US 9,151,793 B1Granted

Method for measuring the complex dielectric constant of a substance

AssigneeKing Fahd University of Petroleum and Minerals

InventorsEssam Eldin Mohamed Hassan, Hassan Ali Ragheb

Life of this patent

Estimating the term…

Abstract

The method for measuring the complex dielectric constant of a substance is based on calculations made in the frequency domain. An open ended rectangular waveguide terminates, at one end, in a non-resonant rectangular cavity, which is then filled with a substance of interest. An electromagnetic wave of known frequency is transmitted through the waveguide and the reflection coefficient at the interface between the waveguide and the cavity is measured by a network analyzer or the like. Another measurement is then made with a slight variation in depth of the cavity or a variation in frequency, and from these two measurements of the reflection coefficient, the complex dielectric constant of the substance of interest can be calculated.

Claims2

Dependent claims

A method for measuring a complex dielectric constant of a substance, comprising the steps of:

(a) providing an open ended rectangular waveguide having a cross-sectional width a and a cross-sectional height b;
(b) providing a non-resonant rectangular cavity in communication with one of the ends of the open ended rectangular waveguide, said non-resonant rectangular cavity having a cross-sectional width g, a cross-sectional height h and a cross-sectional length d, said non-resonant rectangular cavity being symmetrically fed by an electromagnetic wave by the open ended rectangular waveguide;
(c) filling the non-resonant rectangular cavity with a substance of interest;
(d) transmitting an electromagnetic wave generated by an electromagnetic wave generator through the open ended rectangular waveguide toward the non-resonant rectangular cavity;
(e) measuring a first reflection coefficient R1 at an interface between the open ended rectangular waveguide and the non-resonant rectangular cavity with a network analyzer;
(f) varying the cross-sectional length d of the non-resonant rectangular cavity by a length ∈;
(g) re-transmitting the electromagnetic wave through the open ended rectangular waveguide toward the non-resonant rectangular cavity;
(h) measuring a second reflection coefficient R2 at the interface between the open ended rectangular waveguide and the non-resonant rectangular cavity with the network analyzer;
