## Abstract

Electric and magnetic currents are essential to describe electromagnetic-stored energy, and the associated antenna Q and the partial directivity to antenna Q-ratio, *D*/*Q*, for arbitrarily shaped structures. The upper bound of previous *D*/*Q* results for antennas modelled by electric currents is accurate enough to be predictive. This motivates us to develop the analysis required to determine upper and/or lower bounds for electromagnetic problems that include magnetic model currents. Here we derive new expressions for the stored energies, which are used to determine antenna Q bounds and *D*/*Q* bounds for the combination of electric and magnetic currents, in the limit of electrically small antennas. In this investigation, we show both new analytical results and we illustrate numerical realizations of them. We show that the lower bound of antenna Q is inversely proportional to the largest eigenvalue of certain combinations of the electric and magnetic polarizability tensors. These results are an extension of the electric only currents, which come as a special case. The proposed method to determine the minimum *Q*-value which is based on the new expressions for the stored energies, also yields a family of current-density minimizers for optimal electric and magnetic currents that can lend insight into antenna designs.

## 1. Introduction

Time harmonic electromagnetic radiating systems do not in general have a finite total energy associated with them. This is well known since the radiated electric and magnetic fields decay as *r*^{−1} and the corresponding energy density hence decay as *r*^{−2}, which is not an integrable quantity for exterior unbounded regions. This non-integrability differs from the singularities of the electromagnetic energy for charged particles (e.g. [1,2]), where the challenge is the finite mass of particles in coupling Maxwell's equation to the dynamics of the charged particles.

To consistently extract a finite stored energy from the energy densities associated with classical time-harmonic energy has been investigated in [3–9]. These stored energies have been based on spherical (and spheroidal) modes, circuit equivalents and on the input impedance for small antennas. In 2010, Vandenbosch [10] proposed a current-density approach to stored energies also applicable to larger antennas. This approach has generated new interest in electromagnetic stored energy that is explored in [11–16]. This ‘stored energy’ is similar to the results of Collin & Rothschild [5], and it also has similarities with the stored energies proposed in [17,18]. The generalization of stored energy in [19] and in this paper allows for the first time the analysis of both electric and magnetic current densities for arbitrary shapes. Antennas embedded in lossy or dispersive material have been considered in [20–22].

The drive to find a well-defined stored energy stems partly from that it is closely related to the antenna quality factor Q. Lower bounds on antenna Q are directly related to the electric size of the antenna, and indirectly to the maximal matching bandwidth that can be obtained. The relation between antenna Q and bandwidth is not trivial; for a discussion and examples, see [9,14,23,24]. An alternative method to derive bandwidth bounds is sum-rules (e.g. [9,25–29]). The approach given here is related to [9,12,14,19,27]. In this paper, we mathematically derive a direct relation between antenna Q (also called the Q-factor) and the electric and magnetic polarizabilities [30–32] as the leading order term for electrically small structures, using the here given expressions of the stored energies. Realizations of antennas that use both electric and magnetic currents are investigated in [33,34].

Our investigation is based on the asymptotic behaviour of stored energy in the electrically small case for both electric and magnetic currents. This result is an extension of the stored energies in [10] and connect to both antenna Q and partial directivity over antenna Q. That scattering properties are related to the polarizabilities is known (e.g. [35,36]), but that polarizability tensors appear directly as the essential factor in antenna Q estimates is a recent result [19,27,37,38].

The current-representation approach to stored energy enables the maximal partial directivity over antenna Q problem to be reduced to a convex optimization problem [15]. It also enables us to consider fundamental limitations for arbitrary geometries. Convex optimization problems are efficiently solvable [39]. From a user perspective, it can be compared with solving a matrix equation. To numerically find the physical bounds on antenna Q or partial directivity over antenna Q, *D*/*Q* is here reduced to tractable problems, solvable with common electromagnetic tools. In this paper, we illustrate how this can be applied to a range of shapes, both numerically and analytically. The here considered minimization problems investigate how different current and charge density combinations yield different lower bounds on antenna Q. For the electrical dipole problem, we show that the minimizing currents result in Q and *D*/*Q* that agree with [8,27,37]. For the case of a generalized electric dipole with both electric charges and magnetic current densities as sources our result agrees with the sphere in [3]. When we allow dual-modes, i.e. both electrically and magnetically radiation dipoles, we find that the result agree with [19,38]. The mathematical framework provided here for the first time easily account for all these different cases with a generic approach. Another result of the here derived method is that we can show that small antennas for a given shape have a family of current densities that realize the associated optimal antenna Q [12].

