Temperature Correction for Cylindrical Cavity Perturbation Measurements

The need for accurate material property measurements using microwave cavities requires a form of compensation to correct for changes in temperature and other external influences. This paper details a method for temperature correcting microwave cavity perturbation measurements by monitoring two modes; one which is perturbed by the sample and one which is not (referred to as a nodal mode). The nodal modes used (TM310 and TE311 for an axial sample in a cylindrical cavity) are subject only to sample-independent influences. To demonstrate this technique, the bulk permittivity of a PTFE rod has been measured under varying temperature conditions. The results show that without correction, the measured temperature-dependent dielectric constant has large variations associated with the stepped and linear temperature ramping procedures. The corrected response mitigates systematic errors in the real part. However, the correction of the imaginary part requires careful consideration of the mode coupling strength. This paper demonstrates the importance of temperature correction in dynamic cavity perturbation experiments.


Temperature Correction for Cylindrical Cavity Perturbation Measurements
Jerome A. Cuenca, Daniel R. Slocombe, and Adrian Porch Abstract-The need for accurate material property measurements using microwave cavities requires a form of compensation to correct for changes in temperature and other external influences.This paper details a method for temperature correcting microwave cavity perturbation measurements by monitoring two modes; one which is perturbed by the sample and one which is not (referred to as a nodal mode).The nodal modes used (TM 310 and TE 311 for an axial sample in a cylindrical cavity) are subject only to sample-independent influences.To demonstrate this technique, the bulk permittivity of a PTFE rod has been measured under varying temperature conditions.The results show that without correction, the measured temperaturedependent dielectric constant has large variations associated with the stepped and linear temperature ramping procedures.The corrected response mitigates systematic errors in the real part.However, the correction of the imaginary part requires careful consideration of the mode coupling strength.This paper demonstrates the importance of temperature correction in dynamic cavity perturbation experiments.
Index Terms-Cavity perturbation, microwave material measurements, temperature correction, temperature dependence.

I. INTRODUCTION
T HE CAVITY perturbation method is very common for characterizing materials at microwave frequencies because of its precision, simplicity, and noncontact nature.This method involves perturbing a resonant electromagnetic field within a cavity by inserting a sample into the required field region and examining the change in the frequency response of the cavity.Samples, such as those containing ferrous oxides [1], [2] carbon nanotubes [3], oil [4] nanodiamond [5]- [8], bacteria [9], [10], PTFE and common plastics [11], [12], and explosives [13], have all been characterized using this method to infer various properties through changes in microwave complex permittivity and permeability.Due to its noncontact nature, it has also become an effective material probe for measuring chemical reactions in hydrogen storage materials in situ [14] and since high-power microwaves can be used for heating, the dependence of the dielectric properties with heating has also been investigated in situ using a 100-W cavity system [15], [16].
With the increasing number of applications for cavity perturbation as a noncontact material probe in chemistry, one must read carefully when dealing with exothermic and endothermic reactions to keep the system stable [17].Since both the unperturbed resonant frequency and bandwidth are functions of the cavity geometry, the cavity response will change due to the natural thermal expansion and temperature-dependent conductivity of the metal cavity.The change in temperature of a sample can consequently cause the cavity to change resonance frequency through radiative heat transfer.Therefore, the observed shifts in frequency are merely systematic errors as opposed to any change in the dielectric or magnetic property.Thus, when temperature becomes a dependent variable, intrinsic material properties need to be decoupled from system temperature fluctuations.This can be achieved in the cavity perturbation system by in situ monitoring of modes where the sample provides minimal perturbation.
Previous studies have already shown that this technique is effective for rectangular type geometries [18], [19].For dielectric resonator setups, the temperature dependence of the external cavity has already been investigated when extracting temperature coefficients of extremely low loss dielectrics [11], [20], [21].This paper, however, addresses the conventional cavity perturbation setup, where the sample does not resonate and merely produces a small perturbation in a mode as a result of the boundary conditions of just the cavity.
The novel contribution in this paper is threefold.First, for the cavity correction modes presented, the sample has a minimal perturbation in both the E-a n dH -fields.The previous studies use modes where the sample still perturbs the H -field in the assumption that the complex permeability is equivalent to air, restricting the measurement to pure dielectrics.Furthermore, the system in this paper has the potential to measure both the temperature-dependent dielectric and magnetic properties.Second, the system is a hollow cylindrical cavity and exploits the fact that the cavity modes are described by Bessel functions.A cylindrical cavity becomes advantageous in measuring temperature-dependent properties of axial samples because with increasing mode number (and thus Bessel function order), the field maxima are pushed toward the perimeter of the cavity.The higher the mode, the less sensitive it is to the axial sample.Also, for dielectric resonators, the use of correction modes is not possible, since the sample is the resonator or perturbs a large portion of the cavity.
Finally, as a development on previous methods which subtract gradients based on linear coefficients, the proposed method does not rely on previously measured data of expansion coefficients of known materials.Instead, it uses an intermodal relationship between cavity modes and so corrects for each individual experiment.
In this paper, we have used a selection of cylindrical cavity resonators to demonstrate multimode correction.Section II presents a derivation of the temperature-dependent cavity parameters and the correction modes.Section III presents data and discussions on temperature experiments on three cylindrical cavities; two aluminum cavities which are used in our laboratory for dielectric measurements using the TM 010 mode and magnetic measurements using the TE 011 mode, respectively, and a copper cavity similar to the TM 010 aluminum cavity but with smaller dimensions.The chosen nodal modes are the TM m10 and TE m11 modes (where m > 1).