(i) calculating a first variable LL as
LL
=
∑
m
=
1
⁢
⁢
∑
n
=
1
⁢
⁢
B
mn
⁡
(
n
⁢
⁢
π
b
)
⁢
I
yg
⁡
(
m
,
n
)
-
1
j
⁢
⁢
ωμ
o
⁢
∑
m
=
1
⁢
⁢
∑
n
=
1
⁢
⁢
A
mn
⁢
γ
mn
⁡
(
m
⁢
⁢
π
a
)
⁢
I
yg
⁡
(
m
,
n
)
-
1
j
⁢
⁢
ωμ
o
⁢
∑
m
=
2
⁢
⁢
A
m
⁢
⁢
0
⁢
γ
mo
⁡
(
m
⁢
⁢
π
a
)
⁢
I
yg
⁡
(
m
,
o
)
,
where m and n are summation indices ranging between 1 and N, where N is an integer selected for stabilization of the summations,
A
mn
=
-
m
⁢
⁢
π
/
a
k
mn
2
⁢
(
4
ab
)
⁢
I
yg
⁡
(
m
,
n
)
,
where
k
mn
=
(
m
⁢
⁢
π
a
)
2
+
(
n
⁢
⁢
π
b
)
2
and
I
yg
⁡
(
m
,
n
)
=
∫
x
=
0
a
⁢
∫
y
=
0
b
⁢
f
y
⁡
(
x
,
y
)
⁢
sin
⁡
(
m
⁢
⁢
π
⁢
⁢
x
a
)
⁢
cos
⁡
(
n
⁢
⁢
π
⁢
⁢
y
b
)
⁢
⁢
ⅆ
x
⁢
⁢
ⅆ
y
,
where
f
y
⁡
(
x
,
y
)
=
sin
⁡
(
π
a
⁢
x
)
1
-
(
y
-
b
2
b
2
)
2
,
x and y being Cartesian coordinates corresponding to width and height, respectively,
B
mn
=
-
(
n
⁢
⁢
π
b
m
⁢
⁢
π
a
)
⁢
(
j
⁢
⁢
ωε
o
γ
mn
)
⁢
A
mn
,
where j is the imaginary number, ω is an angular frequency of the electromagnetic wave, ∈o is the constant permittivity of free space, and
γ
mn
=
k
mn
2
-
k
o
2
,
where
k
0
=
ω
⁢
μ
0
⁢
ε
o
,
where μ0 is the constant magnetic permeability of free space, wherein
A
m
⁢
⁢
0
=
-
2
m
⁢
⁢
π
/
a
⁢
(
1
ab
)
⁢
I
yg
⁡
(
m
,
0
)
for m≠1;
(j) establishing a first set of estimated values for a real part of a dielectric constant associated with the substance of interest, ∈i, where i ranges between 1 and a pre-selected integer S;
(k) calculating a second variable Mi for i ranging between 1 and S as
M
i
=
∑
p
=
1
⁢
⁢
∑
q
=
1
⁢
⁢
C
pq
⁡
(
q
⁢
⁢
π
h
)
⁢
cos
⁡
(
α
pq
⁢
d
)
⁢
I
yc
⁡
(
p
,
q
)
+
1
j
⁢
⁢
ωμ
o
⁢
∑
p
=
1
⁢
⁢
∑
q
=
1
⁢
⁢
D
pq
⁢
α
pq
⁡
(
p
⁢
⁢
π
g
)
⁢
cos
⁡
(
α
pq
⁢
d
)
⁢
I
yc
⁡
(
p
,
q
)
+
1
j
⁢
⁢
ωμ
o
⁢
∑
p
=
1
⁢
⁢
D
p
⁢
⁢
0
⁢
α
po
⁡
(
p
⁢
⁢
π
g
)
⁢
cos
⁡
(
α
p
⁢
⁢
0
⁢
d
)
⁢
I
yc
⁡
(
p
,
o
)
,
where p and q are summation indices ranging between 1 and N,
D
pq
=
-
p
⁢
⁢
π
/
g
k
pq
2
⁢
sin
⁡
(
α
pq
⁢
d
)
⁢
(
4
gh
)
⁢
I
yc
⁡
(
p
,
q
)
,
where
k
pq
=
(
p
⁢
⁢
π
g
)
2
+
(
q
⁢
⁢
π
h
)
2
and
α
pq
=
k
i
2
-
k
pq
2
,
⁢
where
k
i
=
ω
⁢
μ
0
⁢
ε
o
⁢
ɛ
i
,
and
C
pq
=
-
(
q
⁢
⁢
π
h
p
⁢
⁢
π
g
)
⁢
(
(
jω
o
⁢
ɛ
i
+
δ
)
α
pq
)
⁢
D
pq
,
where
I
yc
⁡
(
p
,
q
)
=
∫
x
=
0
a
⁢
∫
y
=
0
b
⁢
f
y
⁡
(
x
,
y
)
⁢
sin
⁡
(
p
⁢
⁢
π
⁢
⁢
x
g
)
⁢
cos
⁡
(
q
⁢
⁢
π
⁢
⁢
y
h
⁢
⁢
ⅆ
x
⁢
⁢
ⅆ
y
)
and
D
p
⁢
⁢
0
=
-
1
p
⁢
⁢
π
g
⁢
sin
⁡
(
α
p
⁢
⁢
0
⁢
d
)
⁢
(
2
g
⁢
⁢
h
)
⁢
I
yc
⁡
(
p
,
0
)
;
