Radiation Efficiency Limits in Direct Antenna Modulation Transmitters
Kurt Schab
physics.app-ph

	MCFlow: Monte Carlo Flow Models for Data Imputation
    Trevor W. Richardson

Wencheng Wu

Lei Lin

Beilei Xu

Edgar A. Bernal




















Rochester Data Science Consortium, University of Rochester
 
260 E. Main St. Suite 6108, Rochester, NY 14604 

{trevor.richardson, wencheng.wu, lei.lin, beilei.xu, edgar.bernal}@rochester.edu

    December 30, 2023
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		Relative bounds on radiation efficiency are established for time-modulated antenna systems where radiation is generated by broadband conduction currents impressed using idealized non-radiating time-varying subsystems, such as those found in direct antenna modulation (DAM) transmitters.  Analytical and numerical examples demonstrate that the condition of quasi-resonance, common in nearly all practical direct antenna modulation transmitters, imposes severe restrictions on the otherwise unbounded gains in effective efficiency theoretically achievable by this class of time-modulated transmitters.
	
	
	
	Index Terms—Antenna theory, antenna efficiency, time-varying circuits, electrically small antennas.
	
	

§ INTRODUCTION

	
	
	The use of time modulation in an electrically small antenna or its matching network affords the possibility to exceed the strict physical bounds on its linear time invariant (LTI) performance.  Notably, direct antenna modulation (DAM) techniques based on the energy-synchronous modulation of time-varying components have been proposed, simulated, and measured as viable strategies for exceeding the conventional bandwidth-efficiency product limitations of small antennas, e.g., the Chu limit <cit.>.  Energy-synchronous DAM methods in the literature are typically specific to a signal type and antenna topology, with examples including on-off-keying for dipoles <cit.>, phase-shift-keying for dipoles <cit.>, and frequency shift keying on small loops <cit.>.  A theme common across these methods is the use of a time-varying matching network (varying on time scales much less than the carrier period) to impress broadband radiating currents onto the LTI portion of the antenna (e.g., metallic areas supporting conduction currents), thereby bypassing the filtering effects of the LTI radiator's impedance bandwidth. Each of the above techniques relies on tuning the LTI radiator to a quasi-resonant state, though other loop-based methods not requiring quasi-resonance have been reported <cit.>.
	
	
	With the efficacy of DAM demonstrated by a variety of means, it now becomes necessary to determine whether or not there exist quantitative limits in performance gains achievable by adopting such techniques over traditional transmitters.  Here
	 we formulate the question: for a fixed electrical size, how much more efficient could a DAM system be compared to an optimal resistively broadbanded conventional transmitter achieving the same effective bandwidth[Due to their dynamic nature, DAM transmitters do not have a classically defined impedance bandwidth.  An effective bandwidth may, however, be defined via a proxy measure such as distortion<cit.>.]?  Defined in this way, it may appear that relative DAM efficiency increases indefinitely with decreasing transmitter size due to trends in radiation Q-factor.  However, upon closer inspection it appears that competing trends in efficiency may limit benefits in DAM efficiency gains in the extreme electrically small limit.  
	
	
	The goal of this letter is to calculate bounds on this relative efficiency gain by adapting tools developed for the analysis of classical LTI systems.  Namely, we examine low-frequency trends in the efficiency and Q-factor of small antennas to assess the relative benefit of idealized DAM systems over a broad range of scenarios (e.g., electrical size, tuning assumptions, conductivity models).  First, in Sec. <ref>, a DAM transmitter is decomposed into non-LTI and LTI components and models are proposed for ideal conventional, ideal non-resonant DAM, and ideal quasi-resonant DAM systems.  Then, in Sec. <ref>, we apply these models to three examples of varying levels of abstraction and generality.  In all cases, we observe that the condition of quasi-resonance, much like that of self-resonance in classical antennas <cit.>, imposes severe efficiency limitations and greatly reduces the potential gains afforded by DAM.
	
	

§ MODELING IDEAL LTI AND DIRECT ANTENNA MODULATION TRANSMITTERS

	
	Throughout this paper we assume that conventional and DAM systems are represented by the block diagrams in Fig. <ref>.  In both systems, radiation is produced only by conduction currents induced on the object .  We do not consider non-conduction-based radiation, e.g., acoustically driven antennas <cit.>.  Whether driven by LTI or non-LTI systems, the physical bounds governing the maximum achievable radiation efficiency by these currents remain the same.  However, when driven by non-LTI systems, the impact of high Q-factor may not impair the realized bandwidth of a DAM transmitter.
	