II. ANALYSIS
In practice, cylindrical cavity measurements are generally achieved by placing a sample in the field in a nondepolarizing or demagnetizing configuration.Hence, this is the reason why modes, such as the TM 010 and TE 011 modes, are used for thin rod shaped samples measurements placed axially [22]- [24].Given that the cavity perturbation method already offers such high sensitivity and minimal sample requirements, the aim of this paper is to demonstrate the measurement of temperature-dependent properties in situ,i na temperature varying environment.Several challenges exist with this approach, as it is clear that even though the cavity is mechanically sturdy, it is prone to the effects of thermal stress and temperature-dependent electrical conductivity, as well as changes in antenna geometry and coupling symmetry.These systematic errors will offset the resonant frequency and the bandwidth.

A. Temperature-Dependent Frequency
In a temperature controlled permittivity measurement, we assume the cavity to have a fixed volume, though in the event where the sample is changing temperature and potentially radiating heat, the resultant convection will inevitably create a temperature increase at the cavity walls.This process depends upon the temperature of the sample, the thermal properties of the cavity medium, and the thermal properties of the conducting walls of the cavity.We can further examine the effect of temperature on the cavity by looking at the wave dispersion relation of an air-spaced cylindrical cavity resonator.For TM modes generally, we have where c is the speed of light, n, m,and p are the mode integers, k nm is the nth root of the mth order Bessel function of the first kind, and a(T ) and l(T ) are the temperature-dependent radius and height of the cavity, respectively.The temperaturedependent dimensions of the cavity can be described by a linear thermal expansion approximation where a 0 and l 0 are the initial dimension lengths, T is the change in temperature, and α c is the linear thermal expansion coefficient of the cavity metal.Using the first-order partial derivatives, the change in resonant frequency due to an increased temperature can be given by Evaluating each partial derivative and taking discrete measurements of frequency and temperature, we find that the fractional shift in frequency is This shows that any fractional increase in frequency with temperature is simply proportional to the linear thermal expansion coefficient of the cavity material.Therefore, if the cavity changes temperature, the change in frequency as a function of temperature will have a gradient equal to −α c f 0 .Correction of TM 010 measurements simply involves removing this gradient [25].However, since the geometry of the cavity is not ideal due to holes for the sample and coupling, α c may be different for each mode and consequently simple subtraction will not suffice.A more reasonable assumption is that the relationship between each of the modes (or the difference between the gradients of each mode) is the same.The novel aspect of this paper is this principle.