(l) establishing a first set of values of ∈i corresponding to the calculated second variable Mi which satisfy the condition
1
-
R
1
1
+
R
1
=
(
LL
-
M
i
)
⁢
(
ab
)
⁢
jωμ
o
2
⁢
⁢
I
yg
2
⁡
(
1
,
0
)
⁢
γ
10
;
(m) establishing a second set of values of ∈i corresponding to the calculated second variable Mi which satisfy the condition
1
-
R
2
1
+
R
2
=
(
LL
-
M
i
)
⁢
(
ab
)
⁢
jωμ
o
2
⁢
⁢
I
yg
2
⁡
(
1
,
0
)
⁢
γ
10
;
(n) selecting a value of ∈i which is a member of both the first set and the second set of values corresponding to the calculated second variable Mi and setting this selected value equal to a variable ∈d;
(o) establishing a second set of estimated values for the real part of the dielectric constant associated with the substance of interest, ∈ri, where i ranges between 1 and S, such that ∈ri=∈i−j∈′, where ∈′ is an imaginary part of the dielectric constant associated with the substance of interest, where ∈r1=∈d−X1 and ∈rS=∈d+x1 and ∈′=x2, where x1 is approximately equal to 0.5 and x2 is a pre-selected value based on expected conductivity of the substance of interest;
(p) re-calculating the second variable Mi for i ranging between 1 and S as
M
i
=
∑
p
=
1
⁢
∑
q
=
1
⁢
C
pq
⁡
(
q
⁢
⁢
π
h
)
⁢
cos
⁡
(
α
pq
⁢
d
)
⁢
I
yc
⁡
(
p
,
q
)
+
1
jωμ
o
⁢
∑
p
=
1
⁢
∑
q
=
1
⁢
D
pg
⁢
α
pq
⁡
(
p
⁢
⁢
π
g
)
⁢
cos
⁡
(
α
pq
⁢
d
)
⁢
I
yc
⁡
(
p
,
q
)
+
1
jωμ
o
⁢
∑
p
=
1
⁢
D
p
⁢
⁢
0
⁢
α
po
⁡
(
p
⁢
⁢
π
g
)
⁢
cos
⁡
(
α
p
⁢
⁢
0
⁢
d
)
⁢
I
yc
⁡
(
p
,
o
)
,
where
k
i
=
ω
⁢
μ
0
⁢
ε
o
⁡
(
ɛ
ri
-
ɛ
′
)
and
C
pq
=
-
(
q
⁢
⁢
π
h
p
⁢
⁢
π
g
)
⁢
(
(
jωɛ
o
⁢
ɛ
ri
+
δ
)
α
p
⁢
⁢
q
)
⁢
D
pq
;
(q) establishing a first set of values of ∈ri corresponding to the re-calculated second variable Mi which satisfy the condition
1
-
R
1
1
+
R
1
=
(
LL
-
M
i
)
⁢
(
ab
)
⁢
jωμ
o
2
⁢
⁢
I
yg
2
⁡
(
1
,
0
)
⁢
γ
10
;
(r) establishing a second set of values of ∈ri corresponding to the re-calculated second variable Mi which satisfy the condition
1
-
R
2
1
+
R
2
=
(
LL
-
M
i
)
⁢
(
ab
)
⁢
jωμ
o
2
⁢
⁢
I
yg
2
⁡
(
1
,
0
)
⁢
γ
10
;
(s) selecting a value of ∈ri which is a member of both the first set and the second set of values corresponding to the re-calculated second variable Mi and setting the selected value equal to a variable ∈rd;
and
(t) re-calculating the imaginary part of the dielectric constant associated with the substance of interest, ∈′, as ∈′=j(∈rd−∈d),
wherein the dielectric constant associated with the substance of interest is determined as ∈rd+j(∈rd−∈d).

