AXIALLY SYMMETRIC TRANSIENT ELECTROMAG- NETIC FIELDS … · solve problems of electromagnetic waves excitation and propagation in waveguides [17{28], cavities [29{33], free space,

Progress In Electromagnetics Research B, Vol. 48, 375–394, 2013

AXIALLY SYMMETRIC TRANSIENT ELECTROMAG-NETIC FIELDS IN A RADIALLY INHOMOGENEOUS BI-CONICAL TRANSMISSION LINE

Bogdan A. Kochetov1, * and Alexander Yu. Butrym2

1Department of Microwave Electronics, Institute of Radio Astronomyof the National Academy of Sciences of Ukraine, 4, KrasnoznamennayaStr., Kharkov 61002, Ukraine2Department of Theoretical Radio Physics, Karazin Kharkov NationalUniversity, 4, Svobody Sq., Kharkov 61022, Ukraine

Abstract—In the present paper a novel mathematical model ofphysical processes of transient electromagnetic waves excitationand propagation in a biconical transmission line with radiallyinhomogeneous magneto-dielectric filling is proposed. The model isbased on time domain mode expansions over spherical waves. Thebasis functions of the mode expansions are calculated analytically.The mode expansion coefficients are governed by Klein-Gordon-Fockequation with coefficients depending on a radial spatial coordinate.The explicit finite difference time domain computational scheme isderived to calculate the mode expansion coefficients. Dependencesof cutoff frequencies of higher modes of TE and TM waves on theline geometry and dielectric filling are studied. In order to calculateelectromagnetic field in the line with higher accuracy, just finitenumber of terms in the mode expansions is required. Electromagneticfield excited by the transient electric ring current is calculated in bothhomogeneous and radially inhomogeneous biconical transmission line.It is shown that there is a possibility to increase the bandwidth of theline via introduction of partial dielectric filling without changing theline geometrical size.

1. INTRODUCTION

Various conical and biconical metallic conducting structures are keybodies for radar applications. A surface of such objects contains

Received 13 January 2013, Accepted 12 February 2013, Scheduled 13 February 2013* Corresponding author: Bogdan A. Kochetov ([email protected]).

376 Kochetov and Butrym

the typical singularity and can be simply described analytically. Infact, many problems of electromagnetic scattering on conical-likebodies have been solved. Some types of antennas based on biconicallines are proposed for transient electromagnetic waves radiating andreceiving [1–6]. In order to improve or develop some biconical antennait is necessary to understand the evolution of electromagnetic fieldsinside it. The paper [7] is one of the first works where the generaltreatment of the biconical antenna theory is given. Further, theelectromagnetic properties and characteristics of the biconical antennahave been investigated in [5, 6, 8]. In papers [9, 10] a problem oftransient TEM wave excitation and diffraction in the biconical antennahas been solved in the time domain using an approach based onmode matching incorporated into the finite difference time steppingscheme for the mode channels. The problems of electromagnetic wavesexcitation in complex structures of slotted cones have been solvedrigorously in [11, 12].

It is known that waveguides with partial dielectric filling havericher electromagnetic properties than hollow ones. One can seethat introducing a partial dielectric filling in the biconical line canlead to improving electromagnetic parameters of the line. In thepaper [13] the eigenmodes in a layered biconical transmission line arefound. The radiation of elementary electric dipole with harmonic time-dependences isolated by a dielectric sphere is calculated with using thecritical sections of inhomogeneous horn conception in [14]. On thebasis of previous results [13, 14] a new type of compact UWB antennasis designed in [15].

In the studies mentioned above the authors consider the excitationof the fundamental TEM wave which has no dispersion in the biconicalline. Nevertheless higher TE and TM modes can be also excited in abiconical transmission line. So, goal of the present paper consists in thestudy of processes of excitation and propagation of the transient TE,TM and TEM electromagnetic waves in a biconical transmission linewith some radial-inhomogeneous magneto-dielectric filling. Generally,there are two suitable approaches in the transient electromagnetictheory: Finite Difference Time Domain method (FDTD) [16] andmethods based on Mode Expansions in Time Domain (METD) [17–44].FDTD is most general and universal approach used in the time domainelectromagnetics. It can be applied to solve problems with arbitrarygeometry. METD is more special method, which is appropriate tosolve problems of electromagnetic waves excitation and propagationin waveguides [17–28], cavities [29–33], free space, some other mediaand structures [34–39]. Also METD is developed for studying thetransient electromagnetic processes in conical transmission lines [40–

Progress In Electromagnetics Research B, Vol. 48, 2013 377

44]. In order to use METD for solving electromagnetic pulse diffractionproblems on a junction of two regular transmission lines, ModeMatching Time Domain Method was developed [9, 10].

