topic 7 -- smith charts
TRANSCRIPT
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Topic 7 ‐‐ Smith Charts Slide 1
EE 4347
Applied Electromagnetics
Topic #7
Smith Charts
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Lecture Outline
Topic 7 ‐‐ Smith Charts Slide 2
• Construction of the Smith Chart
• Admittance and impedance
• Circuit theory
• Determining VSWR and • Impedance transformation
• Impedance matching
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Construction of the Smith Chart
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Polar Plot of Reflection Coefficient
Topic 7 ‐‐ Smith Charts 5
The Smith chart is based on a polar plot of the voltage reflection coefficient . The outer boundary corresponds to || = 1. The reflection coefficient in any passive system must be|| ≤ 1.
je
radius on Smith chart
angle measured CCW from right side of chart
Normalized Impedance
Topic 7 ‐‐ Smith Charts 6
All impedances are normalized. This is usually done with respect to the characteristic impedance of the transmission line Z0.
0
ZzZ
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Normalized Reflection Coefficient
Topic 7 ‐‐ Smith Charts 7
We can write the reflection coefficient in terms of normalized impedances.
0
0 0 0
00
0 0
1
1
L
L L
LL L
ZZZ Z Z Z z
ZZZ Z zZ Z
Derivation of Smith Chart:Solve for Load Impedance
Topic 7 ‐‐ Smith Charts 8
Solving the previous equation for load impedance, we get
1
1
1 1
1
1
1 1
1
1
L
L
L L
L L
L L
L
L
z
z
z z
z z
z z
z
z
1
1Lz
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Derivation of Smith Chart:Real and imaginary parts
Topic 7 ‐‐ Smith Charts 9
The load impedance and reflection coefficient can be written in terms of real and imaginary parts.
L L L r iz r jx j
1
11
1
1
1
L
r iL L
r i
r iL L
r i
z
jr jx
j
jr jx
j
Substituting these into the load impedance equation yields
Derivation of Smith Chart:Solve for rL and xL
Topic 7 ‐‐ Smith Charts 10
We solve or previous equation for rL and xL by setting the real and imaginary parts equal.
2
2 2
2 2
2 2
2 2
2 2
2 2
2 22 2
1
1
1 1
1 1
1 1 1 1
1
1
1
1 2
1
1 2
1 1
r iL L
r i
r i r i
r i r i
r r r i i r i
r i
r i r i i r i i
r i
r i i
r i
r i i
r i r i
jr jx
j
j j
j j
j j
j j j j
j
j
2 2
2 2
2 2
1
1
2
1
r iL
r i
iL
r i
r
x
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Derivation of Smith Chart:Rearrange equation for rL
Topic 7 ‐‐ Smith Charts 11
We rearrange the equation for rL so that it has the form of a circle.
2 2
2 2
2 22 2
222 2
222 2
2 2 2 2
2 2
2 2
can be factored
1
1
11
11 0
12 1 0
2 1 0
2 1 1 1 0
21
r iL
r i
r ir i
L
irr i
L L L
irr r i
L L L
L r L r r L i i L
L r L r L i L
L rr i
L
r
r
r r r
r r r
r r r r
r r r r
r
r
10
1L
L
r
r
2 2
2
2 2
2
2 22
2 2
2 2 22
2 2
2
22
10
1 1 1
1
1 1 1
1 1
1 1 1
1
1 1 1
1
1 1
L L Lr i
L L L
L L Lr i
L L L
L LL Lr i
L L L
L L Lr i
L L L
Lr i
L L
r r r
r r r
r r r
r r r
r rr r
r r r
r r r
r r r
r
r r
Derivation of Smith Chart:Rearrange equation for xL
Topic 7 ‐‐ Smith Charts 12
We rearrange the equation for xL so that it has the form of a circle.
2 2
2 2
2 2
swap termscan be factored
22
2
2
1
21
21 0
1 11 0
iL
r i
ir i
L
r i iL
r iL L
x
x
x
x x
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Derivation of Smith Chart:Two families of circles
Topic 7 ‐‐ Smith Charts 13
Constant Resistance Circles
2 2
2 1
1 1L
r iL L
r
r r
Constant Reactance Circles
2 2
2 1 11r i
L Lx x
These have centers at These have centers at
01
Lr i
L
r
r
1
1 r iLx
Radii Radii
1
1 Lr1
Lx
Derivation of Smith Chart:Putting it all together
Topic 7 ‐‐ Smith Charts 14
+ + =
Lines of constant resistance
Lines of constant inductive reactance
Lines of constant capacitive reactance
Superposition
We ignore what is outside the || = 1 circle.
We don’t draw the constant || circles.
This is the Smith chart!
Lines of constant reflection coefficient
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Alternate Way of Visualizing the Smith Chart
Topic 7 ‐‐ Smith Charts 15
Lines of constant reactanceLines of constant resistance Reactance Regions
L
C
opencircuit
shortcircuit
3D Smith Chart
Topic 7 ‐‐ Smith Charts
The 3D Smith Chart unifies passive and active circuit design.
2D 3D
16
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Summary of Smith Chart
Topic 7 ‐‐ Smith Charts 17
Impedanceand
Admittance
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Admittance Coordinates
Topic 7 ‐‐ Smith Charts 19
We could have derived the Smith chart in terms of admittance.
You can make an admittance Smithchart by rotating the standardSmith chart by 180.
Impedance/AdmittanceConversion
Topic 7 ‐‐ Smith Charts 20
The Smith chart is just a plot of complex numbers. These could be admittance as well as impedance.