1,084 more elements

An apparatus to measure a complex dielectric constant of a substance, comprising:

an open ended rectangular waveguide having a cross-sectional width a and a cross-sectional height b adapted to receive an electromagnetic wave through an open end thereof;
an electromagnetic wave generator for generating the electromagnetic wave;
a non-resonant rectangular cavity in communication with one of the ends of the open ended rectangular waveguide at an interface between the open ended rectangular waveguide and the non-resonant rectangular cavity, said non-resonant rectangular cavity having a cross-sectional width g, a cross-sectional height h and selectively adjustable a cross-sectional length d, said non-resonant rectangular cavity adapted to be symmetrically fed the electromagnetic wave by the open ended rectangular waveguide, the non-resonant rectangular cavity adapted to be filled with a substance of interest;
a network analyzer;
and
a system to determine the complex dielectric constant of the substance of interest, the system including a processor and a memory, the processor executing instructions of a program stored in the memory to implement the determination of the complex dielectric constant, the program directing the processor to perform the following:
(a) calculate a first variable LL as
LL
=
∑
m
=
1
⁢
∑
n
=
1
⁢
B
mn
⁡
(
n
⁢
⁢
π
b
)
⁢
I
yg
⁡
(
m
,
n
)
-
1
jωμ
o
⁢
∑
m
=
1
⁢
∑
n
=
1
⁢
A
mn
⁢
γ
mn
⁡
(
m
⁢
⁢
π
a
)
⁢
I
yg
⁡
(
m
,
n
)
-
1
jωμ
o
⁢
∑
m
=
2
⁢
A
m
⁢
⁢
0
⁢
γ
mo
⁡
(
m
⁢
⁢
π
a
)
⁢
I
yg
⁡
(
m
,
o
)
,
where in and n are summation indices ranging between 1 and N, where N is an integer selected for stabilization of the summations,
A
mn
=
-
m
⁢
⁢
π
/
a
k
mn
2
⁢
(
4
a
⁢
⁢
b
)
⁢
I
yg
⁡
(
m
,
n
)
,
where
k
mn
=
(
m
⁢
⁢
π
a
)
2
+
(
n
⁢
⁢
π
b
)
2
and
I
yg
⁡
(
m
,
n
)
=
∫
x
=
0
a
⁢
∫
y
=
0
b
⁢
f
y
⁡
(
x
,
y
)
⁢
sin
⁡
(
m
⁢
⁢
π
⁢
⁢
x
a
)
⁢
cos
(
n
⁢
⁢
π
⁢
⁢
y
b
⁢
)
⁢
ⅆ
x
⁢
⁢
ⅆ
y
,
where
f
y
⁡
(
x
,
y
)
=
sin
⁡
(
π
a
⁢
x
)
1
-
(
y
-
b
2
b
2
)
2
,
x and y being Cartesian coordinates corresponding to width and height, respectively,
B
mn
=
-
(
n
⁢
⁢
π
b
m
⁢
⁢
π
a
)
⁢
(
jωε
o
γ
mn
)
⁢
A
mn
,
