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The Smith Chart
Antennas and Propagation
Chapter 1Antennas and Propagation Slide 2
Introduction
Origin
1939 P.H. Smith. Graphical method for
performing transmission line calculations.
(Well before pocket calculators or computers)
Today
Useful for displaying information: shows reflection coefficient and
impedance/admittance simultaneously
Helps engineer gain intuition about using transmissions lines /
creating matching circuits.
Chapter 1Antennas and Propagation Slide 3
Basics
Principle
Waves on a transmission line
Chapter 1Antennas and Propagation Slide 4
Basics (2)
Smith Chart Shows
1. Reflection coefficient
2. Impedance / Admittance
|Γ|
∠∠∠∠Γ
ImΓ
ReΓ
Resistance
Values
Reactance
Values
Chapter 1Antennas and Propagation Slide 5
Bascis (3)
Smith chart is a graphical
implementation of function:
Reflection ⇔⇔⇔⇔ Impedance
Using same function for admittance
Just rotate by 180 degrees
Chapter 1Antennas and Propagation Slide 6
Typical Smith Chart
Information
Constant resistance
Constant reactance
Transmission Line
Length (wavelengths)
ββββl / 2π
Polar angle (for Γ)
Polar radius (for |Γ|)
Chapter 1Antennas and Propagation Slide 7
Combination
Smith Chart
Two smith charts
Rotated by 180o
Can read Y / Z at
once
Chapter 1Antennas and Propagation Slide 8
Examples:
Z,Y ⇔⇔⇔⇔ Γ
Example 1
ZL = 100 + j50 Ω
Z0 = 50 Ω
zL = 2 + j1
From Chart
G = 0.45 ∠ 26o
Exact
G = 0.447 ∠ 26.5o
26o
0.45
Chapter 1Antennas and Propagation Slide 9
Examples:
Z ⇔⇔⇔⇔ Y
Example 2
ZL = 100 + j50 Ω
Z0 = 50 Ω
zL = 2 + j1
What is y?
Rotate by 180o
y = 0.4 – j0.2
Exact:
(Is exact value)
Chapter 1Antennas and Propagation Slide 10
Impedance Transformations
Idea1. Plot load impedance on Smith chart
This gives Γ0
2. Can find gamma at any point on transmission line with
Just means rotation on Smith chart
Which way do we move with increasing len?
3. Can read new impedance value looking into that point.
Note: 1 wavelength (λ), βl = 2π = 360o,
But on Smith chart, 0.5 λ means 360o shift in Γ.
Why?
Chapter 1Antennas and Propagation Slide 11
Examples:
Imp. Transform
Example 3
zL =
0.4 – j0.5
TLine: l/8 = 0.125
zin = 0.32 + j0.25
Exact:
zin = 0.332 + j0.248
0.418
0.418
+0.125
=0.543
Chapter 1Antennas and Propagation Slide 12
Transmission Line Stubs
Idea
Length of (lossless) transmission line
Open or shorted
Presents a reactance / suceptance
Recall
Open stub (ZL = ∞)
Short stub (ZL = 0)
Can get any reactance we like with proper length l
Chapter 1Antennas and Propagation Slide 13
Examples:
Stub Len.
Example 4
Want zstub =
-j1.4
Length?
Shorted stub:
0.35λ
Open stub:
0.1λ
Chapter 1Antennas and Propagation Slide 14
Matching
Goal
Get Γ(z) = 0 (get to origin)
Also means zin = 1 (Zin = Z0)
Operations
Clockwise rotation
Adding transmission line
Moving on constant r circle
Adding/subtracting reactance (series stub / reactance)
Moving on constant g circle
Adding/substracting suceptance (shunt stub / reactance)
Chapter 1Antennas and Propagation Slide 15
Examples:
Stub Match
Example 5
zant =
0.4 – j0.6
l = 0.174λ
+ 0.094λ
= 0.268λ
xstub = -1.4
Chapter 1Antennas and Propagation Slide 16
Examples:
Stub Match
Example 5
zant =
0.4 – j0.6
l = 0.016λ
ystub = -1.3
Chapter 1Antennas and Propagation Slide 17
Other
Examples?
Chapter 1Antennas and Propagation Slide 18
Other
Examples?
Chapter 1Antennas and Propagation Slide 19
Other Uses
Smith Chart also Useful For
Gain / stability analysis of amplifiers
Gain with respect to source / load impedance
Constant gain becomes a circle on chart
Stability circles. Stable inside, unstable outside
Noise figure analysis
Constant noise figure circles
Chapter 1Antennas and Propagation Slide 20
Conclusion
Smith Chart
Graphical tool for doing simple transmission line computations
Transmission line transformations / matching
For this class
Useful mainly for visualization
Also, smith chart gives valuable intuition
“See” how transmission line works without doing computation