Comparative study of multi-physics generated small dipoles in conducting media

In this paper we present the results of a study of electronically and mechanically generated transverse magnetic (TM) and transverse electric (TE) dipoles in a lossy environment, so that antenna design guidelines may be established at the system level. At far-zone, the ratio |EH|:=η0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|\frac{E}{H}|:= \eta _0$$\end{document} is the intrinsic impedance, and they are identical for the TM and its dual TE dipoles. Nonetheless, the ratio in near-zone behaves drastically different between the TM and dual TE. We derived closed form expressions of the antenna Ohmic loss in a spherical lossy shell (SLS) for the first time, yielding precise radiation efficiency ηr\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\eta _r$$\end{document} and accurate computations. For electrically small dipole of normalized half dipole-length |ka|≪1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|ka|\ll 1$$\end{document}, analytic results show that ηr\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\eta _r$$\end{document} is proportional to |ka|3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|ka|^3$$\end{document} for TM dipole, and |ka| for TE dipole, respectively. Consequently, efficiency ηr\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\eta _r$$\end{document} of TE can be better than TM in two to three orders of magnitude for under seawater communication. The time-domain energy flow velocity (EFV) patterns show that the TE dipoles are always radiation-dominating, in either lossless or lossy medium. Numerical results reveal that mechanically spinning dipole is smaller in size and weight but it requires more operation power, compared to its electromagnetic counter-partners. Finally, design, tuning and impedance matching of low-profile TE dipole antenna are outlined.


Introduction
Small dipole antennas play important roles in electrically lossy media such as underwater communication, microwave remote sensing and biomedical telemetry.A dipole generated by an oscillating electric current line in free-space is called Hertz dipole.Hertz dipole is electric dipole, and is also called transverse magnetic dipole or TM dipole, from polarization point of view.By duality principle, a dipole produced by electric loop current is referred to as the magnetic dipole, or transverse electric dipole, or TE dipole in brief.Depicted in Fig. 1b and c are the TM dipole and TE dipole in the software FEKO.

Historic review
Based upon spherical harmonics, L. J. Chu published a historic paper [1] on the fundamental limits for electrically small antennas, which is known as the Chu-Fano theorem and it laid the foundation for small antennas in free-space at relatively low frequencies.Over the past half century, Wheeler, Harrington, Collin, et al. have extended and elaborated the Chu-Fano criterion [2][3][4].The physical bounds of gain and Q-factor of linearly polarized antennas are well studied and presented in [5].Current distribution in conducting medium can be approximated by integral equation [6].Propagation behavior of electromagnetic (EM) waves in lossy media has been studied using Hertzian potential integrals and dispersion relation [7][8][9].

Motivation
In recent years, commercial and scientific interests have moved upward toward higher frequencies in microwave, millimeter wave, and optical regions [10][11][12][13][14][15], due to internet and wireless communication advancement.Nonetheless, low frequency EM signals can penetrate through conductive media such as human tissue, seawater, soil, rock, and building materials.As a result, the study of small dipoles in conducting media is inspired by ground penetrating radar, underground water, and biomedical applications, e.g., artificial cardiac pacemaker [16].
Recently Troy Olsson launched a DARPA project, A MEchanically Based Antenna (AMEBA), to penetrate RF requirements with an antenna system that has the highest theoretical efficiency possible, and minimize Ohmic losses which are intrinsic due to the lossy environment.Targeted RF carrier frequencies are less than 30 kHz, and the goal is to achieve a low-size, weight, and power (SWaP) manportable system: Deliver sufficient magnetic field strength for low data rate communications at relevant distances, and efficiently modulate the RF carrier at useful data rates.The current paper was motivated by the DARPA AMEBA [17].