where j is the imaginary number, ω is an angular frequency of the electromagnetic wave, ∈o is the constant permittivity of free space, and
γ
mn
=
k
mn
2
-
k
o
2
,
where
k
0
=
ω
⁢
μ
0
⁢
ε
o
,
where μ0 is the constant magnetic permeability of free space, wherein
A
m
⁢
⁢
0
=
-
2
m
⁢
⁢
π
/
α
⁢
(
1
a
⁢
⁢
b
)
⁢
I
y
⁢
⁢
g
⁡
(
m
,
0
)
for m≠1;
(b) establish a first set of estimated values for a real part of a dielectric constant associated with the substance of interest, ∈i, where i ranges between 1 and a pre-selected integer S;
(c) calculate a second variable Mi for i ranging between 1 and S as
M
i
=
∑
p
=
1
⁢
∑
q
=
1
⁢
C
pq
⁡
(
q
⁢
⁢
π
h
)
⁢
cos
⁡
(
α
pq
⁢
d
)
⁢
I
yc
⁡
(
p
,
q
)
+
1
j
⁢
⁢
ωμ
o
⁢
∑
p
=
1
⁢
∑
q
=
1
⁢
D
pq
⁢
α
pq
⁡
(
p
⁢
⁢
π
g
)
⁢
cos
⁡
(
α
pq
⁢
d
)
⁢
I
yc
⁡
(
p
,
q
)
+
1
j
⁢
⁢
ωμ
o
⁢
∑
p
=
1
⁢
D
p
⁢
⁢
0
⁢
α
po
⁡
(
p
⁢
⁢
π
g
)
⁢
cos
⁡
(
α
p
⁢
⁢
0
⁢
d
)
⁢
I
yc
⁡
(
p
,
o
)
,
where p and g are summation indices ranging between 1 and N,
D
pq
=
-
p
⁢
⁢
π
/
g
k
pq
2
⁢
sin
⁡
(
α
pq
⁢
d
)
⁢
(
4
g
⁢
⁢
h
)
⁢
I
yc
⁡
(
p
,
q
)
,
where
k
pq
=
(
p
⁢
⁢
π
g
)
2
+
(
q
⁢
⁢
π
h
)
2
and
α
pq
=
k
i
2
-
k
pq
2
,
where
k
i
=
ω
⁢
μ
0
⁢
ε
o
⁢
ɛ
i
,
and
C
pq
=
-
(
q
⁢
⁢
π
h
p
⁢
⁢
π
g
)
⁢
(
(
j
⁢
⁢
ωɛ
o
⁢
ɛ
i
+
δ
)
α
pq
)
⁢
D
pq
,
where
I
yc
⁡
(
p
,
q
)
=
∫
x
=
0
a
⁢
∫
y
=
0
b
⁢
f
y
⁡
(
x
,
y
)
⁢
sin
⁡
(
p
⁢
⁢
π
⁢
⁢
x
g
)
⁢
cos
⁡
(
q
⁢
⁢
π
⁢
⁢
y
h
)
⁢
ⅆ
x
⁢
ⅆ
y
and
D
p
⁢
⁢
0
=
-
1
p
⁢
⁢
π
g
⁢
sin
⁡
(
α
p
⁢
⁢
0
⁢
d
)
⁢
(
2
g
⁢
⁢
h
)
⁢
I
yc
⁡
(
p
,
0
)
;
(d) establish a first set of values of ∈i corresponding to the calculated second variable Mi which satisfy the condition
1
-
R
1
1
+
R
1
=
(
LL
-
M
i
)
⁢
(
ab
)
⁢
j
⁢
⁢
ωμ
o
2
⁢
⁢
I
yg
2
⁡
(
1
,
0
)
⁢
γ
10
,
where R1 is a first reflection coefficient of a transmitted electromagnetic wave measured at the interface between the open ended rectangular waveguide and the non-resonant rectangular cavity at a first cross-sectional length d by the network analyzer;
(e) establish a second set of values of ∈i corresponding to the calculated second variable Mi which satisfy the condition
1
-
R
2
1
+
R
2
=
(
LL
-
M
i
)
⁢
(
ab
)
⁢
j
⁢
⁢
ωμ
o
2
⁢
⁢
I
yg
2
⁡
(
1
,
0
)
⁢
γ
10
,