In the present paper a new mathematical model based on METDis proposed to study the processes of transient electromagnetic wavesexcitation and propagation in an asymmetrical biconical transmissionline with radial-inhomogeneous magneto-dielectric filling [42, 43]. Inthe framework of the proposed model the process of transient TE waveexcitation by the ring electric current in both homogeneous and radiallyinhomogeneous biconical line is analyzed. Also transformations ofsignal spectrum in the line are considered. It is shown that introductionof a partial dielectric filling in the biconical line increases its excitationefficiency. Some results of the present work were previously reportedin [44].

2. PROBLEM STATEMENT

A regular asymmetrical biconical transmission line is considered inFigure 1. Two circular cones with the common vertex and axisform this line. In the spherical coordinate system the surfaces ofthe upper and lower cones are defined via the equations θ = θ1

and θ = θ2, respectively. On the surface of the cones the boundaryconditions are imposed that correspond to perfect electric conductor(PEC). The space between cones is filled with a radially inhomogeneousmagneto-dielectric medium. It means that the relative permittivityand permeability depend on the radial coordinate only ε = ε(r),µ = µ(r). The constraints for permittivity and permeability are givenas: ε ≥ 1, µ ≥ 1. The electromagnetic field in the line is excited

PEC

z

x

y

ε (r)

µ (r)

θ

θ

1

2

Figure 1. An asymmetrical radially inhomogeneous biconical line.


with certain transient electric ~J (~r, t) and magnetic ~J (~r, t) currentsand electric ρ(~r, t) and magnetic ρ(~r, t) charges. The electric ~E(~r, t)and magnetic ~H(~r, t) field strengths in the line are to be found.

The electromagnetic fields are governed by Maxwell equations:

∇× ~H =∂ ~D∂t

+ ~J , ∇× ~E = −∂ ~B∂t− ~J , ∇· ~D = ρ, ∇· ~B = ρ, (1)

with taking into account the constitutive equations:~D (~r, t) = ε0ε (r) ~E (~r, t) , ~B (~r, t) = µ0µ (r) ~H (~r, t) . (2)

The boundary conditions on the cone surfaces are:

~ϕ0 · ~E∣∣∣L

= 0, ~θ0 · ~H∣∣∣L

= 0, ~r0 · ~E∣∣∣L

= 0. (3)

Here ε0 and µ0 are permittivity and permeability of free space,respectively; ~r0, ~ϕ0 and ~θ0 are orts of the spherical coordinate system.

3. PROBLEM SOLUTION

In order to find the electromagnetic field in the line Mode Basis Method(O. A. Tretyakov method) for conical transmission lines [42–44] isapplied here. In the framework of the method, the electric ~E(~r, t) andmagnetic ~H(~r, t) field strengths are presented as the field expansionsover spherical TE, TM and TEM waves:

~E (~r, t) = ~E (~r, t) + ~r0Er (~r, t) ,

~H (~r, t) = ~H (~r, t) + ~r0Hr (~r, t) ,

√ε0

~E (~r, t) =1r

(∑m

eHm (r, t) ~EH

m (θ, ϕ)+∑

n

eEn (r, t) ~EE

n (θ, ϕ)

+∑

k

eTk (r, t) ~ET

k (θ, ϕ)

),

√µ0

~H (~r, t) =1r

(∑m

hHm (r, t) ~HH

m (θ, ϕ)+∑

n

hEn (r, t) ~HE

n (θ, ϕ)

+∑

k

hTk (r, t) ~HT

k (θ, ϕ)

),

√ε0Er (~r, t) =

1r2

∑n

ern (r, t) qnΦE

n (θ, ϕ) ,

√µ0Hr (~r, t) =

1r2

∑m

hrm (r, t) pmΦH

m (θ, ϕ).

(4)


Vector and scalar functions that depend on angular coordinates θ, ϕare the basis functions. Scalar functions that depend on the radialcoordinate r and time t are coefficients of the expansions. Thesecoefficients are often called as mode amplitudes. The superscriptsindicate a type of wave. The index H shows that the basis functionsand mode amplitudes correspond to the transverse electric (TE) waves.Similarly, the index E indicates the transverse magnetic (TM) wavesand index T corresponds to the transverse electromagnetic (TEM)waves. The quantities pm and qn are spectral parameters of eigenvalueboundary problems [43]. For simplicity we consider symmetricalelectromagnetic fields and sources that do not depend on the angularcoordinate ϕ.

TE-waves. The basis functions for TE waves have the form:

ΦHm (θ)= CH

m [Pνm (cos θ) + AmPνm (− cos θ)] ,

~EHm(θ)= − ~ϕ0C

Hm

√νm+1

νm

1sin θ

[Pνm+1 (cos θ)− cos θPνm (cos θ)

−Am (Pνm+1 (− cos θ) + cos θPνm (− cos θ))] ,

~HHm (θ)= ~θ0C

Hm

√νm + 1

νm

1sin θ

[Pνm+1 (cos θ)− cos θPνm (cos θ)

−Am (Pνm+1 (− cos θ) + cos θPνm (− cos θ))] ,

(5)

where spectral parameters νm are the roots of the equation:

dPνm (cos θ)dθ

∣∣∣∣θ=θ1

dQνm (cos θ)dθ

∣∣∣∣θ=θ2

− dPνm (cos θ)dθ

∣∣∣∣θ=θ2

dQνm (cos θ)dθ

∣∣∣∣θ=θ1

= 0 (6)

and the normalized constants are defined as follow:

Am = −dPνm (cos θ)

dθ

∣∣∣θ=θ1

dPνm (− cos θ)dθ

∣∣∣θ=θ1

,

CHm =

√√√√√√2

θ2∫θ1

[Pνm (cos θ) + AmPνm (− cos θ)]2 sin θdθ

.

