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Lecture 4 Smath Chart

1. What is the Smith Chart?

2. Derivation of the Smith Chart Equations

3. The Real Smith Chart

4. Getting Familiar with the Smith Chart

5. Admittance Smith Chart − Y Chart

6. ZY Chart

7. Smith Chart Examples

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1. What is the Smith Chart?

Zin vs Γ calculation diagram

Invention

- Philip Smith, 1939, Electronics magazine

- Mizuhashi Tosaku, 1936, Journal of Electrical Communications Engineers of Japan

Applications

- Impedance matching: 5 − j 210 Ω → 50 + j 0 Ω

- Impedance calculation

ARINST VR 23-6200 ($510)Smith chart

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Zin vs Γ

0

0

0

in 0

0

0

2

0

0

( ) [1 ( )]

( ) [1 ( )]

1 ( ) (input impedance)1 ( )

= (reflection coefficient)

( )

j

j

jL

LL

L

V V e

VI eZ

zZ Z Zz

Z ZZ Z

e

Z ZZ Z

0 ,Z ( )V

( )I

0z z

LZ

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2. Derivation of the Smith Chart Equations

0

2 22 22

2 2

22

2 2

1 1 (normalized impedance)1 1 ( )

(reflection coefficient)

1 1 (circles of constant )1 1(1 )

2 1 1( 1)(1 )

j

Z u jvz r jxZ u jv

u jv e

u v rr u v rr ru v

vx u vx xu v

2

(circles of constant )r

5

6

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3. The Real Smith Chart

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4. Getting Familiar with the Smith Chart

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10

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12

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14

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5. Admittance Smith Chart − Y Chart

0 00

1 1 1 1 ( ) 11 1 ( ) 1

Y Y Z ZZ Z

Usign Z chart for admittance

0 0 0

(rotate 180 )

1/Z Y Z

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Using the Y Smith chart- Rotate the Z Smith chart by 180°

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6. ZY Chart

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Calculating Γ on a transmission line

2j dL e

Re

Im

L

L

L

2L d

2 d

( )I d

( )V d

z d

0z z

LZ

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Γ with a quarter-wave TL and a half-wave TL

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Short-circuited transmission line

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Open-circuited transmission line

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