矿井瞬变电磁矩形回线源瞬变电磁响应研究

为研究均匀全空间介质中任意点（除发射框边）处矩形回线源瞬变电磁响应，从全空间磁偶极子的切向电场和水平圆形回线发射框中心磁场出发，采用互易原理和积分法推导了全空间任意点处的瞬变响应。通过对全空间瞬变场磁场Bz及其时间导数?Bz/?t的核函数Y(ω)和Y’(ω)特征的研究，将全程瞬变响应分为早期、中期和晚期。Y(ω)的早期和晚期阶段，其内层误差函数采用渐近展开式，外层积分采用直接多项式积分；中期阶段，内层误差函数采用有理契比雪夫展开式，外层积分采用龙贝格积分。Y’(ω)的晚期和非对角区域的早期采用误差函数渐近展开式；Y’(ω)的中期和对角区域的早期采用龙贝格积分。采用理论模型计算验证了公式的精度。结果表明，该算法求取的数值解，其Bz核函数Y(ω)的相对误差可以控制在1×10-6%以内；而?Bz/?t的核函数Y’(ω)的相对误差可以控制在1×10-4%以内。早期阶段，Bz和?Bz/?t在发射框内部的极值中心沿着与短边呈45°夹角方向向发射框内部移动，并且短边处的两个极值中心先行合并，然后再沿长边方向合并；?Bz/?t极值向发射框内部和发射框外部移动的速度大于Bz相应的移动速度。

Time domain transient electromagnetic responses from a rectangular loopin homogeneous whole-space

In this paper, whole-space transient electromagnetic(TEM) responses of any point except on the loop path, was derived based on reciprocity and integration, by use of the known homogeneous whole-space tangential electric field excited by magnetic doublet and response equation of central loop. According to the behaviors of the kernel function Y’(ω) and Y(ω) respectively for Bz and ?Bz/?t, the TEM field was divided into early-time, middle-time and late-time period. In late and early stages asymptotic series of Y (ω) was used. For middle stage, the error function was estimated by rational Chebyshev approximations, and the definite integral along the loop frame was also calculated by Romber integral method. For Y’(ω) of late stage and early stage in the non-diagonal area was calculated by asymptotic series of error function, while Y’(ω) in the middle stage was calculated by Romber integral method. The precision of this method was tested by the model calculation. The results indicate that the precision of Y(ω) is less than 1×10-6% and the precision of Y’(ω) less than 1×10-4% using this method. During early-time, extremum of Bz and ?Bz/?t within the loop was moved inward over time along the direction of 45°to the short edge of rectangular loop. Two extrema at both ends of the short edge within the loop were merged firstly, then the rest of extema were merged along the long edge. The velocity of ?Bz/?t was more than that of Bz in early-time.