Smith chart basics

Transmission Lines
© Amanogawa, 2000 - Digital Maestro Series 137
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 simply a
representation of all possible complex impedances with respect to
coordinates defined by the reflection coefficient.
The domain of definition of the
reflection coefficient is a circle of
radius 1 in the complex plane. This
is also the domain of the Smith chart.
Im(Γ )
Re(Γ )
1

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 the load impedance)
( )
( )
( )
( )0
1
( )
1
V d d
Z d Z
I d d
+ Γ
= =
− Γ
This provides the complex function ( ) ( ){ }( ) Re , ImZ d f= Γ 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
( ) ( )
( )0
1
( )
1
Z d d
z d
Z d
+ Γ
= =
− Γ

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)
( ) ( ) ( )Re Imz d z j z r jx= + = +
Let’s represent the reflection coefficient in terms of its coordinates
( ) ( ) ( )Re Imd jΓ = Γ + Γ
Now we can write
( ) ( )
( ) ( )
( ) ( ) ( )
( )( ) ( )
2 2
2 2
1 Re Im
1 Re Im
1 Re Im 2Im
1 Re Im
j
r jx
j
j
+ Γ + Γ
+ =
− Γ − Γ
− Γ − Γ + Γ
=
− Γ + Γ

Transmission Lines
The real part gives
( ) ( )
( )( ) ( )
( )( ) ( )( ) ( ) ( )
( )( ) ( )( ) ( ) ( )
( ) ( ) ( )
( )
( ) ( )
( ) ( ) ( )
2 2
2 2
2 2 2 2
2 2 2
2
2 2
2
2 2
2
1 Re Im
1 Re Im
1 1
Re 1 Re 1 Im Im 0
1 1
1 1
Re 1 Re 1 1 Im
1 1
1
1 Re 2 Re 1 Im
1 11
1
Re Im
1 1
r
r r
r r
r r
r r
r r
r r
r rr
r
r r
− Γ − Γ
=
− Γ + Γ
Γ − + Γ − + Γ + Γ + − =
+ +
Γ − + Γ − + + + Γ =
+ +
+ Γ − Γ + + + Γ =
+ ++
⇒ Γ − + Γ =
+ +
 
  
 
 
 
 
  
= 0
Add a quantity equal to zero
Equation of a circle

Transmission Lines
The imaginary part gives
( )
( )( ) ( )
( )( ) ( ) ( )
( )( ) ( ) ( )
( )( ) ( ) ( )
( )( ) ( )
2 2
22 2
2 2
2 2
2 2
2 2
2
2
2
2 Im
1 Re Im
1 Re Im 2 Im 1 1 0
2 1 1
1 Re Im Im
2 1 1
1 Re Im Im
1 1
Re 1 Im
x
x x
x x x
x x x
x x
Γ
=
− Γ + Γ
− Γ + Γ − Γ + − =
− Γ + Γ − Γ + =
− Γ + Γ − Γ + =
⇒ Γ − + Γ − =
 
 
 
 
 
  
 
  
= 0
Multiply by x and add a
quantity equal to zero
Equation of a circle

Transmission Lines
The result for the real part indicates that on the complex plane with
coordinates (Re(Γ), Im(Γ)) all the possible impedances with a given
normalized resistance r are found on a circle with
{ } 1
, 0
1 1
r
r r+ +
Center = Radius =
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 )
Re(Γ )
r = 0
r →∞
r = 1
r = 0.5
r = 5

Transmission Lines
The result for the imaginary part indicates that on the complex
plane with coordinates (Re(Γ), Im(Γ)) all the possible impedances
with a given normalized reactance x are found on a circle with
{ }1 1
1 ,
x x
Center = Radius =
As the normalized reactance x varies from -∞ to ∞ , we obtain a
family of arcs contained inside the domain of the reflection
coefficient | Γ | ≤ 1 .
Im(Γ )
Re(Γ )
x = 0
x →±∞
x = 1
x = 0.5
x = -1
x = - 0.5

Transmission Lines
Basic Smith Chart techniques for loss-less transmission lines
Given Z(d) ⇒ Find Γ(d)
Given Γ(d) ⇒ Find Z(d)
Given ΓR and ZR ⇒ Find Γ(d) and Z(d)
Given Γ(d) and Z(d) ⇒ Find ΓR and ZR
Find dmax and dmin (maximum and minimum locations for the
voltage standing wave pattern)
Find the Voltage Standing Wave Ratio (VSWR)
Given Z(d) ⇒ Find Y(d)
Given Y(d) ⇒ Find Z(d)

