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. To accomplish this goal, a quasi-static modeling algorithm was developed to simulate finite-length, permeable-core, rod antennas. This quasi-static analysis is applicable for low frequencies where eddy currents are negligible, and it can handle solid or hollow cores with winding insulation thickness between the antenna's windings and its core. The theory was programmed in Matlab, and the modeling code has the ability to predict the TX antenna's gain, maximum magnetic moment, saturation current, series inductance, and core series loss resistance, provided the user enters the corresponding complex permeability for the desired core magnetic flux density. In order to utilize the linear modeling code to model the effects of nonlinear core materials, it is necessary to use the correct complex permeability for a specific core magnetic flux density. In order to test the modeling code, we demonstrated that it can accurately predict changes in the electrical parameters associated with variations in the rod length and the core thickness for antennas made out of low carbon steel wire. These tests demonstrate that the modeling code was successful in predicting the changes in the rod antenna characteristics under high-current nonlinear conditions due to changes in the physical dimensions of the rod provided that the flux density in the core was held constant in order to keep the complex permeability from changing.
 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.
 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.
 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.
 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  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….”
 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  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.
 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  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  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.
 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.
 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.
2. Theory of a Prolate Spheroidal Rod Antenna
2.1. Introduction to the Rod Antenna Analysis
 The finite length rod-core antenna was represented by a prolatespheroidal rod antenna, Figure 1, to allow the analysis in the prolate spheroidal coordinate system. Maxwell's equations were first used to find eigenfunction expansions for the static magnetic fields within the various regions of the prolate spheroidal space. Associated Legendre functions were employed as the eigenfunctions in both the radial and angular dimensions. Boundary conditions were then enforced on the tangential magnetic field intensities and the normal magnetic flux densities within each region of the rod antenna in order to determine the coefficients in the eigenfunction expansions. The resulting static magnetic field expressions were then used to develop a Matlab program that computes the gain, the rod antenna power consumption, the rod antenna weight (wire and core), the battery weight and the maximum moment for a given saturation field. A summary of the magnetostatic analysis is given below.
2.2. Derivation of the Magnetic Field Equations
 Since we are operating at low frequencies (on the order of 1 kHz), we employ a quasi-static magnetic field analysis (therefore we neglect the effect of eddy currents), where we define the magnetic field intensity as the gradient of the magnetic scalar potential
Thus the Laplacian of the magnetic scalar potential must be a solution to Laplace's equation
 The prolate spheroidal coordinate variables η and δ can be related to the cylindrical coordinates ρ and z by ρ = f[(1 − δ2)(η2 − 1)]1/2 and z = fηδ, where f is defined as the focal length, which is half of the length of the prolate spheroid when η = 0. Note that the ranges on the prolate spheroidal coordinate variables are given by 1 ≤ η ≤ ∞ and −1 ≤ δ ≤ 1. Furthermore, the prolate spheroidal metric coefficients are defined by
 Following the procedure in the work of Wait [1953b], we find that the general solution to (2) involves the Associated Legendre functions, Pn and Qn, which are multiplied by the constants A, B, C, and D, i.e.,
Note that the solutions are independent of the angular variable, ϕ, since both the geometry and the source are independent of this variable.
