Antennas and Propagation

The Smith Chart Introduction

Origin 1939 P.H. Smith. Graphical method for performing calculations. (Well before pocket calculators or computers)

Today Useful for displaying information: shows and impedance/ 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 () βββl / 2π Constant resistance

Constant reactance

Polar (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 ( 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