IEEE TAP 2011 a New Formulation of Pock Ling Tons EQ for Thin Wires Using Exact Kernel

8/3/2019 IEEE TAP 2011 a New Formulation of Pock Ling Tons EQ for Thin Wires Using Exact Kernel

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A New Formulation of Pocklingtons Equation forThin Wires Using the Exact Kernel

Ebrahim Forati, Member, IEEE, Aaron D. Mueller, Member, IEEE, Parisa Gandomkar Yarandi, Member, IEEE,and George W. Hanson, Fellow, IEEE

AbstractPocklingtons integro-differential equation for thinwires with the exact kernel is reformulated using a secondderivative formula for improper integrals. This allows for an-alytical evaluation of the second derivatives, resulting in a pureintegral equation, in a similar manner to what is done whenusing the approximate kernel. However, as opposed to using theapproximate kernel, the resulting integral equation developedhere is numerically stable even with a simple pulse function/pointmatching solution. Good convergence for the current is obtainedusing pulse functions, and the severe unphysical oscillations ofthe current that are encountered when using pulse functions with

the approximate kernel are avoided.Index TermsElectromagnetic theory, integral equations.

I. INTRODUCTION

POCKLINGTONS integro-differential equation is a stapleof thin-wire antenna analysis, and appears in most antenna

text books [1]-[4], as well as forming the basis of antennasimulation codes such as the Numerical ElectromagneticsCode (NEC) [5]. As typically presented, for a wire of length2L centered and oriented along the z axis, Pocklingtonsequation is

14j k2 + 2z2L

L

K(z z) I(z) dz = Eiz (z) (1)

for L z L, where I(z) is the wire current (A), k2 =2 with , characterizing the homogeneous space outsideof the wire, Eiz is the incident field, and

K(z z) = 12

ejkR

Rd (2)

=1

2

ejk(zz)2+4a2sin2(/2)

(z z)2 + 4a2sin2 (/2)d.

is the so-called exact kernel. This equation applies to a

perfectly-conducting wire of radius a that is electrically thin(i.e., a , where is the free-space wavelength). Becauseof the perfect electrical conductor (PEC) approximation, allcurrent resides on the wire surface, and due to the electricalthinness assumption, small radial and circumferential currentshave been ignored, as has the circumferential variation of theaxial current.

Pocklingtons equation (1) is somewhat difficult to workwith since the second derivative cannot be simply brought

The authors are with the Department of Electrical Engineering, Uni-versity of Wisconsin-Milwaukee, Milwaukee, WI 53211 USA (e-mail:[email protected]).

inside the integral (the resulting integral over z is non-convergent). For this reason, (1) is often converted to Hallnsequation [4] for numerical solution. Alternatively, it is com-mon to work with Pocklingtons equation directly and makethe approximation [4]

K(z z) Ka(z z) = ejk(zz)2+a2

(z z)2 + a2, (3)

where Ka(z z

) is known as the approximate kernel. Usingthe approximate kernel, the second derivatives can be broughtinside the integral and 2Ka(z z)/z2 can be evaluatedexplicitly, resulting in a simple formulation involving properintegrals that are easy to program.

However, there are both theoretical and numerical prob-lems associated with the approximate kernel. Theoretically,Pocklingtons equation with the approximate kernel has nosolution [6]-[9], whereas Pocklingtons equation with the exactkernel is well-posed, at least in suitable function spaces [10]-[12], [7]. Despite this important fact, Pocklingtons equationwith the approximate kernel often yields numerical results thatagree fairly well with results obtained using the exact kernel.

However, a numerical problem arises when using sub-domainfunctions for the current expansion, such as pulse functions

I(z) =

Nn=1

anfn(z), (4)

where

fn(z) =

1 zn 2 < z zn + 20 otherwise

are the usual pulse functions and ans are the unknowncoefficients to be determined, and where = 2L/N, andzn =

L + (n

1/2). A well-known problem of the

approximate kernel is that one must ensure /a > 1 [9],otherwise severe and non-physical oscillations in the currentappear in the numerical solution (although this is often notmentioned in elementary textbooks). Thus, a value of Nmust be chosen that is large enough to yield a good resultbut small enough so that the condition /a > 1 is notviolated. This is a rather unsatisfactory solution, and a similarproblem exists even for entire-domain basis functions [13]. Itis possible to use the exact kernel and bring one derivativeinside the integral, integrate by parts to move the derivativeonto the current, and then bring the second derivative insidethe integral. However, in solving the resulting equation one


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needs to use suitably smooth expansion functions, and a pulse-function solution is not possible.