Transmission Lines
Given Z(d) ⇒ Find Γ(d)
1. Normalize the impedance
( )
( )
0 0 0
d
d
Z R X
z j r j x
Z Z Z
= = + = +
2. Find the circle of constant normalized resistance r
3. Find the arc of constant normalized reactance x
4. The intersection of the two curves indicates the reflection
coefficient in the complex plane. The chart provides
directly the magnitude and the phase angle of Γ(d)
Example: Find Γ(d), given
( ) 0d 25 100 with 50Z j Z= + Ω = Ω

Transmission Lines
1
-1
0 0.2 0.5 5
0.2
-0.2
21
-0 5
0 5
-3
3
2
-2
1. Normalization
z (d) = (25 + j 100)/50
= 0.5 + j 2.0
2. Find normalized
resistance circle
r = 0.5
3. Find normalized
reactance arc
x = 2.0
4. This vector represents
the reflection coefficient
Γ (d) = 0.52 + j0.64
|Γ (d)| = 0.8246
∠∠ Γ (d) = 0.8885 rad
= 50.906 °
50.906 °
1.
0.8246

Transmission Lines
Given Γ(d) ⇒ Find Z(d)
1. Determine the complex point representing the given
reflection coefficient Γ(d) on the chart.
2. Read the values of the normalized resistance r and of the
normalized reactance x that correspond to the reflection
coefficient point.
3. The normalized impedance is
( )dz r j x= +
and the actual impedance is
( ) ( )0 0 0 0(d) dZ Z z Z r j x Z r j Z x= = + = +

Transmission Lines
Given ΓR and ZR ⇐⇒ Find Γ(d) and Z(d)
NOTE: the magnitude of the reflection coefficient is constant along
a loss-less transmission line terminated by a specified load, since
( ) ( )d exp 2 dR RjΓ = Γ − β = Γ
Therefore, on the complex plane, a circle with center at the origin
and radius | ΓR | represents all possible reflection coefficients
found along the transmission line. When the circle of constant
magnitude of the reflection coefficient is drawn on the Smith chart,
one can determine the values of the line impedance at any location.
The graphical step-by-step procedure is:
1. Identify the load reflection coefficient ΓR and the
normalized load impedance ZR on the Smith chart.

Transmission Lines
2. Draw the circle of constant reflection coefficient
amplitude |Γ(d)| =|ΓR|.
3. Starting from the point representing the load, travel on
the circle in the clockwise direction, by an angle
2
2 d 2 d
π
θ = β =
λ
4. The new location on the chart corresponds to location d
on the transmission line. Here, the values of Γ(d) and
Z(d) can be read from the chart as before.
Example: Given
025 100 50RZ j Z= + Ω = Ωwith
find
( ) ( ) 0.18Z d d dΓ = λand for

Transmission Lines
θ
1
-1
0 0.2 0.5 5
0.2
-0.2
21
-0 5
0 5
-3
3
2
-2
ΓR
zR
∠ ΓR
θθ = 2 β d
= 2 (2π/λ) 0.18 λ
= 2.262 rad
= 129.6°
z(d)
Γ (d)Γ(d) = 0.8246 ∠-78.7°
= 0.161 – j 0.809 z(d) = 0.236 – j1.192
Z(d) = z(d) × Z0 = 11.79 – j59.6 Ω
Circle with constant | Γ |

Transmission Lines
Given ΓR and ZR ⇒ Find dmax and dmin
1. Identify on the Smith chart the load reflection coefficient
ΓR or the normalized load impedance ZR .
amplitude |Γ(d)| =|ΓR|. The circle intersects the real axis
of the reflection coefficient at two points which identify
dmax (when Γ(d) = Real positive) and dmin (when Γ(d) =
Real negative)
3. A commercial Smith chart provides an outer graduation
where the distances normalized to the wavelength can be
read directly. The angles, between the vector ΓR and the
real axis, also provide a way to compute dmax and dmin .
Example: Find dmax and dmin for
025 100 ; 25 100 ( 50 )R RZ j Z j Z= + Ω = − Ω = Ω

Transmission Lines
1
-1
0 0.2 0.5 5
0.2
-0.2
21
-0 5
0 5
-3
3
2
-2
ΓR
ZR
∠ ΓR
2β dmin = 230.9°
dmin = 0.3207λ
2β dmax = 50.9°
dmax = 0.0707λ
Im(ZR) 0
Z j ZR 25 100 500 ( )

Transmission Lines
1
-1
0 0.2 0.5 5
0.2
-0.2
21
-0 5
0 5
-3
3
2
-2
ΓR
ZR
∠ ΓR
2β dmin = 129.1°
dmin = 0.1793 λ
2β dmax = 309.1°
dmax = 0.4293 λ
Im(ZR) 0
Z j ZR 25 100 500 ( )