	The tradeoff between optimal efficiency and bandwidth for current distributions confined to the design region [By proxy, this includes all possible antennas confined to the region  <cit.>.] may be rigorously calculated via the multiobjective optimization methods described in <cit.>.  Examples of these tradeoffs, in the form of Pareto fronts <cit.>, are shown in Fig. <ref>. Two Pareto fronts are plotted with and without the requirement of self-resonance, with the latter being less restrictive.  Points in bandwidth-efficiency space above each Pareto front are infeasible by LTI transmitters.  The self-resonant and non-resonant Pareto fronts meet at the coordinate (η_Q, (η_Q Q_lb)^-1) corresponding to the minimum achievable radiation Q-factor.  From this point both Pareto fronts may be extended toward higher bandwidth via resistive loading, represented by the curve segment B = (η,(η Q_lb)^-1) with η≤η_Q.  For a prescribed bandwidth B_0, the maximum efficiency η_0 achievable by an LTI antenna is readily obtained by finding the intersection of the line B = B_0 and the chosen Pareto front, as shown in Fig. <ref>.  Using this graphical representation of antenna bounds, we now describe models for ideal LTI transmitters, as well as two models for ideal DAM transmitters.
	
	
	
	
	
	

 §.§ Ideal Conventional LTI Transmitter

	We assume that a conventional antenna is matched to a real source impedance via single resonance tuning, see Fig. <ref>.  This tuning may be carried out either via antenna shaping or through the use of external lumped elements.  In either case, we assume that the tuning network is constructed of the same materials as the antenna itself and that the tuning network is confined to the design region of the antenna.  Under these assumptions, we observe that the maximum efficiency possible for a conventional LTI antenna is given by the intersection of the B = B_0 line and the self-resonant Pareto front, i.e.,
	
    η_Conv. = min{η_0,η^sr_max},

	where η_0 is the efficiency at the intersection and minimization explicitly enforces the right boundary of the self-resonant Pareto front in Fig. <ref>.
	
	

 §.§ Ideal Non-resonant DAM Transmitter

	Using non-LTI matching and loading networks, it may be possible to impress a broadband signal onto the maximally efficient current distribution feasible within the LTI design region .  If this to be (at least hypothetically) true, then an ideal DAM transmitter with no constraint on resonance or quasi-resonance would the achieve arbitrary bandwidth B_0 with efficiency η_DAM^nr = η^nr_max.  The actual topology of the antenna and non-LTI system that would achieve this performance is unknown, but this proposed model serves as a first upper bound on DAM performance.
	
	

 §.§ Ideal Quasi-resonant DAM Transmitter

	The aforementioned non-resonant upper bound on radiation efficiency is known to be loose.  Additionally, many DAM transmitter architectures rely on resonant tuning to achieve quasi-resonance, e.g., <cit.>, as part of their broadbanding strategy.  Here we incorporate these features by assuming that an ideal quasi-resonant DAM transmitter may realize arbitrary bandwidth B_0 by impressing a broadband version of the maximum self-resonant efficiency current distribution on the design region , i.e., η_DAM^qr = η^sr_max.  The example data in Fig. <ref> representatively demonstrate that the self-resonant efficiency bound is lower than the non-resonant bound <cit.>.
	
	

§ EFFECTIVE EFFICIENCY GAINS

	
	With models for the ideal efficiency of conventional, non-resonant DAM, and quasi-resonant DAM transmitters established, we proceed by studying the relative efficiency of each DAM transmitter with respect to the conventional case, i.e., η_DAM^nr / η_Conv. and  η_DAM^qr / η_Conv..  These quotients represent the highest possible relative gain in efficiency feasible by each class of DAM transmitter over a conventional LTI system. In the remainder of this section we study these ratios for three examples with unique characteristics.  
	