B. Temperature-Dependent Bandwidth
A similar derivation for bandwidth can be achieved using the well-known quality factor equation, where the power that is lost is caused by the electrical resistance in the cavity metal where Q 0 is the unloaded quality factor, R s is the surface resistance of the cavity metal, S c and V c are the surface area and volume of the cavity, respectively, and H cav is the spatially dependent magnetic field.Evaluating the numerator of ( 6) gives where U mnp is the volume fraction of the cavity that is occupied by the squared magnetic field.Evaluating the denominator of (6) gives Surface resistance is defined as where ρ( T ) is the temperature-dependent resistivity of the cavity walls.Substituting ( 7) and ( 8) into ( 6) yields an equation for the theoretical unloaded bandwidth This equation defines the minimum theoretical bandwidth for a cylindrical cavity resonator constructed out of a material with an electrical resistivity ρ(T ); an aluminum cavity designed for aT M 010 mode at 2.495 GHz (a = 46 mm and l = 40 mm), for example, would give a theoretical minimum bandwidth of approximately 190 kHz (Q 0 ≈ 1.3 × 10 4 ).In practice, however, the aluminum surface quality is not perfect and roughness coupled with a reduction in the electrical quality due to dislocations will greatly increase this value.Using the same approach as the temperature-dependent frequency derivation Assuming that ρ( T ) is a linear function of temperature such that (∂ρ/∂ T ) ∼ = β c , the partial derivatives give the following equation-dependent upon geometrical expansion and resistivity: Thus, theoretical unloaded bandwidth linearly increases with increasing resistivity and decreases with geometrical expansion of the volume.Typically for metals, the coefficient of resistivity is three orders of magnitude greater than thermal expansion, hence the contribution from α is much smaller (for aluminum, β c ≈ 4 × 10 −3 /°C while α c ≈ 22 × 10 −6 /°C).This equation, however, does not consider effects due to coupling strength into the resonator and its symmetry.The cavity coupling strengths are also complex functions of temperature, since the geometry of the coupling structures will expand and change resistance.Because of this, the difference in the bandwidth gradients for each mode may differ greatly.However, if the difference between each mode is constant and known then correction can still be achieved.It must also be stipulated that the variation in bandwidth may be as high 1 kHz, as is in our system described later, which gives a variation of 5 × 10 −3 /°C, a value as large as β c .This means that only large changes in resistivity can be measured while the effects of the temperature-dependent coupling may dominate.More on the reliability of bandwidth correction is explained in Section III-D.

