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