Information Technology Reference
In-Depth Information
1.4 Impedance Polar Diagrams
The dependence of the impedances upon orientation of the measurement axes may
be displayed graphically by polar diagrams. No structural or frequency limitations
are required in constructing the impedance polar diagrams.
1.4.1 Polar Diagrams of the Impedance Tensor
This techniques has been suggested in (Berdichevsky, 1968; Berdichevsky et al.,
1993).
Let the tensor [
Z
] be obtained on the measurement axes
x
,
y
. We will introduce
new axes
x
,
y
rotated through a clockwise angle
. In view of (1.27), (128)
Z
yy
(
2)
=
|
|
Z
xx
(
|
=
|
,
)
+
/
Z
2
+
Z
3
sin 2
+
Z
4
cos 2
Z
xy
(
)
=
Z
yx
(
2)
=
|
|
,
+
/
Z
1
+
Z
3
cos 2
−
Z
4
sin 2
.
arg
Z
xy
(
)
=
arg
Z
yx
(
2)
=
Im (
Z
1
+
Z
3
cos 2
a
−
Z
4
sin 2
a
)
+
/
arctan
Re (
Z
1
+
Z
3
cos 2
a
−
Z
4
sin 2
a
)
(1
.
87)
Plot these values on the
x
-axis. As
, the resultant points
describe closed curves known under the name of
impedance polar diagrams
.The
changes from 0 to 2
diagrams of
Z
xx
,
Z
xy
are amplitude polar diagrams. The diagram of
arg
Z
xy
is a phase polar diagram. One can see from (1.28) that the amplitude and phase
polar diagrams are antisymmetric about any straight line passing through the
origin.
The conditions for extrema of polar diagram radii are
d
Z
xy
(
)
d
arg
Z
xy
(
)
d
|
Z
xx
(
)
|
=
0
,
=
0
,
=
0
.
d
d
d
This yields equations of degree 4 in tan
. Therefore, the interval 0
≤ ≤
2
arg
Z
xy
. Clearly, the
can contain four maxima and four minima of
|
Z
xx
|
,
|
Z
xy
|
,
impedance polar diagrams may have, at most, four petals.
Examples of impedance polar diagrams for 1D, 2D and 3D-models are shown in
Fig. 1.7. Configuration of the impedance polar diagrams is a good indicator of the
dimensionality of geoelectric structures.
In the 1-D model the diagram of
|
Z
xx
|
degenerates into a point, while the dia-
and
arg
Z
xy
are circles of radii
|
Z
xy
|
|
|
|
|
grams of
Z
and
arg
Z
, where
Z
is
Tikhonov-Cagniard's impedance.