This paper is based on the here given stored energies for both electric and magnetic current densities and formulate upper and lower bounds for *D*/*Q* and *Q*, respectively, as mathematical optimization problems, which we solve. Another approach to these energies and associated bounds are given in [19] and provide a more heuristic approach. We investigate the small antenna limit and illustrate how antenna Q and related optimization problems behave for electric and magnetic currents for a range of antenna shapes. These results are based on the leading order terms of the stored energies as the electric size of the domain approach zero. One of the advantages here is that the bounds on *Q* and *D*/*Q* are known once the polarizability tensors are determined for a given shape. We use this knowledge to sweep shape parameters to illustrate how *Q* and *D*/*Q* depend on the shape of the antenna. Analytical expressions for the electrically small case provide physical insight into limiting factors for *Q* and *D*/*Q*. These more general results are shown to reduce to the analytically known cases in [19,37,40].

In §2, we recall the definitions of key antenna and energy quantities. Using an asymptotic expansion of the electric and magnetic currents in §3, we give the explicit leading order current-density representation of the radiated power, stored energies and the radiation intensities. Analytical and numerical examples for antenna Q and *D*/*Q* under different constraints are given in §4. In §5, we formulate the problem as a convex optimization problem and determine *Q* for some shapes. The conclusion and appendix A end the paper.

## 2. Antenna Q and partial directivity

Let *J*_{e} and *J*_{m}, respectively (figure 1). The support *V* is here assumed to be bounded and connected. Through the continuity equations, we define the associated electric and magnetic charge densities *ρ*_{e} and *ρ*_{m}. The time-harmonic Maxwell's equations with electric and magnetic current densities in free-space take the form
*e*^{jωt}, which is suppressed. In this paper, we let *ε*=*ε*_{0}, *μ*=*μ*_{0} and ** E** is the electric field and

**is the magnetic field. The dispersion relation between the wavenumber,**

*H**k*, and the angular frequency,

*ω*, is

*t*is time.

The field energy densities are *ε*|** E**|

^{2}/4 and

*μ*|

**|**

*H*^{2}/4. Here we are interested in stored electric

*W*

_{e}and magnetic

*W*

_{m}energies, which are more challenging to define. We define stored electric and magnetic energies as

*F*_{E},

*F*_{H}are the far-fields, i.e.

**denote a vector in**

*r**r*=|

**| and corresponding unit vector**

*r*Given these stored energies, we define the two main antenna parameters that appear in the physical bounds. The antenna quality factor *P*_{rad} is the radiated power of the system described by (2.1) and (2.2). Defined as

The partial directivity *D*/*Q*, which with the above notation is

The goal in this paper is to optimize and investigate *Q* and *D*/*Q* in terms of the electric and magnetic current densities, in the small antenna limit. We hence express these quantities in terms of the current densities. The stored energies are given in appendix A as a quadratic form in terms of the electric and magnetic current densities. While these calculations are straightforward, they are also rather lengthy, see e.g. [10] for a similar effort, see also [38,43]. The stored energies are
*W*_{e,corr} and *W*_{m,corr} are explicitly given in appendix A, where they also are shown to be perturbation terms for the small antenna case. Note that *W*_{e}−*W*_{e,corr} and *W*_{m}−*W*_{m,corr} are symmetric in the current densities and a natural extension of the electric only current case, *J*_{m}=0. This current-density explicit representation of the stored energies given in (2.8) and (2.9) is new, and, as illustrated below, useful to increase our understanding of stored energies as exemplified in the below obtained bounds on *Q* and *D*/*Q*. The associated operators in (2.8) and (2.9) are
*G*(** r**) is Green's function,

*e*

^{−jkr}/(4

*πr*) and * indicates the complex conjugate, see also figure 1.

The radiation intensity, *J*_{m}=0, these expression agree with e.g. [10,12].

To find the total radiated power *P*_{rad}, in terms of its current-density representation, we can integrate (2.12) over the unit sphere. A more direct route to *P*_{rad} is based on (2.5) and the observation that the electric far-field, *F*_{E}, have the representation
** R**=

*r*_{1}−

*r*_{2},

*R*=|

**|,**

*R**j*

_{n}(

*x*) is the spherical Bessel function of order

*n*[44].

## 3. Electrically small volume approximation

The assumption of that the volume of the antenna is electrical small simplify the above energy and power-related expressions *P*_{rad} and subsequently *Q* and *D*/*Q*. An object is electrically small if its wavelength normalized size is small enough. A common method to quantify this is to enclose the antenna volume in a sphere of radius *a* and let the object be electrically small if *ka*<1. The physical size, here characterized by *a*, appears implicitly in the stored energies, as a consequence we adopt the ordo-notation *ka* is bounded by *Cak*, as *C*. This abuse of the notation conforms to fact that for a given fixed *a*, we have that

To expand the above quantities like the stored energies in terms of small *ka*, we assume that the currents have the asymptotic behaviour
*J*^{(0)}_{e}, *J*^{(0)}_{m}, *J*^{(1)}_{e} and *J*^{(1)}_{m} are all *k*-independent and the two latter correspond to a lowest order static charge through the continuity equation. The expansion of the stored energies are given in appendix A.

