Antennas and Propagation
The Smith Chart 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.
Antennas and Propagation Slide 2 Chapter 1 Basics
Principle Waves on a transmission line
Antennas and Propagation Slide 3 Chapter 1 Basics (2)
Im{Γ} Smith Chart Shows 1. Reflection coefficient ∠∠∠Γ |Γ| Re{Γ}
Resistance 2. Impedance / Admittance Values
Reactance Values
Antennas and Propagation Slide 4 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 Transmission Line Length (wavelengths) βββl / 2π Constant resistance
Constant reactance
Polar angle (for Γ)
Polar radius (for |Γ|)
Antennas and Propagation Slide 6 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
o Example 1 26
ZL = 100 + j50 Ω Z0 = 50 Ω zL = 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 Examples: Z ⇔ Y
Example 2
ZL = 100 + j50 Ω Z0 = 50 Ω zL = 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 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 = 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 0.418 +0.125 zL = =0.543 0.4 – j0.5 TLine: l/8 = 0.125 zin = 0.32 + j0.25 Exact: zin = 0.332 + j0.248
0.418 Antennas and Propagation Slide 11 Chapter 1 Transmission Line Stubs
Idea Length of (lossless) transmission line Open or shorted Presents a reactance / suceptance
Recall
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 zstub = j1.4 Length? Shorted stub: 0.35 λ
Open stub: 0.1 λ
Antennas and Propagation Slide 13 Chapter 1 Matching
Goal Get Γ(z) = 0 (get to origin)
Also means zin = 1 (Zin = 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 zant = 0.4 – j0.6 l = 0.174 λ + 0.094 λ = 0.268 λ
xstub = 1.4
Antennas and Propagation Slide 15 Chapter 1 Examples: Stub Match
Example 5 zant = 0.4 – j0.6 l = 0.016 λ
ystub = 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