## 3 Magnetic Field Intensity Measurement
### The Geometrical Correction  
The numbers resulting from the induced EMV measurement in the sensor coils represented the values of the EMF intensity vector as projections in the directions of the axes of the sensor coils. Because these directions had small deviations from the directions in orthogonal Cartesian space, it was necessary to correct the measured values. Given that the three measured numbers (for the induced EMV in the sensor coils) represented the intensity of the EMF (\(H_{\text{BX}}\), \(H_{\text{BY}}\), \(H_{\text{BZ}}\)), it was necessary to determine the components of these vectors in the Cartesian orthogonal space. For the case of the Z component, the \(H_{\text{BZ}}\) vector together with the deviations from the Z axis, \(\epsilon_{\text{zx}}\) and \(\epsilon_{\text{zy}}\), are shown in Figure 18.  
The measured value of \(H_{\text{BZ}}\) represented the modulus of the vector:  
\[\overset{\longrightarrow}{H_{\text{BZ}}}=A_{Z}\overset{\rightarrow}{i}+B_{Z} \overset{\rightarrow}{j}+C_{Z}\overset{\rightarrow}{k} \tag{16}\]  
with \(i\), \(j\), and \(k\) being unit vectors in the Cartesian system. The following relationships could be written:  
\[H_{\text{BZ}}=\sqrt{A_{\text{Z}}^{2}+B_{\text{Z}}^{2}+C_{Z}^{2}} \tag{17}\]  
\[A_{\text{Z}}=C_{Z}tg\epsilon_{ZX} \tag{18}\]  
\[B_{\text{Z}}=C_{Z}tg\epsilon_{ZY} \tag{19}\]  
\[H_{\text{BZ}}=C_{Z}\sqrt{1+tg^{2}\epsilon_{ZX}+tg^{2}\epsilon_{ZY}} \tag{20}\]  
The coordinates of the \(H_{\text{BZ}}\) vector peaks were thus obtained depending on the deviation angles:  
\[C_{\text{Z}}=\frac{H_{\text{BZ}}}{\sqrt{1+tg^{2}\epsilon_{ZX}+tg^{2}\epsilon_ {ZY}}} \tag{21}\]  
Figure 18: Preview of the EMF vector Z component. \[A_{Z}=\frac{H_{BZ}tg\varepsilon_{ZX}}{\sqrt{1+tg^{2}\varepsilon_{ZX}+tg^{2} \varepsilon_{ZY}}} \tag{22}\]  
\[B_{Z}=\frac{H_{BZ}tg\varepsilon_{ZY}}{\sqrt{1+tg^{2}\varepsilon_{ZX}+tg^{2} \varepsilon_{ZY}}} \tag{23}\]  
Similarly, for the vectors \(H_{BX}\) and \(H_{BY}\), the coordinates in Cartesian space were (the relations were valid if the angles were negative): \(A_{X}\), \(B_{X}\), \(C_{X}\), \(A_{Y}\), \(B_{Y}\), and \(C_{Y}\). Because the vectors \(H_{BX}\), \(H_{BY}\), and \(H_{BZ}\) represented the projections of the EMF intensity vector \(H\) on the sensor coils axes, the \(H\) vector peak was at the intersection of the planes perpendicular to the vectors \(H_{BX}\), \(H_{BY}\), and \(H_{BZ}\) and containing their peaks. A plane perpendicular to any vector, represented by Cartesian coordinates, can be written as follows:  
\[\overset{\longrightarrow}{H_{\text{BZ}}}=A_{Z}\overset{\rightarrow}{i}+B_{Z} \overset{\rightarrow}{j}+C_{Z}\overset{\rightarrow}{k} \tag{16}\]0  
and a plane containing the points (\(x_{0}\), \(y_{0}\), \(z_{0}\)) can be represented by  
\[\overset{\longrightarrow}{H_{\text{BZ}}}=A_{Z}\overset{\rightarrow}{i}+B_{Z} \overset{\rightarrow}{j}+C_{Z}\overset{\rightarrow}{k} \tag{16}\]1  
By implementing mathematical logistics and applying this equation for the vectors \(H_{BX}\), \(H_{BY}\), and \(H_{BZ}\), the following system of linear equations was obtained (the free terms are the module square of each vector):  
\[\left\{ \begin{aligned} & A_{X}x+B_{X}y+C_{X}z=D_{X}\\ & A_{Y}x+B_{Y}y+C_{Y}z=D_{Y}\\ & A_{Z}x+B_{Z}y+C_{Z}z=D_{Z}\end{aligned} \right. \tag{26}\]  
The solution of this linear equation system represented the coordinates in the Cartesian space of the EMF intensity vector \(H\) at the placement point of the sensor:  
\[H_{X}=\frac{\Delta_{X}}{\Delta}\;H_{Y}=\frac{\Delta_{Y}}{\Delta}\;H_{Z}=\frac {\Delta_{Z}}{\Delta} \tag{27}\]  
where \(\Delta\), \(\Delta_{X}\), \(\Delta_{Y}\), and \(\Delta_{Z}\) are the determinants of the system (Cramer rule). The EMF of intensity \(H\) is, in fact, the field that produces the electromotive voltages in the sensor coils. This is different from the \(H_{B}\) intensity field, which results from measurements when the deviations of the coil axes are not considered.