In this work, we show that the use of a rigorous method(similar to a Leibnitz rule [14]) for interchanging the secondderivative and integral in (1) leads to a purely integral formu-lation of Pocklingtons equation with the exact kernel that issimple and easy to program. Even for pulse functions, stableresults are obtained with no restriction on the ratio

/a.

Other papers, such as [15]-[21] also discuss rigorous meth-ods involving the exact kernel, and utilize either expansionsof the kernel, efficient methods of treating the singularity,the use of suitable basis functions, or some combination ofthese techniques. A comparison among the various methodsis beyond the scope of this work, although it can be notedthat the method presented here is straightforward in conceptand simple to implement.

II. THEORY AND FORMULATION

The starting point for the Pocklington equation derivationis the relationship between current density and electric field,

which, in a homogeneous medium, is [22]-[24]

Es(r) =

1

j

k2 + lim

0

VV(r)

g(r, r)J(r)dV,

(5)where V(r) is an exclusion volume [22] as depicted in Fig.1 (when r is not in the source region the limiting procedureis not necessary),

g(r, r) =ejk|rr|4 |r r| =

ejkR

4R. (6)

is the free-space Greens function, and R =

2 + 2 2 cos( ) + (z z)21/2 . Assumingan electrically thin wire a oriented along the z-axis, the small radial and circumferential currents and thecircumferential variation of the axial current can be ignored,such that J(,,z) zJz(, z).

V-

Figure 1. Solid wire geometry showing exclusion volume V .

In the following sections two different cases for thin wiresare considered, a solid wire model and an infinitesimally-thin shell (hollow tube) model. For the PEC case the solid

wire model is not needed, but it is included here since a) forgenerality we consider the more realistic imperfect conductorcase, and b) it utilizes a known rigorous method to pass thesecond derivatives through the principle-value type volumeintegral in (5) (this is related to the well-known depolarizingdyadic problem [22]). For the hollow tube model we presenta new two-dimensional second-derivative formula for thisinterchange of operators. Both the solid wire and hollow tubecases result in the same expression, which is then used to formPocklingtons equation.

A. Case 1: Thin Solid Wire Model

In order to pass the second derivative through the volumeintegral we use the second derivative formula [24, see (1.59)],[25, p. 248; in this reference the static case is treated],

2

xixjlim0

VV(r)

s(r)g(r, r)dV

= s(r) S g(r, r)

xj xi ndS (7)+ lim

0

VV(r)

[s(r) s(r)] 2g(r, r)

xixjdV,

If the wire is thin (a , where is the skin depth), onecan ignore the radial variation of the axial current, so thatJz (, z) I(z) /a2. Moreover, since g(r, r) varies littleover the radius of the wire (the singularity is excluded byisolation in the z-direction as depicted in Fig. 1) we can set = = a and = 0 in R. Using (7), the z-component of(5) can be written as

Esz(r) = I1 + I2 + I3 (8)where

I1 =k2

4jPV

LL

K(z z)I(z)dz, (9)

I2 = I(z)4j

z[K(z L)K(z + L)] , z = L (10)

= 0, z = L

I3 = 14j

PV

LL

[I(z) I(z)] 2

K(z z

)z2

dz. (11)

The term I1simply comes from the k2 term in (5). The secondtwo terms arise from (7). Note that I2, which comes from thefirst term on the right side of (7), is zero for z = L. Toexplain this, consider that in (7), S is the surface of V V(r). This consists of the cylindrical surface (which doesntcontribute because of the unit vector xi = z), the two end-caps, and two end-cap-like surfaces on the top and bottom ofV(r). These latter two contributions cancel each other andgive zero contribution. That leaves the two actual end-caps.