To determine admittance from impedance (or the other way around)…
1. Plot the impedance point on the Smith chart.2. Draw a circle centered on the Smith chart that passes through the point (i.e.
constant VSWR).3. Draw a line from the impedance point, through the center, and to the other side of
the circle.4. The intersection at the other side is the admittance.
impedance admittance
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Visualizing Impedance/Admittance Conversion
Topic 7 ‐‐ Smith Charts 21
Example #1 – Step 1Plot the impedance on the chart
Topic 7 ‐‐ Smith Charts 22
0.2 0.4z j
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Example #1 – Step 2Draw a constant VSWR circle
Topic 7 ‐‐ Smith Charts 23
0.2 0.4z j
Example #1 – Step 3Draw line through center of chart
Topic 7 ‐‐ Smith Charts 24
0.2 0.4z j
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Example #1 – Step 4Read off admittance
Topic 7 ‐‐ Smith Charts 25
0.2 0.4z j
1.0 2.0y j
Example #2 – Step 1Plot the impedance on the chart
Topic 7 ‐‐ Smith Charts 26
0.5 0.3z j
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Example #1 – Step 2Draw a constant VSWR circle
Topic 7 ‐‐ Smith Charts 27
0.5 0.3z j
Example #2 – Step 3Draw line through center of chart
Topic 7 ‐‐ Smith Charts 28
0.5 0.3z j
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Example #2 – Step 4Read off admittance
Topic 7 ‐‐ Smith Charts 29
1.0 2.0y j
0.5 0.3z j
DeterminingVSWR and
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Determining VSWR
Topic 7 ‐‐ Smith Charts 31
1. Plot the normalized load impedance on the Smith chart.2. Draw a circuit centered on the Smith chart that intersections this point.3. The VSWR is read where the circle crosses the real axis on right side.
Example: 50 line connected to 75+j10 load impedance.
0
75 101.5 0.2
50LZ j
z jZ
impedance
VSWR = 1.55
1
VSWR
Example #1 –What is the VSWR?
Topic 7 ‐‐ Smith Charts 32
50
3.3157 nH
1.9894 pF
in
inin
0
0.4 0.
20 40
20 40
0 8
5
Z j
Z jz
Zj
VSWR 4.3
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Example #1 –What is the reflection coefficient?
Topic 7 ‐‐ Smith Charts 33
0.62
Impedance Transformation
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Normalized Impedance Transformation Formula
Topic 7 ‐‐ Smith Charts 35
Our impedance transformation formula was
0in 0
0
tan
tanL
L
Z jZZ Z
Z jZ
We can write this in terms of the reflection coefficient.
00in 0 0
0 0
0 00 00 0
0 0 0
0.5 0.5cos sin
cos sin 0.5 0.5
j j j jLL
j j j jL Z
j jj j j jL LL L
j j j j jL L L L
Z e e Z e eZ jZZ Z Z
Z jZ Z e e Z e e
Z Z e Z Z eZ e Z e Z e Z eZ ZZ e Z e Z e Z e Z Z e Z
0
0
20
0 0 20
0
11
11
j
jL
j jL
j jL
jL
Z e
Z Z e
Z Z e eZ Z
Z Z e e
Z Z e
We normalize by dividing by Z0.
2
in 2
1
1
j
j
ez
e
Interpreting the Formula
Topic 7 ‐‐ Smith Charts 36
The normalized impedance transformation formula was
2
in 2
1
1
j
j
ez
e
Recognizing that = ||ej, this equation can be written as
22
in 2 2
1 1
1 1
jj j
j j j
e e ez
e e e
Thus we see that traversing along the transmission line simply changes the phase of the reflection coefficient.
As we move away from the load and toward the source, we subtract phase from . On the Smith chart, we rotate clockwise (CW) around the constant VSWR circle by an amount 2l. A complete rotation corresponds to /2.
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Impedance Transformationon the Smith chart
Topic 7 ‐‐ Smith Charts 37
1. Plot the normalized load impedance on the Smith chart.2. Move clockwise around the middle of the Smith chart as we move away from the
load (toward generator). One rotation is /2 in the transmission line.3. The final point is the input impedance of the line.
Example #2 – Impedance Trans. Normalize the parameters
Topic 7 ‐‐ Smith Charts 38
0 50 Z 50 25 LZ j
0.67
1 0.5 Lz j
0.67
in
1 0.5 tan 2 0.67tan1.299 0.485
1 tan 1 1 0.5 tan 2 0.67L
L
j jz jz j
jz j j
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Example #2 – Impedance Trans. Plot load impedance
Topic 7 ‐‐ Smith Charts 39
0.67
1 0.5 Lz j
Example #2 – Impedance Trans. Walk away from load 0.67
Topic 7 ‐‐ Smith Charts 40
0.67
1 0.5 Lz j
Since the Smith chart repeats every 0.5, traversing 0.67 is the same as traversing 0.17.
Here we start at 0.145 on the Smith chart.
We traverse around the chart to 0.145 + 0.17 = 0.315.
0.145
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Example #2 – Impedance Trans. Determine input impedance
Topic 7 ‐‐ Smith Charts 41
0.67
1 0.5 Lz j
Reflection at the load will be the same regardless of the length of line.
Therefore the VSWR will the same.
The input impedance must lie on the same VSWR plane.
inZ
in 1.3 0.5z j
Example #2 – Impedance Trans. Denormalize
Topic 7 ‐‐ Smith Charts 42
0.67
1 0.5 Lz j
To determine the actual input impedance, we denormalize.
inZ
in 0 in 50 1.3 0.5 65 25 Z Z z j j