## 2 Materials and Methods
### Measurement of Complex Impedances  
In this research, a measurement hardware based on a simple vector-network analyser circuit was used. This type of circuit is popular among amateur radio enthusiasts and has good performance given its low costs and complexity. For this reason, this hardware type is not only used for measurement tasks in civil engineering [19], but also in other research applications, such as cross-section and velocity measurement of hot wire as well as roll gap and forward slip measurement in rolling mills [20; 21; 22; 23; 24; 25]. This simplified vector-network analyser circuit features a single detector setup, which makes it suitable for one-port networks. The impedance of a device under test (DUT) is measured by determining the complex scattering parameter \(\underline{\mathbb{S}}_{11}\) (reflection coefficient). Figure 8 shows the main structure of the measurement hardware and the connections between the sub-circuits.  
The measurement hardware uses a microcontroller to control the generation of a sinusoidal radio frequency (RF) and local oscillator (LO) signal by two synchronised DDS circuits. The RF signal is used to excite the Wheatstone bridge circuit, which is connected to the DUT. The LO signal sets which component of the I/Q demodulation is the present output of the detector circuit. A fast 24-bit ADC is used to quantise and sample the I/Q components. By using a multiplexer, it is possible to switch between three onboard calibration probes (open, short, match) and the DUT. The microcontroller is directly connected to a PC running MATLAB(r) for data acquisition and visualisation. Table 1 shows an overview of the main specifications and features of this measurement hardware.  
Figure 8: Scheme of the functional blocks of the measurement hardware.  
Figure 7: Flat coil sensor embedded in the motor mould.  
The impedance measurement of a DUT is achieved by using the deflection method in a Wheatstone bridge. It is composed of three identical ohmic resistors, \(\underline{Z}_{0}=50\)\(\Omega\), and one unknown impedance \(\underline{Z}\) of the DUT, as shown in Figure 9.  
The bridge voltage \(\underline{U}_{G}\) can be calculated by Equation (2). Due to the high frequencies of the RF signal, the transformer can be neglected in the following calculation:  
\[\underline{U}_{G}=\frac{1}{2}\cdot U_{RF}\cdot\frac{\underline{Z}-\underline{Z }_{0}}{\underline{Z}+\underline{Z}_{0}} \tag{2}\]  
The resistance ratio in this equation corresponds to the complex scattering parameter \(\underline{S}_{11}\). Because of this relation, this type of Wheatstone bridge is also called a return loss bridge or reflection coefficient bridge. Accordingly, the scattering parameter \(\underline{S}_{11}\) can be derived from the bridge voltage \(\underline{U}_{G}\), as shown in Equation (3):  
\[\underline{S}_{11}=\frac{\underline{Z}-\underline{Z}_{0}}{\underline{Z}+ \underline{Z}_{0}}\quad\rightarrow\quad\underline{U}_{G}=\frac{1}{2}\cdot U _{RF}\cdot\underline{S}_{11}\quad\rightarrow\quad\underline{S}_{11}=2\cdot \frac{\underline{U}_{G}}{\underline{U}_{RF}}. \tag{3}\]  
By using the scattering parameter \(\underline{S}_{11}\), the impedance of the DUT can be calculated using Equation (4):  
\[\underline{Z}=\underline{Z}_{0}\cdot\frac{\underline{S}_{11}+1}{1-\underline{ S}_{11}} \tag{4}\]  
The measurement of the scattering parameter \(\underline{S}_{11}\) is affected by the parasitic influences of the real components, such as electrical parts, PCB layout, cables and connectors. The one-port error can be used to determine the influences and correct them for each measuring frequency.