We apply the small *ka* approximation, and (3.1) and (3.2) to the partial radiation intensity and the far-field *F*_{E} in the form of (2.13). We first note that
*J*^{(0)}_{e,m}=0. Here *J*^{(n)}_{e,m}, indicate that the expression is valid for *J*^{(n)}_{e} and *J*^{(n)}_{m}, *n*=0,1. It follows that the partial radiation intensity (2.12), for a wave with polarization *P*^{(0)} reduces to

To find the total radiated power in (2.5), we start with inserting the expansion (3.3) into the far-field (2.13) to find the small *ka* approximation of the far-field:
*P*_{e} and terms that radiate as magnetic dipoles with power *P*_{m}. The total radiated power can alternatively be expressed as
*m*_{m}. Here

A check that the above expressions agree with what is known for small antennas that radiate as dipoles is obtained by comparing the maximal partial directivity, i.e. *P*^{(0)}_{rad}. We consider two cases: fixed generalized electric dipole moments and no generalized magnetic dipole moment (3.7), i.e. *π*_{m}=0 and *π*_{e}≠0 (or vice versa) and fixed non-zero *π*_{m},*π*_{e}:
*π*_{m} we find that appropriately oriented combinations of *π*_{e} and *π*_{m} can have a directivity of 3, corresponding to a Huygens source, e.g. [47].

The small electric volume stored energies follow directly from their integral representation (2.8), we find that *Q* and *D*/*Q* are based.

## 4. Minimal antenna Q and analytical and numerical illustrations

One of the goals with the expansion in the small antenna limit in §3 is that they should give us some insight into limitations of *Q* and *D*/*Q* and antenna design. It is reasonable to ask the question of what shapes that give low antenna Q. Similarly we investigate which charge and current densities that gives low antenna Q. Another goal with the expressions is to derive *a priori* bounds of antenna Q and *D*/*Q*. Partial answers are given in this section, that extends the relation that a large charge-separation ability in the domain imply a small antenna Q (e.g. [12,19,27,37,40]). Similarly we may think of a shape with low antenna Q, as a structure that supports a large ‘current loop area’ for a magnetic dipole-moment. One of the new results here is that the generic shape results in [12] for *D*/*Q* is extended to lower bounds on antenna Q.

An often studied case is the electric-dipole case [10,15,27,37,40], here represented by the electric charges only and we illustrate below how an optimization problem is used to determine the minimal *Q*. We continue and show that the method and its associated eigenvalue problem extend to the more general case of both electric and magnetic currents that radiate as an electrical dipole. Here we also find that the magnetic polarizability enters in the lower bounds on *Q*. A short review of polarizability tensors are given in [48, App. B].

Consider the minimization problem for finding the lower bound on antenna Q:
*ρ*^{(1)}_{m}. One of the interesting cases in antenna design is when the antenna radiate as an electrical dipole, i.e. when *P*_{m} is negligible and *W*_{m,0}≤*W*_{e,0}. Once the optimal (*P*_{e},*W*_{e,0}) is determined we tune the antenna with a tuning circuit to make the antenna resonant, i.e. *W*_{m,0}=*W*_{e,0}. Thus, we start with the optimization problem for a pure (*W*_{e,0},*P*_{e})-case. The ‘dual-mode’ case, where both *P*_{e} and *P*_{m} are comparable is considered in §4d. Before we consider the general case, let us start with the easier case of an electric dipole when we have only *J*^{(0)}_{m}=0.

### (a) Antenna Q for an electric dipole, i.e. *P*_{m}=0

Different approaches to lower bounds of this antenna Q case have also been investigated in e.g. [10–12,15,27,37,40]. However, one of the goals here is to arrive at a generic method that works for different cases of current-density sources, and the first step towards this goal is to verify that this method indeed gives the previously derived result on the lower bound (e.g. [8,12,19,27,37]). The electric dipole is here equivalent to the assumption *P*_{m}=0 and *W*_{e,0}≥*W*_{m,0}, which yields that *Q*=*Q*_{e} and that we have an optimization problem that depend only on the electric charge-densities *ρ*_{e}. Once the design is determined we can tune the antenna with a tuning circuit to make *W*_{e,0}=*W*_{m,0}. This case is the classical electrical dipole case. Let *V* , i.e. *c* is the speed of light. Hence, (4.2) is accompanied with the constraint of total zero charge,

The associate problem to maximize *D*/*Q* in the small electric volume limit for arbitrary *ρ*_{e}, see [12] corresponds to
*D*/*Q* problem has the simplification in that the integrand in

The method that we apply below to (4.2) works on both problems (4.2) and (4.3) and yield the same result as in [12] where it is applied to (4.3). The final result is similar to the result in [19,37], but obtained with different methods. Note that both (4.2) and (4.3) remain unchanged under the scaling, *p*_{e}. We rewrite (4.2) as the minimization problem as
*V* that realize the same minimum, e.g. for spheres and shapes with appropriate symmetries [32]. To explicitly find the minimum, we use the method of Lagrange multipliers (e.g. [51, §4.14]) and define the Lagrangian _{1} and λ_{2} are Lagrange multipliers, and we use the short-hand notation *ρ*=*ρ*_{e}. Variation of _{1} and λ_{2} gives the two constraints above. Taking the variation of *ρ**, or equivalently, taking a Fréchet derivative of *ρ*. Accompanied with the constraints we find three equations (4.8), (4.5) and (4.6) and three unknown *ρ*, λ_{1} and λ_{2}.

