FIGURE 9.34 = Rubber thickness gauging in an open cavity resonator. 9.22 Thickness gauging using microwave resonant sensors. The resonant fre- quency of microwave cavities is directly proportional to the amount of dielectric material present in the cavity. This allows for simple gauging of sheet products such as rubber, paper, plastics, and fabrics. Consider a cavity that has been "opened" to allow a sheet of rubber to pass between the two halves of the metal cavity, as shown in Figure 9.34 (the height of the cavity is 2a +tmax, where tmax is the opening size through which the rubber sheet passes and is the maximum thickness of rubber expected). The connections to couple energy into the cavity and to measure the resonant frequency are not shown. The lowest resonant ft Cavity Cavity a a = 30 cm b = 50 cm c = 28 cm t max Rubber = 0.5 cm frequency is used for measurements. Assume that the resonant frequency acts as if the interior of the cavity is a mixture of rubber and air (by volume): a. Calculate the resonant frequency as a function of rubber thickness. The relative permittivity of rubber is 4.5 and that of air is 1 (dry air). b. Calculate the sensitivity of the sensor to thickness variations. c. If the frequency of the sensor can be reliably measured down to 250 Hz, what is the resolution of the sensor for thickness measurements at a given rubber thickness of 2.5 mm? d. What is the error in the resonant frequency when measuring a 2.5 mm thick rubber sheet if the air inside the cavity is at 100% relative humidity (the relative permittivity of air at 100% humidity is 1.00213)? Note: Although the cavity is open to allow passage of the rubber sheet, as long as the opening is small with respect to the wavelength, the cavity will resonate as if it were a closed cavity.