Here m = 1, 2, 3, . . . are numbers of TE-waves; Pν(·) and Qν(·) areLegendre functions of the first and second kinds respectively andpm =

√νm(νm + 1). The basis functions satisfy the orthogonality


conditions [43]:

12

θ2∫

θ1

ΦHm (θ)ΦH

m′ (θ) sin θdθ = δmm′ ,

12

θ2∫

θ1

~EHm (θ) · ~EH

m′ (θ) sin θdθ = δmm′ ,

12

θ2∫

θ1

~HHm (θ) · ~HH

m′ (θ) sin θdθ = δmm′ .

(7)

The mode amplitudes are governed by the following system ofevolutionary equations:

ε(r)µ(r)∂2eH

m

∂τ2− ∂2eH

m

∂r2+

dµ

dr

1µ(r)

∂eHm

∂r+

p2meH

m

r2= JH

1 (r, τ) ,

∂hHm

∂τ= − 1

µ(r)∂eH

m

∂r+ JH

2 (r, τ) ,

∂hrm

∂τ= − 1

µ(r)eHm + JH

3 (r, τ) .

(8)

Known sources of the electromagnetic field have form:

JH1 (r, τ) = −µ(r)

∂

∂r

r

µ(r)

√ε0

pm

14π

2π∫

0

θ2∫

θ1

(∇⊥ · ~

J)

ΦHm (θ) sin θdθdϕ

+14π

2π∫

0

θ2∫

θ1

{r

∂

∂τ

(µ(r)

√µ0

[~J × ~r0

])−√ε0∇⊥Jr

}

· ~HHm (θ) sin θdθdϕ,

JH2 (r, τ) =

r

µ(r)

√ε0

pm

14π

2π∫

0

θ2∫

θ1

(∇⊥ · ~

J)

ΦHm (θ) sin θdθdϕ,

JH3 (r, τ) = − r2

µ(r)

√ε0

pm

14π

2π∫

0

θ2∫

θ1

JrΦHm (θ) sin θdθdϕ.

Here τ = ct, c is the light velocity in vacuum, the dot and thecross indicate the dot product and cross product respectively, ∇⊥ =


~θ0∂θ + ~ϕ0(sin θ)−1∂ϕ is a part of Hamilton operator “nabla” in thespherical coordinate system.

TM-waves. The basis functions have the form:ΦE

n (θ) = CEn [Pχn (cos θ) + BnPχn (− cos θ)] ,

~EEn (θ) = ~θ0C

En

√χn + 1

χn

1sin θ

[Pχn+1 (cos θ)− cos θPχn (cos θ)

−Bn (Pχn+1 (− cos θ) + cos θPχn (− cos θ))] ,

~HEn (θ) = ~ϕ0C

En

√χn + 1

χn

1sin θ

[Pχn+1 (cos θ)− cos θPχn (cos θ)

−Bn (Pχn+1 (− cos θ) + cos θPχn (− cos θ))] ,

(9)

where the spectral parameters χn are the roots of the equation:Pχn (cos θ1) Qχn (cos θ2)− Pχn (cos θ2)Qχn (cos θ1) = 0 (10)

and the normalized constants are defined as follow:

Bn = − Pχn (cos θ1)Pχn (− cos θ1)

,

CEn =

√√√√√√2

θ2∫θ1

[Pχn (cos θ) + BnPχn (− cos θ)]2 sin θdθ

.

Here n = 1, 2, 3, . . . are numbers of TM-waves and qn =√

χn(χn + 1).The basis functions satisfy the orthogonality conditions:

12

θ2∫

θ1

ΦEn (θ)ΦE

n′ (θ) sin θdθ = δnn′ ,

12

θ2∫

θ1

~EEn (θ) · ~EE

n′ (θ) sin θdθ = δnn′ ,

12

θ2∫

θ1

~HEn (θ) · ~HE

n′ (θ) sin θdθ = δnn′ .