Upon multiplying (4.8) with *ρ** and integration over *V* , using the zero total charge constraint, we find that *Q*_{e} in (4.2) is equivalent with
_{1}, depends implicitly on *ρ* and λ_{2}. The lower bound of the minimization problem (4.2) is hence determined by the unknown Lagrange multiplier λ_{1} times a constant. Another property of the solution appears if we apply the Laplace operator on (4.8), for ** r**∉∂

*V*we have that

*ρ*(

**)=0. Thus, we reduce (4.8) to**

*r**ρ*

_{s}is the surface charge density, i.e. we have formally the relation that

*ρ*d

*V*=

*ρ*

_{s}d

*S*. A similar result for

*D*/

*Q*was shown in [12].

Using the constraint

To solve equation (4.11), we make first a few observations: any solution *ρ*_{s} of (4.11) for given right-hand sides, yields an associated potential that solves an electrostatic boundary-value problem (cf. [48, App. B]). Such solutions are restricted in their asymptotic behaviour by the electric polarizability tensor *γ*_{e}, which depends only on the shape of *V*. To make this restriction explicit, we note that the electric polarizability tensor *γ*_{e} is defined through [48, eqn (88)] and *γ*_{e} is given, once the shape *V* is known, we thus have a constraint on *ρ*_{s} and its associated potential to comply with the polarizability tensor. The constraint is that
*γ*_{e}. Here we have used that

We conclude that critical points of (4.2) correspond to solutions *Q*_{e} in (4.9), it follows that the largest eigenvalue, (*γ*_{e})_{3} of the polarizability matrix *γ*_{e} yields the minimum *Q*_{e}, i.e.
*Q*_{e} for the electric dipole to finding eigenvalues of *γ*_{e}. This result have large similarities to [19,37], derived with different methods. We conclude that *Q*(*ak*)^{3} in the small volume size only depend on the shape expressed through the normalized electric polarizability *γ*/*a*^{3}. The physical interpretation connects large polarizability eigenvalues to the ability of the structure to separate charge under an external static field in a given direction. The polarizability *γ*_{e} is associated with the scalar Dirichlet problem of the Laplace operator and depends only on the shape of the object [30]. We note also that *γ*_{e} is identical to the high-contrast electric polarizability in e.g. [27].

We note that the low-frequency magnetic charge-density and electric charge-density antenna Q are dual-similar, and hence if we consider a case with either a *ρ*_{e}-term or a *ρ*_{m}-terms both of these problems result in identical minimization problems with a lower bound on antenna Q given by (4.13).

To compare with the *D*/*Q* problem, we note that the constraint *Q*_{e}-case above, we find that *ν*_{1} is connected to *γ*_{e} through the relation *Q* without concern of polarization direction of the antenna, whereas *D*/*Q* assume a fixed *a priori* knowledge about the optimal polarization direction of the structure or alternatively the principal eigenvalue of *γ*_{e} associated with a given structure, we select *Q*_{e} and *D*/*Q*_{e}. This *D*/*Q* result is similar to the sum-rule in [27] for electric sources. With the *Q*_{e} result and the observation of principal directions of *γ*_{e}, we see that these three approaches illustrate closely connected results here with a common energy principle method to obtain them.

To illustrate the result, we begin with a sphere: *γ*_{e}=4*πa*^{3}** I**, where

**is the three times three unit tensor, and all eigenvalues of**

*I*

*γ*_{e}are identical. Note that to these degenerate eigenvalues there are three orthogonal eigenvectors, and the corresponding charge densities in (4.10) for a given amplitude of the dipole-moment

*p*

_{e}. This degeneracy is due to the geometrical symmetries of the shape. Thus even when we remove the scaling invariance, we may have multiple

*ρ*that yield the same lower bound on

*Q*. Note also that for any arbitrary optimizer

*ρ*

^{(1)}

_{e}, here

*J*^{(1)}

_{e}, i.e.

*ρ*

^{(1)}

_{e}. It allows a potentially large design freedom that does not change

*Q*

_{e}in the quasi-static limit. This case is analogous to the case discussed in [12].