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However, e.g., ifz = L then V(r) is centered on z = L and sothere is no end-cap contribution from that point, which wouldotherwise contribute the K(zL) term. In this case the onlycontribution from S is the other end cap, which contributesthe non-singular term K(z + L) = K(2L) term in I2. SinceI(L) = 0 then (10) gives no contribution. It follows that (10)contributes for all z = L.

In both (9) and (11) the notation PV indicates the usualprinciple-value type integral that omits the singular point z =z. Note that in (11) we cannot break up the integral into twointegrals, since for this to be a valid operation the resultingtwo integrals must be convergent, which is not the case.

B. Case 2: Hollow Tube Model

This model is probably more familiar for thin-wire antennaanalysis. In this case we assume that the wire is electricallythin (a ) yet thick compared to a skin depth (a ), sothat we can model the wire as an infinitesimally-thin hollowshell. Then, the z-component of (5) simplifies to

Esz(a, 0, z) =1

j

k2 +

2

z2

lim0

ss(r)

g(r, r)I(z)

2adS

(12)where S is the open surface of the hollow tube (with rim C)and S is an exclusion surface (with rim C), as shown in Fig.2.

n

Ca

C

C

C

n

n

nS-S

Figure 2. Hollow tube geometry showing exclusion surface, S . The rim Cof the surface S S , and the rim C of exclusion surface S , are depictedby the lines on the right side.

In the appendix, a new second derivative formula (30) isderived that is the two-dimensional, open-surface analogue of(7). Using this, we can write (12) as Esz(z) = I1 + I2 + I3,where

I1 =k2

jlim0

ss(r)

g(a, 0, z , a , , z)I(z)

2aaddz, (13)

I2 = I(z)2ja

C

n z

zg(a, 0, z , a , , z)ad, (14)

I3 =1

jlim0

ss(r)

I(z)

2a I(z)

2a

2

z2g(a, 0, z , a , , z)ad

(15)

In (14), C is the boundary of the open surface S (i.e., the topand bottom rims shown in Fig. 1), which are not related to

the open exclusion surface S). One can easily show that (13)-(15) are the same as (9)-(11). As a result, the same equationfor Esz(z) is obtained for both cases.

C. Pocklingtons Integral Equation

To form Pocklingtons integral equation, for generalitywe will assume an imperfectly-conducting wire and use animpedance boundary condition, so that at the surface of thewire

Etotalz (z) = Esz(z) + E

iz(z) = I(z)zs, (16)

where zs is the surface impedance (/m) of the thin wire.

Writing Es

z = I1 + I2 + I3 we have, for L < z < L,k2PV

LL

K(z z)I(z)dz I(z) z

[K(z L)K(z + L)]

(17)

+ PV

LL

[I(z) I(z)] 2K(z z)

z2dz = 4j

I(z)zs Ez

This is the main result of the paper. This integral equation isgeneral, and can be solved using any suitable basis functions.Here we use pulse functions since they are simple, and providean example where their use in the approximate/reduced kernelequation leads to poor results.

Note that for (17) we specify L < z < L rather thanL z L; for the second term on the left side (I2) thisensures that the point z = L is excluded (see 10). Thisterm is easy to evaluate numerically since it does not involveintegration. The first term on the left side (I1) is easy toevaluate numerically, being an undifferentiated potential (thePV notation is somewhat superfluous, and simply indicates thatthe integral is improper (yet convergent)). The third term onthe left side (I3) is also easy to evaluate numerically due to thecurrent difference I(z)I(z). In particular, in a pulse functionpoint matching solution (matching points zm, m = 1, 2, . . . N )

the third term is

I3 =

LL

Nn=1

anfn(z) am

2

z2K(z z)z=zmdz (18)

=Nn=1

n=m

(an am)zn+/2

zn/2

2

z2K(z z)z=zmdz.