(11)


ε(r)µ(r)∂2hE

n

∂τ2− ∂2hE

n

∂r2+

dε

dr

1ε(r)

∂hEn

∂r+

q2nhE

n

r2= JE

1 (r, τ) ,

∂eEn

∂τ= − 1

ε(r)∂hE

n

∂r+ JE

2 (r, τ) ,


∂ern

∂τ= − 1

ε(r)hE

n + JE3 (r, τ) . (12)

Known sources of the TM electromagnetic waves have form:

JE1 (r, τ) = −ε(r)

∂

∂r

r

ε(r)

√µ0

qn

14π

2π∫

0

θ2∫

θ1

(∇⊥ · ~J

)ΦE

n (θ) sin θ dθdϕ

+14π

2π∫

0

θ1∫

θ1

{r

∂

∂τ

(ε(r)

√ε0

[~r0 × ~

J])−√µ0∇⊥Jr

}

· ~EEn (θ) sin θdθdϕ,

JE2 (r, τ) =

r

ε(r)

√µ0

qn

14π

2π∫

0

θ2∫

θ1

(∇⊥ · ~J

)ΦE

n (θ) sin θdθdϕ,

JE3 (r, τ) = − r2

ε(r)

√µ0

qn

14π

2π∫

0

θ2∫

θ1

JrΦEn (θ) sin θdθdϕ.

TEM-waves. The basis functions have form:

~ET (θ) =

√√√√ 2

ln(

1−cos θ21−cos θ1

sin θ1sin θ2

)~θ0

sin θ, ~HT (θ) =

√√√√ 2

ln(

1−cos θ21−cos θ1

sin θ1sin θ2

) ~ϕ0

sin θ.

(13)The basis functions satisfy the orthogonality conditions:

12

θ2∫

θ1

~ET (θ) · ~ET (θ) sin θdθ=1,12

θ2∫

θ1

~HT (θ) · ~HT (θ) sin θdθ=1. (14)


ε(r)µ(r)∂2eT

k

∂τ2− ∂2eT

k

∂r2+

dµ

dr

1µ(r)

∂eTk

∂r=

14π

2π∫

0

θ2∫

θ1

{~f1 (r, θ, ϕ, t)

r2

−~f2 (r, θ, ϕ, t)

r2ε(r)−µ(r)

∂

∂r

(1

r2ε(r)µ(r)

) r∫

0

~f2

(r′, θ, ϕ, t

)dr′

· ~HTk sin (θ) dθdϕ,


∂hTk

∂τ=− 1

µ(r)∂eT

k

∂r+

1r2ε(r)µ(r)

14π

2π∫

0

θ2∫

θ1

r∫

0

~f2

(r′, θ, ϕ, t

)dr′

· ~HTk sin (θ) dθdϕ.

(15)

where ~f1(r, θ, ϕ, t) = r3∂τ (µ‖[µ1/20

~J×~r0])−r2∇⊥ε1/20 Jr, ~f2(r, θ, ϕ, t) =

r2[~r0 ×∇⊥ε−1/20 ρ]− ∂r(r3ε‖ε

1/20

~J).

4. NUMERICAL RESULTS

In the current section the proposed mathematical model is applied forstudying a process of transient TE waves excitation in the biconicalline with a radial-inhomogeneous dielectric filling. In order to excitethe electromagnetic waves in the line a transient ring electric currentis used. The current has the form:

~J (~r, t) = ~ϕ0δ (r −R) δ(θ − π

2

)f (t) . (16)

Here δ(·) is the Dirac delta function, f(t) an arbitrary time-depending function, and R the ring radius. From right hand sidesof Equations (8), (12) and (15) one can see that the current (16) reallyexcites TE waves only. In subsequent calculations the time-dependingfunction in a form of Laguerre pulse is chosen:

f (t) =(

t

T

)2 (1− t

3T

)exp

(− t

T

)η (t) . (17)

where η(t) is the Heaviside step function, T is a scaling parameter. Theform of the Laguerre pulse for T = 33.36 ps is presented in Figure 2.Also the spectrum of this signal is shown in Figure 3.

Figure 2. The Laguerre pulse. Figure 3. The spectrum ofLaguerre pulse.


The authors of work [6] recommend using the biconical antennawith flare angle of 60◦ as a reference antenna for ultrawideband 2–12GHz applications. The antenna length should be approximatelyequal to one wavelength of the central frequency. For this reason theparameters of the line and the excitation current are chosen as follow:θ1 = 60◦, θ2 = 120◦, T = 33.36 ps and R = 0.5 cm.

The discussion about convergence of expansions over sphericalwaves (4) is given in the paper [45] in detail. It was found that sphericalexpansions have a fast convergence and number of main terms in theseries (4) is defined by upper frequency in the signal spectrum and thegeometrical sizes of electromagnetic field sources.

