# Detecting Grounding Grid Orientation: Transient Electromagnetic Approach
## 3 Transient Electromagnetic Method  
The transient electromagnetic method (TEM) is widely used for geological exploration of underground minerals [27; 28; 29]. It is an effective method for determining the electrical resistivity of underground layers [30], as well as the fault diagnosis of the grounding grid [16; 20]. Illustrated in Figure 4 is a typical TEM system that includes a transmitter-receiver pair to transmit and receive magnetic fields. The primary magnetic field is produced by injecting the pulse current of the ramp wave in the transmitter coil. The time varying primary magnetic field induces eddy currents in the grounding conductors. The secondary magnetic field due to the induced eddy currents is recorded above the surface by the receiver coil. This coil is located in the center of the transmitter coil. Inversion calculation of the induced electromotive force (emf) in the secondary coil is obtained utilizing equivalent resistivity imaging equations based on the smoke ring concept [31]. The location of grounding conductors is determined from the equivalent resistivity and magnetic field distribution. High equivalent resistivity and low magnetic field indicate the presence of the grounding conductor and vice versa. The vertical component of the secondary magnetic field in the center of transmitter coil is expressed as [32]:  
\[H_{z}=\frac{I_{tc}}{2r_{tc}}\left[\frac{3}{\sqrt{\pi}u}e^{-u}+(1-\frac{3}{2u^ {2}})erf(u)\right] \tag{1}\]  
where \(r_{tc}\) is the radius of the transmitter coil, \(I_{tc}\) is the magnitude of transmitter current, \(u\) is the transient magnetic field parameter, and \(erf(u)\) is the error function expressed as:  
\[erf(u)=\frac{2}{\sqrt{\pi}}\int_{0}^{u}e^{-u^{2}}dt \tag{2}\]  
The induced electromotive force \(E(t)\) is obtained as [32]:  
\[E(t)=\frac{I_{tc}}{\sigma r_{tc}^{3}}\left[3 erf(u)-\frac{2}{\sqrt{\pi}}u(3+2u^ {2})e^{-u^{2}}\right] \tag{3}\]  
Figure 4: A typical transient electromagnetic method (TEM) system probing the underground grid. The primary magnetic field due to the transmitter coil interacts with the grid buried in the soil and induces eddy currents. Induced eddy currents produce a secondary magnetic field that travels upward to the Earth’s surface and collected by the receiver coil placed in the center of the transmitter coil. where \(\sigma\) is the conductivity of the underground medium. The transient magnetic field parameter \(u\) is expressed as:  
\[u=\sqrt{\frac{\mu\sigma r_{tc}^{2}}{4t}} \tag{4}\]  
The conductivity \(\sigma\) is obtained from (4) as:  
\[\sigma=\frac{4u^{2}t}{\mu r_{tc}^{2}} \tag{5}\]  
Inserting \(\sigma\) in (3), \(E(t)\) becomes:  
\[E(t)=\frac{I_{tc}\mu}{4u^{2}tr_{tc}}\Bigg{[}3 erf(u)-\frac{2}{\sqrt{\pi}}u(3+2u^ {2})e^{-u^{2}}\Bigg{]} \tag{6}\]  
A function \(F(u)\) is setup using (6):  
\[3 erf(u)-\frac{2}{\sqrt{\pi}}u(3+2u^{2})e^{-u^{2}}-\frac{4u^{2}tr_{tc}tE(t)}{ \mu I_{tc}}=0 \tag{7}\]  
\[F(u)=3 erf(u)-\frac{2}{\sqrt{\pi}}u(3+2u^{2})e^{-u^{2}}-\frac{4u^{2}tr_{tc}tE(t )}{\mu I_{tc}} \tag{8}\]  
As resistivity \(\rho\) is reciprocal to conductivity \(\sigma\), so the apparent resistivity in terms of \(u\) is given by:  
\[\rho(t)=\frac{\mu r_{tc}^{2}}{4u^{2}t} \tag{9}\]  
Employing the iterative method in (8), the transient magnetic field parameter \(u\) is determined. The vertical depth \(d(m)\) and downward velocity \(v(m/s)\) of the induced eddy currents can be calculated as [31]:  
\[erf(u)=\frac{2}{\sqrt{\pi}}\int_{0}^{u}e^{-u^{2}}dt \tag{2}\]0  
\[erf(u)=\frac{2}{\sqrt{\pi}}\int_{0}^{u}e^{-u^{2}}dt \tag{2}\]1  
where \(t\) is the sampling time and \(\mu\) is the permeability of the medium. Downward velocity \(v(m/s)\) between two consecutive time samples is expressed as:  
\[erf(u)=\frac{2}{\sqrt{\pi}}\int_{0}^{u}e^{-u^{2}}dt \tag{2}\]2where \(t_{i}\) and \(t_{i+1}\) are the two consecutive time samples and \(d_{i}\) and \(d_{i+1}\) are the corresponding vertical depths.