# The Smith Chart Say we wish to map a line

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## The Smith Chart

Re{ } 2

′ =

Im{ }

′ = −1

Γ =i 0

r i

2r 2i

i

Re{ }z Im{ }z

r =2

x =-1

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r

r

= 2

= 0 5

2 2

2 1

1 r 1

r i

r r

⎛ ⎞ ⎛ ⎞

Γ − + Γ =

⎜ + ⎟ ⎜ + ⎟

⎝ ⎠ ⎝ ⎠

c

2 +

c

2 =

2

c c c

= =

r

Γ

0

r

Γ =

Γ =

(3)

Γ =i 0

1 1 r

= +

Γr

Γi

1

0 3 Γ =

r = − .

r = 0 0.

r = 0 3.

r = 1 0.

r =3 0.

r = −5 0.

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r

1. If

r

1

Γ =

2. If

r

3. If

r

Γ =1

4. If

r = ∞

Γ = Γ =r 1, i 0

Γ = 1

j0

= ∞

= ∞

=

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i

= −2

= 0 5

r 1

2 i 1 2 12

i i

⎛ ⎞

Γ − + Γ −⎜ ⎟ =

⎝ ⎠

i

1 1

c r i

i

⎛ ⎞

Γ = Γ =

⎜ ⎟

⎝ ⎠

Γ =r 1

1

i

=

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i

Γ = Γ =r 1, i 0

1. If

i

Γ > 0

r

Γi

### 1 Γ =

x = 0 5.

x = 3 0.

x = −0 5.

x = 2 0. x = 1 0.

x = −3 0.

x = −2 0. x = −1 0.

r 1 Γ =

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2. If

i

Γ <i 0

3. If

i

i 0

Γ =

4. If

i = ±∞

Γ = Γ =r 1, i 0

Γ =1

j0

= ∞ or

= −∞

= ∞

### , the impedance is infinite (an open circuit), regardless of what the value of the resistive component r is.

5. Note also that much of the circle formed by mapping

= i

Γ =1

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= r

= i

Re{ }Γ Im{ }Γ

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= r

= i

Γ

= r

= i

= r

= i

x =1

x = 0

x = −1 r = 0

r =1

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## References

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