 Eigenfunction expansions can now be constructed in the four regions of our prolate spheroidal antenna Figure 2. Region 0 is defined to be inside the hollow prolate spheroid, region 1 is the prolate spheroidal core space, region 2 is the winding insulation thickness between the prolate spheroidal core and the current coil, and region 3 is the area around the antenna. The general solution of (6) is modified for the four regions of the prolate spheroidal antenna by looking at cases where Pn and Qn blow up. For example since Qn(δ) approaches infinity as δ goes to ±1, and δ can be equal to ±1 in all four regions, this eigenfunction is excluded from all potential solutions. Furthermore, in region 0, η can be equal to 1, thus Qn(η) is excluded since Qn(1) blows up. In region 3, η can be equal to ∞, thus Pn(η) is excluded since Pn(∞) blows up. Therefore, the potentials in the four regions can be expressed as
By following the same procedure as above, the tangential components of the other magnetic field intensities are
 Boundary conditions are now enforced in order to solve for the unknown constants. At the source-free interfaces η = ηc and η = ηb the tangential components of H are equal and the normal components of B are equal. At the interface between regions 2 and 3 (i.e., η = ηa), the normal components of B are equal, the tangential components of H are discontinuous by the current density within the coil, which will be modeled as a surface current density. We will define a uniform current density, Jsϕ, along the arc length, larc, at ηa for −δI ≤ δ ≤ δI, where Nturns is the number of turns and Icurr is the peak-to-peak drive current level
2.3. Finding the Rod Antenna's Maximum Magnetic Moment
 In order to quantify the magnetic moment for our rod antenna, we will compare its magnetic field to the magnetic field produced by an air core planar loop of current (i.e., IS, where S represents the loop area). Essentially, we will enforce the equivalence theorem to determine the necessary moment for an air core planar loop of current that will produce the same external magnetic field as our rod antenna. The magnetic moment will be determined by using the broadside magnetic field component, where θ= 90° in cylindrical coordinates and δ = 0 in prolate spheroidal coordinates, i.e.,
Solving for the equivalent magnetic moment, IS, then gives
Furthermore, using (17), the maximum achievable moment is given by
where Bsat/Bmax renormalizes the moment so that saturation has been taken into account. Bsat is the magnetic saturation of the prolate spheroid core and Bmax is the maximum value of the B field within the prolate spheroid core for a given input current, I. Note that the magnetic flux is approximately uniform across the cross section of the high-permeability rod core. Therefore (23) gives the maximum moment that can be achieved when the maximum magnetic flux density is equal to the saturation field of the core material. The variables Rfarobs, Kfarobs, δfarobs and ηfarobs refer to an observation point far away from the antenna, e.g., 100 times greater than ηa. This ensures that only the first term in the magnetic field's series is necessary. Their values are equal to ηfarobs = 100ηa, δfarobs = 0, Rfarobs = fηfarobs and
In order to determine Bmax, we will use the B field value at the midpoint along the length of the rod (i.e., at the location δobs = 0) and ηobs = ηb as the maximum value that will occur in the finite-length cylindrical rod antenna. Note that only the tangential B field component is used, since its value is much greater than the normal component, i.e.,
2.4. Finding the Rod Antenna's Gain and Saturation Current
 In order to determine the gain of the prolate spheroidal core antenna with a distributed winding over an air-core loop antenna with the same diameter and number of turns, we match the boundary conditions again, but use a Jsϕ which is located at δ0 = 0, i.e.,
 We also set our permeability of the core to μ0. Our resulting new set of constants and magnetic fields will be referred to by placing “air core loop” in the constant's superscript. The original set of constants and magnetic fields solved for a distributed winding will be referred to by placing “prolate spheroidal core” in the constant's superscript.
 The gain is defined as the ratio of the H field at ηfarobs = 100ηa and δfarobs = 0 due to the prolate spheroid core antenna normalized by the H field at the same observation location for the air-core loop antenna. It is important to note that only the first term in both H field series, i.e., (17), is needed since the observation point is far enough away from the antenna, i.e.,
Finally, the saturation current necessary to make Bmax equal to Bsat is given by
2.5. Finding the Rod Antenna's Series Inductance and Resistance
 To determine the series Inductance and Resistance in the equivalent model for the rod antenna, we must find the complex flux flowing through the core. Following the analysis for toroids in the work of Harrington  we find that the equivalent inductance and resistance for the rod antenna are given by
where ψ represents the magnetic flux
3. Rod Antenna Modeling Analysis
3.1. Prolate Spheroidal Core: 1% of Length Covered by Coil Winding
 To understand the magnetic field induced in a core we first investigate the case where the coil winding is centered on the core about z = 0 and covers one percent of the rod length. The tangential and normal B field components are plotted in Figure 3 on the outer surface of the core, i.e., η = ηb. For this case, the relative permeability is equal to 500, the number of turns in the coil winding is 200, the drive current level is 5 A peak to peak, there is no winding insulation thickness between the coil and core and the number of terms in the B field modal series expansion is 1601. Here we see that the tangential magnetic field, Bδ, is much stronger than the normal magnetic field, Bη. This is understandable since the coil induces a magnetic field in the tangential direction which then flows along the length of the rod. The edge of the coil winding is also plotted in Figure 3. It is interesting to note that the tangential field is strong directly underneath the center of the coil and then rapidly decreases in strength toward the edge of the coil. The reduction in the tangential field along the rod can be attributed to leakage into the external region of the rod. By looking at the physical nature of the antenna's end in Figure 1, it can be seen that any remaining tangential magnetic field will be transformed into a normal magnetic field so that the field lines can exit the antenna into the external region and connect in closed loops outside of the antenna. This explains the sudden rise in the normal magnetic field at the end of the antenna. It is also interesting to note that the normal B field is equal to zero at the center of the rod (because of symmetry). It then quickly increases until the edge of the coil where it reverses and quickly decreases after the edge of the coil. The strong normal magnetic field components and rapidly decaying tangential magnetic fields at the center and the end of the rod can be attributed to relatively large leakage of the B field into the antenna's external region.