For the sphere, we find (*ka*)^{3}*Q*_{e}=3/2 and for a disc (*ka*)^{3}*Q*_{e}=9*π*/8 for the electrical dipole case ([48, App. D]). If we instead study *γ*_{e} of a rectangular plate of size ℓ_{2}×ℓ_{1} and sweep the ratio ℓ_{1}/ℓ_{2} we find that the two non-zero eigenvalues depicted as the two curves with highest value, marked with (E), in figure 2*a*, and corresponding *Q* in figure 2*b* marked with (E). Note that *a*, for appropriate directions

The corresponding, electric charge maximization problem of *D*/*Q* is solved in [12,27,40]. We have thus the solution to both the

### (b) Antenna Q for an electric current magnetic dipole

Analogous to how the electric dipole, *ρ*^{(1)}_{e}, and the magnetic dipole with *ρ*^{(1)}_{m} yield the same optimization problem in the previous section, we see that an electric *J*^{(0)}_{e} or a magnetic *J*^{(0)}_{m} current density result in identical optimization problems. We associate a magnetic dipole moment *J*^{(0)}_{e} here denoted ** J**, to find the minimization problem:

**∈**

*J**X*

_{0}. Here

*P*

_{e}=0, and

*W*

_{e,0}≤

*W*

_{m,0}. Once the optimization is done, we can tune the antenna with a tuning circuit to reach resonance

*W*

_{e,0}=

*W*

_{m,0}.

We apply once again the method in (4.2)–(4.7) to the minimization of (4.15). Scaling invariance is broken by the assumption that ** J**∈

*X*

_{0}. The associated critical point equation is

*** and integrate over**

*J**V*to find that

*Q*

_{m}is determined by λ

_{1}.

**d**

*J**V*=

*J*_{s}d

*S*, and (4.17) reduces to

*V*.

Similar to the electric case (4.11), we compare this with the definition of the magnetic polarizability tensor, *γ*_{m} in [48, App. B] where *V* is given. The *γ*_{m})_{3} is the largest eigenvalue of *γ*_{m}. The analogous case for *D*/*Q* is given in [12]. The sphere has the magnetic polarizability tensor 2*πa*^{3}** I**, which yields (

*ka*)

^{3}

*Q*

_{m}=3 (cf. [8,47]). Here

**is a unit three times three tensor.**

*I*The electric and magnetic polarizabilities of a rectangular plate are depicted in figure 2*a* marked with (E) and (M), respectively. The polarizability tensors are diagonal for geometries with two orthogonal reflection symmetries and co-aligned coordinate systems [31,52] and for planar structures we have only one eigenvalue of *γ*_{m}, orthogonal to the plane. We can physically think of large *γ*_{m}-eigenvalues as that the region *V* supports a large loop current for the corresponding dipole-moment. Note that planar structures have one non-zero eigenvalue in *γ*_{m} which is associated with the normal-to-the-surface dipole-moment with the planar ‘current loop area’. We observe that the magnetic polarizability tensor is connected to the scalar Neumann problem of Laplace equation ([48, App. B]). Note that the magnetic polarizability corresponds to the permeable case of *γ*_{m}; however, we note that λ_{1}≥0 in (4.21), see (4.15).

A similar current loop-area argument is illustrated in figure 3 for a flat ellipse and a thin ellipsoid. The eigenvalues of the polarizability tensor of an ellipsoid are known ([48, App. D]), and they are depicted in figure 3*a*,*b*. The two curves marked with (M) in figure 3*c* correspond to *Q*_{m}, the upper one, marked (M), is for an ellipse of zero thickness and only one *γ*_{m}-eigenvalue corresponding to a current loop-area over the surface. The other marked (M, thick) corresponds to an ellipsoid identical to the flat one, but where the radius normal to the paper is *h*/100 where *h* is the height of the ellipse. The two transverse eigenvalues of *γ*_{m} are ignored by *Q*_{m} until the width, *w*, is *h*/100, where equivalent current loop-area spanned by the height and normal (out of the paper) dominates the transverse current loop-area and *Q*_{m} changes slowly for *w*/*h*<10^{−2} since this area is essentially preserved.

### (c) Lower bound on antenna Q for both electric charge and magnetic currents

The common electric and magnetic dipoles cases above agree with previously derived results [14,19]. We here extend these results to include both the electric charge density *ρ*_{e} and the magnetic current density *J*_{m}, i.e. the components making up a generalized electric dipole-moment *π*_{e} (3.7). We once again consider the case where the antenna radiates as an electrical dipole, i.e. *P*_{m}=0 and where the stored energy is mainly electric, *W*_{m,0}≤*W*_{e,0}. After the optimization, we tune the antenna to make the stored electric and magnetic energies equal. Optimizing for the (*P*_{m},*W*_{m,0})-case is identical to the (*P*_{e},*W*_{e,0})-case up to a sign and the free-space impedance normalization of the currents. Similar to the above discussion in §4a,b of electric and magnetic dipoles, we optimize
** J**=

