The dipole uniform current I flowing over an elemental length h is the dual of a ‘‘magnetic current’’ MzS = Ih and the surface area is S = h/k. The fields due to the infinitesimal loop are then found from the vector and scalar potentials.

Vector and Scalar Potentials. The wave equation, in the form of the inhomogeneous Helmholtz equation, is used here with most of the underlying vector arithmetic omitted; see Refs. 10 to 12 for more details. For a magnetic current element source, the electric displacement D is always solenoidal (the field lines do not originate or terminate on sources), that is, in the absence of source charges the divergence is zero,

V D = 0 (1)

and the electric displacement field can be represented by the curl of an arbitrary vector F,





Loop antennas, particularly circular loops, were among the first radiating structures analyzed, beginning as early as 1897 with Pocklington’s analysis of a thin wire loop excited by a plane wave (4). Later, Hallen (5) and Storer (6) studied driven loops. All these authors used a Fourier expansion of the loop current, and the latter two authors discovered nu­merical difficulties with the approach. The difficulties could be avoided, as pointed out by Wu (7), by integrating the Green’s function over the toroidal surface of the wire. The present author coauthored an improved theory (8,9) that spe­cifically takes into account the finite dimension of the loop wire and extends the validity of the solution to thicker wires than previously considered. Additionally, the work revealed some detail of the loop current around the loop cross section. Arbitrarily shaped loops, such as triangular loops and square loops, as well as loop arrays can be conveniently analyzed us­ing numerical methods, such as by the moment method (3).

The Infinitesimal Loop Antenna

The infinitesimal single turn current loop consists of a circu­lating current I enclosing an infinitesimal surface area S, and is solved by analogy to the infinitesimal dipole. The fields of an elementary loop element of radius b can be written in terms of the loop enclosed area, S = nb2, and a constant exci­tation current I (when I is rms, then the fields are also rms quantities). The fields are ‘‘near’’ in the sense that the dis­tance parameter r is far smaller than the wavelength but far larger than the loop dimension 2b. Hence, this is not the close near-field region. The term kIS is often called the loop mo­ment and is analogous to the similar term Ih associated with the dipole moment. The infinitesimally small loop is pictured in Fig. 1(a) next to its elementary dipole analog [Fig. 1(b)].

D = e0E = V x F

where F is the vector potential and obeys the vector identity V • V X F = 0. Using Ampere’s law in the absence of electric sources

V xH = jox0E

and with the vector identity V X (—V<^) = 0, where Ф repre­sents an arbitrary scalar function of position, it follows that

H = —V Ф — jo)F

and for a homogeneous medium, after some manipulation we get

V2F + k2F = — e0M + V(V F + >M0е0Ф) (5)

where k is the wave number and k2 = ш2^0е0. Although Eq. (2) defines the curl of F, the divergence of F can be indepen­dently defined and the Lorentz condition is chosen:

ja^0Є0Ф = — V F

We define V2 as the Laplacian operator

2 d2 d2 d2

V =——— 1— 1——-

dx2 dy2 dz2



Substituting the simplification of Eq. (6) into Eq. (5) leads to the inhomogeneous Helmholtz equation

V 2F + k2F = — M


Figure 1. Small-antenna geometry showing (a) the parameters of the infinitesimal loop moment, and (b) its elementary dipole dual. [Source: Siwiak (2).]

Similarly, by using Eqs. (6) and (4) it is seen that

V 2Ф + ^Ф = 0 (9)

(a) (b)

Using Eq. (4) and the Lorentz condition of Eq. (6) we can find the electric field solely in terms of the vector potential F. The utility of that definition becomes apparent when we consider a magnetic current source aligned along a single vector direc­tion, for example, M = zMz for which the vector potential is F = zFz, where z is the unit vector aligned with the z axis, and Eq. (8) becomes a scalar equation.

