Analytical solutions for flexible circular spiral eddy current array coils inside a conductive tube under different operation modes

The eddy current induction of flexible eddy current array probe composed of thin curved circular spiral coils is studied using the second-order vector potential formalism. The analytical expressions of self-and mutual impedance change of coil array element inside a conductive tube under three standard eddy current array operation modes are derived. They are self-transmitting and receiving, absolute and differential transmit-receive, respectively. Further derivation shows that the coefficient matrices of characteristic equations in these modes are the same regardless of whether the element is placed inside or outside a tube, but the non-homogeneous terms are different. Therefore, the solution to another case can be obtained quickly through the transformation relationship. The experimental results show that the theoretical predictions are consistent with the experiment over the 5 kHz–1 MHz excitation frequency range. At the same time, the real and imaginary part errors of impedance change are analyzed, and the application of small inductance flexible coil in three operation modes is discussed. Finally, these expressions provide academic support for the eddy current testing of pipeline defects and their significance in the inversion of tube electromagnetic properties.