3.2 COAXIAL-LINE REFLECTION METHOD
3.2.2 Coaxial probes terminated into layered
In the above discussion, the coaxial probe is termi- nated by a homogeneous semi-infinite medium. In some situations, the semi-infinite requirement can- not be satisfied, and sometimes the samples under study may be of multilayer structure. So the knowl- edge about the admittance of coaxial probe termi- nated into layered dielectric materials is important for materials property characterization.
3.2.2.1 Admittance of coaxial probe terminated into layered materials
The admittance of a coaxial probe terminated into layered materials has been studied by lumped parameter approach and quasi-static approximation approach (Fanet al. 1990; Andersonet al. 1986).
Here, we introduce a general formulation for an open-ended coaxial transmission line terminated by a multilayered dielectric (Bakhtiariet al. 1994).
As shown in Figure 3.11, we assume that the open end of a coaxial transmission line is connected to a perfectly conducting infinite flange, and the inner and outer diameters of the coaxial line are 2a and 2b respectively. As shown in Figure 3.12, the multilayer composite may be terminated into an infinite half-space or backed by a conductor.
y
x 2a z
2b
r
f
Infinite flange
Figure 3.11 Coaxial line with inner diameter 2a and outer diameter 2bopening onto a perfectly conducting infinite flange. Source: Bakhtiari, S. Ganchev, S. I. and Zoughi, R. (1994). “Analysis of radiation from an open-ended coaxial line into stratified dielectrics”,IEEE Transactions on Microwave Theory and Techniques,42 (7), 1261–1267.2003 IEEE
Coax
(a) (b)
Coax
z0=0
m0,erc m0,erc
zN=1 zN=1
dN−1
d2
d1 d1 d2 dN
erN−1erN erN s = ∞
er2
er1 er1 er2
z1 z0=0 z1 zN
Figure 3.12 Cross sections of coaxial line radiating into layered media. (a) The multilayer material is terminated into an infinite half-space and (b) cross section of a coaxial line radiating into a layered media terminated into a perfectly conducting sheet. (Bakhtiari et al. 1994). Modified from Bakhtiari, S. Ganchev, S. I. and Zoughi, R.
(1994). “Analysis of radiation from an open-ended coaxial line into stratified dielectrics”, IEEE Transactions on Microwave Theory and Techniques,42(7), 1261–1267.2003 IEEE
If only the fundamental transverse electromag- netic (TEM) mode propagates inside the coaxial line, the terminating admittance of the line can be obtained using the continuity of the power flow across the aperture (Bakhtiariet al. 1994):
ys=gs+jbs= εr1
√εrcln(b/a) ∞
0
[J0(k0ξ b)−J0(k0ξ a)]2
ξ F (ξ )dξ (3.40) where gs and bs are the normalized aperture conductance and susceptance, εrc is the relative permittivity of the dielectric filling inside the coaxial line,J0is the zero-order first kind of Bessel function, and the function F (ξ )is given by
F (ξ )= 1 εr1−ξ2
1+ρ1 1−ρ1
(3.41)
For anN-layer medium, the value ofρ1 may be calculated from the following recurrence relations:
ρi = 1−κiβi+1 1+κiβi+1
e−j2k0zi√
εri−ξ2 (3.42)
where
κi =εr(i+1)
εri ã
εr−ξ2
εr(i+1)−ξ2 (3.43) βi+1 =1−ρi+1ej2k0zi√
εr(i+1)−ξ2
1+ρi+1ej2k0zi√
εr(i+1)−ξ2 (3.44) zn=
n
i=1
di (3.45)
In Eq. (3.45), if the Nth layer is infinite in +z- direction, 1≤n≤(N–1); if the Nth layer is backed by conducting sheet, 1≤n≤N.
If theNth layer is infinite in+z-direction,
ρN =0 (3.46)
If the Nth layer is terminated into a conduct- ing sheet,
ρN =e−j2k0zN√
εrN−ξ2 (3.47) The above calculation must start from i=N−1 and carried out backward to i=1. The value of
ρN is chosen from Eq. (3.46) or (3.47) depending on whether the Nth medium is an infinite half- space or is of finite thickness backed by a conducting sheet.
3.2.2.2 Applications in materials property characterization
The conclusions for the aperture admittance of coaxial probe terminated into stratified dielectrics can be used to modify the conventional reflec- tion method using coaxial dielectric probe. In the following, we discuss several examples of modifi- cations often used in materials property character- ization.
Dielectric samples with finite thickness backed by metal plate
For a laminar sample, its thickness is usually not thick enough for semi-infinite requirement.
