Simultaneous Maximization of Voltage and Power Efficiencies in Magneto-Mechanical Transmitters

Magneto-mechanical resonator arrays (MMRAs) have emerged as a promising transmitter solution for compact ultralow frequency (ULF) wireless communication systems and can be extremely power-efficient compared to traditional electrical antennas in the ULF range. The efficiency of ULF signal generation using magneto-mechanical transmitters (MMTs) is dictated by multiphysical effects from mechanical, magnetic, and electrical domains, leading to an interesting trade space. In this work, we show that an MMT’s most power-efficient and most voltage-efficient driving frequencies always differ, forcing designers to sacrifice one efficiency for the other. To address this issue, we propose an efficiency optimization method that minimizes the total impedance of the MMT at the most power-efficient driving frequency, by means of a compensation capacitor added to the electromagnetic actuation coil system. Our experimental results show excellent agreement with our analytical model, and we demonstrate that our approach enables simultaneous maximization of voltage and power efficiencies of an MMT at the same driving frequency. We additionally describe how to apply this optimization method on multiresonator magneto-mechanical arrays and present numerical analysis that predicts much greater improvement factors in systems having larger net magnetic moments and drive coils with larger sizes.


Simultaneous Maximization of Voltage and Power Efficiencies in Magneto-Mechanical Transmitters
Jiheng Jing , Graduate Student Member, IEEE, and Gaurav Bahl , Senior Member, IEEE Abstract-Magneto-mechanical resonator arrays (MMRAs) have emerged as a promising transmitter solution for compact ultralow frequency (ULF) wireless communication systems and can be extremely power-efficient compared to traditional electrical antennas in the ULF range.The efficiency of ULF signal generation using magneto-mechanical transmitters (MMTs) is dictated by multiphysical effects from mechanical, magnetic, and electrical domains, leading to an interesting trade space.In this work, we show that an MMT's most power-efficient and most voltage-efficient driving frequencies always differ, forcing designers to sacrifice one efficiency for the other.To address this issue, we propose an efficiency optimization method that minimizes the total impedance of the MMT at the most power-efficient driving frequency, by means of a compensation capacitor added to the electromagnetic actuation coil system.Our experimental results show excellent agreement with our analytical model, and we demonstrate that our approach enables simultaneous maximization of voltage and power efficiencies of an MMT at the same driving frequency.We additionally describe how to apply this optimization method on multiresonator magnetomechanical arrays and present numerical analysis that predicts much greater improvement factors in systems having larger net magnetic moments and drive coils with larger sizes.

I. INTRODUCTION
W IRELESS data transfer plays a valuable role in our daily activities.However, wireless communications are particularly important for marine and underground activities, including search and rescue, locator beacons, and environmental monitoring.Unfortunately, radio signals cannot propagate through conductive media such as seawater, metal, rock, and soil due to significant signal attenuation [1].In this context, ultralow frequency (ULF, <3 kHz) communication systems have been shown as a good alternative due to the very large skin depth at ULF, which significantly enhances the range.The difficulty arises with traditional electrical antennas since, when scaled for ULF, they require a very large area [2] and prohibitive power for generating sufficient signal levels [3], [4], making them impractical for many real-world applications.
Recently, a fundamentally new approach to building ULF transmitters has been developed [5], [6], [7], [8], [9], [10], [11], [12], commonly referred to as magneto-mechanical transmitters (MMTs).In an MMT, the magnetic field carrier signal is generated through the angular motion of permanent magnetic dipoles, on which data can be encoded using amplitude or frequency modulations.Since the permanent magnetic dipole brings the power requirement for generating the magnetic field to exactly zero, the power dissipation now only appears in the driving mechanism [5], the mechanical losses in the suspension system [13], and the eddy current losses in the magnetic materials [14].Notably, as the mechanical oscillation frequency decreases, the associated mechanical loss and eddy current loss also decrease rapidly.Therefore, MMTs provide a power-efficient alternative to traditional electrical antennas for ULF transmitters.A typical oscillatory MMT consists of a drive coil and a magneto-mechanical resonator array (MMRA) [14] as shown in Fig. 1(a).The MMRA can be designed with one or more rotors and stators, with the stators providing a restoring torque for resonant operation [5], [6].Ac current provided to the coil produces an alternating magnetic field, generating an alternating torque on the rotors, which in turn leads to oscillatory motion.The resulting mechanical motion of the permanent dipoles produces a time-varying magnetic field at all points in space around the MMT, whose amplitude is directly related to the oscillatory amplitude of the rotors [5].The driving frequency ω of the source is always set to be close to the mechanical resonant frequency of the MMRA [5], [6], [14] in order to achieve higher voltage efficiency, which can be expressed in terms of the time-varying magnetic field signal strength at the receiver normalized to the voltage level applied on the drive coil.We can also define power efficiency as the time-varying magnetic field signal strength normalized to the power dissipated in the MMT system.Both efficiency metrics are key for determining the practicality of MMTs, and both are functions of the driving frequency ω due to the resonance behavior of the MMRA as well as the coil [5].In this article, we observe that an MMT's most power-efficient and most voltage-efficient driving frequencies always differ, i.e., their resonances do not align, forcing designers to necessarily sacrifice one efficiency for the other.To address this issue, we propose an efficiency optimization method that minimizes the total impedance of the MMT at B, which varies with the angular position of the rotor magnet.The scalar magnetic field sensor is placed in the x direction to measure the projected magnetic field B meas (t) along the x direction.
the most power-efficient driving frequency.Using experiments and analysis, we demonstrate that this method allows us to simultaneously maximize the voltage and power efficiencies of an MMT at the same driving frequency.