Main problem under investigation and potential novelty
Under seawater sensing and communication are dominated by sonar via acoustic waves because seawater is too lossy for RF signals.However, acoustic signals cannot communicate with a satellite in space or a radio station in land.If electromagnetic (EM) waves are used for under seawater, the frequency must be very low, because skin depth (measure of penetration of EM waves in conducting media) is inversely proportional to √ f .Our paper investigates simple, compact, and low-power methods for generating extremely low-frequency radio signals in 300-30 KHz.Our paper is motivated by the prohibitively large size and low efficiency of conventional antennas operating in this frequency range.We conducted studies of generating electromagnetic waves in very low frequency by mechanically rotating magnets and electrets, which has been experimentally implemented in many other papers [18,19].
We begin with the equations of Hertz dipole in free-space, which represents the lowest radiation mode [20].The Hertz dipole is an ideal case of elemental dipole with infinitesimal wire diameter and infinitesimal feeding gap.We extend the model to electrically lossy media in both frequency and time domains.Both electric and magnetic dipoles are studied in air and lossy seawater environment with no passive structure.Numerical results of commercial software FEKO showed that such an extension is valid and accurate for electric dipoles of large gap dimension and fat wire diameters (see Fig. 3).For electric small antennas, the reactive power in near field is dominant and significant to device functionality [21].In time-domain, we calculated the radial energy flow velocity at specific time to compare the reactive energy in air and lossy medium.For EM waves in ELF (extremely low frequency) and VLF (very low frequency) bands, the free-space wavelengths range from tens to thousands of kilometers, resulting in extremely small |ka|, where k is the wavenumber and a is one half of the dipole length.Our analytical model reveals that in electrically lossy media the radiation efficiency for electrically small antennas is proportional to |ka| 3 for TM (electric) and |ka| for TE (magnetic) radiator, making magnetic dipoles more efficient than electric ones over two to three orders of magnitude.
In this paper, more attention is paid to the magnetic dipole because of its better performance and rear appearance in the literature.Magnetic current based antennas (MCA) are less popular than electric counterpartners by far [22][23][24][25][26].The contra-wound toroidal helix antenna (CTHA) was studied numerically and experimentally at the VHF band [23,24].Studies reported there were for CTHA with air core without electrets.McDonald has published a series MCA papers in tensor forms [25].The near zone field analyses of MCA were also reported for physical therapy [27].Magnetic antennas with dielectric core or ferromagnetic laminate composite materials are referred to as permeable antenna [28].
Another option to achieve a low-size, weight, and power (SWaP) man-portable system is the dipole due to mechanically spinning of an electret or magnet, which is smaller in size and weight compared to its electromagnetic counterpartners.However, it requires more power to maintain the rotation and a flywheel is required [29].

Frequency domain formulation of dipoles
In this paper, we use the SI unit system, and time convention of e j t is assumed and suppressed.Under the lowest mode assumption, the field components of electric dipole (TM antenna) are [20] (1) where I m is magnetic current.As shown in Fig. 1a, the antenna is inside air-filled sphere of radius a = 0.3 m and surrounded by seawater and operating at f = 1000 Hz.The parameters of a = 0.3 m, f = 1 kHz will be assumed through out the entire paper.
We extend (1) and ( 2) to more general cases below: (a) Air or free-space with k = k 0 , (b) Lossy medium with com- plex permittivity, and (c) Seawater (highly lossy).

In air environment (lossless)
I n e q u a t i o n s ( 1 ) a n d ( 2 ) , w e u s e k = k 0 = ∕c = 2.094 × 10 −5 m −1 , w a v e l e n g t h = 0 = c∕f = 3 × 10 5 m, wave impedance 0 = 377Ω .The magnitude ratio of the normalized E-to H-field as a function of normalized distance r∕ is plotted in Fig. 2a.The far-zone field components are those proportional to the 1/r term in Eq. ( 1) and (2).It is seen that the far-zone can be considered when distance r ≥ 1.0 .It is obvious that in the far- zone |E∕H| TM = |E∕H| TE = 0 = 377Ω , free-space intrinsic impedance, which is the asymptotic value as r → ∞ .On the contrary, the two ratios in the near zone have opposite trends when distance shrinks toward zero, as seen in Fig. 2a.When distance r → 0 , impedance curve grows rapidly for electric dipole, while the other curve fast decreases for magnetic dipole.Figure 2a is identical to the plot in [25] which was derived in the Gaussian CGS units.