3.2. Maximum Moment and Gain as a Function of Coil Winding Width
Figure 4 depicts the maximum moments and gains that are achieved for a rod antenna with a varying distributed current winding width. The characteristics of the antenna are the same as in section 3.1. In addition, we have assumed that the test material has a value of Bsat is equal to 1.6 teslas. The coil is centered on the rod shell at z = 0 and its width is changed by 5% increments of the overall length of the rod shell while maintaining the same number of turns, i.e., 200. The maximum moments are calculated by adjusting the current through the coil until it produces a maximum magnetic field that is equal to the saturated magnetic field at η = ηb and δ = 0 (the location where the strongest magnetic field occurs in an actual rod antenna with a centered distributed coil winding). Essentially, the wider the coil, the more evenly the B field is induced along the length of the core. Since the field is more evenly distributed along the rod's length, it allows for more current to be driven into the coil before the largest value of the magnetic field reaches the saturation point within the core. This in turn allows for a larger achievable moment. However, the increase in maximum moment does come at the cost of a lower gain. As can be seen in Figure 4 both the maximum moment and gain scales range between 60 and 160. It is apparent that if maximum moment is more desirable, then the cost of a 30 percent reduction in the gain for a 230% increase in maximum moment (from 1% to 96% coil width/rod length) is a good trade. If gain is more desirable, a significant loss in the maximum moment will have to be accepted. Since we are more interested in maximizing the moment, we will assume that the coil is wound over 90% of the rod in our future studies.
3.3. Prolate Spheroidal Core: 90% of Length Covered by Coil Winding
 The characteristics of the antenna that is simulated in this section are identical to those in section 3.1 except that the number of terms in the B field modal series expansion is 201, and the coil that is centered on a prolate spheroidal core and covers 90% of its length. Figure 5 illustrates the nature of the magnetic fields on the outer surface of the core, i.e., η = ηb for this case. The edge of the coil winding is also plotted in Figure 5. It is interesting to note that before the edge of the coil, the tangential field is approximately constant and the normal magnetic field is increasing at a relatively stable rate, but once the edge of the coil is reached, then the tangential field decreases and the normal field increases rapidly. There are two reasons for this. First, it is due to the fact that the field lines are exiting the core region covered by the coil and they are starting to fringe around the end of the coil. Second, due to the nature of the coordinate system, the tangential magnetic field component is transforming into a normal magnetic field component and exiting the core into the external region.
 In comparison to the B field along the core for a 1% winding (Figure 3), the B field is much more evenly distributed along the core's length for the 90% winding case. Therefore, the drive current at which core saturation occurs will be much larger for the 90% winding case since the B field at z = 0 is significantly lower. This larger saturation current results in a larger achievable moment. Furthermore, fewer terms in the modal series expansion are required to model the 90% winding case than the 1% case since the field distribution is more uniform in the core [Jordan, 2008].
