A Differential Microwave Sensor Loaded With Magnetic-LC Resonators for Simultaneous Thickness and Permittivity Measurement of Material Under Test by Odd- and Even-Mode

A differential microwave sensor for simultaneously retrieving the thickness and dielectric constant of material under test (MUT) based on a magnetic- LC (MLC) resonator is proposed in this manuscript. The procedure for designing the proposed microwave sensor is illustrated as follows: 1) designing a splitter/combiner; 2) separately etching an MLC resonator under every microstrip branch of the splitter/combiner; and 3) the symmetry axis of the electric wall of each resonator is moved by an offset distance away from the center of every microstrip branch, so the odd- and even-mode will be produced. One resonator is regarded as a reference, the other is used as a test. From the numerical simulation, the electric fields of odd- and even-mode are mainly concentrated at the center arm and two sides of the resonator, respectively. The area of the resonator is regarded as a sensing region. The equivalent circuit model is adopted to illustrate the operating principle of the proposed sensor. The mathematical model of the relative frequency shifts of two resonant modes with respect to thickness and permittivity of loaded MUT can be established by simulated data. In measurement, the unloaded odd/even-mode resonant frequency and quality factor are 2.5875/5.625 GHz and 251/77.67, respectively. The maximum errors for retrieving permittivity and thickness are about 4.02% and 8.4%, respectively. Besides, the sensitivities for odd- and even-mode are separately calculated as 2.21% and 1.38%, if permittivity and thickness of MUT are set as 6 and 0.1 mm, respectively. If the lowest sensitivity of even-mode is set as 0.31%, then the minimum size of MUT is about $3.4\times 10\times0.1$ mm. In comparison to traditional microstrip sensors, the proposed sensor can simultaneously retrieve thickness and permittivity of MUT by odd- and even-mode, and it can reduce interferences of the external environment by virtue of differential structure. The proposed microwave sensor has a compact size of about ${{0}.{43}}{\lambda }_{{{0}}}{{ \times {0}.{436}}}{\lambda }_{{{0}}} {(}{\lambda }_{{{0}}}$ is the wavelength in free space at 2.5875 GHz), and it is a good candidate in the field of characterizing solid materials with some superiorities.