Lossy medium (complex permittivity)
In a lossy medium of relative permittivity r and conductivi t y , t h e c o m p l e x p e r m i t t i v i t y , complex propagation constant ∶= + j = jk , and complex wave impedance c = 0 ( + j )∕(|k| 2 c).
Using complex valued k = − j and = c in Eqs. ( 1) and (2), the average power outflows, passing a fictitious (2) sphere of radius r for the electric and magnetic-dipole, are respectively

In seawater environment (highly lossy,
≫ !" 0 " r ) For highly lossy seawater, conductivity = 4 S/m and relative permittivity r = 80 .Thus, complex permittivity, c = 0 80 − j  of the E-field to H-field is plotted in Fig. 2b, where the far-zone field components are those proportional to the 1/r terms in Eqs. ( 1) and (2).It is obvious that far-zone can be considered when distance r ≥ 1.0 .In the far-zone 0444Ω , independent of r.On the contrary, as r → 0 , the two ratios in the near field zone have opposite trends, as seen in Fig. 2b.Contrary to Fig. 2a, the two impedance curves are monotonically varying with distance in lossy environment.

Software validation
We use commercial software, Feko, to conduct the simulation.A small TM dipole is centered in a hollow sphere of r = 0.3 m, which is submerged in a seawater sphere of outer radius R = 400 m.Seawater conductivity = 4 S/m and relative permittivity r = 80 , rendering = jk = 0.1256(1 + j) m −1 , = ≈ 0.1256 m −1 .Depicted in Fig. 1b is a TM antenna with conducting rod of radius = 0.02 m, length = 0.2 m for each arm, and gap g = 0.1 m in the middle for source exci- tation.This fat dipole with large gap is deliberately exaggerated to test the validity of analytic model of Eqs. ( 1) and ( 2) in lossy environment.Figure 1c illustrates the Feko model of a TE antenna with loop radius 0.1 m, wire radius 0.1 mm.Again, the TE antenna is contained in an air-filled sphere of r = 0.3 m, which is submerged in seawater.The Feko evalu- ated total E-field and H-field at various distances from the antenna are compared with analytic solutions in Fig. 3 and excellent agreement is observed.The L2 errors are only 0.31% for TM and 0.12% for TE between the closed form equations and MoM numerical approximations in lossy medium for the first time.

Time domain formulation: energy flow velocity (EFV)
The main purpose of the underwater antenna is to conduct short-range communication, and EFV is an important quantity in studying the reactive energy in the near-field of an antenna [21].In this section, we will study antenna's EFV in both free space and lossy seawater.The electromagnetic energy f low velocity is, v = S∕w , where S = E × H is the Poynting vector and w = 1 2 |E| 2 + 1 2 |H| 2 is the stored electromagnetic energy.

Lossless medium
The time-domain electromagnetic fields of TM antenna in free space can be found in [30] Similarly, for TE case Here, we use time dependent cur rent source, i(t) = i 0 sin ( 0 t)u(t) , where u(t) is the unit step function which is turned on at t = 0.
In free space (or lossless medium), the time-domain normalized radial EFV within one wavelength is shown in Fig. 4a and b for TE and TM respectively.Our TM result of Fig. 4a repeated the pattern in [30].We recognize EFV is highly related to radiation efficiency.For TE antenna in lossless medium, Fig. 4b, its EFV is identical to TM EFV because both of them has efficiency of 100%.And both patterns are showing as orange lobes.When displaying ( 5)
On the contrary, Fig. 5a and b illustrate the EFVs for TE and TM dipoles in highly lossy seawater.They clearly demonstrate that the radial EFV of TE dipole is mainly in the radiation mode (showing as orange lobes), while TM dipole is essentially in cavity mode (showing as onion layers).Time-domain quantities are highly intriguing, and here we can see why radiation efficiency for TE is much better than TM from a qualitative and intuitive point of view.The super radiation efficiency of the TE over TM in a lossy medium is revealed by EFV pattern in time-domain.Figure 5b of TE dipole in lossy medium indicates that radiation EFV pattern is like orange lobes, while Fig. 5a of TM dipole pattern like onion layers.
When displaying the process in real time video, Fig. 5b behaves like creation of soap bubbles, similar to Fig. 4. In contrast, the corresponding time domain video of Fig. 5a shows standing wave pattern of concentric face-to-face hemi-balloons.

