# Isotropic IoT-Based Magnetic Flux Density Meter Implementation for ELF Field Measurements +
## 2 Magnetic Flux Density Meter Operating Principles  
The induced electromotive force in any closed circuit is equal to the negative time rate of change in the magnetic flux contained in the circuit, according to Faraday's Law of Electromagnetic Induction. Consequently, when magnetic flux flows through a coil, a voltage is generated across it, the magnitude of which is dependent on the field's intensity rate and the enclosed surface area. In mathematical terms,  
\[E=-\frac{d\Phi}{dt}=-A\ N\frac{dB}{dt} \tag{4}\]  
where \(\Phi\) is the magnetic flux passing through the coil, \(A\) is the area enclosed by each turn of the coil, \(N\) is the total number of turns of the coil, \(B\) is the magnetic flux density of the field (magnetic field), and \(t\) is the time. Essentially, the time derivative of the magnetic flux density determines the rate of change in the magnetic flux density (a derivative of the magnetic flux concerning time). It is considered that in space there is a harmonic (sinusoidal) magnetic field of the following form:  
\[B(t)=B_{o}\cos(2\pi ft) \tag{5}\]  
where \(B(t)\) is the instantaneous value of the magnetic flux density (function of time), \(B_{o}\) is the maximum value of the oscillating field, and \(f\) is the frequency of oscillation. From (4) and (5), we conclude that if a coil is placed into the harmonic field, a harmonic voltage \(E(t)\) will be induced at its terminals, which will be equal to the following:  
\[E(t)=2\pi fANB_{0}\sin(2\pi ft) \tag{6}\]  
We observe that the induced voltage is proportional to the geometric characteristics of the coil (\(A\), \(N\)), the frequency f, and the field strength \(B_{o}\). The harmonic term \(\sin(2\pi ft)\) indicates that voltage is a harmonic function of time and has a phase difference of \(90^{\circ}\) concerning the field. From (6), we observe that if the induced voltage \(E\) can be measured, the \(B\) field can be calculated, as long as the exact frequency \(f\) is known. In practice, when measuring the magnetic field of an arbitrary (unknown) source, its frequency is undetermined. The unknown frequency problem is resolved by multiplying both sides of Equation (7) by the absolute value of the \(G/G2\pi f\) term, where \(G\) is a constant for any specific frequency \(f\), corresponding to amplifier voltage gain. Then, we will have:  
\[\left|\frac{G}{2\pi f}\right|E(t)=G\ A\ N\ B_{0}\ \sin(2\pi ft) \tag{7}\]  
By setting:  
\[V(t)=\left|G/2\pi f\right|\ E(t)\]  
hence,  
\[V(t)=G\ ANB_{0}\sin(2\pi ft)=G\ ANB(t) \tag{8}\]  
where \(V(t)\) is the instantaneous value of the harmonic voltage induced in the coil as a product by the absolute value of the \(G/2\pi f\) term. Referring to RMS terms (Root Mean Squared values) the _RMS_ voltage (\(V_{rms}\)) is proportional to the RMS magnetic field density (\(B_{rms}\)) and independent of the frequency [31; 32]:  
\[V_{rms}=G\ A\ N\ B_{rms} \tag{9}\]  
Therefore, it is easy to calculate the magnetic flux density (\(B_{rms}\)) by measuring the voltage value (\(V_{rms}\)) with an analogue electronic device, i.e., a magnetic flux density meter. Figure 3 illustrates the block diagram of the proposed low-cost IoT magnetic flux density meter providing cloud-based services, as a comparative advantage among similar high-cost measurement equipment [13; 14]. The proposed magnetic flux density meter's electronic circuit is straightforward to implement, and the components have a low tolerance for measurement error minimization. A detector coil with well-defined geometric characteristics as a field sensor, an amplifier with voltage gain equal to \(G/2\pi f\), an \(RMS\) detector, and a voltage display are required for a real circuit that makes use of Equation (9). In the proposed implementation, the Arduino UNO Wi-Fi Rev.2 board [33] is used instead of common voltmeters at analog outputs of the device, providing network connectivity for data transferring to the Blynk cloud [34], as a complete IoT EMF solution.