In the framework of the proposed model, the electromagneticfield in the biconical line is calculated (see Equations (4)–(5)) via thefollowing relations:

~E (r, θ, t) =−~ϕ0√

ε0r sin θ

∞∑

m=1

eHm (r, t) CH

m

√νm + 1

νm[Pνm+1 (cos θ)

− cos θPνm (cos θ)−Am (Pνm+1 (− cos θ)+cos θPνm (− cos θ))] , (18)

~H (r, θ, t) =~θ0√

µ0r sin θ

∞∑

m=1

hHm (r, t) CH

m

√νm + 1

νm[Pνm+1 (cos θ)

− cos θPνm (cos θ)−Am (Pνm+1 (− cos θ)+cos θPνm (− cos θ))] , (19)

Hr (r, θ, t) =1√µ0r2

∞∑

m=1

hrm (r, t) CH

m

√νm (νm + 1) [Pνm (cos θ)

+AmPνm (− cos θ)] . (20)

The mode amplitudes eHm(r, t), hH

m(r, t) and hrm(r, t) satisfy to

Equation (8). Equation (8) is written in the convenient form to derivean explicit finite-difference scheme. A uniform grid u(r, t) → u|ji :=u(i∆r, j∆t) are used. Here ∆r and ∆t are steps in radial spatialcoordinate and time, respectively. The derivatives are calculatedapproximately via central finite differences:

∂ u|ji∂r

≈ u|ji+1 − u|ji−1

2∆r,

∂2 u|ji∂r2

≈ u|ji+1 − 2 u|ji + u|ji−1

∆r2,

∂ u|ji∂t

≈ u|j+1i − u|j−1

i

2∆t,

∂2 u|ji∂t2

≈ u|j+1i − 2 u|ji + u|j−1

i

∆t2.

(21)

Taking into account the current (16), Equation (21) and conditionµ(r) = 1, we arrive to the following an explicit finite-difference scheme


for system of differential Equations (8):

eHm

∣∣j+1

i=αm

i eHm

∣∣ji+ βi

(eHm

∣∣ji+1

+ eHm

∣∣ji−1

)

− eHm

∣∣j−1

i+ Fm (i∆r, j∆τ) ,

hHm

∣∣j+1

i= hH

m

∣∣j−1

i− c∆t

∆r

(eHm

∣∣ji+1

− eHm

∣∣ji−1

),

hrm|j+1

i = hrm|j−1

i − 2c∆t eHm

∣∣ji.

(22)

where Fm(r, t) = c√

µ0∆t2

2ε(r) rδ(r −R)df(t)dt CH

m

√νm+1

νmPνm+1(0)(1−Am),

αmi = 2− c2∆t2

∆r2 ε|i (2 + νm(νm+1)i2

), βi = c2∆t2

∆r2 ε|i .In order to verify the developed computational scheme we solve a

simple problem about radiation of the ring current (16) in a vacuum.In this case ε(r) = 1, νm = m, CH

m =√

2m + 1, Am = 0 and from (18)the solution for electric field has form:

~E (r, θ, t) =~ϕ0√

ε0r sin (θ)

∞∑

m=1

√m (m + 1)2m + 1

eHm (r, t) [Pm−1 (cos (θ))

−Pm+1 (cos (θ))] . (23)

The mode amplitudes eHm(r, t) are calculated numerically using the

finite-difference scheme (22). This problem also has the analyticalsolution:

~E (r, θ, t)= ~ϕ0µ0R

2

4π

2π∫

0

f ′(

t−√

r2+R2+2rR sin(θ) cos(ξ)

c

)

√r2+R2+2rR sin (θ) cos (ξ)

cos(ξ)dξ, (24)

where the prime in Equation (24) means differentiation on theargument.

In Figure 4 the time-dependences of electric field magnitude in thefree space are presented. They are calculated at two different spacepoints by means of both mode basis method (23) and Equation (24).In Figure 4 as well as in the figures below an axis of abscissa indicatesa real time multiplied by the light velocity in vacuum. In order tocalculate the time-dependences, the first ten nonzero terms are takeninto account in series (23) only. One can see that keeping the firstten spherical waves provides a high accuracy in the calculations ofelectromagnetic fields in the free space (bi-conical line is absent). Asmentioned in [44, 45], mode expansions (18)–(20) of the field in thebiconical line converge if they converge for same sources of the fieldin the free space. Thus the first ten nonzero spherical waves are more


Figure 4. Time-dependences of electric field magnitude in vacuum.

Table 1. Spectral parameter.

m νm m νm

1 2.627061 11 32.512523 8.545464 13 38.5105965 14.527461 15 44.5091857 20.519652 17 50.5081059 26.515296 19 56.507252

than enough to calculate electromagnetic fields in the biconical linewith high accuracy and we are assured that the computational schemeon the basis of (18)–(20), (22) is correct.

In order to use the solution for electromagnetic field in thebiconical line (18)–(20) it is necessary to calculate spectral parameter ofthe line νm for desired angles via Equation (6). For example, in Table 1the spectral parameter νm for the first ten odd spherical waves (theeven spherical waves are not excited by the current (16)) is presentedfor biconical line with angles θ1 = 60◦ and θ2 = 120◦.