where R2 is a second reflection coefficient measured at the interface between the open ended rectangular waveguide and the non-resonant rectangular cavity at a second cross-sectional length d different from the first cross-sectional length d by the network analyzer;
(f) select a value of ∈i which is a member of both the first set and the second set of values corresponding to the calculated second variable ∈i and set this selected value equal to a variable ∈d;
(g) establish a second set of estimated values for the real part of the dielectric constant associated with the substance of interest, ∈ri, where i ranges between 1 and S, such that ∈ri=∈i−j∈′, where ∈′ is an imaginary part of the dielectric constant associated with the substance of interest, where ∈r1=∈d−x1 and ∈rS=∈d+x1 and ∈′=x2, where x1 is approximately equal to 0.5 and x2 is a pre-selected value based on expected conductivity of the substance of interest;
(h) re-calculate the second variable Mi for i ranging between 1 and S as
M
i
=
∑
p
=
1
⁢
∑
q
=
1
⁢
C
pq
⁡
(
q
⁢
⁢
π
h
)
⁢
cos
⁡
(
α
pq
⁢
d
)
⁢
I
yc
⁡
(
p
,
q
)
+
1
j
⁢
⁢
ωμ
o
⁢
∑
p
=
1
⁢
∑
q
=
1
⁢
D
pq
⁢
α
pq
⁡
(
p
⁢
⁢
π
g
)
⁢
cos
⁡
(
α
pq
⁢
d
)
⁢
I
yc
⁡
(
p
,
q
)
+
1
j
⁢
⁢
ωμ
o
⁢
∑
p
=
1
⁢
D
p
⁢
⁢
0
⁢
α
po
⁡
(
p
⁢
⁢
π
g
)
⁢
cos
⁡
(
α
p
⁢
⁢
0
⁢
d
)
⁢
I
yc
⁡
(
p
,
o
)
,
where
k
i
=
ω
⁢
μ
0
⁢
ε
o
⁡
(
ɛ
ri
-
ɛ
′
)
and
C
pq
=
-
(
q
⁢
⁢
π
h
p
⁢
⁢
π
g
)
⁢
(
(
j
⁢
⁢
ωɛ
o
⁢
ɛ
ri
+
δ
)
α
pq
)
⁢
D
pq
;
(i) establish a first set of values of ∈ri corresponding to the re-calculated second variable Mi which satisfy the condition
1
-
R
1
1
+
R
1
=
(
LL
-
M
i
)
⁢
(
ab
)
⁢
j
⁢
⁢
ωμ
o
2
⁢
⁢
I
yg
2
⁡
(
1
,
0
)
⁢
γ
10
,
(j) establish a second set of values of ∈ri corresponding to the re-calculated second variable Mi which satisfy the condition
1
-
R
2
1
+
R
2
=
(
LL
-
M
i
)
⁢
(
ab
)
⁢
j
⁢
⁢
ωμ
o
2
⁢
⁢
I
yg
2
⁡
(
1
,
0
)
⁢
γ
10
,
(k) select a value of ∈ri which is a member of both the first set and the second set of values corresponding to the re-calculated second variable Mi and setting the selected value equal to a variable ∈rd;
and
(l) re-calculate the imaginary part of the dielectric constant associated with the substance of interest, ∈′, as ∈′=j(∈rd−∈d),
wherein the dielectric constant associated with the substance of interest is determined as ∈rd+j(∈rd−∈d).