*J*^{(0)}

_{m}/

*η*, and

**=0 to account for the Gauge freedom of the associated vector potential. To include this Gauge freedom into the optimization problem, we restrict the current-density space to**

*J*The minimization problem is scaling invariant under transformations (*ρ*,** J**)↦(

*ρ*,

**)**

*J**α*for any complex valued scalar

*α*. By assuming that the denominator has a given value

_{1},λ

_{2}, we define the Lagrangian

_{1}and λ

_{2}gives the constraints. The variation with respect to

*ρ** and

*** yields**

*J**V*, and we recognize λ

_{2}as a way to ensure that the total charge is zero. To investigate the properties of these Euler–Lagrange equations, we first note that the inner product of these equations with

*ρ** and

***, respectively, and that their sum can be rewritten as the original problem**

*J*_{1}for

*ρ*,

**that solves (4.28) and (4.29).**

*J*Similar to the charge-density case (4.9), we note that λ_{1} implicitly depend on *ρ* and ** J** through the Euler–Lagrange equations. Another property of the minimization problem appears if we for

**∉∂**

*r**V*operate with

*Δ*and with ∇×∇× on (4.28) and (4.29), respectively. We find that

*ρ*and

**only have support on the boundary, and we use the notation**

*J***d**

*J**V*=

*J*_{s}d

*S*and

*ρ*d

*V*=

*ρ*

_{s}d

*S*. We hence find the Euler–Lagrange equations

*r*_{1}∈∂

*V*. We have here introduced the electric and magnetic dipole-moments for the current and charge-distribution that solve (4.31) and (4.32):

**and**

*m***are presently unknown apart from the constraints that**

*p*To determine λ_{1}, we recall the definitions of the electric polarizability tensor *γ*_{e} and magnetic polarizability tensor *γ*_{m} in [48, App. B]. We compare (4.31) and (4.32) with the corresponding electric and magnetic boundary integrals [48, eqns (88), (92)]. The polarizability tensors *γ*_{e} and *γ*_{m} are known, once *V* is given, and they impose constraints on λ_{1} and *γ*_{e}+*γ*_{m}. Furthermore, *γ*_{e}+*γ*_{m} of unit length. Thus, we have found that in this case the lower bound on *Q* is given by
*γ*_{e}+*γ*_{m})_{3} is the largest eigenvalue of the *γ*_{e}+*γ*_{m} tensor. The corresponding *ρ*_{s},*J*_{s} are hence the solution of (4.31) and (4.32), where *γ*_{e}+*γ*_{m}≥0.

The minimization procedure also establishes that there exists a λ_{1}≥0 such that
*ρ* and ** J** that satisfy the bi-condition

**∈**

*J**X*

_{0}and

*ρ*and

**satisfy the Euler–Lagrange equations above, yielding 1/λ**

*J*_{1}=(

*γ*_{e}+

*γ*_{m})

_{3}. An equivalent formulation of this result is

*Q*

_{e}or

*Q*

_{m}. The inequality for the (

*P*

_{m},

*W*

_{m,0})-case is obtained identically with the above described case starting from

*P*

_{m}and

*W*

_{m,0}with the substitution of

**=−**

*J*

*J*^{(1)}

_{e}and

*ρ*=

*cρ*

^{(1)}

_{m}/

*η*giving (4.22) with

*r**ρ*+

**×**

*r***/2 of the integrand in the denominator.**

*J*#### (i) Comparisons and numerical examples for the *Q*-lower bound for the dual-mode case (4.34)

We note that for a sphere where both electric and magnetic currents contribute to the generalized electric dipole-moment we find that (*ka*)^{3}*Q*_{e}=1 [3]. The *Q*-lower bound for the flat ellipse and the thin ellipsoid are depicted in figure 3. For planar structures, we note that there is only one non-zero eigenvalue of *γ*_{m}, in the direction normal to the surface and hence perpendicular to the non-zero direction of *γ*_{e}. For a rectangular plate, this eigenvalue is depicted in figure 2. We conclude that in planar structures *γ*_{e} and *γ*_{m} do not couple to improve the antenna *Q*. As is clear from the case where we add a small thickness of the domain as in figure 3*c*, we see that there is a rather small reduction of *Q* when compared with the flat case.

The polarizability tensors for spheroidal shapes are known ([48, App. D], figure 4*a*,*b*). We depict *Q* for spheroidal bodies as a function of the ratio between height and diameter in figure 4*c*. Here, the curves marked with (+) correspond to *Q* given in (4.34) are shown for both the prolate (dashed lines) and oblate cases (solid lines).