	

 §.§ Electrically Small Spherical Shell

	
	As a first example, we analytically study the optimal performance of a spherical shell carrying a TM_10 (dipole) modal current distribution.  Asymptotic forms for the Q-factor and dissipation factor[Asymptotic scaling rules for loss for this example are much clearer when expressed in terms of dissipation factor δ, related to efficiency via , see <cit.>.] of this structure in the electrically small (ka→ 0) limit are <cit.>
	
	
		
    Q_rad^nr = 3/2(ka)^3,   Q_rad^sr = 1/(ka)^3,

		
    δ^nr = R_s/Z_09/4(ka)^2,  δ^sr = R_s/Z_03/(ka)^4,

		
	
	
	where R_s and Z_0 are the shell's surface resistivity and the impedance of free space, respectively.  The non-resonant case is optimal in non-resonant dissipation factor, while the self-resonant case (where the TE_10 mode is used for tuning) represents both minimum Q-factor and minimum self-resonant dissipation factor.  It can be shown that the Pareto front between these parameters is linear in δ-Q_rad space between these points <cit.>.  
	
	Pareto fronts for shells of three electrical sizes are shown in Fig. <ref>.  For this example calculation, a non-dispersive surface resistivity R_s = 0.0014 Ω/□ is used, corresponding to thick copper at 30 MHz <cit.>, and an objective bandwidth is set at B_0 = 0.005.  The maximal efficiencies η^nr_max and η^sr_max are denoted by square and triangular markers on each Pareto front. The intercept between each Pareto front and the  line are also marked. These data show that for all three electrical sizes, the non-resonant DAM efficiency is much higher than the maximally efficient conventional transmitter.  This is not the case for DAM transmitters with the constraint of quasi-resonance, which have efficiency that reduces rapidly with decreasing electrical size, eventually passing to the left of the Pareto intercept point (η_0,B_0).
	
	
	The trends observed in Fig. <ref> indicate that the “effective efficiency gain” η / η_Conv in a DAM system depends highly on whether or not the condition of quasi-resonance is enforced.  At moderately small electrical sizes, gains in both forms of DAM are substantial.  For extremely small electrical sizes, the rapid increase in losses imposed by resonance overtakes the necessary efficiency sacrifice required to resistively broadband a conventional transmitter to the objective bandwidth B_0.  This is further demonstrated in Fig. <ref>, where the effective efficiency gain η/η_Conv. is plotted as a function of electrical size ka for two objective bandwidths.  There we clearly observe that gains for non-resonant DAM transmitters are unbounded in the electrically small limit, whereas quasi-resonant DAM transmitters provide an efficiency benefit only over a limited size range.  Naturally, the span of this range depends on the target bandwidth, with more extreme broadbanding leading to gains over broader ranges.
	
	
	
	

 §.§ Optimal Substructure Embedded Antenna

	
	As a more complex example utilizing the generality of the Pareto front calculation methods in <cit.>, an embedded antenna is considered where only a portion of the current support is considered controllable.  The controllable region _c over which currents may be optimized is defined as a rectangular region of dimensions h ×αℓ.  The uncontrollable scatterer _u is located a distance g below the controllable region and has dimensions h ×ℓ, as shown in Fig. <ref>.  As in the previous example, a non-dispersive surface resistance  R_s = 0.0014 Ω/□ is used.  
	

	Effective efficiency gains for several different values of the parameter α are shown in Fig. <ref>, where we observe the same trends as in the spherical transmitter.  Interestingly, the low-frequency trend in quasi-resonant DAM efficiency gain appears to be only weakly dependent on the geometry parameter α.  This can be understood as a rather weak dependence of the maximally efficiency tuning current Q-factor Q_L on this parameter, consistent with the competing trends in increasing electrical size and decreasing aspect ratio for standalone rectangular plates <cit.>.
	
	
	

 §.§ Driven Wire Dipole

	As a final example, we step away from Pareto-optimal current analysis and calculate the relative efficiencies of a driven antenna.  Here we explicitly require conventional and DAM systems to radiate via the dipole's driven current distribution, though DAM systems may be designed to impress this current distribution over a broad bandwidth, bypassing the dipole's narrow impedance bandwidth (see <cit.>).  The impedance  and radiation efficiency η_a of a center-fed copper wire dipole with length 2 m and wire radius 1.6 mm are calculated using NEC <cit.> over the frequency range 0.2 – 70 MHz.  
	