For the pulse function point matching solution, (17) be-comes the matrix equation

[Zmn]NN [an]N1 = [Vn]N1 (19)


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where

Zmm = k2

zm+2

zm2

K(zm z)dz

Nn=1

n=m,

zn+2

zn2

2

z2K(z z)z=zmdz

z

[K(z L)K(z + L)]z=zm j4zs,

Zmnm=n

=

zn+2

zn2

(k2 +2

z2)K(z z)z=zmdz, (20)

and Vm = j4Eiz(zm).

III . RESULTSIn all results we assume that a unit-amplitude plane wave

is normally-incident on the wire, f = 10 GHz, a = 0.005,and we vary L and N. In all plots = 5.9 107 S/m.

Fig. 3 shows the current for L = /2 and N = 61 (/a =3.28), so that the approximate kernel is expected to performreasonably well. As can be seen from the figure, the currentdetermined by using the approximate kernel (3) in (1) and thecurrent obtained from the new integral equation (17) agreevery well.

1 0.5 0 0.5 10

10

20

30

40

50

z/L

|I(z)|A

(1) with approx. kernel(17) with exact kernel

Figure 3. Magnitude of the current obtained by using the approximate kernelin (1) and by the new integral equation (17) for L = /2 and N = 61 . Since/a > 1 the approximate kernel is expected to yield good results.

The new integral equation (17) was also verified by compar-ing to Hallns equation using the exact kernel, where excellentagreement was found. Table I shows the current at the center ofthe wire for the geometry considered in Fig. 3. The last column

is the percent difference between the two results. It can beseen that the Pocklington form converges faster than the Hallnform; for the Pocklington form the percent difference betweenthe N = 100 and N = 5000 results is 0.06%, whereas for theHalln form it is 0.3%.

Regarding computer times, both the new Pocklington formand the Halln form only depend on z z, and thus havea Toeplitz form. Only the first row of the matrix need befilled, and thus both methods are very efficient. Computing thenew Pocklington form takes approximately twice as long asthe Halln form since it involves numerical integration of twodifferent integrands (K and 2K/z2), whereas the Hallnform involves only K. On a standard PC matrix fill time foreither method is under a second for N = 300.

Fig. 4 shows the current for L = /20 and N = 81 (/a =

Table ICOMPARISON OF |I(0)|(A) BETWEEN THE NEW POCKLINGTON FORM

AN D HALLNS EQUATION WITH EXACT KERNEL FOR THE CASECONSIDERED IN FIG . 3 .

N Pocklington Halln %

100 40.0187 40.0390 0.365300 40.0177 40.0117 0.150500 40.0171 40.0134 0.092

1000 40.0167 40.0148 0.0462000 40.0165 40.0155 0.0243000 40.0164 40.0157 0.01574000 40.0163 40.0159 0.01185000 40.0163 40.0159 0.0094

0.25), such that the approximate kernel is not expected toprovide accurate results. Indeed, using the approximate kernelnon-physical oscillations in the current appear at the wire ends,as discussed in [9] (for a delta-gap generator these oscillationsoccur near the wire center). However, the current obtainedfrom (17) does not exhibit these non-physical oscillations, aswe would expect when using the exact kernel.

Fig. 5. shows a convergence study for the current at thewire center and near the wire ends obtained from the twomethods, for L = /20. The approximate kernel yields aresult that slowly converges to a value that is approximately12% too large, and which diverges near the wire ends as N isincreased, as expected. The new formulation (17) yields resultsthat converge quickly at all points on the wire.

IV. CONCLUSIONS

Pocklingtons integro-differential equation for thin wireswith the exact kernel has been reformulated into a pureintegral equation using a second derivative formula for im-proper integrals. Solid wire and hollow tube models havebeen considered. It was shown that a simple pulse-functionpoint matching solution of the resulting integral equation leadsto results that are stable and show good convergence, andwhich avoids the severe and non-physical oscillations of thecurrent that arise from the same numerical procedure used inconjunction with the approximate kernel.

Acknowledgment: The authors would like to thank DanielaGomez, UWM Electrical Engineering, for help with this


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1 0.8 0.6 0.4 0.230

20

10

0

10

20

30

40

z/L

Re{I(z)}

(nA)

(1) with approx. kernel

(17) with exact kernel

Figure 4. Real part of the current for L = /20 and N = 81, where/a = 0.25, such that the approximate kernel is not expected to provide

accurate results. The new integral equation (17) uses the exact kernel andavoids the problem of current oscillations near the wire ends arising from useof the approximate kernel.