As shown in Figure 3.13, the dielectric proper- ties of a laminar sample backed by a conducting plane can be measured by using an open-ended coaxial probe. Actually, it is a special case of Figure 3.12(b) withN=1. The terminating admit- tance of the probe can be calculated according to Eq. (3.40), and the functionF (ξ )can be calculated according to Eq. (3.41) withρ1 given by
ρ1 =e−j2k0d1√
εr−ξ2 (3.48)
Metal plate
d1 er
Figure 3.13 Geometry for the measurement of a laminar dielectric sample backed by a conducting plane
Free space
er2
d1 er1
Figure 3.14 Geometry for the measurement of a laminar dielectric sample backed by free space Dielectric samples with finite thickness backed by free space
As shown in Figure 3.14, in the measurement of a laminar sample, the sample can also be backed by free space. This is a special case of Figure 3.12(a) when N=2. The second layer is infinite in +z- direction. The explicit form ofF (ξ )is given by
F (ξ )= 1
εr1−ξ2 ã κ1+j tan(k0d1
εr1−ξ2) 1+jκ1tan(k0d1
εr1−ξ2) (3.49) with
κ1 = εr2 εr1 ã
εr1−ξ2 εr2−ξ2
(3.50)
where d1 is the thickness of the laminar sample under test, εr1 is the relative permittivity of the sample, andεr2 is the relative permittivity of the half-infinite sample backing the sample. For a sample backed by free space,εr2=1.
Two-layered media terminated by metal plate In the measurement of dielectric properties using coaxial probes, good contact between the coaxial aperture and the sample surface is required. If there is an air gap between the sample and the probe, the discontinuity of the electric field causes a large error in the calculation of permittivity. In actual measurements, the air gap between the coaxial
aperture and the sample surface is inevitable. To minimize the effect of the air gap, Baker-Jarvis et al. developed a model for the electromagnetic response of a coaxial probe with liftoff (Baker- Jarviset al. 1994).
Figure 3.15 shows a coaxial probe terminated by a two-layer dielectric sample backed by a conducting plane, and the dielectric permittivity values of the two layers areεr1andεr2respectively.
It is a special case of Figure 3.12(b) whenN=2.
The explicit form ofF (ξ )is given by
F (ξ )= j εr1−ξ2 ã
tan(k0d1
εr1−ξ2) tan(k0d2
εr2−ξ2)−κ1
κ1tan(k0d1
εr1−ξ2) +tan(k0d2
εr2−ξ2) (3.51) For the case of coaxial probe with liftoff, the first layer with thickness d1 is air (εr1=1), and the second layer with thickness d2 is the sample under test. Probes with liftoff are quite useful for nondestructive material property testing and material thickness testing. However, in order to increase the interaction of the field with the material under test, either larger diameter probes or higher frequencies need to be used.
The configuration shown in Figure 3.15 can also be used in the measurement of high-dielectric- constant materials. When a dielectric probe is used to measure materials with high dielectric constant, for example, higher than 50, because of the serious
Metal plate
d1 d2 er1 er2
Figure 3.15 Geometry for the measurement of two- layer dielectric sample backed by a conducting plate
impedance mismatch between the sample and the coaxial line, only a small part of the microwave signal is radiated into the sample, and so the measurement accuracy and sensitivity are low. To increase the measurement accuracy and sensitivity, we may introduce a matching layer with known dielectric properties between the probe aperture and the sample, so that more microwave signal can be radiated into the sample under test.
Two-layered media terminated by free space Figure 3.16 shows a coaxial probe terminated by a two-layer sample backed by free space. The free space can be taken as a third layer with half-infinite thickness; so the medium terminating the probe can be taken as a structure consisting of three layers with relative permittivity εr1, εr2, and εr3 respectively and thickness d1, d2, and d3 respectively. The explicit expression for F (ξ ) when the third layer is an infinite half-space extending in the z-direction with a dielectric constant εr3 is given by (Ganchevet al. 1995)
F (ξ )= 1
εr1−ξ2 ã1+ρ1
1−ρ1
(3.52)
where
ρ1= κ1−β2 κ1+β2
e−j2k0d1√
εr1−ς2 (3.53)
Free space
d1 d2 er1 er2 er3
Figure 3.16 Geometry for the measurement of two- layer dielectric sample backed by free space
κ1 = εr2
εr1 ã
εr1−ς2 εr2−ς2
(3.54)
β2 = 1−ρ2ej2k0d1√
εr2−ς2
1+ρ2ej2k0d1√
εr2−ς2
(3.55) ρ2= κ2−1
κ2+1ãe−j2k0(d1+d2)√
εr2−ς2 (3.56) κ2 = εr3
εr2 ã
εr2−ς2
εr3−ς2 (3.57)
Similar to the case of two-layer sample backed by a conducting plane, the above model can be used in improving the conventional coaxial dielectric probe in two ways. One is to minimize the uncertainties caused by the inevitable air gap between the dielectric probe, and the other is to measure samples with high dielectric constants.