Power relation for EM generated dipoles
Low frequency propagation of EM wave in conducting media plays an important role in low data rate communication under seawater.In highly lossy seawater, source generated power is mostly dissipated in near-zone region for electrically small antennas of ≪  .In this section, we shall derive a closed form expression of Ohmic loss in a lossy sphere, and use it to study radiation efficiency of small electric and magnetic dipoles.
The power outflow consists of radiated power and Ohmic losses.Setting = 0 ⇒ = k , the general power equations of (3) and (4) reduce to lossless case; Meanwhile, both square brackets in (3) and (4) reduce to unity, meaning the net power flow is the radiated power.Therefore, term 1 in the brackets of (3) and ( 4) is the radiated power, while other higher-order terms represent power loss due to the evanescent fields.
For high lossy media, e.g., seawater, �k� = √ 2 = √ 2 , which reduces the general power of Eq. ( 3) and (4) into In the near zone, the last terms in (3) and ( 11) are dominating, implying the Ohmic loss is proportional to the |ka| −3 for electric dipole.Likewise, the last terms in ( 4) and (12) (11) are dominating, implying the Ohmic loss is proportional to the |ka| −1 for magnetic dipole antenna.
To provide a first hand intuition, we calculated the numerical values of high-order terms in Eqs. ( 3) and ( 4), at 0.3m (near zone) and at 100 m (far zone), respectively.In reference to the radiation term of unity, Tables 1 and 2 listed numerical values of the high-order terms at 0.3 m (near zone) and at 100 m (far zone), respectively.

Exact analytic ohmic loss in closed form and spherical shell
It is well-known that in an open lossless medium, the Sommerfeld radiation boundary condition must be enforced [31], namely where u(r, , ) can be any component of E-or H-field.Physically, Eq. ( 13) forces the far zone field to decay in the rate of 1 4 r , which is due to energy splitting along spherical surfaces.
In a homogeneous lossy medium, however, there is additional decay factor of e −2 r in (3) and (4) (and ( 11) and ( 12)), representing power loss owing to thermal dissipation.Comparing to lossless, decay in lossy environment will be more severely and rapidly.To quantitatively describe radiation efficiency of a dipole in a homogenous lossy medium, spherical lossy shell (SLS) is introduced.In [16] an SLS was defined, whose interior radius r = a is source border and exterior radius r = b was evaluated approximately such that 90% power has been dissipated.Nevertheless, this definition introduced a 10% uncertainty into a deterministic antenna problem.Our closed form expression removed math difficulty and preserved pin-point accuracy up to 4th decimal point.Consequently, we modified the percentage into 100% to emphasize the entire Ohmic loss is considered.
Employing Poynting theorem, we have derived the Ohmic loss in closed form of Eq. ( 14) for the first time.It says that the antenna thermal loss in an SLS is equal to the difference of power flow at its interior and exterior borders, namely, where S(r) for TM and TE were given in (3) and (4); no more integrations are needed when evaluating the power loss in an SLS.
Numerical data strongly support our derivations and we quote them below.Using numerical and integration by parts, the total Ohmic losses are (13) Comparing to Poynting power ( 3) and ( 4) to the Ohmic losses (15) and ( 16) one can see

Antenna radiation efficiency
Radiation efficiency of an antenna is defined as the ratio of radiated power to total source power (sum of radiated and ohmic power), namely where P loss (b) is power loss in the SLS of r = b defined in (14), and P rad (b) is radiated power passing sphere of r = b which is the 1st term of (3) or ( 4), (or ( 11) and ( 12)).Notice that r (b) depends on the field point, i.e., radius of the exte- rior border of the SLS.As radiation is concerned, b is in the far-zone.From Figs. 2a and b, b ≥ .Before evaluating r (b) , let us consider the power ratio in (21).For a highly lossy medium, the TM case, Similarly, for TE case Let a = 0.3 m and b = 100 m, the ratios above can be approximated by its dominant term, yielding ( 15)

Finally we estimate (21) as
From discussions above, we conclude that the radiation efficiency of TM mode is proportional to (|k|a) 3 and TE mode is proportional to |k|a.It is seen that the efficiency of TM is 1.42106 × 10 −15 and TE is 5.00451 × 10 −13 .In term of antenna efficiency, the TE case is about 352 times better than TM case.