In Figure 5 the time-dependences of electric field magnitude in thehollow biconical line with angles θ1 = 60◦ and θ2 = 120◦ are presentedat different distances from the current (16). The dependences arecalculated directly at θ = 90◦. Figure 5 shows the pulse waveformtransformation with increasing of the radial spatial coordinate. Thewaveform transformation appears due to two factors. The first one is aspherical attenuation of the wave as 1/r. The second one consists in the


existence of TE wave’s dispersion. The spherical attenuation followsfrom the energy conservation law and exists at any space point. Butaction of the dispersion attenuation appears near the sources of waves.In the upper panel of Figure 5 one can see that wave attenuationoccurs faster than 1/r, as well as the qualitative transformation ofthe pulse waveform takes place. As can be seen in the lower panelof Figure 5 after distance r ≈ 4R the pulse waveform is transformed

Figure 5. Time-dependences of electric field magnitude in biconicalline without filling.

Figure 6. Transformation of the electric field spectrum in thebiconical line without filling.


due to spherical attenuation only. It means that in space r > 4R allspherical modes excited by current (16) become propagating waves.The transformation of the pulse waveform leads to transformationof pulse spectrum also. In Figure 6 the spectra of the electric fieldmagnitudes in the biconical line are shown. The calculated spectracorrespond to TE pulse waves which are presented in Figure 5. As canbe seen in Figure 6 the transformation of the spectrum also confirmsthe previous result where essential influence of dispersion TE wavesappears only in the bounded space domain near the field source. Asseen in the lower panel of Figure 6 the spectrum magnitude nearfrequency 60GHz goes to zero. It means that directivity pattern ofthe biconical line have zero at direct θ = 90◦ at this frequency. Zerosof the directivity pattern depend on the angles θ1, θ2 and the linefilling.

It is obviously that the line filling with a dielectric leads toincreasing electrical sizes without changing the geometrical ones. Thusthe dielectric introducing in the line is a way to improve its radiationcharacteristics. Below we consider the biconical line with radial-inhomogeneous dielectric filling. The permittivity has a piecewiseconstant profile:

ε(r) ={

3, r ≤ 6R1, r > 6R

. (25)

The time-dependence of the electric field in the radial-inhomogeneous biconical line with the dielectric filling (25) and sameparameters of the line and excited current as in previous calculations isshown in Figure 7. The field is calculated at r = 11R and θ = 90◦. Theradiated field consists of series of separated decreasing pulses. Theyappear due to reflections at the dielectric interface and in the vicinityof vertex of the cones.

The first radiated pulse from dielectric ball is shown in the upperpanel of Figure 8 in the details. In the lower panel of Figure 8 its

Figure 7. Time-dependence of electric field magnitude in the radial-inhomogeneous biconical line.


Figure 8. The first radiated pulse from dielectric ball in biconical lineand its spectrum.

Figure 9. Comparison of radiated pulses in biconical line withdifferent dielectric filling and its spectra.

spectrum is presented. As can be seen in Figure 8 the first pulse hasonly uniform spherical attenuation as 1/r.

In the upper panel of Figure 9 the radiated pulse in the biconicalline without filling ε(r) = 1 is compared with the first radiated pulsein the line with radial-inhomogeneous filling having form (25). Thetime-dependences of electric field magnitudes are calculated at samespace point (r = 7R, θ = 90◦). In the lower panel of Figure 9 the


spectra of such pulses are shown. The first radiated pulse (some partof total radiated field) from dielectric ball has higher amplitude thanthat radiated one in the hollow biconical line. It is due to in thedielectric ball excited pulse has amplitude up to

√ε =

√3 times higher

than in the vacuum. Also the first pulse has expected time delay5R(

√ε− 1) = 0.018 m. In Figure 9 the curves show a possibility to

increase a radiation efficiency of the biconical line by means of partialdielectric filling.

5. CONCLUSIONS

In the present paper a new mathematical model of processesof excitation and propagation of axially symmetrical transientelectromagnetic fields in the biconical line with radial-inhomogeneousmagneto-dielectric filling is proposed. On the basis of the modelan efficient computational scheme is obtained. The validity of thescheme is verified for the limit case of free space. In the frameworkof the scheme the transient TE electromagnetic wave is calculatedboth in the hollow biconical transmission line and in the line withpiecewise constant dielectric filling along radial direction. A possibilityto increase the radiation efficiency of compact electromagnetic fieldsources in the biconical line by means of partial dielectric filling isshown. The developed computational scheme can be used to find anoptimal magneto-dielectric filling for asymmetrical biconical lines andantennas based on such lines.

REFERENCES

1. Schantz, H., The Art and Science of Ultrawideband Antennas,Artech House Publishers, 2005.

2. McLean, J., U. Trucchi, J. Sivaswamy, and R. Sutton,“Development of a precision biconical antenna for broadbandmetrology applications,” Proc. IEEE International Symposium onElectromagnetic Compatibility, 529–534, Washington, DC, USA,Aug. 21–25, 2000.

3. Palud, S., F. Colombel, M. Himdi, and C. L. Meins, “A novelbroadband eighth-wave conical antenna,” IEEE Transactions onAntennas and Propagation, Vol. 56, No. 7, 2112–2116, Jul. 2008.

4. Zhou, S.-G., J. Ma, J.-Y. Deng, and Q.-Z. Liu, “A low-profileand broadband conical antenna,” Progress In ElectromagneticsResearch Letters, Vol. 7, 97–103, 2009.

