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SJSU EE 172 - cavity_colin

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Colin Constant EE172 Spring 2011 Final Project Paper Ray Kwok Cavity Resonator for Complex Permittivity Measurements Introduction A cylindrical aluminum cavity resonator was used to measure the complex permittivity of various dielectric materials by means of perturbation theory. The resonant frequencies of the empty cavity can be compared to the resonant frequencies of a cavity containing a dielectric material, to find the complex permittivity of the material. The change in the frequency can be used to find the real part of the material’s complex permittivity, and the change in the quality factor (Q), of the frequency’s peak, can be used to find the imaginary part. This method of complex permittivity evaluation is a use of the perturbation theory in conjunction with the complex eigenfrequency of resonance. Theory An electrical signal having voltage and current can be represented in terms of the wave propagation of its electric field (E) and magnetic field (H). When an electric field enters a cavity of a conducting material, the E-field must drop to zero at the walls of the cavity in accordance with Maxwell’s equations. The energy carried by the wave must be conserved, forcing the cavity to store this energy in standing waves which oscillate at nodes on the boundaries of the cavity, where the electric field is zero. In an unbounded medium, the electromagnetic (EM) wave will oscillate as transverse electric and transverse magnetic (TEM), since there is no boundary to delineate which wave is not propagating transversely. In the bounded cavity, however, the standing waves will propagate such that only the E-field or H-field are transverse to the direction of propagation. This limits the waves in the cavity to two modes of propagation: transverse electric (TE), and transverse magnetic (TM). The propagation patterns for some of the lowest resonant frequencies, in these modes, are shown below. Figure 1: propagation of EM waves in a cylinder [Pozar]In a cylindrical cavity the E-field and H-field are best realized by their cylindrical coordinate components; r, ϕ, and z. With Maxwell’s equations and some algebra, all non-z components of the E-filed and H-field can be expressed in terms of their z component as shown. Figure 2: cylindrical component representation by z components [Kwok] Evaluating Ez using Poisson’s equation, and relating it to arbitrary functions of r and ϕ, it can be shown that Ez reduces to the Bessel equation. Figure 3: Ez in terms of the Bessel function [Kwok] With Ez found, all cylindrical components in Figure 2 can be solved. Cavity perturbation theory says that the difference in the resonant complex eigenfrequency of a cavity, due to a perturbation, can be expressed by the difference in the complex permittivity of the transmission mediums, the original and the one causing the perturbation. The equation below shows the exact relationship. Figure 4: underlying equation of cavity perturbation theory [Meng]The value f~ is the complex eigenfrequency, which has the frequency as its real part and the inverse of its quality factor as its imaginary part. Using a dielectric sample to perturb a dielectric propagation medium, no change will be seen in the permeability, and the term goes to zero. If a small enough region of the cavity is perturbed, then it can be assumed that E and H have similar magnitudes and the above equation can be simplified. Figure 5: simplified equation of cavity perturbation theory [Meng] If the cavity is filled with air, or a material of relative permittivity equal to one, then the complex permittivity of the perturbing material can be expanded, and the above equation can be written as shown below. Figure 6: expansion of previous equation showing complex permittivity [Meng] Relating the real and imaginary parts of the equation above, to the real and imaginary parts of the complex eigenfrequency, the equations below can be solved for to find the real and imaginary parts of the perturbing material’s complex permittivity.Figure 7: equations to find complex permittivity Subscripts of zero, in all equations above, refer to the unperturbed frequencies and quality factors. Those values without subscripts refer to frequencies and quality factors due to perturbations. Eint refers to the E-field inside the sample perturbing the cavity. By recording the frequency and quality factors with and without perturbing materials, the values recorded can then be used in the above set of equations to find the complex permittivity. The E-field values can be found for a cylindrical cavity, by using the identities given earlier for cylindrical components of an E-field. The direction of the E-field, giving E-field components present, can be seen by analyzing the propagation patterns in Figure 1. Quality factors can be calculated in a number of ways. The simplest and most common is the full-width half-max method. In this method, the frequencies at -3dB to the left and right of the peak frequency are recorded and their difference taken. Dividing the peak frequency, gives the quality factor at that frequency. A ratio of the energy stored, to the energy dissipated. A resonant cavity given two ports acts as a symmetrical two port network. This simplifies positioning of ports, when measuring frequencies. Procedure An aluminum cylinder with radius 0.9025” and length 2.25” was used as the resonant cavity. Mounting holes were drilled to place electrical couplers at the top and bottom, and 1800 from each other on the side walls.Figure 8: cavity showing position of mounting holes The electrical couplers were placed in three configurations to read resonant frequencies below 6GHz using a network analyzer measuring S21. The first configuration placed the horizontal couplers 1800 apart from each other, on the side walls of the cavity, and measured TE111 and TM011 modes. Mode TM011 was not used to calculate complex permittivity. The second configuration placed the couplers vertically at the top and bottom of the cylinder. In this configuration TM010 and TM011 were seen. Again TM011 was not used. In the final configuration the couplers were placed with one coupler at the top of the


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