Concepts and Features
R&S
®
ZNA
97
User Manual 1178.6462.02 ─ 12
Example: Reflection coefficients in the Smith chart
If the measured quantity is a complex reflection coefficient
Γ
(e.g. S
11
, S
22
), then the
unit Smith chart can be used to read the normalized impedance of the DUT. The coor-
dinates in the normalized impedance plane and in the reflection coefficient plane are
related as follows (see also: definition of matched-circuit (converted) impedances):
Z / Z
0
= (1 +
Γ
) / (1 –
Γ
)
From this equation, it is easy to relate the real and imaginary components of the com-
plex resistance to the real and imaginary parts of
Γ
:
,
)
Im(
)
Re(
1
)
Im(
)
Re(
1
)
/
Re(
2
2
2
2
0
Z
Z
R
2
2
0
)
Im(
)
Re(
1
)
Im(
2
)
/
Im(
Z
Z
X
According to the two equations above, the graphical representation in a Smith chart
has the following properties:
●
Real reflection coefficients are mapped to real impedances (resistances).
●
The center of the
Γ
plane (
Γ
= 0) is mapped to the reference impedance Z
0
,
whereas the circle with |
Γ
| = 1 is mapped to the imaginary axis of the Z plane.
●
The circles for the points of equal resistance are centered on the real axis and
intersect at Z = infinity. The arcs for the points of equal reactance also belong to
circles intersecting at Z = infinity (open circuit point (1, 0)), centered on a straight
vertical line.
Examples for special points in the Smith chart:
●
The magnitude of the reflection coefficient of an open circuit (Z = infinity, I = 0) is
one, its phase is zero.
●
The magnitude of the reflection coefficient of a short circuit (Z = 0, U = 0) is one, its
phase is –180 deg.
Screen Elements
