## 3 Magnetic Field Intensity Measurement
### Slow Variation Compensation  
Determining the EMF distribution required that we not change the WPT emitting coil voltage or the measurement process parameters during the scanning process. Figure 15: Signal diagram for determining the EMF sense. Given the large number of measurement points (>100 k pts. ), the duration of the scanning process could be estimated at several hours. For a short lapse of time (for example, during the switch to measure the field components in the three directions), we could see that the process parameters were constant. However, during the full scanning process we observed their modification, mainly due to temperature variations or/and the supply voltage. These changes were considered slow variations, and during the scanning process these deviations had to be compensated for by adjustment. It is known that slow temperature fluctuations can affect the operation of amplifiers, rms-to-dc converters, and digital-to-analog converters. For the amplifiers, due to their AC mode of operation, the variations were due to the thermal drift of the resistors that determine the amplification. For the resistors used herein, with a metal film, the temperature coefficient was +50 ppm/\({}^{\circ}\)C. However, the amplification did not change, as it was determined by the ratio of implied resistance (with a similar temperature coefficient). The data sheet of the rms-to-dc converter specified a temperature variation within the conversion accuracy of \(+/-1\) mV \(+/-0.005\%\) of the converted value for each Celsius degree, while the AD converter data sheet showed the gain variation pattern presented in Figure 16.  
Consequently, the converters could introduce (for rated voltage values between 0 and 2 V) global variations of less than 0.03% at a full-scale rate (FSR). These errors fell into the range of slow variations that were compensated for by applying the algorithm. The solution adopted in this paper was to make both linearity correction and slow variation compensations after the scanning process. The field sense detection involved in two steps, with the switch I in Figure 15 open and then closed. As the duration of these steps was very short, and the measured values were stored, the block diagram could be functionally redrawn, as in Figure 17 (including the analog-to-digital conversion). In the diagram, \(U_{G}\) is the voltage at the terminals of the emitting coil of the WPT, \(N_{G}\) is the number associated with the measurement of \(U_{G}\), \(U_{I}\) is the induced EMV in the sensor coil, and \(N\) is the number associated with the \(U_{I}\) measurement. Figure 16: Gain variation versus temperature for ADC [17]. Figure 17: Block diagram for signal processing. Both \(N\) and \(N_{G}\) were corrected for linearity using the algorithm described above, resulting in the new \(N_{CG}\) and \(N_{C}\) numbers. Thus, they were proportional to \(U_{G}\) and \(U_{I}\). Hence,  
\[N_{CG}=g_{G}U_{G} \tag{8}\]  
\[N_{C}=gU_{I} \tag{9}\]  
where \(g_{G}\) and \(g\) are the proportionality constants. Due to the physical phenomena that brought about slow variations, the numbers \(N_{CG}\) and \(N_{C}\) were modified in the scanning process, i.e., they could be expressed as:  
\[N_{CG}=g_{G}U_{G}f_{G} \tag{10}\]  
\[N_{C}=gU_{I}f \tag{11}\]  
where \(f_{G}\) and \(f\) are slow variation factors that were affected by the phenomena. If the conversion circuits from the block diagram in Figure 17 were from the same series and mounted in the same enclosure (same operating temperatures), then it could be approximated that \(f_{G}=f\). The induced EMV \(U_{I}\) depended on the sensor coil placement in the EMF and, proportionally, the current fed into the WPT emitting coil. Hence,  
\[U_{I}=pI_{B} \tag{12}\]  
where \(I_{B}\) is the current and \(p\) is the proportionality factor derived from the position of the sensor coil.