 When finding the maximum moment we will utilize the B field at the center of the rod (z = 0) to test for field saturation. Based on Islam , we believe that the prolate spheroidal rod provides a good model for the gain associated with a finite length rod, and it will also provide a good estimate of the saturation field for a uniform shell thickness provided we utilize the field at (z = 0).
3.4. Maximum Moment and Gain as a Function of the Winding Insulation Thickness
Figure 6 investigates the effect that the winding insulation thickness between the coil winding (η = ηa) and core (η = ηb) has on the maximum moment and gain. Since actual antenna designs will unavoidably include this winding insulation thickness, it is necessary to determine its effect on the crucial antenna properties such as the maximum moment and gain. The winding insulation thickness can be substantial if a large number of turns are wrapped around the core to reduce the required drive current level necessary for the desired moment. The winding insulation thickness/outer core radius percentage is defined as
 The characteristics of the antenna that are simulated in this section are identical to those in section 3.1. There are two important conclusions to be made from Figure 6. First, the gain decreases rapidly at first and seems to approach a gain limit of about 20, a worst case gain reduction of 400%. Second, the maximum moment only slightly increases, which is due to the small increasing air-core moment contribution. The relative insensitivity of the maximum moment can be attributed to the fact that the same amount of core material still saturates, but at a higher drive current, hence the lower gain.
3.5. Maximum Moment as a Function of the B Field Saturation Level, Bsat
 If maximum moment is the most desirable trait of the antenna, then the saturation level of the core material (Bsat) is an extremely important factor. Ferrite “28” material saturates at 0.33 teslas and Purified Armco iron, which is used in common applications such as relays, electromagnets and other dc apparatus, saturates at 2.15 teslas [Bozorth, 1993; Chen, 1977]. The maximum moment is directly proportional to the B field saturation value of the permeable material that is used to construct the core. If the B field saturation level is doubled, the maximum moment also doubles. Furthermore, maximum moment is directly proportional to the core volume. If the core volume doubles, there is twice as much material to produce magnetic moment before core saturation is reached.
4. Experimental Rod Antenna Results
 In order to test the modeling code, we demonstrate that it can accurately predict changes in the electrical parameters associated with variations in the rod length and the core thickness for cores made out of low carbon steel wire, provided that the magnetic flux density is held constant so that we can apply the linear code to model nonlinear rod antennas. For our length test, two 014 rod antenna cores, were constructed from 0.014” diameter Malin Co. black oxide low-carbon steel wire where a physical scale factor of 2 exists between the rod core lengths. Low-carbon steel wire was used as the core material for several reasons. First, it possesses a high saturation value (≈2.0 tesla). Furthermore, it has a black oxide coating on it to help break up eddy currents and the wire can be manipulated to manufacture any desired core dimensions with relative ease compared to metal foil tapes such as Metglas.
 The longer of the two antennas possesses the following characteristics: core weight is 750 g, length is 0.510 m, core outer radius is 1.27 cm, core inner radius is 0.635 cm, number of turns is 322, DC resistance is 0.273 ohms. To fabricate this rod antenna, six steel pins were burned into each of the ends of a 1.27 cm diameter wooden dowel and the steel wire was wound around the ends. Wrapping the wires along the longitudinal direction ensures good magnetic coupling along the rod and breaks up induced eddy currents in the circumferential direction. Once the overall core diameter was built-up, the entire rod was covered with shrink-wrap, the ends of the core were cutoff and the winding pins were pulled from the remaining wooden dowel ends. We then wound 322 turns of 14AWG wire around the rod in two flat layers, thereby covering 90% of the core.
 The shorter of the two antennas is half of the length of the first, it is wrapped with half the number of coil windings and it possesses the following characteristics: core weight is 375 g, length is 0.255 m, core outer radius is 1.27 cm, core inner radius is 0.635 cm, number of turns is 158, DC resistance is 0.134 ohms. The same manufacturing procedure was followed to create the second B.O. L.C.S. rod antenna. The use of the long and short B.O. L.C.S. 014 rod antennas will test the modeling code's capability of changing the antenna length, i.e., the length test.