C. Nodal Mode Correction
From ( 5), all resonant frequencies of the cavity will drift up and down with temperature by the same fractional amount.The frequency and bandwidth measurement of the TM 010 mode whilst the temperature is varying is in error by a gradient which is dependent on thermal expansion and resistivity and so a simple subtraction of this gradient is required.A more robust method is to use a mode where the sample has minimal perturbation, referred to in this paper as a nodal mode.Therefore, the corrected frequency shift (attributed to just the properties of the sample) is simply the fractional frequency shift of the sample measurement mode f / f sample minus the fractional frequency shift of the nodal mode ( f / f node ).T h e fractional frequency shift of the measurement and nodal modes with temperature can be modeled as follows: where the fractional frequency shift denoted by k is the shift caused by the perturbation, κ 010 and κ m10 are given as the differences in gradient from the intrinsic thermal expansion coefficient of the metal and the observed mode, and κ err is a systematic thermal expansion error in the experiment.Subtracting ( 13) from ( 12) the thermal expansion error in the experiment is eliminated although, the gradient terms remain.These are simply removed using subtraction with the unperturbed state where u and s denote the temperature-dependent fractional frequency shift of the unperturbed and perturbed cavity, respectively.Through cavity perturbation, the change in permittivity with respect to a reference temperature is, therefore, where synonymously, the imaginary part can be obtained from the corrected bandwidth or quality factor This approach has a distinct advantage over a simple gradient subtraction in that it corrects for any geometrical expansion at the measurement time.In the simple case, while it is reasonable to assume that the cavity homogenously changes temperature due to the high thermal conductivity of the metal, it is also assumed that the structural thermal expansion happens instantaneously for a given temperature increase.This is of course not the case and there is a finite time required for the cavity to expand which the nodal correction method does take into account since the correction is performed at the time of measurement.For bandwidth, however, it is not as simple, since each mode will be affected by its own mode-dependent coupling strength.This can be mitigated somewhat by correcting for unloaded bandwidth [26].
There are numerous modes, which have a node on the axis.This can be realized by recalling that all E-a n dH -field components in cylindrical modes are dependent on mth order Bessel functions of the first kind [27].Taking r to be the radial distance from the center of the cavity, any mode where J m (r = 0) and J ′ m (r = 0) are equal to zero will yield a possible nodal measurement mode.Examples of these modes are shown in Fig. 1.As m increases, the central node spot gets broader and the mode is, therefore, less sensitive to the sample, although as mode number increases, interference from other modes starts to become considerable.
The existence of these nodal modes depends upon whether the coupling structures excite them.Open circuit coaxial probes (E-field antennas) are used to excite TM modes and short circuited coaxial loops (H -field antennas) are used to excite TE modes.Placing these antennas on the end plates and near the axis of the cavity allows coupling into the TM 0np and TE 0np modes for the E-a n dH -field antennas, respectively.In this position, some of the nodal modes are also excited.However, since the field maxima tends toward the perimeter with increasing m, much higher order modes are not excited.If the coupling antennas are located on the side of the cavity (i.e., H -field coupling to the TM 010 mode or E-field coupling to the TE 111 mode) then this configuration will excite the higher nodal modes.The cavities used in this paper are shown in Fig. 2; an E-field coupled TM cavity, an H -field coupled TE cavity, and an H -field coupled TM cavity.The first two nodal modes of the TM cavities are the TM 210 and TM 310 modes.For the H -field TE cavity, the first two nodal modes are the TE 211 and TE 311 modes.

III. EXPERIMENT AND DISCUSSION
The experiments were conducted with the two aluminum cavities and one copper cavity, schematics shown in Fig. 2. The first cavity is designed to excite TM modes using E-field antennas and the second is designed to excite TE modes using H -field coupling.The copper cavity excites both TE and TM modes due to its side H -field antennas and was used to demonstrate the measurements of the thermal properties of a different material to aluminum.A photograph of the experimental setup is shown in Fig. 3. Temperature ramps were achieved by placing the cavity inside of a Memmert UF30 laboratory oven.The temperature profiles (stepped and ramped) were programmed using software provided by the manufacturer.The resonant frequency and bandwidth were recorded using standard transmission measurements (i.e., S21), obtained using a Keysight Technologies VNA (PNA-L N5232A) with RF cables provided by RF COM.The temperature of the cavity was obtained using two Omega PT100 temperature sensors placed on the walls of the cavity and a National Instruments (NI) cDAQ-9171.The VNA and cDAQ were interfaced with the PC and a program written using NI LabVIEW provided automated data recording over long periods.

