## 2 Measuring Method and Measuring System  
### Method  
For a quasi-static problem, the voltage \(U(t)\) of the conductor is linear to the electric field \(E(t)\) caused by it, which can be expressed as:  
\[U(t)=kE(t). \tag{1}\]  
Normally, the voltage waveform of a transient process includes the power-frequency component and the transient component. Since the amplitude of the power-frequency voltage is known for a power grid, by comparing the amplitudes of the power-frequency voltage and the measured power-frequency electric field, the coefficient \(k\) in (1) can be given. In an actual AC power system, the transmission line contains three phase conductors, which unavoidably leads to the coupling effect for the non-contact measuring technique as illustrated in Figure 1. Beneath each phase conductor, an electric-field sensor is installed denoted as Sensor A, Sensor B, and Sensor C for the corresponding phase. Sensors are arranged at the height level with enough air insulation clearance against corresponding conductors. As conductors are energized, the potentials of three-phase conductors produce electric fields at sensor's locations denoted by \(P_{\text{SA}}\), \(P_{\text{SB}}\), and \(P_{\text{SC}}\) in Figure 1. Taking \(P_{\text{SA}}\) as an example, the electric field at \(P_{\text{SA}}\) is a compound electric field contributed by \(\mathbf{E_{\text{AA}}}\), \(\mathbf{E_{\text{BA}}}\), and \(\mathbf{E_{\text{CA}}}\) which can be decomposed into the components in the x-axis (\(E_{\text{AA-x}}\), \(E_{\text{bA-x}}\), \(E_{\text{CA-x}}\)) and z-axis (\(E_{\text{aA-z}}\), \(E_{\text{bA-z}}\), \(E_{\text{CA-z}}\)) respectively. The total electric field at \(P_{\text{SA}}\) (\(\mathbf{E_{\text{A}}}\)) can be given by:  
\[\mathbf{E}_{A}(t) =E_{A-x}(t)+E_{A-z}(t)i \tag{2}\] \[=\left[ \begin{array}{cc}k_{aA-x}&k_{bA-x}&k_{cA-x}\end{array}  \right]\left[ \begin{array}{c}U_{a}(t)\\ U_{b}(t)\\ U_{c}(t)\end{array} \right]+\left[ \begin{array}{cc}k_{aA-z}&k_{bA-z}&k_{cA-z} \end{array} \right]\left[ \begin{array}{c}U_{a}(t)\\ U_{b}(t)\\ U_{c}(t)\end{array} \right]i,\]  
where \(k\) is a coefficient equalling to the ratio of electric field detected to the origin potential. The first letter in the subscript of \(k\) represents the phase conductor producing the aimed electric field, and the second letter represents the phase under investigation. To integrate three phases, the electric field matrix can be written as follows:  
\[\left[ \begin{array}{c}\mathbf{E}_{A}\\ \mathbf{E}_{B}\\ \mathbf{E}_{C}\end{array} \right]=\left[ \begin{array}{c}E_{A-x}\\ E_{B-x}\\ E_{C-x}\end{array} \right]+\left[ \begin{array}{c}E_{A-z}(t)\\ E_{B-z}(t)\\ E_{C-z}(t)\end{array} \right]i=\left[ \begin{array}{cc}k_{aA-x}&k_{bA-x}&k_{cA- x}\\ k_{bA-x}&k_{bB-x}&k_{cB-x}\\ k_{cC-x}&k_{cC-x}&k_{cC-x}\end{array} \right]\left[ \begin{array}{c}U_{a}(t) \\ U_{b}(t)\\ k_{cC-z}&k_{bC-z}&k_{cC-z}\end{array} \right]\left[ \begin{array}{c}U_{a}(t) \\ U_{b}(t)\\ U_{c}(t)\end{array} \right]i. \tag{3}\]  
Equation (3) can be divided into  
\[\left[ \begin{array}{c}E_{A-x}(t)\\ E_{B-x}(t)\\ E_{C-x}(t)\end{array} \right]=\left[ \begin{array}{cc}k_{aA-x}&k_{bA-x}&k_{ CA-x}\\ k_{bA-x}&k_{bB-x}&k_{CB-x}\\ k_{cC-x}&k_{bC-x}&k_{CC-x}\end{array} \right]\left[ \begin{array}{c}U_{a}(t) \\ U_{b}(t)\\ U_{c}(t)\end{array} \right] \tag{4}\]  
and  
\[\left[ \begin{array}{c}E_{A-z}(t)\\ E_{B-z}(t)\\ E_{C-z}(t)\end{array} \right]=\left[ \begin{array}{cc}k_{aA-z}&k_{bA-z}&k_{ CA-z}\\ k_{bB-z}&k_{bB-z}&k_{CB-z}\\ k_{cC-z}&k_{bC-z}&k_{CC-z}\end{array} \right]\left[ \begin{array}{c}U_{a}(t) \\ U_{b}(t)\\ U_{c}(t)\end{array} \right]. \tag{5}\]  
Figure 1: Schematic diagram of the measuring principle. \(U_{a}\), \(U_{b}\), \(U_{c}\)—potentials on phase conductors; \(P_{\text{SA}}\), \(P_{\text{SB}}\), \(P_{\text{SC}}\)—positions where electric field sensors are set; \(\mathbf{E_{\text{AA}}}\), \(\mathbf{E_{\text{bA}}}\), \(\mathbf{E_{\text{CA}}}\)—electric field produced by \(U_{a}\), \(U_{\text{b}}\), and \(U_{c}\), respectively at position \(P_{\text{SA}}\).