20 40 60 80 100 120 140 160 180 200 220

2.6

2.8

3

3.2

Number of Segments N

|I(z=0)|(A)

20 40 60 80 100 120 140 160 180 200 220

0

50

100

150

|I(z=L)|(A)

(1) with approx kernel

(17) with exact kernel

Figure 5. Magnitude of the current at the wire center and at the wire endsobtained from the two methods for L = /20. The new formulation (17)yields results that converge very well at all points on the wire.

project. The authors would like to thank Ric Ancel, UWM

Department of Mathematics, for helpful discussions concern-ing the two-dimensional divergence theorem.

V. APPENDIX

To obtain the second derivative formula similar to (7) butfor open surfaces, consider the quantity

(r) = (a,,z) = lim0

SS

s (r) g (r, r) dS (21)

where S is the surface of a hollow, infinitesimally-thin tubeand S is an exclusion surface, depicted in Fig. 2. The quantity

(21) defines a single-layer potential with source density s.It is well-known that a tangential derivative of (21) can bepassed through the integral (normal derivatives generate a terms (, z) /2, where (, z) is a point on the surface [26]-[27]).Therefore,

z

(r) =

lim0 SS s (r

)

zg (r, r) dS

= lim0

SS

s (r)

zgr, r

dS

SS

zsr

gr, r

dS

= lim0

C+C

n zsr

gr, r

dl +

SS

s (r)

z gr, r

dS

.

(22)

where we use g (r, r) /z = g (r, r) /z in the first lineand the product rule for derivatives to obtain the second line,and where the third line comes from the divergence theorem

in two dimensions,S

AdS =C

n Adl, (23)

where C is the contour of the open surface S. The two-dimensional divergence theorem is usually applied to opensurfaces in the plane. However, it is valid for any compactdifferential manifold that is equipped with an inner productat each point; the inner product at a point on a tube in R3

is simply the dot product on R3 restricted to vectors tangentto the tube at the given point. In this case, these will be unitvectors parallel to the axis of the tube pointing away from the

tube at the tube rims, i.e.,n

= z

.It can be easily shown that

lim0

C

s (r) g (r, r) z ndl = 0, (24)

where it should be noted that C comprises both the top andbottom rims of S, which are taken in opposing directions.Therefore, (22) converts to

z (r) =

C

n zs (r) g (r, r) dl (25)

+ lim0 SS

s (r)

z g (r, r) dS ,

which may be differentiated again to yield

2

z2 (r) =

C

n zs (r)

zg (r, r) dl

lim0

SS

(s (r) S(r))z

g (r, r)

z dS

,(26)


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where we have used

(s (r) S(r))z

=s (r)

z . (27)

Using the two-dimensional divergence theorem, the secondintegral of (26) may be rewritten as

SS

(s (r) S(r))z

g (r, r)

z dS =

SS

(s (r) S(r)) 2g (r, r)

z 2dS

+

C+C

n zg (r, r)

z (s (r) s (r)) dl. (28)

As with (24), it can be shown that

C n z (s (r) s (r)) g (r, r

)

z dl = 0, (29)

so that we have

2

z2 (r) = s (r)

C

n zg (r, r)

zdl

+lim0

SS

(s (r) S(r)) 2g (r, r)

z2dS, (30)

the desired expression. Note that the source density should beHlder-continuous [27] in order to pass a derivative through

the surface integral. This is not an issue, since we will expandthe current into functions that are differentiable in the regionof the singularity. This holds even for pulse functions, sincea pulse function will be centered on the singularity, and overthe singularity the pulse function will be constant.

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IEEE TAP 2011 a New Formulation of Pock Ling Tons EQ for Thin Wires Using Exact Kernel

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Transcript of IEEE TAP 2011 a New Formulation of Pock Ling Tons EQ for Thin Wires Using Exact Kernel