Environmental Engineering Reference
In-Depth Information
coefficient along the whole transmission chain (i.e., starting from the instrument
output connector, up to the probe end). As a direct consequence, the resulting re-
sponse is inevitably affected by all the parasitics and losses introduced by the cable,
connections, section transitions, and probe-head portion. A calibration procedure
allows reducing the influence of these parasitics.
As well known, the aim of the calibration procedure is to assess the three scat-
tering parameters that characterize the 'error network', depicted in Fig. 5.5, which
models all systematic errors effects between the instrument output connector and
the section corresponding to the beginning of the MUT [1]. Three short circuits
were applied at different sections along the probe; in fact, these loads can be easily
applied at whichever distance from the probe head.
For this specific configuration, the first short was directly applied to section A,
whereas the other two shorts were applied at distances
l
1
and
l
2
from the probe head,
respectively. When a short circuit condition is realized at a generic distance
l
from
the probe head, the reflection coefficient at section A (
S
11
sc
,
A
) can be expressed as
follows:
e
−
2iβ
l
S
11
sc
,
A
=
−
(5.14)
2
is the propagation constant in air. The 'error network',
which is characterized by its scattering parameters
S
11
,
E
;
S
12
,
E
;and
S
22
,
E
, transforms
such reflection coefficient at section A in a new one, namely
S
11
sc
,
B
, measured at the
TDR connector (section B):
1
/
where
β
=
2
π
f
(
μ
0
ε
0
)
S
12
,
E
e
−
2iβ
l
S
11
sc
,
B
=
S
11
,
E
−
(5.15)
1
+
S
22
,
E
e
−
2iβ
l
Therefore, if the values of
S
11
sc
,
B
corresponding to three different positions of the
short circuit (namely,
l
=0,
l
l
2
) are measured, a system of three
equations in three unknowns is obtained, from which the three error parameters
S
11
,
E
;
S
12
,
E
;and
S
22
,
E
can be retrieved. In particular, after appropriate simplifica-
tions within the Mathematica
TM
=
l
1
,and
l
=
software, the following expressions are obtained
for the error network parameters:
i
S
11
sc0
,
B
S
11
SC
1
,
B
S
11
sc2
,
B
S
11
sc2
,
B
S
11
sc0
,
B
−
S
11
sc1
,
B
cot
=
−
−
−
(
β
l
1
)
S
11E
+
A
S
11
SC
1
,
B
S
11
sc0
,
B
−
S
11
sc2
,
B
cot
(
β
l
2
)
+
(5.16)
A
2i
S
11
sc0
,
B
−
S
11
sc1
,
B
S
11
sc0
,
B
−
S
11
sc2
,
B
S
12E
=
×
A
2
S
11
sc1
,
B
−
S
11
sc2
,
B
csc
(
β
l
1
)
csc
(
β
l
2
)
sin
(
β
(
l
1
−
l
2
))
×
(5.17)
A
2
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