Carta De Smith

Transmission Lines

Smith Chart
The Smith chart is one of the most useful graphical tools for high
frequency circuit applications. The chart provides a clever way to
visualize complex functions and it continues to endure popularity,
decades after its original conception.
From a mathematical point of view, the Smith chart is a 4-D
representation of all possible complex impedances withrespect to
coordinates defined by the complex reflection coefficient.
Im(Γ )

1
Re(Γ )

The domain of definition of the reflection
coefficient for a loss-less line is a circle of
unitary radius in the complex plane. This
is also the domain of the Smith chart.
In the case of a general lossy line, the
reflection
coefficient
might
have
magnitude larger than one, due to the
complexcharacteristic
impedance,
requiring an extended Smith chart.

© Amanogawa, 2006 - Digital Maestro Series

165

Transmission Lines

The goal of the Smith chart is to identify all possible impedances on
the domain of existence of the reflection coefficient. To do so, we
start from the general definition of line impedance (which is equally
applicable to a load impedance when d=0)

V(d)
1+Γ(d)
Z( d ) =
= Z0
1 − Γ(d)
I(d)
This provides the complex function Z( d ) = f {Re ( Γ ) , Im ( Γ )} that
we want to graph. It is obvious that the result would be applicable
only to lines with exactly characteristic impedance Z0.
In order to obtain universal curves, we introduce the concept of
normalized impedance

Z (d ) 1+ Γ(d )
zn ( d ) =
=
Z0
1− Γ(d )
© Amanogawa, 2006 -Digital Maestro Series

166

Transmission Lines

The normalized impedance is represented on the Smith chart by
using families of curves that identify the normalized resistance r
(real part) and the normalized reactance x (imaginary part)

zn ( d ) = Re ( zn ) + j Im ( zn ) = r + jx
Let’s represent the reflection coefficient in terms of its coordinates

Γ ( d ) = Re ( Γ ) + j Im ( Γ )
Nowwe can write

1 + Re ( Γ ) + j Im ( Γ )
r + jx =
1 − Re ( Γ ) − j Im ( Γ )
=

1 − Re2 ( Γ ) − Im2 ( Γ ) + j 2 Im ( Γ )

© Amanogawa, 2006 - Digital Maestro Series

(1 − Re ( Γ ) )

2

+ Im2 ( Γ )
167

Transmission Lines

The real part gives

r=

1 − Re

2

Add a quantity equal to zero

( Γ ) − Im2 ( Γ )

( 1 − Re ( Γ ) )2 + Im 2 ( Γ )

(

2

 r ( Re ( Γ ) −1 )2 +



( Re

r ( Re ( Γ ) − 1 ) + Re
2

(Γ) − 1
2

=0

) + r Im

(Γ) − 1

)

2

( Γ ) + Im ( Γ ) +
2

1
1+ r

−

1
1+ r

=0

 + ( 1 + r ) Im 2 ( Γ ) = 1
+

1 + r
1+ r
1

2
2

r
r
1
2
( 1 + r )  Re ( Γ ) − 2 Re ( Γ )
+
+ ( 1 + r ) Im ( Γ ) =
2
1 + r (1 + r ) 
1+ r


⇒

 Re ( Γ ) −




1 + r
r

2

© Amanogawa,2006 - Digital Maestro Series

+ Im

2

(Γ) =

()
1

2

Equation of a circle

1+ r

168

Transmission Lines

The imaginary part gives

x=
x

2

Multiply by x and add a
quantity equal to zero

2 Im ( Γ )

( 1 − Re ( Γ ) )2 + Im2 ( Γ )

=0

( 1 − Re ( Γ ) ) 2 + Im2 ( Γ )  − 2 x Im ( Γ ) + 1 − 1 = 0



( 1 − Re ( Γ ) ) 2 + Im 2 ( Γ )  − 2 Im ( Γ ) + 1 =1

x
2
2
x
x
2
1
1
2
2
( 1 − Re ( Γ ) ) +  Im ( Γ ) − Im ( Γ ) + 2  = 2
x

x x
⇒

2

1
 Im ( Γ ) −
=
( Re ( Γ ) − 1 ) +
2



x
x
2

© Amanogawa, 2006 - Digital Maestro Series

1

Equation of a circle

169

Transmission Lines

The result for the real part indicates that on the complex plane with
coordinates (Re(Γ), Im(Γ)) all the possibleimpedances with a given
normalized resistance r are found on a circle with

r

,0
Center = 
 1+ r 

1
Radius =
1+ r

As the normalized resistance r varies from 0 to ∞ , we obtain a
family of circles completely contained inside the domain of the
reflection coefficient | Γ | ≤ 1 .
Im(Γ )

r=1

r=5

r=0
Re(Γ )

r = 0.5

© Amanogawa, 2006 - Digital Maestro Series

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