5. Amert, A. K. and K. W. Whites, “Miniaturization of the biconical


antenna for ultrawideband applications,” IEEE Transactions onAntennas and Propagation, Vol. 57, No. 12, 3728–3735, Dec. 2009.

6. Kudpik, R., N. Siripon, K. Meksamoot, and S. Kosulvit, “Designof a compact biconical antenna for UWB applications,” Proc.International Symposium on Intelligent Signal Processing andCommunications Systems (ISPACS), 1–6, Chiang Mai, Thailand,Dec. 7–9, 2011.

7. Tai, C. T., “On the theory of biconical antennas,” Journal ofApplied Physics, Vol. 19, 1155–1160, 1948.

8. Makurin, M. N. and N. P. Chubinskiy, “Calculation of propertiesof biconical antenna by the method of partial domains,”Radiotekhnika i Electronika, Vol. 52, No. 10, 1199–1208, Oct. 2007(in Russian).

9. Butrym, A. Y., B. A. Kochetov, and M. N. Legenkiy, “Numericalanalysis of simply TEM conical-like antennas using modematching in time domain,” Proc. 3rd European Conference onAntennas and Propagation (EuCAP 2009), 1–4, Berlin, Germany,Mar. 23–27, 2009.

10. Legenkiy, M. N. and A. Y. Butrym, “Method of mode matching intime domain,” Progress In Electromagnetics Research B, Vol. 22,257–283, 2010.

11. Semenova, E. K. and V. A. Doroshenko, “Electromagneticexcitation of PEC slotted cones by elementary radial dipoles —A semi-inversion analysis,” IEEE Transactions on Antennas andPropagation, Vol. 56, No. 7, 1976–1983, Jul. 2008.

12. Doroshenko, V. A., A. P. Blishun, Y. D. Shimuk, and N. G. Zuev,“Singular integral equations method in mathematical modeling ofspecific open conical structure excitation,” Proc. 8th InternationalConference on Antenna Theory and Techniques (ICATT 2011),254–256, Kyiv, Ukraine, Sep. 20–23, 2011.

13. Mitrokhin, V. N. and A. Y. Polishchuk, “Eigenmodes oflayered biconical waveguide,” Bulletin of Moscow State TechnicalUniversity, No. 4 (37), 80–89, Priborostroenie, 1999 (in Russian).

14. Mitrokhin, V. N. and A. Y. Polishchuk, “Electric dipole in adielectric sphere,” Antennas, No. 8 (54), 41–47, 2001 (in Russian).

15. Bey, N. A., V. N. Mitrokhin, and A. Y. Polishchuk, “CompactUWB antennas,” Electrodynamics and Microwave Technology ofUHF, EHF and Optical Frequencies, Vol. 10, No. 2 (34), 154–159,2002.

16. Tafove, A. and S. Hagness, Computational Eletrodynamics:The Finite-difference Time-domain Method, 2nd Edition, Artech


House, 2000.17. Kisunko, G. V., Electrodynamics of Hollow Systems, VKAS,

Leningrad, USSR, 1949 (in Russian).18. Tretyakov, O. A., “Evolutionary waveguide equations,” Ra-

diotekhnika and Elektronika, Vol. 34, No. 5, 917–926, 1989 (inRussian).

19. Borisov, V. V., Transients in Waveguides, Publishing House ofLeningrad State University, Leningrad, 1991 (in Russian).

20. Tretyakov, O. A., Analytical and Numerical Methods inElectromagnetic Wave Theory, 572, Science House Co, Ltd, Tokyo,1993.

21. Butrym, A. Yu., Y. Zheng, and O. A. Tretyakov, “Transientdiffraction on a permittivity step in a waveguide: Closed-formsolution in time domain,” Journal of Electromagnetic Waves andApplications, Vol. 18, No. 7, 861–876, 2004.

22. Aksoy, S. and O. A. Tretyakov, “Evolution equations for analyticalstudy of digital signals in waveguides,” Journal of ElectromagneticWaves and Applications, Vol. 17, No. 12, 263–270, 2004.

23. Wen, G., “A time-domain theory of waveguide,” Progress InElectromagnetics Research, Vol. 59, 267–297, 2006.

24. Kochetov, B. A. and A. Y. Butrym, “Calculation of pulsewave propagation in a quasi-TEM line using mode expansionin time domain,” Proc. 4th International Conference onUltrawideband and Ultrashort Impulse Signals (UWBUSIS’08),222–224, Sevastopol, Ukraine, Sep. 15–19, 2008.

25. Butrym, A. Y. and B. A. Kochetov, “Time domain mode basismethod for a waveguide with transverse inhomogeneous multi-connected cross-section. 1. The general theory of method,” RadioPhysics and Radio Astronomy, Vol. 14, No. 2, 162–173, 2009 (inRussian).

