smith Chart: Plotting Impedance and Reading Out Γ
0.5 1.0Lz j
0.5Lr
1Lx
| |
83
| | 0.62
What is ZL if Z0 = 50 Ω?0.135
R
getting |Γ| with a ruler:
2) measure 1) measure
3) | | / R
R
83
ElecEng4FJ4 9
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ElecEng4FJ4 10
The Smith Chart: Tracking Impedance Changes with L
20
1at generator: ,
1g j L
in gg
Z Z e
2
0 211
j L
in j LeZ Ze
relation #2: input impedance versus the TL length L
compare with1at load: 1Lz
2
211
j L
in j Leze
on the Smith chart, the point corresponding to zin is rotated by −2βL (decreasing angle, clockwise rotation) with respect to the point corresponding to zL along an SWR circle (toward generator)
one full circle on the Smith chart is 2βLmax = 2π, i.e., Lmax = λ/2; this reflects the π-periodicity of zin
(see L08, sl. 4)
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r
i1j
0 1
0.5 0.5Lz j
1 1inz j
toward generator
toward load
/ 4L
11
The Smith Chart: Tracking Impedance Changes with L – 2
for Z0 = 50 Ω, the quarter-wavelength TL transforms a load of
25 25 LZ j to an input impedance of
50 50 inZ j
check and see whether
0 L inZ Z Z
For a frequency-independent load ZL, what would be the direction of the locus of Zin as frequency increases?
SWR circle
ElecEng4FJ4
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ElecEng4FJ4 12
The Smith Chart: Read Out Distance to Load• unknown distance to load
in terms of λ/nD D
• known load ZL
75 75 LZ j
• known Z0
0 50 Z
1.5 1.5Lz j A
• measured Zin
23 34 inZ j B
0.46 0.68inz j
toward generator 0.194AL
0.394BL
0.22n B AD L L n
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ElecEng4FJ4 13
The Smith Chart: Reading Out SWR
, 1 1L Az j
, 2.6L Br
, 2.6L BSWR r
SWR circle
A
B
A BSWR SWR
,
,
11
L BB
L B
rr
,
1 | |1 | |
BB
B
B L B
SWR
SWR r
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ElecEng4FJ4 14
The Admittance Smith Chart• normalized load admittance
11 1 1 1
1 1 1
j
L jLy ee
z
• normalized input admittance (at generator)2
12
11
j L
in in j Ley ze
• the relation between yin and yL is the same as that between zin and zL– one can get from load to generator (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 Smith chart but rotated by 180° [see eq. (*) ] in the complex Γ plane
( )
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ElecEng4FJ4 15
Reading Out Normalized Conductance and Susceptance Values
00
1LL L
L
Yy Y ZY z
YL – load admittanceY0 – characteristic admittance
L L LY G jB
conductance susceptance
L L Ly g jb
• normalized admittance
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ElecEng4FJ4 16
Conductance and Susceptance Circles in Admittance Smith Chart combined impedance and conductance Smith Charts
open circuit0
( )1
LL
YZ
short circuit
( 0)1
LL
YZ
conductance circles resistance circles
reactance circlessusceptance circles
positive (capacitive) susceptance
negative (inductive) susceptance
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ElecEng4FJ4 17
Switching Between Impedance and Admittance on Smith Chart
• impedance values from a standard Smith chart can be easily converted to admittance by rotation along a circle by exactly 180°
• rotation by 180° on the impedance Smith chart corresponds to impedance transformation by a quarter-wavelength TL
24
1 1( / 4)11
11
j
in j
L
ez Le
z
1( / 4)in LL
z L yz
• in impedance Smith chart, the point diametrically opposite from an impedance point shows its respective “admittance” value
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18
Switching Between Impedance and Admittance: Example
Check whether in this example the yL found from the Smith chart satisfies 1
LL
yz
r
i1j
0 1
0.5 0.5Lz j
1 1inz j
toward generator
toward load
/ 4L 1 1Ly j
same as
ElecEng4FJ4
Example: L-network matching
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ElecEng4FJ4 19
Quarter-wave Transformer Revisited
from L08, sl. 18:20
/4in LL
ZZZ
for impedance match at the input terminals of the λ/4 TL, Zin = ZG*
0 G LZ Z Z⇒ ⇒
inZ
0 / 4L
0( , )ZLZ
GZ
GV
loss-free line
1G Lz z ⇒ G Lz y
TL must be designed to have this specific Z0
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20
Quarter-wave Transformer Revisited – 2 the impedance match with the λ/4 transformer holds perfectly at
one frequency only, f0, where L = λ0/4
this impedance-match device is narrow-band0 0
00 0
tan( ) 2( ) , where tan( ) 4 2
Lin
L
Z jZ L fZ f Z LZ jZ L f
0
0
( )| ( ) |( )
in
in
Z f ZfZ f Z
0
100 50 70.71
L
G
ZZZ
perfect match
ElecEng4FJ4
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ElecEng4FJ4 21
Optimal Power Delivery: Review (Homework) at the generator’s terminals, a loaded TL is equivalently represented
by its input impedance Zin
GZ
inZinV
inI
GV
active (or average) power delivered to the loaded TL (this is also the power delivered to the load ZL if the line is loss-free)
22 21 1 1 1( ) Re | | Re | | Re
2 2 2in
in av in in in in GG in in
ZP V I V Y VZ Z Z
22 2
1( ) | |2 ( ) ( )
inin av G
in G in G
RP VR R X X
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ElecEng4FJ4 22
optimal matching is achieved when maximum active power is delivered to the load Zin – what is this optimal value of Zin?
assume generator’s impedance ZG = RG + jXG is known and fixed
opt max ( )in
in in inZZ P Z
find the optimal Rin and Xin by obtaining the respective derivatives
2 2 20 ( ) 0inG in in G
in
P R R X XR
0 ( ) 0inin in G
in
P X X XX
maximum power is delivered to the load under conditions of conjugate match
opt opt opt and in G in G in GR R X X Z Z
Optimal Power Delivery: Review (Homework)
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ElecEng4FJ4 23
the impedance Smith chart depicts a normalized load impedance as a point in the complex Γ plane
the load impedance is normalized with respect to the characteristic impedance Z0 of the TL
the admittance Smith chart depicts a normalized load admittance yLas a point in the complex Γ plane
the admittance Smith chart is rarely used because the impedance Smith chart can be readily used as an admittance chart as well – the sense of rotation with increasing TL length is the same
resistance/reactance impedance values are determined from the resistance/reactance circles
the input impedance zin of a TL loaded with a known zL is found by following the SWR circle, starting from zL, and completing an angle of 2βL in a clockwise direction (on chart: “toward generator”)
Summary
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ElecEng4FJ4 24
if the input impedance zin of a TL is known but the load zL is not, zL is determined by starting from zin , following the SWR circle and completing an angle of 2βL in a counter-clockwise direction (on chart: “toward load”)
the normalized admittance yL of a given impedance zL is found by reading out the value of the point diametrically opposite to zL on the Smith chart
many more applications of the Smith chart will be shown during the tutorial
Summary – 2