 Using the same construction technique, we also constructed two black oxide low carbon steel 023 rod antennas, using 0.023” diameter black oxide low carbon steel from Malin Co., to further verify the necessity for holding magnetic flux density constant in order to use the linear code to model nonlinear rod antennas. A scale factor of 2 exists between the two rod antenna core thicknesses. The thicker of the two antennas possesses the following characteristics: core weight is 750 g, length is 0.510 m, core outer radius is 1.27 cm, core inner radius is 0.635 cm, number of turns is 322, DC resistance is 0.297 ohms. The thinner of the two antennas possesses the following characteristics: core weight is 468 g, length is 0.510 m, core outer radius is 1.27 cm, core inner radius is 0.953 cm, number of turns is 322, DC resistance is 0.290 ohms. The thinner B.O. L.C.S. rod antenna was constructed with a 1.905 cm diameter wooden dowel, as opposed to using a 1.27 cm diameter wooden dowel. The use of the thick and thin B.O. L.C.S. 023 rod antennas will test the modeling code's capability of changing antenna's core thickness, i.e., the thickness test.
4.1. Method for Measuring the Antenna's Properties
 The experimental setup shown in Figure 7 was used to measure the antenna saturation curves, core magnetic flux densities, high-powered rod input impedances, and magnetic moments. The saturation curve is obtained by doubling the current in the TX antenna and measuring the change in the RX signal. The drive current is determined by measuring the voltage across a 0.1 ohm series resistor. Isolation and attenuation (in order to not exceed the network analyzers input voltage threshold) is accomplished with the Techtronix A6902B isolation amplifier. The received signal is measured with a 120 turn loop air core antenna connected to a Stanford Research 560 differential amplifier. The onset of core saturation occurs when a doubling of drive current creates less than a doubling of received signal.
 Core magnetic flux density is calculated by wrapping NB-field turns around the rod antenna and measuring the induced voltage, Vinduced, i.e.,
where Acore is the rod core cross-sectional area and FF is the filling factor expressed as a decimal. The filling factor is the percentage of the actual high-permeable core material that is present in the rod core cross section. For isolation purposes, a precision battery powered voltmeter is used to measure the induced voltage. This approximation works only if the core material's permeability is large, since it is assumed that the majority of the magnetic field is contained in the core material.
 At the present, we are unaware of any commercial low-frequency, high-power impedance analyzers that are able to drive up to 10 A of drive current. Therefore this new system was developed, since measuring high-power antenna inductance and core loss is crucial for designing a practical and efficient transmitter. In order to avoid DC contact resistance from the connection clips, the output voltage is monitored at the same location where the current is driven into the antenna, i.e., a four-terminal measurement. The output voltage is attenuated and isolated with the other channel of the Techtronix A6902B isolation amplifier. The system is calibrated with a noninductive resistive load at the frequency of interest, where the resistance is similar to that of the DUT. The system components can be seen in Figure 7.
 In order to measure the transmitted magnetic moment at the frequency of interest, the input current, Icurr, is driven through the antenna and its transmitted signal is measured at a known distance via a 120 turn loop air-core antenna. Next an air-core antenna with a known moment is driven with the same current, Icurr, and its transmitted signal is again measured at the same known distance. By using the ratio of the two antenna's received signals, the moment for the rod antenna under test is determined.
4.2. Experimental Verification of the Modeling Code in the High-Power Region of the Saturation Curve
 An important limitation of permeable materials is their saturation level. This means that there is a point at which increasing the intensity of the magnetic field within the material will not increase the magnetic flux density. When the antenna's permeable core saturates, data can no longer be transmitted efficiently. Therefore it is necessary to study the effects of core saturation.
 Deviations from the expected linear 6 dB change in the received signal corresponding to a doubling of the input current allows for the characterization of the rod core and is crucial in determining saturation current level. If doubling the drive current results in a change of more than 6 dB, then this indicates that the permeability of the core material has increased since increasing the permeability of the core increases the gain of the antenna. The amount of overshoot in the saturation curves (i.e., the nonlinearity of the antenna) is dependent upon the core material's initial permeability, the maximum permeability and the rod's length-to-diameter ratio. Our analysis defines antenna saturation as occurring when the received field increases less than two times for a twofold increase in transmitter current.
