The Smith Chart Antennas and Propagation - Faculty.jacobs ...

Antennas and Propagation
The Smith Chart

Introduction
Origin
Today
1939 P.H. Smith. Graphical method for
performing transmission line calculations.
(Well before pocket calculators or computers)
Useful for displaying information: shows reflection coefficient and
impedance/admittance simultaneously
Helps engineer gain intuition about using transmissions lines /
creating matching circuits.
Antennas and Propagation Slide 2
Chapter 1

Basics
Principle
Waves on a transmission line
Antennas and Propagation Slide 3
Chapter 1

Basics (2)
Smith Chart Shows
1. Reflection coefficient
2. Impedance / Admittance
Reactance
Values
Antennas and Propagation Slide 4
Im{Γ}
|Γ|
∠Γ
Re{Γ}
Resistance
Values
Chapter 1

Bascis (3)
Smith chart is a graphical
implementation of function:
Reflection ⇔ Impedance
Using same function for admittance
Just rotate by 180 degrees
Antennas and Propagation Slide 5
Chapter 1

Typical Smith Chart
Information
Constant resistance
Constant reactance
Polar angle (for Γ)
Polar radius (for |Γ|)
Antennas and Propagation Slide 6
Transmission Line
Length (wavelengths)
βl / 2π
Chapter 1

Combination
Smith Chart
Two smith charts
Rotated by 180 o
Can read Y / Z at
once
Antennas and Propagation Slide 7
Chapter 1

Examples:
Z,Y ⇔ Γ
Example 1
Z L = 100 + j50 Ω
Z0 = 50 Ω
z L = 2 + j1
From Chart
G = 0.45 ∠ 26 o
Exact
G = 0.447 ∠ 26.5 o
0.45
Antennas and Propagation Slide 8
Chapter 1
26 o

Examples:
Z ⇔ Y
Example 2
Z L = 100 + j50 Ω
Z0 = 50 Ω
z L = 2 + j1
What is y?
Rotate by 180 o
y = 0.4 – j0.2
Exact:
(Is exact value)
Antennas and Propagation Slide 9
Chapter 1

Impedance Transformations
Idea
1. Plot load impedance on Smith chart
This gives Γ 0
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 = 360 o ,
But on Smith chart, 0.5 λ means 360 o shift in Γ.
Why?
Antennas and Propagation Slide 10
Chapter 1

Examples:
Imp. Transform
Example 3
z L =
0.4 – j0.5
TLine: l/8 = 0.125
z in = 0.32 + j0.25
Exact:
z in = 0.332 + j0.248
0.418
+0.125
=0.543
0.418
Antennas and Propagation Slide 11
Chapter 1

Transmission Line Stubs
Idea
Recall
Length of (lossless) transmission line
Open or shorted
Presents a reactance / suceptance
Open stub (Z L = ∞)
Short stub (Z L = 0)
Can get any reactance we like with proper length l
Antennas and Propagation Slide 12
Chapter 1

Examples:
Stub Len.
Example 4
Want z stub =
-j1.4
Length?
Shorted stub:
0.35λ
Open stub:
0.1λ
Antennas and Propagation Slide 13
Chapter 1

Matching
Goal
Get Γ(z) = 0 (get to origin)
Also means z in = 1 (Z in = Z 0)
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)
Antennas and Propagation Slide 14
Chapter 1

Examples:
Stub Match
Example 5
z ant =
0.4 – j0.6
l = 0.174λ
+ 0.094λ
= 0.268λ
x stub = -1.4
Antennas and Propagation Slide 15
Chapter 1

Examples:
Stub Match
Example 5
z ant =
0.4 – j0.6
l = 0.016λ
y stub = -1.3
Antennas and Propagation Slide 16
Chapter 1

Other
Examples?
Antennas and Propagation Slide 17
Chapter 1

Other
Examples?
Antennas and Propagation Slide 18
Chapter 1

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
Antennas and Propagation Slide 19
Chapter 1

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
Antennas and Propagation Slide 20
Chapter 1