Fabrication Processes for RF and Microwave Circuits
3.6 Characterization of Materials for RF and Microwave Circuits
3.6.1 Measurement of Dielectric Loss and Dielectric Constant
3.6.1.2 Dielectric Characterization by Cavity Perturbation
k k
3.6 Characterization of Materials for RF and Microwave Circuits 107
Example 3.11 Determine the frequency range over which only the TE01 mode will propagate through a helical waveguide whose internal diameter is 50 mm.
Comment: The first root of J′0(x) occurs when x=3.8317, and the second root when x=7.0156.
Solution Using Eq. (1.86):
Cut-off wavelength of TE01mode:𝜆c,01= 2𝜋×25
3.8317 mm=41.00 mm.
Cut-off wavelength of TE02mode:𝜆c,02= 2𝜋×25
7.0156mm=22.39 mm.
Then,
fc,01= c 𝜆c,01
=3×108
41.00 Hz=7.32 GHz, fc,02= c
𝜆c,01
= 3×108
22.39 Hz=13.40 GHz.
Frequency range: (7.32–13.40) GHz.
k k 𝜀′′=
(1 Q1 − 1
Qo )
× Vc 4Vo
, (3.30)
where
fo=resonant frequency of cavity without sample f1=resonant frequency of cavity with sample Qo=Qof cavity without sample
Q1=Qof cavity with sample Vs=volume of sample within cavity Vc=volume of empty cavity
Example 3.12 The dielectric properties of a cylindrical rod of insulating material, diameter=1 mm, were mea- sured using a cavity perturbation technique, with the sample positioned as shown in Figure 3.21 within a rectangular waveguide cavity supporting the TE101mode. The cavity cross-sectional data was:a=22.86 mm andb=10.19 mm. The following measurement data were obtained:
Cavity empty: Resonant frequency=12 GHz Q=8704
Cavity+rod: Resonant frequency=11.23 GHz Q=7538
Determine the dielectric constant and loss tangent of the dielectric rod.
Solution
At 12 GHz the length of the cavity is𝜆g/2 for the TE101mode where:
1 𝜆2o
= 1 (2a)2 + 1
𝜆2g
⇒ 1
(25)2= 1
(2×22.86)2+ 1 𝜆2g
⇒ 𝜆g=29.86 mm, l= 29.86
2 mm=14.93 mm.
Volume of rod within cavity=Vs=𝜋×1×10.19 mm=32.02 mm.
Volume of empty cavity=Vc=(22.86×10.19×14.93) mm3=3477.84 mm3. Using Eq. (3.29):
𝜀′ =𝜀r=12.00−11.23
2×12 ×3477.84
32.02 +1=4.48.
Using Eq. (3.30):
𝜀′′= ( 1
7538− 1 8704
)
× 3477.84
4×32.02 =4.83×10−4. Loss tangent:
tan𝛿=𝜀′′
𝜀′ = 4.83×10−4
4.48 =1.08×10−4. Summary:
Dielectric constant=4.48 and loss tangent=1.08×10−4.
Although Eqs. (3.29) and (3.30) are widely used for dielectric characterization at low microwave frequencies they do involve a basic approximation in that it is assumed that the fields within the sample-loaded cavity are uniform. Recent work by Orloff et al. [24] showed significant improvements in measurement accuracy by making some corrections to the basic method to account for the non-uniform fields created by the presence of the sample within the cavity. The details of
k k
3.6 Characterization of Materials for RF and Microwave Circuits 109
the corrections proposed by Orloff are outside the scope of this textbook, but the details are well-explained in [24], which also presents data on measured samples of a known quartz dielectric to verify the new technique.
One of the difficulties that are encountered with the cavity perturbation technique is determining the precise volume of the dielectric specimen, which is necessarily kept small so as to not significantly disturb the fields within the cavity.
Equations (3.29) and (3.30) can be rearranged to give tan𝛿=𝜀r−1
2𝜀r
fo fo−f1
( 1 Q1− 1
Qo )
, (3.31)
which means that in situations where the dielectric constant is already known the loss tangent can be determined with- out the need to know volume of the sample within the cavity. Since the dielectric constant of materials does not change substantially with frequency up to the low microwave region this is a useful expression for situations where the dielectric constant is often known from low-frequency measurements, and where the primary interest is in determining the loss in material.
An alternative to using a thin rod of the dielectric material under test is to use a thin membrane of the material and insert it through a slit in the waveguide cavity wall. Providing that the slit does not significantly interrupt the current flow there will be negligible radiation. Figure 3.22 shows how non-radiating slits may be cut in a rectangular waveguide cavity (supporting TE10nmodes) and an over-moded circular waveguide cavity (supporting TE01nmodes). In both cases the long dimension of the slit is parallel to the wall currents, and for narrow slits there is very little interruption to the current paths.
In the case of the circular waveguide cavity shown in Figure 3.22b, a section of the cavity is made from helical waveguide to act as a mode filter and suppress the unwanted modes. It should also be noted that the circular cavity is fed through an off-centre coupling iris to excite the TE01mode; the position of the coupling iris corresponds to the maximum of the electric field.
The cavity perturbation techniques discussed so far have used homogeneous dielectric samples. But there is a need at RF and microwave frequencies to be able to characterize layers of dielectric that are not self-supporting but need to be printed or deposited on a rigid base. An example is thick-film dielectric which has to be printed on top of a supporting substrate, normally alumina. Consequently, there is a need to characterize dielectric materials that form two-layer samples. One technique to characterize a thick-film layer is to partially coat a thin low-loss substrate with the material under test, as shown in Figure 3.23. Typically the substrate would be 250μm thick high purity alumina.
l
b Non- radiating slit
l
D Non-radiating
slit Mode filter
Coupling
iris Coupling
iris
(a) (b)
Figure 3.22 Non-radiating slits in waveguide cavities: (a) rectangular waveguide cavity and (b) circular waveguide cavity.
Printed thick-
film Printed
thick-film
Supporting substrate
Supporting substrate
(a) (b)
Figure 3.23 Two-layer thick-film test samples: (a) substrate partially coated with thick-film layer under test and (b) substrate sawn to produce a two-layer test specimen and a reference sample.
k k having the same dimensions and properties as that supporting the thick-film layer. The resonant frequency andQof the
cavity are then measured with the uncoated tile in place to form the reference values,foandQo. The reference tile is then replaced with the two-layer test specimen and the measurements repeated to enable the dielectric constant and loss tangent of the thick-film layer to be calculated using the previous theory. The method assumes that there is no variation in dielectric properties across the original alumina tile, and also that the reference piece is located precisely in the same position as the tile supporting the thick-film layer under test.