 The saturation curves for the previously discussed short and long B.O. L.C.S. 014 antennas (where a scaling factor of 2 exists between the two rod antenna's length and number of turns) at 100 Hz can be seen in Figure 8. As a first step in validating the use of the modeling code with nonlinear magnetic cores, the longer B.O. L.C.S. rod antenna with a drive current of 2.4 A was further investigated. This drive current was chosen since it is the first point on the saturation curve after which a doubling in current no longer produces at least a doubling in receiver signal. The magnetic flux density, inductance, core series loss resistance and magnetic moment were measured at this drive level. Next, the shorter B.O. L.C.S. rod antenna was driven to the same B field (≈0.80 tesla) as the longer B.O. L.C.S. rod antenna and the inductance, core series loss resistance and magnetic moment were once again measured. The saturation curves are shown in Figure 8. The first observation to be made is that the two drive currents possess approximately the same change in receiver signal. Also, the overshoot for the longer rod antenna is larger than the shorter antenna since its length-to-diameter ratio is greater. Since both rod antennas are now operating at the same magnetic flux intensity and at the same point on the B-H curve, the permeability (derivative of B-H curve) of the two core materials are nearly identical.
 The next step in proving that the modeling code has predictive capabilities in the high-power region is to use the measured data from one rod antenna and use it to predict the antenna parameters of another. Since the longer rod is more sensitive to changes in core material permeability, we use its inductance and core series loss resistance in the modeling code to back-calculate the relative complex permeability as 445-j*230. It is important to note that since the core is built with circular wire, the measured fill factor (cross-sectional-area-of-metal/total-model-core-cross-sectional-area) of ≈0.50 must be taken into account. This is done by multiplying the true operating B field of the core material (≈0.80 tesla) by the fill factor. The effective core B field is used in place of Bsat since we are using the code to predict antenna parameters for a magnetic flux density before saturation. Using the back-calculated complex permeability, 100 Hz drive frequency, the effective B field and the physical dimensions of the shorter B.O. L.C.S. rod antenna, we then predict the inductance, core series loss resistance, drive current and magnetic moment. The comparison between the measured and theoretical (using long B.O. L.C.S. rod antenna back-calculated complex permeability) antenna parameters is shown in Table 1. The antenna parameters were predicted within ±10% of the measured values. Note that the core series loss resistance is fairly small and we are reaching the limitations of the measurement system.
Via comparing measurements on short B.O. L.C.S. rod antenna and to theoretical values calculated from modeling code using long B.O. L.C.S. rod antenna back-calculated complex permeability approximately equal to 445-j*230, B field approximately equal to 0.80 tesla, 100 Hz.
 Next, the complex permeabilities were back calculated at 200Hz (445-j*225) and 300Hz (445-j*225). Then the same comparisons between theory and measurements were performed at 200Hz and 300Hz. The percent differences between theory and measurement, for the changing core length test, are summarized in Figure 9. These results demonstrate that the code was successful in predicting the antenna parameters for a changing core length, even when the rod is operating in a nonlinear region.
 In the next test, the complex permeabilities for the thin B.O. L.C.S. rod antenna were back-calculated at 100 Hz (425-j*267), 200 Hz (405-j*285), and 300 Hz (385-j*290), for a core magnetic flux density of ≈0.80 tesla. These back-calculated permeabilities were used to theoretically predict the antenna parameters for the thick B.O. L.C.S. rod antenna and were then compared to measurements. The percent errors, for the changing core thickness test, are summarized in Figure 10. The antenna parameters were predicted within ±10% of the measured values. The code was successful in predicting antenna parameters for a changing core thickness, even when the rod is operating in a nonlinear region.
 We verified that the skin effect is negligible for this core, at the frequencies and steel wire diameters used here, using the following test. We constructed cores with diameters of 0.014,” 0.023,” 0.041” and 0.080.” Cores made with the largest diameter steel wire have decreasing Ls and large Rs as frequency increases over the range of interest. For the two smallest diameter steel wires, the Ls is flat and the Rs is small as frequency increases over the range of interest. We used only the small-diameter steel wires in Figures 8–10. Further research will be conducted in order to determine when eddy currents lead to complications in the modeling. Determining where the code breaks down for 0.014” and 0.023” diameter B.O. L.C.S. is not so much of an issue since material losses become quite large after 300 Hz. Thus, we have proven the code's predictive capabilities are valid up to these frequencies.