1,094 more elements

Dependencies are inferred from the claim text.

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Abstract

A method for measuring a complex dielectric constant of a substance, comprising the steps of:

(a) providing an open ended rectangular waveguide having a cross-sectional width a and a cross-sectional height b;
(b) providing a non-resonant rectangular cavity in communication with one of the ends of the open ended rectangular waveguide, said non-resonant rectangular cavity having a cross-sectional width g, a cross-sectional height h and a cross-sectional length d, said non-resonant rectangular cavity being symmetrically fed by an electromagnetic wave by the open ended rectangular waveguide;
(c) filling the non-resonant rectangular cavity with a substance of interest;
(d) transmitting an electromagnetic wave generated by an electromagnetic wave generator through the open ended rectangular waveguide toward the non-resonant rectangular cavity;
(e) measuring a first reflection coefficient R1 at an interface between the open ended rectangular waveguide and the non-resonant rectangular cavity with a network analyzer;
(f) varying the cross-sectional length d of the non-resonant rectangular cavity by a length ∈;
(g) re-transmitting the electromagnetic wave through the open ended rectangular waveguide toward the non-resonant rectangular cavity;
(h) measuring a second reflection coefficient R2 at the interface between the open ended rectangular waveguide and the non-resonant rectangular cavity with the network analyzer;
(i) calculating a first variable LL as
LL
=
∑
m
=
1
⁢
⁢
∑
n
=
1
⁢
⁢
B
mn
⁡
(
n
⁢
⁢
π
b
)
⁢
I
yg
⁡
(
m
,
n
)
-
1
j
⁢
⁢
ωμ
o
⁢
∑
m
=
1
⁢
⁢
∑
n
=
1
⁢
⁢
A
mn
⁢
γ
mn
⁡
(
m
⁢
⁢
π
a
)
⁢
I
yg
⁡
(
m
,
n
)
-
1
j
⁢
⁢
ωμ
o
⁢
∑
m
=
2
⁢
⁢
A
m
⁢
⁢
0
⁢
γ
mo
⁡
(
m
⁢
⁢
π
a
)
⁢
I
yg
⁡
(
m
,
o
)
,
where m and n are summation indices ranging between 1 and N, where N is an integer selected for stabilization of the summations,
A
mn
=
-
m
⁢
⁢
π
/
a
k
mn
2
⁢
(
4
ab
)
⁢
I
yg
⁡
(
m
,
n
)
,
where
k
mn
=
(
m
⁢
⁢
π
a
)
2
+
(
n
⁢
⁢
π
b
)
2
and
I
yg
⁡
(
m
,
n
)
=
∫
x
=
0
a
⁢
∫
y
=
0
b
⁢
f
y
⁡
(
x
,
y
)
⁢
sin
⁡
(
m
⁢
⁢
π
⁢
⁢
x
a
)
⁢
cos
⁡
(
n
⁢
⁢
π
⁢
⁢
y
b
)
⁢
⁢
ⅆ
x
⁢
⁢
ⅆ
y
,
where
f
y
⁡
(
x
,
y
)
=
sin
⁡
(
π
a
⁢
x
)
1
-
(
y
-
b
2
b
2
)
2
,
x and y being Cartesian coordinates corresponding to width and height, respectively,
B
mn
=
-
(
n
⁢
⁢
π
b
m
⁢
⁢
π
a
)
⁢
(
j
⁢
⁢
ωε
o
γ
mn
)
⁢
A
mn
,
where j is the imaginary number, ω is an angular frequency of the electromagnetic wave, ∈o is the constant permittivity of free space, and
γ
mn
=
k
mn
2
-
k
o
2
,
where
k
0
=
ω
⁢
μ
0
⁢
ε
o
,
where μ0 is the constant magnetic permeability of free space, wherein
A
m
⁢
⁢
0
=
-
2
m
⁢
⁢
π
/
a
⁢
(
1
ab
)
⁢
I
yg
⁡
(
m
,
0
)
for m≠1;
(j) establishing a first set of estimated values for a real part of a dielectric constant associated with the substance of interest, ∈i, where i ranges between 1 and a pre-selected integer S;
(k) calculating a second variable Mi for i ranging between 1 and S as
M
i
=
∑
p
=
1
⁢
⁢
∑
q
=
1
⁢
⁢
C
pq
⁡
(
q
⁢
⁢
π
h
)
⁢
cos
⁡
(
α
pq
⁢
d
)
⁢
I
yc
⁡
(
p
,
q
)
+
1
j
⁢
⁢
ωμ
o
⁢
∑
p
=
1
⁢
⁢
∑
q
=
1
⁢
⁢
D
pq
⁢
α
pq
⁡
(
p
⁢
⁢
π