Radiation from a Magnetic Current Element. The solution to the wave equation, Eq. (8), presented here, with the details suppressed, is a spherical wave. The results are used to de­rive the radiation properties of the infinitesimal current loop as the dual of the infinitesimal current element. The infini­tesimal magnetic current element M = zMz located at the ori­gin satisfies a one-dimensional, hence scalar form of Eq. (8). At points excluding the origin where the infinitesimal current element is located, Eq. (8) is source-free and is written as a function of radial distance r,

Equations (15) and (16) for the magnetic fields Hr and He of the infinitesimal loop have exactly the same form as the electric fields Er and Ee for the infinitesimal dipole, while Eq. (17) for the electric field of the loop Eф has exactly the same form as the magnetic field Hф of the dipole when the term kIS of the loop expressions is replaced with Ih for the infinitesi­mal ideal (uniform current element) dipole. In the case for which the loop moment kIS is superimposed on, and equals the dipole moment Ih, the fields in all space will be circu­larly polarized.

Equations (15) to (17) describe a particularly complex field behavior for what is a very idealized selection of sources: a simple linear magnetic current M representing a current loop I encompassing an infinitesimal surface S = nb2. Equations (15) to (17) are valid only in the region sufficiently far (r > kS) from the region of the magnetic current source M.

The Wave Impedance of Loop Radiation. The wave imped­ance can be defined as the ratio of the total electric field di­vided by the total magnetic field. We can study the wave im­pedance of the loop fields by using Eqs. (15) to (17) for the infinitesimal loop fields, along with their dual quantities for the ideal electric dipole. Figure 2 shows the loop field wave impedance as a function of distance kr from the loop along the direction of maximum far-field radiation. The wave im­pedance for the elementary dipole is shown for comparison. At distances near kr = 1 the wave impedance of loop radiation exceeds = 376.73 П, the intrinsic free-space impedance, while that of the infinitesimal loop is below 376.73 П. In this region, the electric fields of the loop dominate.

The Radiation Regions of Loops. Inspection of Eqs. (15) to (17) for the loop reveal a very complex field structure. There are components of the fields that vary as the inverse third power of distance r, inverse square of r, and the inverse of r. In the near-field or induction region of the idealized infinites­imal loop, that is, for kr < 1 (however, r > kS for the loop and r > h for the dipole), the magnetic fields vary as the inverse third power of distance.

The region in which kr is nearly unity is part of the radiat­ing near field of the Fresnel zone. The inner boundary of that

! ЭFz (r) dr

1 3

r2 dr

V 2FZ (r) + k2Fz (r) =

+ k2Fz (r) = 0 (10)


which can be reduced to

d2Fz(r) 2 dFz(r) 2

—гъ 1 — j -k Fz(r) = 0

dr2 r dr


Since Fz is a function of only the radial coordinate, the partial derivative in Eq. (10) was replaced with the ordinary deriva­tive. Equation (11) has a solution





There is a second solution in which the exponent of the pha — sor quantity is positive; however, we are interested here in outward traveling waves so we discard that solution. In the static case the phasor quantity is unity. The constant C1 is related to the strength of the source current and is found by integrating Eq. (8) over the volume including the source, giv­ing

Сл = -9-kIS 4n


and the solution for the vector potential is in the z unit vector direction,


kis г

4n r




Figure 2. Small loop antenna and dipole antenna wave impedances compared. [Source: Siwiak (2).]

which is an outward propagating spherical wave with increas­ing phase delay (increasingly negative phase) and with ampli­tude decreasing as the inverse of distance. We may now solve for the magnetic fields of an infinitesimal current element by inserting Eq. (14) into Eq. (4) with Eq. (6) and then for the electric field by using Eq. (2). The fields, after sufficient ma­nipulation, and for r > kS (see Ref. 10), are

я = kIS e-jkrk2 Г 2ж V (kr)2 ‘ (kr)3

J + 1


(15) (17)


kIS _

Hr, = e

6 4 7t



where — tyj = c^0 = 376.730313 is the intrinsic free-space im­pedance, c is the velocity of propagation in free space (see Ref. 13 for definitions of constants), and I is the loop current.

zone is taken by Jordan and Balmain (12) to be r2 > 0.38D3/A, and the outer boundary is r > 2D2/A, where D is the largest dimension of the antenna, here equal to 2b. The outer boundary criterion is based on a maximum phase error of п/

8. There is a significant radial component of the field in the Fresnel zone.

The far field or Fraunhofer zone is the region of the field for which the angular radiation pattern is essentially inde­pendent of distance. That region is usually defined as ex­tending from r < 2D2/A to infinity, and the field amplitudes there are essentially proportional to the inverse of distance from the source. The far-zone behavior is identified with the basic free-space propagation law.

