Implementation of Cavity Perturbation Method for Determining Relative Permittivity of Non Magnetic Materials

Techniques for the cavity measurement of the electrical characteristics of the materials are well established using the approximate method due to its simplicity in material insertion and fabrication. However, the exact method which requires more comprehensive mathematical analysis as well, owing to the practical difficulties for the material insertion, is not mostly used while performing the measurements as compared to approximate method in most of the works. In this work the comparative analysis of both the approximate as well as Exact method is performed and accuracy of the Exact method is established by performing the measurements of non-magnetic material Teflon within the cavity.

form are enlisted in [3]. On careful modification in Q equation for extremely low-loss dielectric samples [6] showed an increased accuracy of resonant perturbation method. Cavity resonators are often modified by making small changes in their shape or by introducing small piece of dielectric or metallic materials [7][8][9][10][11][12]. For the measurement of complex permittivity of dielectric materials with arbitrary size and shape, [11]

DESIGN PROBLEM
The cylindrical cavity is a section of cylindrical waveguide of length 'd' and radius 'a' with the short circuiting plates at the ends as shown in Fig. 1. Both magnetic and electric fields exist inside the cavity and total energy corresponding to these fields is stored in the cavity. The dissipation of this energy is either in the cavity metallic walls or in the dielectric cavity filling as well [13]. Thus the dielectric filling will affect the resonant frequency as well as the quality factor Q of the cavity.
When the cavity is tuned to frequency of unknown source, it absorbs the maximum power from the input A step by step design procedure is described in [14].
Owing to the field distribution of current TM 0n0 cylindrical cavities are widely used and the TM 010 mode is most widely used that has the lowest cutoff frequency, also called as the dominant mode. The resonant frequency of this mode is independent upon the resonator length.
Here q is the number of half cycle variations in the direction of propagation, a is the cavity radius and d is the cavity length where x np is the zeroth of Bessel Function.

SIMULATIONS AND ANALYSIS ON HFSS
TM 010 mode excitation was chosen for the design purpose as the field pattern suggests that the maximum change in resonant frequency will take place after the sample insertion, thus enabling accurate measurement of the electrical characteristics of the inserted sample [17][18].
The mode can be excited in the cavity using a metallic loop. Simulations on HFSS are performed to confirm the proper mode excitation at the desired frequency.
The cylindrical cavity excited for TM 010 mode through the loop coupling is shown in Fig. 2  The current consists of the conduction current along the conducting surface and displacement current between the upper and the lower walls as shown in Fig. 4(b). With all these simulations (Figs. 2-4), it is confirmed that TM 010 mode is properly excited in the empty cavity and resonant nearly at the desired frequency.

FIG. 3. S 11 LOOP COUPLED CYLINDRICAL CAVITY
The loop-coupled cavity is perturbed with Teflon which is placed at the centre of the bottom conducting plate [19][20]. The electric field is maximum at this location to ensure the maximum change in the resonant frequency.
The sample size inserted is comparable to the cavity size to ensure the accuracy of the method as shown in Fig. 5  The strength of the electric and magnetic field has increased due to the sample introduction. The electric field is still maximum at the centre and minimum at the boundary as per the boundary conditions as shown in Fig. 7(a).
The magnetic field has also increased at the boundary after the sample positioning and is minimum at the centre as shown in Fig. 7(b).
The surface current density will converge on the top surface and diverge at the bottom surface owing the direction of the magnetic as shown in Fig. 7(c).

PHYSICAL IMPLEMENTATION AND ACTUAL MEASUREMENTS
The cavity obtained has the following dimensions a=107.75 mm, which using Equation (1) gives f 010 =1.067285 GHz. The HFSS Model which is used for actual cavity resonator is shown in Fig. 8. The test bed and the Measurement Setup in the lab are shown in Fig. 9. The observations are recorded with an Agilent Network Analyzer.
The cylindrical cavity resonator of the desired dimension is designed and excited through the M 010 . The sample under test is placed exactly at the centre of the cavity.
The response of the cavity is measured through the network analyzer and resonant frequency observed is 1.0629 GHz as shown in Fig. 10(a). After perturbation with the Teflon, the new resonant frequency is measured on the network analyzer and is recorded to be 1.03732 GHz as shown in Fig. 10(b).
The actual and the measured resonant frequency are given as ω = 1.03732 GHz while ω 0 = 1.0629 GHz. And  Since the excitation mode is TM 010 , the resonant frequency which is independent upon the cavity length, the cavity is cut into half so the sample can be properly placed in the centre along its length as shown in Fig. 13. The resonant frequency is independent upon the cavity length and measured to be 1.0621GHz as shown in Fig. 14.
The response of the cavity after the perturbation by placing the sample along the entire length of the cavity is ω = 1.055GHz. Here ω = 1.055010246 GHz and ω 0 =

CONCLUSION
The cavity perturbation method was employed to determine the relative permittivity of non-magnetic material (Teflon). Simulations were performed to confirm the proper mode excitation and later physical cavity was designed to measure the actual response and the actual permittivity. However the percentage error found using the method was beyond the acceptable limits, so Exact Method was employed in the actual measurements and recalculation were done for the permittivity. There was a substantial improvement in the permittivity. Thus this method described can be established as accurate and more precise for the unknown permittivity measurements of the materials.