Rohde & Schwarz 1145.1010.04/05/06, Operating Manual

Page: 132 / 813

Screen Elements R&S
®
ZVA/ZVB/ZVT
1145.1084.12 3.22 E-1
The basic properties of the Smith chart follow from this construction:
•
The central horizontal axis corresponds to zero reactance (real impedance). The center of the
diagram represents Z/Z
0
= 1 which is the reference impedance of the system (zero reflection). At
the left and right intersection points between the horizontal axis and the outer circle, the
impedance is zero (short) and infinity (open).
•
The outer circle corresponds to zero resistance (purely imaginary impedance). Points outside the
outer circle indicate an active component.
•
The upper and lower half of the diagram correspond to positive (inductive) and negative
(capacitive) reactive components of the impedance, respectively.
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 Γ + Γ −
Γ − Γ −
= =
Z Z R
[ ]
,
) Im( ) Re( 1
) Im( 2
) / Im(
2
2
0
Γ + Γ −
Γ ⋅
= =
Z Z X
in order to deduce the following properties of the graphical representation in a Smith chart:
•
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.