ekS - Instytut Agrofizyki im. Bohdana Dobrzańskiego PAN w Lublinie ...

assumed that the conductivity of soil solid phase and air is negligible). Therefore
the soil dielectric permittivity is a complex value with the real part conditioned by
its water content and imaginary part by its salinity. The presented discussion
refers to the conditions when the dielectric measurement of soil water content is
not influenced by the conductivity of the electrolyte in it. The objective of the
considerations below is the determination of the frequency of the electric field
applied to the soil that would allow interpreting its reaction only on the basis of
water and solid phase in it.
3.4.1. Frequency dispersion of the electrolyte dielectric permittivity
Dielectric permittivity, ε, of an electrolyte is a complex value dependent on
the frequency of the applied electric field, f. Its real, Re(ε), and imaginary Im(ε)
parts have the form (Chełkowski [14], Hasted [37]):
Re( ε ) = ε '
(14)
ECw
Im( ε ) = ε"
+
ωε
0
(15)
where: ε’ and ε” are real and imaginary parts of the complex dielectric
permittivity of water, respectively, EC w – electrical conductivity of the electrolyte
[Sm -1 ], ω - angular frequency of the applied electric field (equals to 2πf)[s -1 ], ε 0
–
is the dielectric permittivity of free space [Fm -1 ].
The real and imaginary parts of the complex dielectric permittivity are
modeled by Cole-Cole formulas [14,37]:
⎡ 1−h
hπ ⎤
( ε
s − ε
∞
)
⎢
1 + ( ωτ ) sin
2 ⎥
ε ' = ε
∞
+
⎣
⎦
(16)
( ) ( 1 h ) 1 h hπ
1
2 −
−
+ ωτ + 2( ωτ ) sin
2
⎡ 1−h
hπ ⎤
( ε
s
− ε
∞
)
⎢( ωτ ) cos
2 ⎥
ε " =
⎣
⎦
(17)
( ) ( 1 h ) 1 h hπ
1
2 −
−
+ ωτ + 2( ωτ ) sin
2
−1
where: ε ∞
is the relative water dielectric permittivity when ω > τ , ε is the
s
relative dielectric permittivity of water when ω = 0, τ is the relaxation time of
26