System Overview
R&S
®
ZVA
70
Quick Start Guide 1145.1090.62 ─ 10
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 coordinates
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 complex
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 inter-
sect 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.
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