	For a conventional transmitter, the antenna may be tuned to resonance at each frequency via an inductance L with Q-factor Q_L, giving rise to the total efficiency 
	
    η =  η_aR_a/R_a-X_a/Q_L.

	As in the previous example, the conventional efficiency is assumed to be tuned and resistively loaded as needed to obtain a target bandwidth B_0, i.e.,
	
    η_Conv. = min{(B_0 Q_Z)^-1,η},

	where Q_Z is the impedance-based Q-factor of the standalone[Calculating Q_Z requires implicit inclusion of a lossless tuning element in conjunction with measured or simulated impedance data.] antenna <cit.>.
		
	An ideal DAM transmitter which requires the same resonant tuning as in the conventional system (see <cit.>) will, at best, achieve the resonant efficiency η_DAM^qr = η, whereas one without the resonance constraint would achieve the standalone antenna efficiency η_DAM^nr = η_a.  Note that the latter is equivalent to assuming lossless inductors are available, though rigorous comparison to the conventional case in that particular scenario requires an alteration to (<ref>), so we do not consider this interpretation further in the present paper.  Taking the ratio of conventional and DAM transmitter efficiencies, with some rearranging, yields
	
		
    η_DAM^qr / η_Conv. = max{Q_Z R_a B_0/R_a - X_a/Q_L,1}

		
    η_DAM^nr / η_Conv. = max{Q_Z B_0,R_a - X_a/Q_L/R_a}.

		
	
	
	
	
	Curves are shown in Fig. <ref> illustrating the efficiency ratios in (<ref>) for two bandwidths B_0 using tuning elements defined by a realistic frequency dependent Q-factor for air-coil inductors <cit.>
	
    Q_L = 150 √(ω / (2π· 30· 10^6)).

	We observe the same characteristic features as in the previous examples, where efficiency gains in the non-resonant system are potentially unbounded in the low frequency limit while gains in the resonant system are limited and present only over a finite bandwidth.  This example indicates that for realistic loss models, physical size, and frequency ranges, the potential benefits of a DAM transmitter are severely limited when some form of resonance (e.g., quasi-resonance) is required.
	
	In the electrically small limit, the dipole antenna's impedance behavior is accurately[Note that the skin depth model is used throughout, which is non-physical in the extreme electrically small limit.] approximated as
	
    R_a≈ R_rad,0ω^2/ω_0^2 + R_Ω,0√(ω/ω_0),   X_a≈1/jω C_0

	where R_rad,0, R_Ω,0, and C_0 are the radiation resistance at frequency ω_0, the ohmic loss resistance at frequency ω_0, and static capacitance, respectively.  Additionally, the antenna's radiation Q-factor may be modeled with the asymptotic cubic scaling in electrical size, i.e., Q_a = Q_a,0ω_0^3/ω^3, and a general form of (<ref>) where Q_L = Q_L,0√(ω/ω_0) may be assumed for the inductor Q-factor.  Determining the frequency range over which the self-resonant DAM efficiency in (<ref>) is greater than unity in this case is best carried out numerically, however the location of the peak value of self-resonant efficiency gain may be estimated as
	
    ω_peak = √(ω_0^5/(6R_rad,0C_0Q_L,0)^2)

	by assuming that at this frequency radiation and inductor losses far outweigh ohmic losses.  In Fig. <ref> we observe that this approximation, which results in a peak location dependent only on low frequency asymptotic coefficients, is extremely accurate (<1% error).  A similar analysis may be carried out by any small dipole radiator characterized by (<ref>).  
	
	
	

§ CONCLUSIONS

	In this paper, an effective efficiency ratio is used to quantify the advantage of DAM transmitters over their conventional counterparts.  By splitting the system into LTI and non-LTI components, many results from optimal LTI antenna theory are applied.  From all reported examples, we observe that the requirement of resonance (e.g., quasi-resonance) severely limits the potential benefits of a DAM transmitter.  This implies that quasi-resonant DAM systems, prevalent in the literature, may have limited applicability at extremely low frequencies.  The same conclusion holds for any loop based methods, regardless of resonance condition, as their radiation efficiency scales inherently as (ka)^4.  Similar concepts may be applied to the analysis of scattering from non-linear loads <cit.>, as the underlying bifurcation of a system into linear and non-linear components is very similar to that used in the antenna problems studied here.
	
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