A. Metal Rod Perturbation
First, to test the effectiveness of the nodal modes, a copper rod with a diameter of 3 mm was placed at the sample location of the aluminum cavities.The differences in resonant frequency after perturbation are given in Table I and the resonant traces are shown in Fig. 4. For the TM cavity, this perturbation essentially destroys the resonance of the TM 010 mode as the maximum E-field along the axis is forced to zero.This completely supresses the mode, since an E-field cannot exist parallel to a metal interface.For the TE cavity, the H -field in the TE 011 mode is forced to zero at the center of the cavity, but the distribution is not grossly affected.In this case, the H -field perturbation by the metal rod represents the maximum possible perturbation of a magnetic sample material, since the rod completely screens the H -field. Thus, a metal rod perturbation will give an indication as to whether the observed mode is a nodal mode if the measured frequency and bandwidth shifts are very small.Using curve fitting, the fractional frequency can be measured with a maximum error across the modes of 2 × 10 −7 (for example, the TE 211 mode has a 1-kHz variation around a 4.83-GHz resonance).For bandwidth, the maximum curve fitting error across all of the modes is approximately 4 × 10 −3 (1-kHz variation for a typical quality factor of 9 × 10 3 ).The results show that TM 210 is partially perturbed although this is only a maximum fractional shift of 5.2 × 10 −5 .This shift is also upward, implying this small perturbation is in the H -field, which is expected for TM modes due to some magnetic field encroaching near to the axis of the cavity, as shown in Fig. 1.The change, however, in TM 310 is much less affected by the sample at only an upward fractional shift of 4 10 −7 , which is approaching the curve fitting measurement error.This is expected, since the energy density fraction where the 3-mm metal rod occupies is at greater in the TM 210 mode as compared with the TM 310 mode (verified with COMSOL).This makes these two modes ideal for the correction of the TM 010 mode.The TE 211 and TE 311 modes yield shifts of −9.8 × 10 −6 and −3 × 10 −7 , respectively.This is expected again, since the energy fraction in the TE 211 mode is greater than for TE 311 mode.The shift downward is implicit of a slight perturbation into the E-field.The fractional changes in bandwidth are much more sensitive to the sample location, though still of the same order of the measurement error, as shown in Table I.Notice that the bandwidth decreases in the TM modes, which is implicit of a decrease in the power losses in the cavity.This can be explained by the fact that the rod is made of copper and has a higher conductivity than aluminum.Hence, the induced surface current in a better conductor caused by the slight H -field perturbation actually increases the quality factor.

B. Ambient Temperature Drift
Both the aluminum TE cavity and the copper cavity were measured over a period of 4 h on separate occasions in the laboratory to study the effects of the ambient temperature on the nodal and sample measurement modes from 3 to 10 GHz.The results in Fig. 5 show that for a given cavity, all modes drift with the same fractional frequency shift with temperature.The bandwidth drift was below the noise floor of the measurements and yielded a stable response.This is because the sensitivity is much less as the fractional variations are with respect to kilohertz as opposed to gigahertz.The temperature differences are also very small and thus the resistivity will not change by much.This experiment demonstrates the sensitivity of the frequency measurement and that all of the resonant frequencies are inherently linked through geometrical expansion, as described previously in (5).

C. Temperature Ramps
A temperature ramp profile of 30 °C-80 °C at 1 °C per minute was imposed on the cavity to prove the nodal mode principle.While a small temperature range has been used owing to laboratory limitations, this cavity thermal expansion correction method will also work over a much wider temperature range, at both lower and higher temperatures.This experiment demonstrates the linear relationship between frequency and temperature.Fig. 6 shows the ramping results while Table II shows the linear fits.The gradients show the approximate values for the linear thermal expansion coefficients of their respective metals as expected from (5).The TM 010 mode of the aluminum TM cavity has a slightly smaller than expected value of 19.3 × 10 −6 /°C, though this is not an issue, since it is the relationship between this mode and the nodal modes that is important.The TM 210 and TM 310 modes yield gradients much closer to expected values.There is also a discrepancy in the copper TM cavity measurements, where the gradient of the TM 210 mode is much smaller than the others.This is most likely due to this mode having a slightly asymmetrical resonance because of three coupling port holes on the cavity.
Resistivity measurements were variable, owing to the issues of sensitivity in measuring the fractional bandwidth shifts combined with the temperature dependence of the coupling antennas.Both the Al TE cavity and the C u TM cavity have loop coupling structures, which will expand as temperature increases, thereby causing a temperature-dependant coupling strength.The only repeatable measurements are those for the TM cavity, as shown in Table II.These values are similar   This experiment demonstrates a pertinent result in that these modes are all linearly dependent on each other through their own thermal expansion coefficient and, therefore, if the volume of the cavity suddenly changes due to temperature, then other modes can be measured to infer this volume change.

