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TECHNIQUE FOR MEASURING THE DIELECTRIC
CONSTANT OF THIN MATERIALS
f
K. Sarabandi and F. T. Ulaby
The University of Michigan
A practical technique for measuring the dielectric constant of vegetation leaves
and similarly thin materials is presented. A rectangular section of the leaf is placed in
the transverse plane in a rectangular wavaguide and the magnitude and phase of the
reflection coefficient are measured over the desired frequency band using a vector
network analyzer. By treating the leaf as an infinitesimally thin resistive sheet, an
explicit expression for its dielectric constant is obtained in terms of the reflection
coefficient. Because of the thin-sheat approximation, however, this approach is valid
only at frequencies below 1.5 GHz. To extend the technique to higher frequencies,
higher order approximations are dedved and their accuracies are compared to the
exact dielectdc-slab solution. For a matedal whose thickness is 0.5 mm or less, the
proposed technique was found to provide accurate values of its dielectric constant up
to frequencies of 12 GHz or higher. The technique was used to measure the 8-12 GHz
dielectric spectrum for vegetation leaves, teflon, and rock samples.
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This work was supported by the National Aeronautics and Space Administration under
Contract NAG5-480.
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I. INTRODUCTION
Prompted by the need for a practical technique for measuring the microwave
dielectric constant of vegetation leaves, solutions were sought for the voltage reflection
coefficient measured at the input of a rectangular waveguide containing a thin slab
placed in a plane orthogonal to the propagation direction (Fig. 1). The slab is modeled
in Section II as a resistive-current-sheet [1,2], which has proved to be an excellent
approach for characterizing the radar-cross-section of a vegetation leaf over a wide
range of moisture conditions (and a correspondingly wide range of the relative
dielectric contstant s).
To evaluate the accuracy of the technique for measuring the real and imaginary
parts of _ from measurements of the complex reflection coefficient F, an exact solution
for T" of the slab will be obtained in Section III and then used to simulate the
measurement process for given values of _. The evaluation is performed in Section IV
by comparing the true value of _ with that predicted by the resistive-current-sheet
expression. It turns out that the resistive-current-sheet solution is identical with the
zeroth-order approximation of the exact solution for T'. One of the attractive features of
the zeroth-order solution is that it provides an explicit expression for _ in terms of F.
The evaluation shows that the zeroth-order solution provides an excellent estimate
for the real part of the dielectric constant, £, if the slab thickness t is sufficiently small
to satisfy the condition _ _;0.05 _.0_r_, where _'0 is the free-space wavelength.
For a typical leaf-thickness of 0.3 ram, this condition is satisfied for any moisture
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condition if the frequency f _; 15 GHz. A much more stringent condition on • is
required in order for the zeroth-order solution to give accurate values for s"; namely
< 0.01 ;L0_ and s"/_' > 0.1, or equivalently, f _; 1.5 GHz for vegetation leaves.
To relax this limitation, alternate solutions for F are obtained in Section III by invoking
approximations that lead to first-order and second-order solutions whose forms are
invertible to explicit expressions for s. Use of the second-order solution is found to
extend the frequency range from 1.5 GHz to 12 GHz for a leaf with a high moisture
content and to higher frequencies for drier leaves.
Section V presents 8-12 GHz spectra of the dielectric constant _ for vegetation
leaves, teflon and rock slices, all measured using the technique developed in this
paper. Where possible, the results are compared with measurements made by other
techniques.
II. MODEL FOR A THIN RESISTIVE SHEET
Consider the rectangular waveguide diagrammed in Fig. 1(a). The guide is
terminated with a matched load, has dimensions a x b, and contains a thin resistive
sheet of thickness t at z = 0. The waveguide dimensions are such that only the TEl0
mode can propagate in the guide.
We seek a relationship between the input voltage reflection coefficient F and
the relative complex dielectric constant of the sheet material _. To this end, we shall
develop expressions for the electric and magnetic fields in Regions I and II and then
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apply the appropriate boundary conditions.
regions I and II, respectively, solutions of the scalar Helmholtz equation
If _1 and _rll are the electric potentials in
(I)
for the TEIO mode leads to [4, pp. 148-149]:
_1 = cos C 1 e + C2 e ; z_> 0
(2)
(-_') C3 ikzz
_11 = cos . e
;z < 0, (3)
where a time factor ei(ot was asssumed and suppressed. The constants C1 and C 2
represent the magnitudes of the incident and reflected waves in Region I, C 3
represents the magnitude of the wave traveling towards the matched load in Region II,
and
=
(4)
The components of E and H may be obtained from (2) and (3) by applying the
relations [4, p. 130].
_ A _ ^ 1 ^
E = -Vx(_z) , H =-i(o_(_z) +--VV .(_z) . (5)
i(ol_
The resistive sheet model [1] treats the sheet in the plane z - 0 as infinitesimally
m
thin carrying an induced tangential electric current 3 that is related to E by
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x _ xE =-RJ, (6)
where _ is the surface normal of the sheet (_ = _' in Region I and _ = - z- in Region II)
and R is the sheet resistivity,
-i 110
R = , ohms per square meter. (7)
k'_ (s- 1)
In the above expression,
k = 2_ / Z0 , _ is the sheet thickness, 110is the free space
intrinsic impedance, and
= ¢' -i_:",
is its relative complex dielectric constant.
(8)
The condition for continuity of the tangential
electric field from Region I to Region I! and the boundary condition for the magnetic field
requires that
ZX - =0, Z x II =
(9)
The unknown coefficients C1, C2, C3 can be obtained by applying the boundary
conditions given by (6) and (9). The complex voltage reflection coefficient is then found
to be
c2 k2t 1) (lo)
C1 k2_(6-1)- 2ik z
from which an explicit expression for s is obtained,
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