# Planar Microwave Resonant Sensors: A Review and Recent Developments
## 4 Prototype Examples
### Differential-Mode Sensors
#### 4.4.3 Volume Fraction Determination in Liquid Solutions  
Obviously, with the sensors reported in Sections 4.4.1 and 4.4.2, determination of the volume fraction in liquid solutions is possible. As this measurement involves only one input variable (the volume fraction), a single output variable suffices. In the sensor of Section 4.4.1, devoted to the measurement of electrolyte concentration in water, the output variable was the maximum of the cross mode transmission coefficient. The frequency position of such maximum slightly varies, but it is possible to obtain the value at a fixed frequency, particularly the resonance frequency of the resonant element. In this case, a single-frequency measurement suffices, and, in this case, the cross-mode transmission coefficient can be inferred by means of a reflective-mode differential measurement, involving a simple two-port measurement, as reported in [69]. The sensor device consists of two pairs of grounded OSRR-loaded lines connected to the coupled ports of a rat race coupler by means of two identical transmission line sections. The input signal is injected to the \(\Delta\)-port, whereas the output port is the \(\Sigma\)-port (see the schematic in Figure 32). Figure 31: Real (**a**) and imaginary (**b**) parts of the complex dielectric constant for the different solutions of isopropanol in DI water. From [64], reprinted with permission. Figure 30: Measured cross-mode transmission coefficients for the different solutions of isopropanol in DI water (**a**) and representation of the output variables as a function of the isopropanol concentration (**b**). From [64], reprinted with permission. The transmission coefficient at \(f_{0}\), the operating frequency of the rat-race coupler, in the structure of Figure 32 can be expressed as  
\[S_{12}\ =\ -\frac{\dot{j}}{2}e^{-j2\dot{\phi}}\ (\rho_{REF}-\rho_{MUT}) \tag{6}\]  
namely, \(S_{12}\) is proportional to the differential reflection coefficient. Therefore, by asymmetrically loading the sensing loads (OSRRs), it follows that differential sensing can be achieved. In Figure 33, the fabricated device, where microfluidic channels have been introduced in order to perform measurements of liquids, is shown. The objective of this sensor was the measurement of volume fraction of isopropanol in DI water. The measured transmission coefficients after subsequently injecting different solutions of isopropanol-water in the MUT channel, with DI water in the REF channel, are depicted in Figure 34. It can be seen that the transmission coefficient at \(f_{0}\) experiences a strong variation as the concentration of isopropanol increases. Moreover, the sensor is able to resolve amounts of isopropanol content as small as 2.5%. Figure 35 depicts the dependence of the transmission coefficient at \(f_{0}\) with the isopropanol concentration, including the calibration curve (with a correlation coefficient of \(R^{2}=0.9987\)), i.e.,  
\[Fv(\%)\ =\ 184.942e^{(\frac{S_{21}(dB)}{5.688})}+81.618e^{(\frac{S_{21}(dB)}{ 48.449})}-34.793 \tag{7}\]  
From this curve, the concentration of volume fraction of isopropanol can be inferred differentially, by means of a single two-port measurement. The main differential aspect of the sensor of Figure 33 is the fact that a simple two-port measurement suffices to infer the differential output variable. Other differential sensor involving two-port measurements have been reported [12; 68], but, in such sensors, neither resonant element is used, nor is operation in reflection mode considered. Nevertheless, note that in the recently presented sensor of [68], based on the phase difference generated in a pair of meandered lines, high sensitivity (11.52 dB/%) and resolution (1% of volume fraction) is obtained for the measurement of isopropanol content in DI water. This high sensitivity and resolution is in part due to the specific arrangement considered in [68], where a pair of rat race couplers is used (see further details in [68]).