Wang and Bell19 developed a quantitative model of the reactance and resistance for whistler mode transmission in plasma, solid lines in Figure 4. Chevalier et al.22 conducted a computer simulation, in open circles, as a part of DSX mission development. The two results agree well with each other. However, both appear to be in conflict with the measurements, especially when noting that their reactance in the upper panel has been multiplied by a factor of 3,000 in order to show them on the plot. Furthermore, in addition to the difference in the magnitude, the overall trend as a function of frequency is in a wrong direction. In their frequency range, the antenna resistance is more than an order of magnitude smaller than the measurements. Therefore, we conclude that Wang and Bell19 and Chevalier et al.22 models are incompatible with the experiment.
Song et al.20 developed a physical model of the plasma sheath surrounding a VLF transmission antenna as part of TNT development study. They reasoned that when the antenna is charged with alternately varying high voltage during transmission, charged particles will move between the two antenna branches in response. When the electromagnetic field oscillates in the whistler mode frequency range, electrons will be repelled from the negatively charged branch of the antenna and attracted to the positively charged one while the ions do not have enough time to respond before the field reverses. Around the negatively charged branch, an ion sheath is formed with net positive charge. Around the positive branch, on the other hand, there is a tendency to form an electron sheath with net negative charges. The overall effect of this process is that the sheath is formed around each branch of the antenna with an oscillating radius, which is achieved mostly by electron movements between the two branches. The corresponding electric current of such electron movements tends to cancel the driving current of the antenna, which produces the difficulty for whistler mode transmission in plasma. The equivalent reactance of the sheath is
$$- {\bar {X}_a}=\frac{1}{{\alpha 2{\pi ^2}f{\varepsilon _0}l}}\left[ {\ln \left( {\frac{{{I_a}}}{{{\pi ^2}fle{N_0}r_{a}^{2}}}+2} \right) - 1} \right]$$
2
where f is the transmission frequency, \(l,{\text{ }}e,{\text{ }}{N_0},{\text{ }}{r_a},{\text{ }}{\varepsilon _0},{\text{ and }}{I_a}\) are the antenna length, elementary electric charge, plasma density, antenna radius, permittivity in vacuum, and antenna current. We have introduced a factor α to Song et al.20 to account for the effect of the antenna structure. The TNT antenna consists of three separate parallel conducting wires, for the purpose of redundancy and robustness, of 0.15 mm each in radius along a circle of diameter of 20 cm. In a highly simplified model, if each wire has a capacitance C’, the three parallel wires could be considered as three capacitors in parallel and the total capacitance would be 3C’. However, the three wires may have some interaction and the net capacitance could be represented as αC’ and α can be estimated during the antenna development. Experiments have indicated α = 2.2 in vacuum of such a three-wire antenna. According to (2), the antenna reactance depends on the plasma density logarithmically. We assume that the plasma density is 2,000/cc28 based on measurements from the RPI instrument on the IMAGE spacecraft. The prediction is shown as black patched of plus signs in the upper panel of Figure 4. Because they are so close together, they form black patches. The model appears to be generally consistent with the measurements.
Also, during the TNT development Song et al.23 developed a model of whistler mode radiation resistance. It is based on the Huygens-Fresnel diffraction theory to derive the radiation from an antenna. The model predicts that the radiation resistance is
$${R}_{rad}\approx \frac{3{\pi }^{3}{\mu }_{0}}{c}{f}_{ce}^{3}{f}_{pe}{f}_{}^{-2}{d}^{2}=3.9\times 1{0}^{-13}{f}_{ce}^{3}{f}_{pe}{f}_{}^{-2}{d}^{2}$$
3
where µ0, c, \({f}_{pe}\), \({f}_{ce}\), and d are permeability of free space, the speed of light, electron plasma frequency, electron gyrofrequency, and half-length of antenna, respectively. For typical conditions of DSX, Rrad is 2.6 kΩ at 10 kHz and is proportional to f−2. The black patches of plus signs in the lower panel of Figure 4 show the prediction of the model. Because the measures of impedance, Xa, and Ra, are consistent with the theoretical predictions given in (2) and (3), we conclude that the antenna sheath model20 and whistler mode radiation model23 approximate the process in reality.
Furthermore, Tu et al.21 conducted full particle simulations to investigate plasma sheath structures around a VLF antenna in plasma. In their simulation, dipole antenna with tip-to-tip length of 200 m and radius of 0.2 m is used. The resulting reactance is scaled based on Song et al.20,23 and is plotted as a triangle on the top panel of Figure 4. From the simulated phase difference between the antenna voltage and current, the antenna resistance is calculated and plotted as a triangle in the bottom panel of Figure 4. Both reactance and resistance from Tu et al.21 are reasonably consistent with the DSX measurements.
The net antenna power output is more complicated by several effects, such as tuner setting and plasma conditions. From (1) and Figure 1b, it can be approximated as
$${P_{out}} \approx \frac{{V_{1}^{2}}}{2}\frac{{{R_a}}}{{{{\left( {{R_a}+{R_1}} \right)}^2}}}$$
4
The approximation neglects the current flowing through C1. In the experiments, each of the tuning bands uses a fixed inductance and increases capacitance C1, corresponding to transmission from higher frequency end to lower frequency end within each band. From the resonance condition, the lower frequency band uses a greater L1 and hence greater R1 as the two are proportional. Since V1 for the data included in Figure 3 varies in a relatively small range, the step-like increase in Pout from the lower band to higher band is due to the decrease in L1 and hence R1 in each band setting. Within each band, the change in R1 is relatively small, but Ra decreases with frequency as frequency increases, resulting in the power decrease within each band.