Nonuniform Transmission Line Model for Electromagnetic Radiation in Free Space

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I. INTRODUCTION
HE electromagnetic radiation problem has been extensively investigated over the past century.Except several issues concerning with the electromagnetic powers and energies [1]- [3], the theoretic aspects of the electromagnetic radiation of antennas are well understood by researchers and engineers.In free space, the radiation fields of a current source distribution in a bounded region can be directly calculated with integrals involving the dyadic Green's function.These integrals generally have to be performed over the source region with some kind of numerical methods.However, in many practical applications, such as designing antennas with better directivity, higher efficiency or wider bandwidth, it may require more insightful information and interpretation of the radiation process than those the numerical solutions can provide.
Consider a source distribution   1 J r in a bounded region s V .
In a spherical coordinate system shown in Fig. 1(a), the vector spherical basis functions form a complete bases for the functional space of the radiation fields of the source [4]- [8].Let 0 S with a radius of 0 r denote the smallest sphere containing the source region.The electric and magnetic fields outside 0 S can be expanded with nm TM mode and nm TE mode, where n is the degree and m the order of the mode, respectively.The electromagnetic fields of the spherical mode can be explicitly expressed with a set of vector eigenfunctions.As can be found in many literatures [4]- [8], the fields of the spherical modes nm TM can be expressed as and for nm TE spherical modes, we have The coefficients m where Y is the harmonic function on the unit sphere, , , At region far away from the source, 1 kr  .Making use of the asymptotic expansions of the spherical Hankel functions for large argument, the far fields of the spherical modes can be approximately considered as spherical TEM waves with radiation patterns described by the corresponding spherical harmonic functions.However, the behavior of the near zone fields has not been revealed so intuitively.By numerically evaluating the amplitudes or some other quantities related to spherical Hankel functions [9]- [12], the fields of the mode with degree n may be roughly considered to decay exponentially with radius r if kr n  is satisfied.When applying fast multipole methods (FMM) for electromagnetic problems [12], the empirical criterion of

TM
modes are analyzed in [13] (where they are respectively referred to as n TE and n TM ).Each mode is equipped with a two-port equivalent circuit model.The input port of the equivalent circuit is 0 S , while the other port is S  , the concentric spherical surface when r   .S  is considered to be terminated with the intrinsic impedance  in free space and is perfectly matched.For 0 n TM modes, the normalized voltage and current are defined on the basis of the electric field and magnetic field on the input port 0 S ((7) in [13]), respectively, where n a is a mode-dependent constant and 0 0 u kr  .The normalized input impedance is then expressed using the ratio of the voltage and current, The 0 n TE modes were addressed in a similar way in [13].The radiation power, energies and Q factors of the mode are evaluated accordingly.Furthermore, the normalized impedance of 0 n TM or the normalized admittance of 0 n TE is expanded with a continuous fraction that can help us to establish an equivalent circuit consisting of cascaded inductances, capacitances and a termination of a unitary resistance.All the elements in the equivalent circuit are normalized with the intrinsic impedance of free space.This circuit model is very popular and has been widely used to predict the minimum radiation Q-factor of an antenna bounded in a small sphere.However, it mainly describes the characteristics on the input port 0 S and does not provide intuitive information about the radiation process from 0 S to S  .In this paper, a nonuniform transmission line (NTL) model for the radiation process from 0 S to S  is developed, in which 0 S serves as the input port and S  the matched terminal port.
The equivalent voltage and current at any concentric sphere with a radius r ( 0 r r    ) are properly defined.The governing differential equation is obtained and found to be similar to the conventional Telegraphers' equation [14] except that the equivalent distribution inductance and capacitance are dependent on kr .It can be interpreted as a kind of nonuniform transmission line or a nonuniform waveguide.The generalized input impedance at any position on the NTL can be obtained, and the normalized impedance defined by Chu is simply the special case at the input port 0 S , where 0 r r  .It will be shown that the NTL model can provide more insightful interpretation for the radiation process from 0 S to S  than conventional theories.