D. Sample Tube Temperature Ramps
The temperature-dependent complex permittivity of a PTFE rod was measured.This is simply to demonstrate the capability to exclusively extract the temperature-dependent dielectric properties of a sample whilst negating any systematic fluctuations.First, to ensure that any measured temperaturedependent response was due to the sample, ramping experiments were carried out on an empty cavity followed by the sample perturbed cavity.Experiments were conducted using the TM aluminum cavity and the PTFE rod (shown in Fig. 7).The complex permittivity was calculated using the usual cavity perturbation approach given in (16) using linear interpolation at each temperature.In order to induce a systematic error related to thermal expansion, both a linear ramp as well as a stepped profile was conducted.As stated in Section II-C, in an ideal case, the change in temperature of the cavity should result in an instantaneous change in frequency and bandwidth which is not the case because while the thermal conductivity is high, the mechanical deformation has an associated delay or "thermal lag."Two linear ramps and two stepped ramps (at 10 °C steps with a dwell time of 1 h) were conducted starting from 30 °C and 70 °C, where the reference frequencies and bandwidths for the correction method are static values taken at the beginning of the experiment.
The measurements of the PTFE ramps are shown in Fig. 8. Without any correction, the observed dielectric constant decreases with temperature, which is congruent to results already reported [11].Interestingly, the stepped temperature ramps show different values to those given in the linear ramp.The stepped response yields a linearly decreasing trend, however, there is a noticeable ripple within the data, associated with the stepped ramp profile; noticeable features can be seen at each 10 °C plateau in the dielectric constant.The linear ramp response shows different absolute values.The reason for this is that the recorded temperature at the aluminum surface does not necessarily mean that the equivalent mechanical thermal expansion of the cavity has occurred yet.Hence, when the perturbation equations are used, the cavity volume of the unperturbed state is not the same as the volume when perturbed.
The corrected results, however, are almost equal with a very little difference between the correction modes used (TM 210 and TM 310 ) and a very little difference between ramping profiles.Again, the absolute values and decreasing trend are similar to the reported values of approximately 2.02 to 2.09 although at a slightly higher frequency of 10 GHz [11].With the correction procedure, it is shown that finer temperature steps can be used and minute fluctuations in resonant frequency can be corrected.This is because the reference frequencies are static values, taken at a known temperature, and therefore, a reference volume.For the dynamic measurements of dielectric constant, it is clear that the profile must be accounted for to avoid erroneous data.
The loss measurements for PTFE, however, are not as well corrected as the dielectric constant.First, it must be stated that this material has a very low loss tangent and the variations in loss with temperature may seem to increase, but the observed changes approach the measurement limit of this system, as shown Fig. 8. Second, while the corrected data are erroneous, it actually allows one to understand the limitations behind this correction method.The correction seemingly increases the error on the measurement, where the original traces show repeatable data and the corrected measurements show the opposite.The reason for this is vastly due to the temperature dependence of the coupling structures.As mentioned previously, the coupling antennas are also functions of temperature, and their effects are more profound as their coupling strength increases.In this experiment, antennas have been designed to excite all three of these modes, however, the coupling strengths (inferred by the magnitude of the resonant peaks in an S 21 measurement) are much stronger for the correction modes.The magnitude at resonance for TM 010 is approximately −30 dB while TM 210 and TM 310 have magnitudes at approximately −12 and −10 dB, respectively.This means that the influences from the coupling structure on the bandwidth measurements are approximately 0.1%, 6%, and 10%, all of which are greater than the measurement errors given in Table I.This is most likely responsible for the response given in Fig. 8 in the imaginary part.Hence, to use the correction method, the coupling strength into each mode must be minimized, although this is a challenging task, since reducing the strength of the nodal modes will inherently reduce the detectable peak of the measurement mode (TM 010 ) toward the noise floor of the VNA.Ultimately, this creates further difficulties in the temperature-dependent measurement of low loss materials.