The approach in [37] provides an antenna Q, *Q*_{V}, depending only on *γ*_{e} and volume *V*. To compare *Q*_{V} with (4.34), we use the inequality [30, 1.5.19]
*γ*_{e}+*γ*_{m}. Equality holds for several cases in particular for ellipsoidal shapes. An alternative approach to antenna Q is given in [19], see also [38]. To illustrate that there is a difference between *Q* and *Q*_{V}, we calculate both antenna Q's for a cylinder. We assume here that the currents radiate as an electrical dipole aligned with the cylinder axis, i.e. the vertical *Q* from (4.34) and *Q*_{V} are shown in figure 5. To demand that a small antenna radiates as an electric dipole in a given direction is equivalent to selecting the corresponding eigenvalue of the polarizability tensor. Such a choice of eigenvalue does not necessarily minimize antenna *Q*.

The above examples illustrate how the shape of a small antenna enters into the antenna Q-bound. The shape characterization in antenna Q is encoded in the respective polarizability tensors. The electric polarizability is a measure on how easy it is to separate charge for a given volume *V* , i.e. to create a large electric dipole-moment. Similarly, the magnetic polarizability measure how easy it is to create a large magnetic dipole moment, i.e. finding a large ‘current-loop area’ in the domain.

If we similarly [19,33,34] associate the magnetic currents with layers/volumes of magnetization or synthesized Amperian current loops, we note that the associated volumes for the electric and magnetic currents do not necessary need to occupy identical volumes/surfaces. In such a case, there are a considerable design freedom for *γ*_{e} and *γ*_{m}, with the performance bounded by the eigenvalues of *γ*_{e}+*γ*_{m} for the total volume *V*.

### (d) Dual-mode antennas

Self-resonant dual-mode antennas where both the electric *P*_{e} and magnetic *P*_{m} dipole radiation contribute significantly to the radiation and *W*_{e,0}=*W*_{m,0} is considered here. Using that the problem decouples, we use the respective electric and magnetic case above with identities (4.35) where λ_{1}≥0 for both *W*_{e,0} and *W*_{m,0}. We hence find that the general case can be bounded by
*Q* is half the value of *Q*_{e} or *Q*_{m} when only electric or magnetic dipole radiation is allowed. The sphere yields (*ka*)^{3}*Q*=1/2, which agrees with the result of the sphere given in [3,54,55] see also [22]. A similar result is given in [19], derived with a different method.^{1} The antenna *Q* for this case is illustrated for spheroidal shapes in figure 4*c*, for curves marked with a (T).

## 5. Convex optimization for optimal currents

Bounds on *D*/*Q* can be expressed as a convex optimization problems [15]. Here, these results are generalized to include electric and magnetic current densities. We consider a volume *V* with electric *J*_{e} and magnetic *J*_{m} current densities. We expand the current densities in local basis-functions
*N* matrix **J**_{v} with elements {*J*_{e,n}} for *n*=1,…,*N* and {*η*^{−1}*J*_{m,n−N}} for *n*=*N*+1,…,2*N* to simplify the notation. The basis functions are assumed to be real valued, divergence conforming, and having vanishing normal components at the boundary [57].

A standard method of moment implementation using the Galerkin procedure computes the stored energies given in appendix A as matrices. For simplicity, we here compute these stored energy matrices **X**_{e} and **X**_{m} only for the leading order term in (2.10) and (2.11), for *ka*≪1, i.e.

We also use the radiated far field, *N*×1 matrix

Using the scaling invariance of *D*/*Q*, we rewrite the maximization of *D*/*Q* into the convex optimization problem of maximization of the far-field in one direction subject to a bounded stored energy [15], i.e.
`CVX` [58].

We consider planar geometries and bodies of revolution to illustrate the bound. The resulting *Q* of (2.4) for a small spherical capped dipole antenna is depicted in figure 6*a* as a function of the angle *θ* for a maximized omnidirectional partial directivity in *θ*=90° and polarized in the *J*_{e}+*J*_{m}, only electric currents *J*_{e}, and only magnetic currents *J*_{m} are analysed. The requirement of electric dipole-radiation implies *P*_{m}=0, *P*_{e}≠0, and that we can use *ρ*_{e} to represent the electric currents *J*_{e}. We observe that the *θ*=90° case corresponds to a spherical shell with the classical [3,8,34,37] bounds *Qk*^{3}*a*^{3}={1,1.5,3} for the *J*_{e}+*J*_{m}, *J*_{e} and *J*_{m} cases, respectively. The reduced *Q* of the combined *J*_{e}+*J*_{m} case is understood from the suppression of the energy in the interior of the structure. This is also shown in figure 6*b*,*c*, where the resulting electric energy density is depicted for the cases to electric currents *J*_{e} and combined electric and magnetic currents *J*_{e}+*J*_{m}. We also note that the potential improvement with combined electric and magnetic currents *J*_{e}+*J*_{m} decreases as *θ* deceases. This can be understood from the increased internal energy as the magnetic current can only cancel the internal field for closed structures. Moreover, the faster increase of *Qk*^{3}*a*^{3} as *θ*→0 for the *J*_{m} case than for the *J*_{e} case is understood from the loop-type currents of *J*_{m}, whereas *J*_{e} is due to charge separation.

