The mutual inductance calculation between circular and quadrilateral coils at arbitrary attitudes using a rotation matrix for airborne transient electromagnetic systems

• We present a mutual inductance calculation of circular and quadrilateral coils.
• The coils can be at arbitrary attitudes and positions.
• The method avoids heavy numerical calculations.
• The method uses the yaw, pitch, roll angle and geometric parameters of the coils.

Performance testing and calibration of airborne transient electromagnetic (ATEM) systems are conducted to obtain the electromagnetic response of ground loops. It is necessary to accurately calculate the mutual inductance between transmitting coils, receiving coils and ground loops to compute the electromagnetic responses. Therefore, based on Neumann's formula and the measured attitudes of the coils, this study deduces the formula for the mutual inductance calculation between circular and quadrilateral coils, circular and circular coils, and quadrilateral and quadrilateral coils using a rotation matrix, and then proposes a method to calculate the mutual inductance between two coils at arbitrary attitudes (roll, pitch, and yaw). Using coil attitude simulated data of an ATEM system, we calculate the mutual inductance of transmitting coils and ground loops at different attitudes, analyze the impact of coil attitudes on mutual inductance, and compare the computational accuracy and speed of the proposed method with those of other methods using the same data. The results show that the relative error of the calculation is smaller and that the speed-up is significant compared to other methods. Moreover, the proposed method is also applicable to the mutual inductance calculation of polygonal and circular coils at arbitrary attitudes and is highly expandable.