The Induction Zone of Loops. We can study the induction zone in comparison to the far field by considering induction zone coupling, which was investigated by Hazeltine (14), and which was applied to low-frequency radio receiver designs of his time. Today the problem might be applied to the design of a miniature radio module in which inductors must be oriented for minimum coupling. The problem Hazeltine solved was one of finding the geometric orientation for which two loops in parallel planes have minimum coupling in the induction zone of their near fields and serves to illustrate that the ‘‘near­field’’ behavior differs fundamentally and significantly from ‘‘far-field’’ behavior. To study the problem we invoke the prin­ciple of reciprocity (see Ref. 10), which states

jT (Eb • Ja — Hb • Ma) dV = jf (Ea • Jb — Ha • Mb) dV (18)

That is, the reaction on antenna (a) of sources (b) equals the reaction on antenna (b) of sources (a). For two loops with loop moments parallel to the z axis we want to find the angle в for which the coupling between the loops vanishes, that is, both sides of Eq. (18) are zero. The reference geometry is shown in Fig. 3. In the case of the loop, there are no electric sources in Eq. (18), so Ja = Jb = 0, and both Ma and Mb are aligned with z, the unit vector parallel to the z axis. Retaining only the inductive field components and clearing common constants in Eqs. (15) and (17) are placed into (18). We require that (Hr + He0)z = 0. Since r ■ z = —sin(e) and в-z = cos(e), we are left with 2 cos2(e) — sin2(e) = 0, for which в = 54.736°. When oriented as shown in Fig. 3, two loops parallel to the x-y plane whose centers are displaced by an angle of 54.736° with respect to the z axis will not couple in their near fields. To be sure, the angle determined above is ‘‘exactly’’ correct for



Figure 3. Two small loops in parallel planes and with в = 54.736° will not couple in their near fields. [Source: Siwiak (2).]


Figure 4. A metal detector employs two loops initially oriented to minimize coupling in their near fields.

infinitesimally small loops; however, that angle will be nomi­nally the same for larger loops. Hazeltine (14) used this prin­ciple, placing the axes of the inductors in a common plane each at an angle of 54.7° with respect to the normal form the radio chassis, to minimize the coupling between the in­ductors.

The same principle can be exploited in the design of a metal detector, as depicted in Fig. 4. The loop a is driven with an audio frequency oscillator. Loop b, in a parallel plane and displaced so that nominally в = 54.7°, is connected to a detec­tor that might comprise an audio amplifier that feeds a set of headphones. Any conductive object near loop a will disrupt the balance of the system and result in an increased coupling between the two loops, thus indicating the presence of a con­ducting object near a.

The Intermediate — and Far-Field Zones of Loops. The loop — coupling problem provides us with a way to investigate the intermediate — and far-field coupling by applying Eq. (18) with Eqs. (15) and (16) for various loop separations kr. In the far — field region only the He term of the magnetic field survives, and by inspection of Eq. (16), the minimum coupling occurs for в = 0° or 180°. Figure 5 compares the coupling (normalized to their peak values) for loops in parallel planes whose fields are given by Eqs. (15) to (17). Figure 5 shows the coupling as a function of angle в for an intermediate region (kr = 2) and for the far-field case (kr = 1000) in comparison with the in­duction zone case (kr = 0.001). The patterns are fundamen­tally and significantly different. The coupling null at в = 54.7° is clearly evident for the induction zone case kr = 0.001 and for which the (1/kr)3 terms dominate. Equally evident is the far-field coupling null for parallel loops on a common axis when the 1/kr terms dominate. The intermediate-zone cou-



в (deg)

Figure 5. Normalized induction zone, intermediate zone, and far zone coupling between loops in parallel planes. [Source: Siwiak (2).]

pling shows a transitional behavior in which all the terms in kr are comparable.

The Directivity and Impedance of Small Loops. The directive gain of the electrically small loop can be found from the far — field radially directed Poynting vector in ratio to the average Poynting vector over the radian sphere: the Q, the quality factor defined in (2), is inversely propor­tional to the third power of the loop radius, a result that is consistent with the fundamental limit behavior for small an­tennas.

