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between these petals are oriented in the longitudinal and transverse directions. The
diagrams of
Z
xy
and
arg
Z
xy
have the form of a regular oval. Their principal
diameters 2
Z
,
2
Z
⊥
and 2
arg
Z
,
2
arg
Z
⊥
are oriented in the longitudinal
and transverse directions.
Similar form of the impedance polar diagrams would be found in the axially
symmetric 3D-models (
ske
w
S
=
0
,
ske
w
B
=
0). Here bisectrices of diagram of
|
Z
xx
|
and principal diameters 2
|
Z
r
|
,
2
|
Z
t
|
and 2
|
arg
Z
r
|
,
2
|
arg
Z
t
|
of diagrams of
Z
xy
and
arg
Z
xy
are oriented in the radial and tangential directions.
If the 3D-model is asymmetric, the regular form of polar diagrams is violated,
and they can take rather whimsical form. But in the special case of quasi-symmetry
(3D,a), when
ske
w
S
=
0 and
ske
w
B
=
0, the
|
Z
xx
|
-diagram is cross-shaped, while
diagrams of
Z
xy
and
arg
Z
xy
are shaped into oblique figures eight with or with-
out small petals. In the general event (3D,b) that
ske
w
=
0 and
ske
w
=
0, all
S
B
diagrams look like oblique figure eight with more or less narrow waist.
1.4.2 Polar Diagrams of H- and E-Polarized Impedances
This technique has been advanced in (Berdichevsky and Logunovich, 2005). It is
based on the decomposition of the electromagnetic field in conjugate and associate
directions suggested by Counil et al. (1986).
The Counil-Le Mouel-Menvielle decomposition is associated with so called
“
induction intensity
” and “
current intensity
”. But this terminology is vulnerable to
criticism. The electric current and electromagnetic induction are interconnected via
the Ampere, Faraday, and Ohm laws. An electric current generates a magnetic field
that in turn induces an electric field producing an electric current. The intensity of
electromagnetic induction depends on the intensity of the inducing current, and the
intensity of the induced current depends on the intensity of electromagnetic induc-
tion. The separation of these phenomena is scarcely constructive and only compli-
cates their mathematical description. The formulation of the problem is significantly
simplified if the construction of polar diagrams involves the formal terminology,
reflecting the mathematical meaning of the values to be determined.
Following Yee and Paulson (1987), we introduce a scalar indicator defined as the
ratio between the Euclidean norms
E
and
H
of electric and magnetic fields
E
(
E
x
,
E
y
) and
H
(
H
x
,
H
y
),the magnetic field being linearly polarized at an angle
H
to the original
x
axis:
|
E
x
+
E
y
E
·
2
E
x
+
E
y
2
E
E
x
|
E
E
y
=
H
=
=
H
y
=
Z
H
(
H
)
+
H
y
H
x
+
2
H
·
H
H
x
H
y
2
|
H
x
|
Z
xx
H
x
+
Z
xy
H
y
+
Z
yx
H
x
+
Z
yy
H
y
2
2
=
+
H
y
2
2
|
H
x
|