Transmission Lines
Given ΓR and ZR ⇒ Find the Voltage Standing Wave Ratio (VSWR)
The Voltage standing Wave Ratio or VSWR is defined as
max
min
1
1
R
R
V
VSWR
V
+ Γ
= =
− Γ
The normalized impedance at a maximum location of the standing
wave pattern is given by
( )
( )
( )
max
max
max
1 1
!!!
1 1
R
R
d
z d VSWR
d
+ Γ + Γ
= = =
− Γ − Γ
This quantity is always real and ≥ 1. The VSWR is simply obtained
on the Smith chart, by reading the value of the (real) normalized
impedance, at the location dmax where Γ is real and positive.

Transmission Lines
3. Find the intersection of this circle with the real positive
axis for the reflection coefficient (corresponding to the
transmission line location dmax).
4. A circle of constant normalized resistance will also
intersect this point. Read or interpolate the value of the
normalized resistance to determine the VSWR.
Example: Find the VSWR for
1 2 025 100 ; 25 100 ( 50 )R RZ j Z j Z= + Ω = − Ω = Ω

Transmission Lines
1
-1
0 0.2 0.5 5
0.2
-0.2
21
-0 5
0 5
-3
3
2
-2
ΓR1
zR1
zR2
ΓR2
z(dmax )=10.4
For both loads
VSWR = 10.4
Circle of constant
conductance r = 10.4

Transmission Lines
Given Z(d) ⇐⇒ Find Y(d)
Note: The normalized impedance and admittance are defined as
( )
( )
( )
( )
1 1
( ) ( )
1 1
d d
z d y d
d d
+ Γ − Γ
= =
− Γ + Γ
Since
( )
( )
( )
( )
4
1
14
4 1
1
4
d d
d
d
z d y d
d
d
λ 
Γ + = −Γ 
 
λ 
+ Γ +  − Γλ   ⇒ + = = =  λ + Γ   − Γ + 
 

Transmission Lines
Keep in mind that the equality
( )
4
z d y d
λ 
+ = 
 
is only valid for normalized impedance and admittance. The actual
values are given by
0
0
0
4 4
( )
( ) ( )
Z d Z z d
y d
Y d Y y d
Z
λ λ   
+ = ⋅ +   
   
= ⋅ =
where Y0=1 /Z0 is the characteristic admittance of the transmission

Transmission Lines
line.
3. The normalized admittance is located at a point on the
circle of constant |Γ| which is diametrically opposite to the
normalized impedance.
Example: Given
025 100 with 50RZ j Z= + Ω = Ω
find YR .

Transmission Lines
1
-1
0 0.2 0.5 5
0.2
-0.2
21
-0 5
0 5
-3
3
2
-2
z(d) = 0.5 + j 2.0
Z(d) = 25 + j100 [ Ω ]
y(d) = 0.11765 – j 0.4706
Y(d) = 0.002353 – j 0.009412 [ S ]
z(d+λ/4) = 0.11765 – j 0.4706
Z(d+λ/4) = 5.8824 – j 23.5294 [ Ω ]
θ = 180°
= 2β⋅λ/4

Transmission Lines
The Smith chart can be used for line admittances, by shifting the
space reference to the admittance location. After that, one can
move on the chart just reading the numerical values as
representing admittances.
Let’s review the impedance-admittance terminology:
Impedance = Resistance + j Reactance
Z R jX= +
Admittance = Conductance + j Susceptance
Y G jB= +
On the impedance chart, the correct reflection coefficient is always
represented by the vector corresponding to the normalized
impedance. Charts specifically prepared for admittances are
modified to give the correct reflection coefficient in correspondence
of admittance.

Transmission Lines
Smith Chart for
Admittances
00.20.55
-0.2
0.2
2 1
0 5
-0 5
3
-3
-2
2
-1
1
Positive
(capacitive)
susceptance
Negative
(inductive)
susceptanceΓ
y(d) = 0.11765 – j 0.4706
z(d) = 0.5 + 2.0

Transmission Lines
Since related impedance and admittance are on opposite sides of
the same Smith chart, the imaginary parts always have different
sign.
Therefore, a positive (inductive) reactance corresponds to a
negative (inductive) susceptance, while a negative (capacitive)
reactance corresponds to a positive (capacitive) susceptance.
Numerically, we have
( )( ) 2 2
2 2 2 2
1
z r j x y g j b
r j x
r j x r jx
y
r j x r j x r x
r x
g b
r x r x
= + = + =
+
− −
= =
+ − +
⇒ = = −
+ +

Smith chart basics

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Smith chart basics