Source requirement
The DARPA AMEBA requires to deliver 100fT magnetic flux density undersea to 100m (far-zone) at 1 kHz.It the source to be TM: Using (15), the required source power TE: Using ( 16), the required source power The numerical values for dipole moment ( I e dl or I m dl ) and required power of the source are tabulated in Table 3.It shows that the required power of magnetic dipole is less than that of electric dipole in two to three orders of magnitude.Since the required current moment, I m dl , is 7.12687 Vm and a linear dipole has an effective length of half physical length, the magnetic current, I m , required to deliver the 100fT to 100 m under sea is 23.7562 V.Under assumption that magnetic field is uniform within the permeable dipole core, and with current loop feeding, the magnetic current moment of an N-turn loop is where I loop is the electric loop current around the magnetic core and A is the cross-section area of the core.

Mechanically spinning dipoles
We study a mechanically spinning dipole from viewpoint of tuning and impedance matching.A mechanically rotating dipole at 1000 cycle per second may radiate EM waves at 1 kHz; no particular resonant circuit need be built to attain the frequency.Although ordinarily it is assumed that antennas with physical size much smaller than the wavelength are difficult to tune and match.Alternatively, permeable antenna of the kind has been recently demonstrated to exhibit good efficiency in low impedance environments [28].
Figure 6 depicts a dipole spinning at angular frequency along x-axis, with its ends tracing a circle of radius h/2.It can be determined from [32] that the current moment produced by the spinning dipole is The effective charge at the ends of a permanently polarized material (magnet or electret) is where P e,m is the polarization density, and A is the cross sec- tion.To radiate at 1 kHz, the dipole may spin at 60,000 rpm.The kinetic energy for such rotational dipole is where the moment of inertia is J k = 1 12 M(h 2 + 3 2 ) for cyl- inder with height h, mass M and radius shown in Fig. 6.
Since the kinetic energy limit is 5000 Joules from AMEBA requirement, the moment of inertia We chose h = 2 for square longitudinal cross-section, yield- ing kinetic energy and the mass density where D is material density in g∕cm 3 .Dividing ( 30) by (31)   For NdFeB magnets D = 7.5 , ceramic electrets D = 3.8 , and carbon fiber D = 1.5 , the corresponding cylinder height h m = 0.0494 m, h e = 0.0566 m, and h c = 0.0682 m.This means the spinning dipole has required height h in the order of 0.05 m and cross-section of 2 = 1.9635 × 10 −3 m 2 , and volume of 9.8175 × 10 −5 m 3 for stored kinetic energy of 5 kJ spinning at 60000 rpm.
For electric dipole, using today's best electrets with P e = 0.034C∕ m 2 , the source and for magnetic dipole with today's best magnet of saturation magnetization M s = 1.15 × 10 6 A/m, the corresponding P m = 0 M s = 1.4451T,Using ( 15) and ( 16), the total source power is 1.081 × 10 −4 W for TM case and 0.2702W for TE case.The data suggest that spinning magnet operating at 1 kHz may produce 100 × 0.8914∕7.12687= 12.51f T magnetic field at 100 m, and this antenna operates within 5 kJ of kinetic energy.Further calculations indicate that it requires mechanical power of 220 W to compensate ball bering friction loss [29].

Numerical examples: TE-dipole based submersed antennas
From previous discussions, we found that electric dipoles encounter too much Ohmic power.Dipole of mechanically spinning electrets/magnets cannot provide required H-field level under pre-specified kinetic energy unless we replace dipole with quadrupoles.Therefore, we decide to use the magnetic dipole, called permeable antenna, which is a ferrite cylinder with loop current as source.The objection of the design is to achieve 100 fT magnetic flux density at 100 m away from source under seawater environment.We present two examples of permeable antenna design together with their source and matching circuit requirements.The first example is the contra-wound toroidal helix antenna (CTHA) with elliptical shaped loop feeding and the second example uses traditional electric solenoidal coil around the inner rod core as input excitation [23,24,33].