g
)
⁢
cos
⁡
(
α
pq
⁢
d
)
⁢
I
yc
⁡
(
p
,
q
)
+
1
j
⁢
⁢
ωμ
o
⁢
∑
p
=
1
⁢
⁢
D
p
⁢
⁢
0
⁢
α
po
⁡
(
p
⁢
⁢
π
g
)
⁢
cos
⁡
(
α
p
⁢
⁢
0
⁢
d
)
⁢
I
yc
⁡
(
p
,
o
)
,
where p and q are summation indices ranging between 1 and N,
D
pq
=
-
p
⁢
⁢
π
/
g
k
pq
2
⁢
sin
⁡
(
α
pq
⁢
d
)
⁢
(
4
gh
)
⁢
I
yc
⁡
(
p
,
q
)
,
where
k
pq
=
(
p
⁢
⁢
π
g
)
2
+
(
q
⁢
⁢
π
h
)
2
and
α
pq
=
k
i
2
-
k
pq
2
,
⁢
where
k
i
=
ω
⁢
μ
0
⁢
ε
o
⁢
ɛ
i
,
and
C
pq
=
-
(
q
⁢
⁢
π
h
p
⁢
⁢
π
g
)
⁢
(
(
jω
o
⁢
ɛ
i
+
δ
)
α
pq
)
⁢
D
pq
,
where
I
yc
⁡
(
p
,
q
)
=
∫
x
=
0
a
⁢
∫
y
=
0
b
⁢
f
y
⁡
(
x
,
y
)
⁢
sin
⁡
(
p
⁢
⁢
π
⁢
⁢
x
g
)
⁢
cos
⁡
(
q
⁢
⁢
π
⁢
⁢
y
h
⁢
⁢
ⅆ
x
⁢
⁢
ⅆ
y
)
and
D
p
⁢
⁢
0
=
-
1
p
⁢
⁢
π
g
⁢
sin
⁡
(
α
p
⁢
⁢
0
⁢
d
)
⁢
(
2
g
⁢
⁢
h
)
⁢
I
yc
⁡
(
p
,
0
)
;
(l) establishing a first set of values of ∈i corresponding to the calculated second variable Mi which satisfy the condition
1
-
R
1
1
+
R
1
=
(
LL
-
M
i
)
⁢
(
ab
)
⁢
jωμ
o
2
⁢
⁢
I
yg
2
⁡
(
1
,
0
)
⁢
γ
10
;
(m) establishing a second set of values of ∈i corresponding to the calculated second variable Mi which satisfy the condition
1
-
R
2
1
+
R
2
=
(
LL
-
M
i
)
⁢
(
ab
)
⁢
jωμ
o
2
⁢
⁢
I
yg
2
⁡
(
1
,
0
)
⁢
γ
10
;
(n) selecting a value of ∈i which is a member of both the first set and the second set of values corresponding to the calculated second variable Mi and setting this selected value equal to a variable ∈d;
(o) establishing a second set of estimated values for the real part of the dielectric constant associated with the substance of interest, ∈ri, where i ranges between 1 and S, such that ∈ri=∈i−j∈′, where ∈′ is an imaginary part of the dielectric constant associated with the substance of interest, where ∈r1=∈d−X1 and ∈rS=∈d+x1 and ∈′=x2, where x1 is approximately equal to 0.5 and x2 is a pre-selected value based on expected conductivity of the substance of interest;
(p) re-calculating the second variable Mi for i ranging between 1 and S as
M
i
=
∑
p
=
1
⁢
∑
q
=
1
⁢
C
pq
⁡
(
q
⁢
⁢
π
h
)
⁢
cos
⁡
(
α
pq
⁢
d
)
⁢
I
yc
⁡
(
p
,
q
)
+
1
jωμ
o
⁢
∑
p
=
1
⁢
∑
q
=
1
⁢
D
pg
⁢
α
pq
⁡
(
p
⁢
⁢
π
g
)
⁢
cos
⁡
(
α
pq
⁢
d
)
⁢
I
yc
⁡
(
p
,
q
)
+
1
jωμ
o
⁢
∑
p
=
1
⁢
D
p
⁢
⁢
0
⁢
α
po
⁡
(
p
⁢
⁢
π
g
)
⁢
cos
⁡
(
α
p
⁢
⁢
0
⁢
d
)
⁢
I
yc
⁡
(
p
,
o
)
,
where
k
i
=
ω
⁢
μ
0
⁢
ε
o
⁡
(
ɛ
ri
-
ɛ
′
)
and
C
pq
=
-
(
q
⁢
⁢
π
h
p
⁢
⁢
π
g
)
⁢
(
(
jωɛ
o
⁢
ɛ
ri
+
δ
)
α
p
⁢
⁢
q
)
⁢
D
pq
;
(q) establishing a first set of values of ∈ri corresponding to the re-calculated second variable Mi which satisfy the condition
1
-
R
1
1
+
R
1
=
(
LL
-
M
i
)
⁢
(
ab
)
⁢
jωμ
o
2
⁢
⁢
I
yg
2
⁡
(
1
,
0
)
⁢
γ
10
;
(r) establishing a second set of values of ∈ri corresponding to the re-calculated second variable Mi which satisfy the condition
1
-
R
2
1
+
R
2
=
(
LL
-
M
i
)
⁢
(
ab
)
⁢
jωμ
o
2
⁢
⁢
I
yg
2
⁡
(
1
,
0
)
⁢
γ
10
;
(s) selecting a value of ∈ri which is a member of both the first set and the second set of values corresponding to the re-calculated second variable Mi and setting the selected value equal to a variable ∈rd;
and
(t) re-calculating the imaginary part of the dielectric constant associated with the substance of interest, ∈′, as ∈′=j(∈rd−∈d),
wherein the dielectric constant associated with the substance of interest is determined as ∈rd+j(∈rd−∈d).

1,084 more elements