II. NONUNIFORM TRANSMISSION LINE MODEL
In order to develop the NTL model for the radiation problem, we will multiply the mode fields with r and find the relationship between the transverse electric field and the rotated transverse magnetic field.Let's consider nm TM mode at first.According to (2) and (3) , we can write π has been applied.The subscript  represents transverse component.Note that both r and u kr  are used in the derivation for the sake of brevity.Taking the derivative of both side of (10) with respect to u , and making use of the relationship [9] of we get Taking the derivative of both side of (11) with respect to u , we obtain Define the equivalent voltage and current on a concentric sphere with radius r as Then the electromagnetic fields can be expressed by It can be verified that the radiation power crossing the sphere can be correctly calculated with the voltage and current, Substituting ( 17) and ( 18) into ( 13) and ( 14) gives the governing differential equation for the Note that equation set (20) are with respect to the phase shift u instead of the radial distance r .It is valid for all 0 u u    .
Obviously, (20) is similar to the Telegraphers' equation that the voltage   V u and current   I u satisfy on a transmission line with distribution inductance   L u and capacitance   Comparing (20) with (21), we can introduce an equivalent distribution inductance and capacitance for nm TM mode as At a given frequency, the capacitance for nm TM is constant, while the inductance varies with u kr  .Therefore, (21) describes the voltage-current relationship on an NTL.The field propagation from 0 S to S  is then transformed to the voltage and current signal transmission on an NTL from 0 u to u  .Note that the inductance and capacitance are only dependent on the degree n of the mode.All nm TM modes of the same degree n share the same NTL model.
We borrow the concept in transmission line theory and define a local characteristic impedance for the NTL as Similarly, a local phase velocity is defined as, We have to take care that the phase velocity defined in ( 24) is with respect to the phase shift u .Since du dt k dr dt  , the conventional phase velocity is obtained to be where c is the light velocity in free space.
A typical plot of the local characteristic impedance and the phase velocity of an NTL for nm TM mode are shown in Fig. 2. We can summarize from Fig. 2 that: i) for u kr  , the characteristic impedance approaches  and the phase velocity approaches the light velocity c.
ii) for   2 1 u n n   , the characteristic impedance and the phase velocity are all real.It is an ordinary transmission line.The voltage and current signals will propagate on the line.However, the line is not a uniform transmission line (UTL) but an NTL.Generally, an NTL can be approximately considered as a section-wise-uniform transmission line, i.e., cascaded by short UTL sections with different characteristic impedances.Therefore, local reflections will inevitably occur [15] [16].
iii) for   2 1 u n n   , the characteristic impedance and the phase velocity are all imaginary, as can be seen from ( 22).We can treat it as a nonuniform transmission line with negative inductance and positive capacitance.For a transmission line with a constant negative inductance and a constant positive capacitance, the general solution of (21 , is real.The voltage and current on the line are evanescent.They will not propagate like waves but decay exponentially.This is similar to the case that has been widely analyzed in handling left-handed materials or transmission lines [17] [18] when there is a negative permittivity or a negative permeability.The transmission line has similar behavior when the negative inductance and positive capacitance are position-dependent.

Propagating zone Evanescent zone
is real  is imaginary  represents the critical state between the propagating and the evanescent state.We can consider the space between 0 S and S  as a two-port radial waveguide.Therefore, it is natural to introduce a cutoff spatial phase shift c u as .Otherwise, it remains to be an evanescent mode till it reaches a larger sphere that meets the propagation condition.Therefore, for a sphere with radius r , we can define a cutoff mode degree c n for nm TM mode, which is the largest integer satisfies, We may take the largest integer smaller than the right-hand side of (29).The S to send the same amount of radiation power to S  , in other words, it is generally less efficient to excite mode with larger degree.
In order to efficiently excite a nm TM mode, we may choose 0 c u u  so that the evanescent zone and the high local reflection zone are excluded in the propagating path of that mode, intuitively speaking, we may put 0 S to the right side of the red line in Fig. 3.The NTL model for nm TE modes can be developed similarly.Starting from (4) and ( 5), we can write It can be verified that Propagating zone Evanescent zone Making use of ( 12), the Telegrapher's equation for Accordingly, the equivalent inductance and capacitance are Note that the capacitance varies with u while the inductance is constant at a given frequency, which is different from that for nm TM mode.
The local characteristic impedance for The curves of the characteristic impedance of nm TE mode for n=5 and n=15 are plotted in Fig. 5.Note that at the cutoff point the characteristic impedance of the nm TM mode is zero while it is infinite for nm TE mode.
It can be readily checked that the normalized impedance (9) defined by Chu [13] is exactly the special case of (38) at the input port ( For nm TE mode, the generalized input admittance at point u is defined as It is reasonable to take the intrinsic impedance  as the reference impedance to define a generalized reflection coefficient at u , where   in Z u is the generalized input impedance of a spherical wave mode.In the evanescent regions, the reflection coefficient is very large.Most of the electromagnetic power carried by the mode will be reflected, so the radiation efficiency tends to be very low.A Hertzian dipole generates 10 TM mode field in the free space.We can check that the equivalent voltage and current on the NTL are respectively, The cutoff radius of  By introducing the cutoff spatial phase shift c c u kr  , the traveling path of a spherical mode can be divided into an evanescent zone and a propagating zone.Although the amplitudes of the mode fields will decay in both regions, the mechanism and behaviors in the two regions are quite different.
The NTL model does not provide information about how the feeding source distributes its electromagnetic power to each spherical mode.However, it can help us to predict which modes can effectively penetrate the barrier in the near region and propagate to far region.