The case of a spheroidal body with the additional radiation constraint corresponding to an electrical dipole along the vertical axis is given in figure 7. It is interesting to compare this constrained result with the minimal *Q* as shown in figure 4*c*, the (+)-curve. Small ℓ_{1}/ℓ_{2} in figure 7 corresponds to small *ζ* as depicted with solid lines in figure 4*c*. We see that in the constrained case *Q* approaches the pure magnetic current case marked *J*_{m}, whereas in figure 4*c*, *Q* marked with (+), approaches the pure *Q*_{e} case (solid line marked (E)), and it is a lower value than the result indicated in figure 7. The cause of this difference is the requirement of the radiation pattern, locking *Q* to a disadvantageous eigenvalue, see figure 4*a* and the vertical polarization direction (solid line). The physical interpretation is clear: for the disc it is easier to excite an electrical dipoles aligned with the surface. The required vertical electric dipole is the cause of the higher *Q* in figure 7. For ℓ_{1}/ℓ_{2} large, we see that both results agree (dashed lines in figure 4*c*, as

## 6. Conclusion

This paper introduces a common mathematical framework for deriving lower bounds on antenna Q to arbitrary shapes for electric and magnetic current densities. For the corresponding cases considered in [19,27], we get identical results for appropriate choices of the ratio of electric and magnetic dipole radiation *P*_{e} and *P*_{m}. This is rather remarkable since the underlying physics and mathematical approaches use widely different ways to arrive to antenna Q and *D*/*Q*. The result also verify that both electric and magnetic current densities are required to reach the classical results for a sphere in e.g. [3,54]. This method also provides a minimization method to determine the minimizing currents, which is attractive for optimization procedures, where antenna Q-related problems can be considered. A few of these minimization problems are demonstrated in the present paper, and extensions analogously to the convex optimization results in [15] follows directly from the explicit results shown here.

In this paper, we derive the antenna Q lower bound for small electric antennas. The lower bound on antenna Q depends symmetrically on both the electric and magnetic polarizabilities, which reflect the dual symmetry of the electromagnetic equations with electrical and magnetic current densities. The explicit lower bound enables *a priori* estimates of antenna Q given the shape of the object in terms of the static polarizability tensors *γ*_{e} and *γ*_{m}. We also determine the antenna Q for planar rectangles, ellipsoids and cylinders. Here we sweep a geometrical shape parameter, to illustrate how the antenna properties *Q* and *D*/*Q* depend on the shape. Low antenna *Q* is associated with low fields inside closed domains, with the present technique we can study objects like the spherical cap to observe how the cancellation of the fields in the interior of an essentially open structures behave for optimal or constrained antenna *Q*.

We conclude that the presented new current-density representation of the stored energy yields explicit analytical expressions on antenna Q and *D*/*Q* in terms of the polarizability tensors. We also illustrate that the polarizabilities and different antenna Q-related optimization problems are straight forward to calculate, given standard software. This follows through the relation of the polarizabilities to the scalar Dirichlet and Neumann problems. The present results are applicable to a range of practical antenna problem, as *a priori* limitations of their antenna Q-performance, and more subtle as explicit current minimizers that might give insight into antenna design problems.

## Data accessibility

This manuscript does not contain primary data and as a result has no supporting material associated with the results presented.

## Author contributions

Both authors contributed to the formulation, did numerical simulations and drafted the manuscript. Both authors gave final approval for publication.

## Funding statement

The authors would like to acknowledge the support of the Swedish Strategic Research Agency (SSF) on the grant ‘Complex analysis and convex optimization for EM design’ AM13-0011 and the Swedish Research Council (Vetenskapsrådet). B.L.G.J. would also like to acknowledge the funding from the Swedish Governmental Agency for Innovation Systems (VINNOVA) in the VINN Excellence Centre Chase under the project NGAA.

## Conflict of interests

We have no competing interests.

## Appendix A. Stored energy: general sources

The stored energies are derived from (2.3) using an approach with potentials. The result consists of a sum of terms each of a given leading *k*^{n}-behaviour for *n*=0,1,… as *k*, as is indicated by the

The second term contains the leading order cross-term
** R**=

*r*_{1}−

*r*_{2},

*R*=|

**| and**

*R**W*

^{(3)}

_{em}term is

The last term *W*^{rest}_{em} is *k* and it is coordinate dependent in certain cases [24]
*W*^{rest}_{em} and *W*^{(3)}_{em} are of the same asymptotic order in *k*. We keep the terms separate due to the sign-change of *W*^{(3)}_{em} in (.1) and since *W*^{rest}_{em} can depend on the coordinate system. We consider the coordinate independent part of these energies as the essential physical quantity of the stored energy.

## Footnotes

- Received November 18, 2014.
- Accepted February 24, 2015.

- © 2015 The Author(s) Published by the Royal Society. All rights reserved.