If we use Eq. (22) and ignore the dipole mode terms and second-order terms in a/b, the unloaded Q of the loop an­tenna is




— ) -2


a ) .


(E xH*) ■ r

D(e,<P) =




4n. 10



|(E x H*) ■ r sin(e) de dф

which for b/a = 6 becomes



kSk 4я г

Pd =


1вП0 =



Rr =



Only the в component of H and the ф component of E survive into the far field. If we use Eq. (16) for He and Eq. (17) for Eф and retain only the 1/kr terms, Eq. (19) yields D = 1.5 sin2(e) by noting that the functional form of the product of E and H is simply sin2(e) and by carrying out the simple inte­gration in the denominator of Eq. (19).

Taking into account the directive gain, the far-field power density Pd in the peak of the pattern is

for the infinitesimal loop of loop radius b.

When fed by a gap, there is a dipole moment that adds terms not only to the impedance of the loop but also to the close near fields. For the geometry shown in Fig. 6, and using the analysis of King and Harrison (15), the electrically small loop, having a diameter 2b and wire diameter 2a, exhibits a feed point impedance given by

for radiated power I2Rr, hence, we can solve for the radiation resistance:

—щ = щ-{кЪ)4


4 яг2



which has the proper limiting behavior for small loop radius. The Q of the small loop given by Eq. (23) is indeed larger than the minimum possible Qmin = (kb)-3 predicted by Siwiak (2) for a structure of its size. It must be emphasized that the actual Q of such an antenna will be smaller than given by Eq. (24) due to unavoidable dissipative losses not represented in Eqs. (22) to (24). We can approach the minimum Q but never go smaller, except by introducing dissipative losses.

The Gap-Fed Loop

The analysis of arbitrarily thick wire loops follows the method in Ref. 8, shown in simplified form in Ref. 9 and summarized here. The toroid geometry of the loop is expressed in cylindri­cal coordinates p, ф, and z with the toroid located symmetri­cally in the z = 0 plane. The relevant geometry is shown in Fig. 6.

Loop Surface Current Density. The current density on the surface of the toroidal surface of the loop is given by

JJ = E E BnpejnFp




+ jn0kb

Zloop=r]o^(kb)4[l + 8(kbf] (l-p)

/8b „ 2 ,, 2

ln(—) — 2+3 (kb)

[1 + 2(kbf]

including dipole mode terms valid for kb < 0.1. The leading term of Eq. (22) is the same as derived in Eq. (21) for the infinitesimal loop. Expression (22) adds the detail of terms considering the dipole moment of the gap fed loop as well as refinements for loop wire radius a. The small loop antenna is characterized by a radiation resistance that is proportional to the fourth power of the loop radius b. The reactance is induc­tive, hence, is proportional to the loop radius. It follows that

n = — TO p = — TO

where the functions Fp are symmetrical about the z axis and are simple functions of cos(n^), where ф is in the cross section of the wire as shown in Fig. 6 and is related to the cylindrical coordinate by z = a sin^). These functions are orthonor­malized over the conductor surface using the Gram-Schmidt method described in Ref. 16, yielding


0 2 nfab


F1 = F0


COS(lIf) — —

1 — (a/2b)2 v 2b


The higher-order functions are lengthy but simple functions of sin(pф) and cos(pф).

Scalar and Vector Potentials. The electric field is obtained from the vector and scalar potentials







Figure 7. Loop radiation resistance.



Ap =


The boundary conditions require that Еф, Еф, and Ep are zero on the surface of the loop everywhere except at the feed gap |ф| < e. Because this analysis will be limited to wire diameters significantly smaller than a wavelength, the boundary condi­tions on Еф and Ep will not be enforced. In the gap Еф = V0/2ep, where V0 is the gap excitation voltage.