Fig. 1
Fig.1 The electromagnetic radiation.(a) a source in a bounded region.(b)antenna with symmetrical structure.


. k is the wavenumber and  is the intrinsic impedance in free space.u kr  this paper.The spherical coordinate r ,  , and  are defined in their usual way, and their unit vectors are respectively denoted by r , θ , and φ .The normalized vector harmonics are expressed asNonuniform Transmission Line Model for Electromagnetic Radiation in Free SpaceGaobiao Xiao, Senior Member, IEEE


is the associated Legendre function.The normalized spherical harmonics form a complete set of orthogonal basis functions, number of spherical modes for a given distance 0 r and an accuracy-related constant  .An equivalent circuit model was proposed by Chu in 1948[13].As shown in Fig.1(b), an antenna with symmetrical structure is completely contained in the sphere 0

Fig. 2
Fig. 2 Typical characteristic impedance and phase velocity of  , at a given frequency, we can accordingly define a cutoff radius c c r u k  associated with a cutoff sphere.The mode nm TM is an evanescent one inside the cutoff sphere and becomes a propagating one outside the cutoff sphere.We can then write the propagation condition as, On the other hand, for an arbitrarily specified concentric sphere with radius r , nm TM is a propagating mode outside the sphere if its degree n satisfies     2 1 n n k r   Fig. 3 NTL model for for nm TM mode.n=5, 15, 25.(a)characteristic impedance; (b) phase velocity.

Fig. 5
Fig. 5 Characteristic impedance of nm TE mode for n=5 and 15.

Fig. 6
mode.Making use of (19) and (38), we obtain that the radiation resistance of nm TM mode.It can be used to predict the radiation capability of the mode.Large radiation capability.The generalized input impedance of nm TM mode for n=5, 15, and 25 are plotted in Fig.6, where the dash-line indicate the cutoff phase shift for the corresponding mode.The input resistances in Fig.6(a) clearly reveal that the radiation of a mode is weak when 0 Generalized input impedance of nm TM mode for n=5, 15 and 25.(a) real part.(b) imaginary part.
Furthermore, (38) and(40)   show that mathematically the normalized input impedance of the nm TM mode equals to the normalized input admittance of the nm TE mode, i.e., is exactly the same expression used by Chu to develop the equivalent circuit model.Therefore, if necessary, the type of Chu's equivalent circuit model can be developed for all nm TM and nm TE mode at any spheres between 0 S and S  .
in Fig.8shows the evanescent region of the 10 TM mode.The fields of the Hertzian dipole in this area will decay exponentially.As is well known, the Hertzian dipole is not an efficient radiator.
have two basic methods to improve the radiation efficiency.The first one is to efficiently guide the electromagnetic power outside the cutoff region using some kind of metal structures.A typical structure is the half wavelength dipole shown in Fig.8.Its end reaches 0 S with 0 4 r   , which is apparently locating in the propagating region of the 10 TM mode.Because of the two metal arms of the dipole, spherical TEM mode can exist within 0 S .The transmission of spherical TEM mode can be modelled using a UTL with cutoff radius of 0 c r  [14].It has the potential to efficiently transfer electromagnetic power from the feeding point to 0 S .The second method is to fill the region surrounding the feeding source with some kind of dielectrics.For a dielectric with relative permittivity of 1 r   , the cutoff radius of all spherical modes will decrease by a factor of r  .The radiation efficiency may be improved due to the shrink of the evanescent region.IV.CONCLUSIONSThe proposed NTL model for electromagnetic radiation of sources in a bounded region can provide an intuitive way for illustrating the propagation process of mode fields.As the NTL model is independent on the order m of the spherical modes,