26. Butrym, A. Y. and B. A. Kochetov, “Time domain mode basismethod for a waveguide with transverse inhomogeneous multi-connected cross-section. 2. Example of numerical implementationof the method,” Radio Physics and Radio Astronomy, Vol. 14,No. 3, 266–277, 2009 (in Russian).

27. Butrym, A. Y. and M. N. Legenkiy, “Charge transport by a pulseE-wave in a waveguide with conductive medium,” Progress InElectromagnetics Research B, Vol. 15, 325–346, 2009.

28. Kochetov, B. A. and A. Y. Butrym, “Rigorous calculation ofultra short pulse propagation in a shielded microstrip line usingcoupled mode expansion in time domain,” Proc. 6th International


Conference on Ultrawideband and Ultrashort Impulse Signals(UWBUSIS’12), 284–287, Sevastopol, Ukraine, Sep. 17–21, 2012.

29. Tretyakov, O. A., “Mode basis method,” Radiotekhnika andElektronika, Vol. 31, No. 6, 1071–1082, 1986 (in Russian).

30. Aksoy, S. and O. A. Tretyakov, “Study of a time variant cavitysystem,” Journal of Electromagnetic Waves and Applications,Vol. 16, No. 11, 1535–1553, 2002.

31. Wen, G., “Time-domain theory of metal cavity resonator,”Progress In Electromagnetics Research, Vol. 78, 219–253, 2008.

32. Tretyakov, O. A. and F. Erden, “Temporal cavity oscillationscaused by a wide-band waveform,” Progress In ElectromagneticsResearch B, Vol. 6, 183–204, 2008.

33. Antyufeyeva, M. S., A. Y. Butrym, and O. A. Tretyakov,“Transient electromagnetic fields in cavity with dispersive doublenegative medium,” Progress In Electromagnetics Research M,Vol. 8, 51–65, 2009.

34. Borisov, V. V., “Electromagnetic field of a current with arbitrarytime dependence distributed on the surface of a sphere,”Radiophysics and Quantum Electronics, Vol. 19, No. 12, 1291–1298, 1976.

35. Shvartsburg, A. B., Impulse Time-domain Electromagnetics ofContinuouse Media, Birkhauser Boston, Basel, Berlin, 1999.

36. Tretyakov, O., A. Dumin, O. Dumina, and V. Katrich, “Modalbasis method in radiation problems,” Proc. Int. Conf. on Math.Methods in Electromagnetic Theory (MMET-2004), 312–314,Dnepropetrovsk, Ukraine, 2004.

37. Dumin, O. M., O. O. Dumina, and V. O. Katrich, “Propagationof spherical transient electromagnetic wave through radiallyinhomogeneous medium,” Proc. Int. Conf. on Ultrawidebandand Ultrashort Impulse Signals (UWBUSIS-2006), 276–278,Sevastopol, Ukraine, Sep. 18–22, 2006.

38. Dumin, A. N., “Radiation of transient localized waves from anopen-ended coaxial waveguide with infinite flange,” Telecommu-nications and Radio Engineering, Vol. 53, No. 6, 30–34, 1999.

39. Tretyakov, O. A. and A. N. Dumin, “Emission of nonstationaryelectromagnetic fields by a plane radiator,” Telecommunicationsand Radio Engineering, Vol. 54, No. 1, 2–15, 2000.

40. Borisov, V. V., “Excitation of nonperiodic fields in a conical horn,”Radiotekhnika and Elektronika, Vol. 30, 443–447, Mar. 1985 (inRussian).

41. Shlivinski, A. and E. Heyman, “Time-domain near-field analysis


of short-pulse antennas. Part I: Spherical wave (multipole)expansion,” IEEE Transactions on Antennas and Propagation,Vol. 47, No. 2, 271–279, Feb. 1999.

42. Butrym, A. Y. and B. A. Kochetov, “Mode basis methodfor spherical TEM-transmission lines and antennas,” Proc.International Conference on Antenna Theory and Techniques(ICATT-07), 243–245, Sevastopol, 2007.

43. Butrym, A. Y. and B. A. Kochetov, “Mode expansion in timedomain for conical lines with angular medium inhomogeneity,”Progress In Electromagnetics Research B, Vol. 19, 151–176, 2010.

44. Kochetov, B. A. and A. Y. Butrym, “Transient wave propagationin radially inhomogeneous biconical line,” Proc. 5th InternationalConference on Ultrawideband and Ultrashort Impulse Signals(UWBUSIS’10), 71–73, Sevastopol, Ukraine, Sep. 6–10, 2010.

45. Kochetov, B. A. and A. Y. Butrym, “About convergenceof the spherical mode expansions in time domain,” No. 883,Bulletin of Karazin Kharkov National University, Radiophysicsand Electronics, No. 15, 41–44, 2009 (in Russian).

Documents

AXIALLY SYMMETRIC TRANSIENT ELECTROMAG- NETIC FIELDS … · solve problems of electromagnetic waves excitation and propagation in waveguides [17{28], cavities [29{33], free space,