Example 1. contra-wound toroidal helix antenna (CTHA)
The CTHA is a low-profile low-directivity surface-mounted antenna with reasonable efficiency at a height of less than ∕30 .It operates at the inductive side of resonance, so that it can be tuned with a high-efficiency capacitor.When feeding the helical windings around the toroidal core with opposite currents, the net horizontal electric fields cancels and the magnetic fields adds up to make the loop field much stronger than the dipole.The geometry of the CTHA is shown in Fig. 7 with toroidal core radius R t = 0.2 m and core cross-section radius of R c = 0.1 m.The antenna is consisted of two spiral windings on a toroid inner core.To achieve the same far field strength of 100 fT magnetic flux density at r = 100 m and frequency of 1 kHz, the required source current can be calculated in Fig. 7 CTHA permeable antenna geometry with feeding point [23] where If N = 10 toroidal loops are used with the NiZn ferrite mate- rial core with permittivity r = 14 − j0.14 and permeability r = 2000 , the required current I n = 0.2892 A. The input impedance to design matching circuit is still missing because there is currently no closed form solution when the core is permeable material.It also ignores fringing effect from the feeding loops.

Example 2: permeable antenna with electric coil feed
The dimensions of permeable dipole antenna with N-turn of solenoid feed are shown in Fig. 8.The input admittance to the antenna can be represented in lumped circuit model [28] ( where The equivalent circuit is shown in Fig. 9, where Y mat repre- sents input admittance caused by internal dipole material, Y an is for wire antenna input admittance, and Y cap for external tuning capacitor.
For the same NiZn ferrite tile material with permittivity r = 14 − j0.14 and permeability r = 2000 , the resulting input is shown in Fig. 10 with 23.7 mF shunt capacitor.The antenna's resonance lies at 1 kHz.
In order to feed the dipole, a solenoid is used.The required current is found from (26) at approximate.It requires turn-current product NI = 14.3658A ⋅ turn , which is moderate current and can be easily achieved from copper wires.For example, the required current is around 1.44 A when using 10 turns of solenoidal feed.The wire of American Wire Gauge (AWG) number 7 has wire diameter of with maximum frequency of 1300 Hz.Increasing loop number can further reduce the current at the feeding.

Conclusion
We investigated multi-physics generated small dipoles including TM, TE dipoles and mechanically spinning of electret and magnet in homogeneous lossy medium at 76-1000 Hz.Using Poynting's theorem and energy conservation, we derived closed form equations for power loss and radiation efficiency r , bypassing previous statistics-like definition of P TM,90% , and related numerical calculations.
The study shows that the radiation efficiency, r , is proportional to |ka| 3 for TM and |ka| for TE dipole, respectively.For electrically small antenna of |ka| << 1 , the required source power for TE is only 1/350 of the TM dipole owing to the higher efficiency of TE.Dipoles of mechanically spinning electrets/magnets are smaller and lighter than the electromagnetically driven antennas in physical size and weight, but it may produce weaker field intensity in the far-zone.In addition, flying wheel is a must for a mechanical spinner at rotation speed of 60,000 cycles per minute.This will complicate the mechanical design and maintenance.From the comparative study, we recommend the use of TE-dipole based antennas.In section 6 of numerical examples, two prototype TE-dipole based antennas are provided, including feeding, tuning and matching circuits.Future work is to extend the homogeneous lossy medium to multi-layered ground-water-air inhomogeneous media, and to biomedical applications.

Fig. 1
Fig. 1 Dipole antenna surrounded by seawater.a Configuration of small dipole.b Feko model of TM dipole.c Feko model of TE dipole

Fig. 2
Fig. 2 Normalized impedance versus distance for E-dipole and H-dipole in: a air and b seawater

Fig. 3
Fig. 3 Comparison of E-and H-field between analytic solution with simulation results for TM and TE dipole antennas

( 26 )Table 3
I m dl = N 0 r I loop A (27) I e,m dl = Q e,m h Required source power for under seawater dipoles TM TE I e dl = 160.4AmI m dl = 7.12687Vm P TM = 6.324 × 10 3 W P TE = 17.2745WFig. 6 Moment of inertia for cylinder rotating along x-axis