## 1. Introduction

[2] It is important to design compact magnetic transmitting antennas for use in geophysical exploration, such as borehole to borehole and surface exploration applications [*Depavia et al.*, 2008; *Holladay and Wilt*, 2002; *Wilt et al.*, 1995, 1991]. A study of rod antennas could also be useful for the design of transmitting antennas for communication application that require very low frequencies. Therefore, the goal of this research is to develop and validate a low-frequency modeling code for high-moment transmitter rod antennas to aid in the design of future low-frequency TX antennas with high magnetic moments.

[3] The magnetic moment quantifies the ability of a loop antenna to induce magnetic fields at distant receiver sites. The magnetic moment for an air-core antenna, such as a multiturn loop antenna, is equal to *M* = *NIS*, where *N, I,* and *S* denote the number of turns, current and surface area, respectively. In order to obtain a large magnetic moment with an air core antenna, a large cross-sectional area, or a large number of turns, is necessary. One benefit of an air core antenna is its linear response, i.e., a doubling of the current leads to a doubling of the field and the antenna's moment.

[4] Another method for obtaining large magnetic moment transmitter antennas involves wrapping windings around a high-permeability core. This approach can lead to gains in the space and power efficiencies, but there is a trade-off with the increase in design complexity. Unlike air cores, high-permeable cores are nonlinear and saturate at high field strengths (e.g., on the order of 0.3–2.0 tesla). Furthermore, since the permeable core material properties not only change with frequency but also with current drive levels, the design of high-power rod-core transmitter antennas is difficult.

[5] An analysis of a loop antenna with a hollow spherical core is relatively straightforward since the boundaries of the spherical shell are located on constant radial surfaces in the spherical coordinate system. This allows the electromagnetic fields to be obtained through the application of the separation of variables technique [*Islam*, 1963]. In the case of cylindrical rod cores, it is also possible to apply the separation of variables technique to a layered cylindrical structure; however, the permeable core would have to be infinitely long. In order to account for the effects of a finite length rod core, researchers typically employ the separation of variables technique in a prolate spheroidal coordinate system. *Islam* [1963] states in his paper that “It is extremely difficult to obtain a solution for the vector potential caused by a current loop and finite cylindrical core, because of the complexities involved at the ends of the cylinder….The nearest approximation to a finite cylinder is a prolate spheroid….”

[6] High-permeability rod cores can provide a large increase in the effective moment and are commonly used as receiver (RX) antennas. The effect of the length-to-diameter ratio on the antenna gain is shown by *Keller and Frischknecht* [1966] for the case of solid rod core antennas. *Wait* [1953a] further verified that the gains achieved by long, solid prolate spheroidal cores approach their relative permeability for large length-to-diameter ratios. Researchers have also investigated the use of hollow prolate spheroidal cores [*Burgess*, 1946; *Wait*, 1953b; *Simpson and Zhu*, 2006]. The results from these studies show that thinner shells reach their maximum gains at smaller length/diameter ratios than thicker shells. Furthermore, thinner shells have lower gains than thicker shells.

[7] The theory presented in this paper not only combines several different aspects of previous researcher's works, but it also makes significant new contributions to the understanding of prolate spheroidal antennas, specifically when used as transmitting antennas. *Wait* [1953a] and *Islam* [1963] both studied rod antenna gain, but did not solve for inductance and core series loss resistance. *Wait* [1953a] studied the effects of solid and hollow core antennas, but neglected the effects of the winding insulation thickness between the coil and the core. *Islam* [1963] included winding insulation thickness in his analysis, but only studied solid cores. Both Wait and Islam also only analyzed single-turn loops placed around the prolate spheroidal antennas. Simpson analyzed a prolate spheroidal core whose entire length is covered by a distributed coil. Another issue that had been neglected up to this point is the problem of core saturation since the previously referenced studies only addressed receiver antennas. In fact, we have not found any published papers that investigate how magnetic field saturation limits the maximum moment that can be achieved with high-permeability cores when they are used in the design of transmitter antennas.

[8] The work presented here has the ability to model the electrical properties of linear (low field level) transmitter antennas when there is negligible skin effects. Furthermore, it can also be applied to high-level transmitter antennas to model changes in the performance that result from changes in the physical rod dimensions, provided that the level of the flux density in the core is constant. We can model solid cores, hollow cores and winding insulation thickness between the coil and core. Our theory can also model the effects of a distributed coil, which can vary anywhere between a concentrated loop at the center of the core to a distributed winding that is spread evenly along the entire antenna's core length.

[9] In order to validate the linear modeling code, we compared measured and predicted antenna parameters, such as magnetic moment, drive current, inductance and core series loss resistance, for antennas operating at both low and high flux density levels, i.e., in both the linear and nonlinear regions. In order to make the necessary impedance measurements in the rod antenna's nonlinear region, a high-power, low-frequency impedance analyzer system was developed.