The components of the vector potential are simply

Jф sin(4i — ф’) dS

A =



and the vector potential is

19^ p 3 ф

JV о 4n k



Ф =

given by


эАф dz 9AP dz

Hp =




H =

дАф A,


G= — У

2 j Z-


1 dAp

p dp

Hz =




where the value of dS = [b + a sin(^)]a d^f. The Green’s func­tion G is expressed in terms of cylindrical waves to match the rotational symmetry of the loop,


Jm p — v)Hm)(p2 — v)e—jZ(z—z’) dZ



— іт(ф—ф’)


v = /k2 + f2 p1 = p — a cos(^) p2 = p + a cos(^)

and where Jm(vp) and HmKvp) are the Bessel and Hankel func­tions.

Matching the Boundary Conditions. Expression (32) is now inserted into Eqs. (29) to (31) and the electric field is then found from Eq. (28) and the boundary condition is enforced. For constant p on the wire

The loop current across a section of the wire is found by inte­grating the function Jф in Eq. (25) around the wire cross sec­tion. The loop radiation impedance is then the applied volt­age V0 in the gap divided by the current in the gap. Figure 7 shows the loop feed radiation resistance, and Fig. 8 shows the corresponding loop reactance, as a function of loop radius kr for a thin wire, П = 15, and a thick wire, П = 10, where П = 2 ln(2nb/a) is Storer’s parameter (6). The thin-wire loop has very sharp resonant behavior compared with the thick-wire loop, especially for a half-wavelength diameter (kb = 0.5) structure. The higher resonances are less pronounced for both loops. Thick-wire loops exhibit an interesting behavior in that over a diameter of about a half wavelength, the reactance is



Ефе^с1ф = — Ъ>^Р11 P PC

This condition is enforced on the wire as many times as there are harmonics in ф. Truncating the index p as described in Ref. 9 to a small finite number P, we force Еф = 0 except in the feeding gap along the lines of constant p on the surface of the toroid. If we truncate to P, the number of harmonics Fp in ф, and to M the number of harmonics in ф, we find the radia­tion current by solving M systems of P by P algebraic equa­tions in Bm, p. In Ref. 9, P = 2 and M in the several hundreds was found to be a reasonable computational task that led to useful solutions.

Loop Fields and Impedance. With the harmonic amplitudes Bmp known, the current density is found from Eq.(1). The elec­tric field is found next from Eq. (2) and the magnetic field is

Table 1. Parameter Y for Various Loop Thicknesses and b = 0.01 Wavelengths






















essentially always capacitive and the total impedance re­mains well behaved.

Small Gap-Fed Loops. The detailed analysis of the thick, gap-fed wire loop, as shown in Refs. 8 and 9, reveals that the current density around the circumference of the wire, angle ф in Fig. 6, is not constant. An approximation to the current density along the wire circumference for a small diameter loop is

Js = — 2cos(0)(M>)2][l +У cos(^r)] (37)

^ 2n a

where Іф is the loop current, which has cosine variation along the loop circumference, and where the variation around the wire circumference is shown as a function of the angle ф. Y is the ratio of the first — to the zero-order mode in ф and is not a simple function of loop dimensions a and b, but can be found numerically [Siwiak (2)] and from the analysis of the preced­ing section. For the small loop Y is negative and of order a/b so Eq. (37) predicts that there is current bunching along the inner contour (ф = 180°) of the wire loop. Table 1 gives representative values for Y as a function of a/b.

This increased current density results in a corresponding increase in dissipative losses in the small loop. We can infer that the cross-sectional shape of the conductor formed into a loop antenna will impact the loss performance in a small loop.

The small loop fed with a voltage gap has a charge accu­mulation at the gap and will exhibit a close near electric field. For a small loop of radius b and in the x-y plane, the fields at (x, y) = (0, 0) are derived in Ref. 9 and given here as numerical codes, such as the numerical electromagnetic code (NEC) described in Ref. 3, and often used in the numerical analysis of wire antenna structures.

When the small loop is used as an untuned and unshielded field probe, the current induced in the loop will have a compo­nent due to the magnetic field normal to the loop plane as well as a component due to the electric field in the plane of the loop. A measure of E field to H field sensitivity is apparent from Eq. (40). The electric field to magnetic field sensitivity ratio of a simple small-loop probe is proportional to the loop diameter. The small gap-fed loop, then, has a dipole moment, which complicates its use as a purely magnetic field probe.

Updated: 02.04.2014 — 20:40