5. Application of the Modeling Code
 Using the modeling code to predict the antenna characteristics at saturation is possible only if the complex permeability is known at Bsat. If the maximum magnetic moment is not the main objective and the user wants to operate the antenna at the optimum efficiency, then the antenna's magnetic flux density will be below the core's saturation level. To use the code at smaller drive levels, Bsat is replaced with the desired core effective B field level (core B field taking into account core material filling factor) and the corresponding complex permeability is needed.
 To determine how to optimize an antenna using a specific core material, a rod antenna with a larger length-to-diameter ratio should be constructed to avoid the gain filtering effect of a small length-to-diameter ratio rod antenna [Wait, 1953a, 1953b; Keller and Frischknecht, 1966]. Inductance and core series loss measurements should be made at the frequency of interest at several drive levels. Each drive level can then be associated with a specific effective core B field. Using the inductance and core series loss resistance, the complex permeabilities can be back-calculated with the modeling code and can be associated with the effective core B field values. The user can now model rod antennas at different drive levels with length-to-diameter ratios that are equal to or less than the initially constructed rod antenna. This allows one to optimize a design by taking the size, weight and power consumption requirements into consideration.
 The goal of this research was 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 antenna with high magnetic moment. To accomplish this goal, a modeling algorithm was developed to simulate finite-length permeable-core rod antennas. This quasi-static analysis is applicable for low frequencies where eddy currents are negligible and it can handle solid or hollow cores with wire insulation thickness between the antenna's windings and its core. The theory was programmed in Matlab and has the ability to predict the antennas' gain, maximum magnetic moment, saturation current and series inductance and core loss resistance, even in the nonlinear regions of the core material.
 To maximize the magnetic moment, the following criteria should be considered. Distribute the coil winding along the entire length of the core material to obtain an evenly distributed B field profile along the rod, thus eliminating premature saturation of the core. Premature saturation leads to a smaller saturation current, thereby limiting the maximum achievable moment. This comes at the cost of lowering the gain. The core material should posses a high B field saturation level since the maximum moment is directly proportional to the B field saturation value of the core material. The core should posses a substantial amount of material since more material allows for a larger saturation current and higher moment.
 It was determined that the linear modeling code can be used with nonlinear magnetic cores, by using the corresponding complex permeability for the desired core B field. The linear modeling code was validated by comparing measured and predicted antenna parameters, such as magnetic moment, drive current, inductance and core series loss resistance. 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. Complex permeabilities were back-calculated from constructed rod antennas and were used to theoretically predict antenna parameters for other rod antennas made from the same core material. Two core materials were tested; 0.014” diameter and 0.023” diameter black oxide low carbon steel wire from Malin Co. Tests were performed at 100 Hz, 200 Hz, and 300 Hz with core magnetic flux densities of approximately 0.80 tesla. All comparisons between theory and measurements were within 10%. It was also recommend that if complex permeabilities are going to be back-calculated from one rod antenna, to model other rod antennas with the same core material, that it possesses a length-to-diameter ratio equal to or greater than the modeled rod antennas to avoid the length-to-diameter gain filtering effect.
 Tests were performed to show that the modeling code was also able to predict inductance and core series loss resistance of rod antennas operating in the linear, low-drive-level region. However, due to space limitations, these results are not presented.
 In summary, we have found that small diameter, black oxide low carbon steel wire is an excellent (high saturation level, cheap, readily available, and easy to use) core material for low frequencies, however, materials that possess much smaller core losses in the frequency range between a few hundred hertz and kilohertz need to be further investigated.
 We are indebted to the Defense Advanced Research Projects Agency, the Air Force Research Laboratory, and Raytheon UTD, who provided the motivation and funding for this research under contract FA8650-06-C-7601 and to the reviewers of this paper for their insightful comments, making this a stronger publication.