IV. CONCLUSION
In conclusion, microwave cavity perturbation measurements can be corrected by monitoring specific resonant cavity modes.The TM 210 and TM 310 modes have been investigated to correct TM 010 measurements due to their insensitivity to the axial sample.The measurements show that the cavity modes are insensitive to the sample location on the axis.The cavity modes are linked geometrically through linear thermal expansion, though these gradients can vary across the modes and offsets may be apparent.The presented results demonstrate that the nodal correction method can effectively account for any external temperature related influences that affect the cavity volume during perturbation measurements.The analysis presented in Section II-C states that the temperature related offsets (κ 010 −κ m10 ) may be for through subtraction.Hence, the biggest limitation in this method is that the unperturbed and perturbed states must be conducted with the same cavity as to eliminate the (κ 010 + κ m10 ) term.For example, one cannot use two cavities to measure the unperturbed and perturbed states at the same time as to save time on the experiment.Even if the cavities are designed to the same dimensions, any manufacturing errors would result in differing κ values: (κ 010,u − κ m10,u ) for unperturbed and (κ 010, p −κ m10, p ) for perturbed.Thus, these terms do not cancel each other out.
While effective for dielectric constant, this correction method must be used with caution on strongly coupled resonators, since the temperature dependence of the coupling structures can dominate the corrected response, leading to erroneous results.With this in mind, it must be advised that any temperature ramping experiments must be approached with caution, and the effects of the probing system need to be accounted for before any concrete data can be presented.

Fig. 1 .
Fig. 1.Field distributions (from left to right) of TM 210 ,T M 310 ,TE 211 ,a n d TE 311 using COMSOL multiphysics (cavity dimensions of a = 46 mm and l = 40 mm).The top and bottom rows depict E-a n dH -field distributions, respectively.Notice that as m increases, the axial node (dark region) gets broader, whilst the field maxima (light regions) tend toward the perimeter owing to the distributions being described by Bessel functions.

Fig. 2 .
Fig. 2. Cut views of the cylindrical cavities used in this paper.Aluminum TM-probe antenna cavity (left) couples into the E-field of TM modes to provide dielectric measurements.Aluminum TE-loop antenna cavity (middle) couples into the H -field of TE modes to provide magnetic measurements.A copper side-loop coupled cavity couples into TM modes to provide dielectric measurements whilst also coupling to some TE nodal modes.

Fig. 3 .
Fig. 3. Photograph of the experimental setup with inset showing the cavities used in this paper (coin is a 50-pence piece).A Keysight Technologies VNA (PNA-L N232A) is used to measure the cavity resonance.A Memmert oven is used change the ambient temperature of the cavity.Temperature sensors are attached to the cavity and interfaced to a PC via a NI-cDAQ-9171.Temperature and S21 data are recorded using a program written using NI-LabVIEW.

Fig. 5 .
Fig. 5. Fractional frequency shifts in the laboratory for 4 h of the copper cavity (top) and the aluminum TE cavity (bottom).Traces have been separated by 1 × 10 −6 for visibility.

Fig. 6 .
Fig. 6.Temperature-dependent change in fractional frequency shifts of (top) aluminum TM cavity (middle) aluminum TE cavity, and (bottom) copper TM cavity.Plots have been separated for visibility.

Fig. 7 .
Fig. 7. Photograph of the rod samples used in this paper.The copper rod is used in the metal rod perturbation experiments while the PTFE rod is used in the sample temperature ramping experiments.

Fig. 8 .
Fig. 8. Calculated temperature-dependent effective complex permittivity of PTFE rod using both stepped and linear ramping profiles.The uncorrected temperature ramps are shown in black, which show varying temperaturedependent behavior owing to the profile used.TM 210 and TM 310 corrected ramps are shown in lighter shades.Regardless of ramp profile, the correction gives similar results for the real part, but variation in the imaginary.

TABLE II OBSERVED
THERMAL EXPANSION COEFFICIENTS to reported of 3.6 × 10 −3 with discrepancies again associated with the temperature dependence of the coupling antennas.