II. POWER AND VOLTAGE EFFICIENCIES OF AN MMT
We consider a single-rotor MMT as shown in Fig. 1(a).The driving voltage, V (t), is a sinusoid at frequency ω.The induced current in the coil, I (t), and the resulting rotor motion, θ(t), can then generally be expressed as I (t) = (1/2)I m e jωt + c.c. and θ (t) = (1/2)θ m e j (ωt+φ) +c.c.where I m and θ m are the current amplitude and the oscillation amplitude, respectively, and φ is some relative phase difference between I (t) and θ (t).The configuration of MMT and receiver (x-oriented) shown in Fig. 1(b) has been previously discussed [5] as being the most optimal for generating the largest time-varying magnetic field at frequency ω.The x-oriented magnetic field measured at the receiver, B meas (t) = B M M R A (t) + B c (t), is comprised of the magnetic fields generated by both the MMRA and the coil.Under approximation that the angular motion of the MMRA dipole is relatively small, we can expand the sinusoidal using Taylor expansion and ignore the higher-order dependencies and simply write where A θ and A I are coefficients that describe the relationship between the magnetic fields and the current and rotor motion, whose values have been discussed in [14] and [15], respectively.The received magnetic signal can be defined as the rms value of B meas (t) The average power consumed by the MMT can be evaluated as where R c is the resistance of the coil and β is the damping coefficient of the MMRA.
To quantify the power efficiency, we normalize the atreceiver magnetic signal strength to the average power consumed through the Field Per square Root Watt (FPRW) metric where H θ/I (ω) is the transfer function from the coil current to the angular position of the rotor.We take the square root of the average power in this expression since FPRW is independent of the input voltage level for a particular MMT.To quantify the voltage efficiency we similarly relate the at-receiver magnetic signal strength to the rms value of the supplied voltage through the field per volt (FPV) metric where H I /V (ω) is the transfer function from the input voltage to the coil current.FPV(ω) also remains constant for an MMT regardless of the input voltage level.Since it is typical to use a voltage source to drive the coil [5], [6], the natural choice for driving frequency is where the FPV(ω) is maximized.We can find this resonance condition using (3), with the simplification that the damping coefficient β of the MMT is very small for most MMTs [5], [6], [14].The denominator of FPRW(ω) is then dominated by the coil resistance R c , implying that the shape of FPRW(ω) closely matches the shape of H θ/I (ω), which exhibits a resonance as we will see in the following paragraph.On the other hand, as can be seen in ( 4), the shape of FPV(ω) depends on both H θ/I (ω) and H I /V (ω).Since H θ/I (ω) also exhibits a resonance, the most power-efficient and most voltage-efficient driving frequencies always differ, forcing the users to necessarily sacrifice one efficiency for the other.One approach to improve the FPV is by adding a stepup voltage transformer.However, adding a step-up transformer will not change the most power-efficient and most voltageefficient driving frequencies and does not provide a solution to the problem.
To maximize FPRW(ω) and FPV(ω) at the same driving frequency, we need to match the resonant frequencies of Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.Fig. 2. Equivalent circuit model of the MMT.The circuit on the left models the driving system, and the circuit on the right models the MMRA.R c and L c are the resistance and inductance of the coil, respectively.C c is the optional capacitance that can be introduced as a part of the driving system.κ, β, and J are the restoring stiffness, damping coefficient, and moment of inertia of the rotor, respectively.Two circuits are coupled through a gyrator [5], [16] where the coupling coefficient is o .The flow variables in the coil and MMT circuits are I (t) and θ(t), respectively.V (t) in the coil circuit is the input voltage.
H θ/I (ω) and H I /V (ω).To accomplish this, we start from an equivalent circuit model [5] that describes the dynamical behavior of the system, as shown in Fig. 2. The circuit equations (all variables are defined in Fig. 2) can be written as Here we have introduced an extra compensation capacitor C c in series with the coil which we will use later to optimize the efficiencies with respect to drive voltage and power consumption.By converting to frequency domain, the transfer functions H I /V (ω) and H θ/I (ω) can be evaluated as and The resonance of H θ/I (ω) can be evaluated readily from (7) as ω r es = (κ/J ) − (β 2 /4J 2 ) ≈ √ (κ/J ) and is the same as the mechanical resonance frequency.We can similarly extract the spectral characteristics of the impedance function H I /V (ω) by examining (6).With the assumption of small β, we can see that the Re{H I /V (ω) −1 } will reach a minimum value R c at the frequency at which Im{H I /V (ω) −1 } = 0.At this frequency, which we define as ω opt , the function H I /V (ω) will hit a resonance condition.Our optimization goal is now to find a compensation capacitor such that the resonance of H I /V (ω) and of H θ/I (ω) are aligned, i.e., when ω opt = ω r es .The value of this series capacitor C c can then be found by setting Im{H I /V (ω r es ) −1 } = 0, resulting in the solution The slight deviation should appear due to the small β approximation that we applied earlier, which is best adjusted via experiments.As we will show later, this is not the same capacitance where the coil resonant frequency matches with the MMRA resonant frequency.It is also worth noting that irrespective of the compensation capacitor, the FPRW of the system is unaffected.

III. EXPERIMENTAL RESULTS AND DISCUSSION
We have experimentally verified this optimization method using a single-rotor MMT.The MMT device is shown in Fig. 3, and values of experimental parameters used for efficiency calculations are measured and listed in Table I (other material properties and geometric parameters are given in Table II).We can now employ (8) to estimate that the required capacitance of the compensation capacitor will be C c = 31.0µF.As a side note, C c = 53.5 µF is the capacitance  required to form an L-C tank resonator with the coil with a resonance that matches the MMRA.
We now experimentally test three important scenarios: 1) using a coil alone as the driving system (i.e., no compensation capacitor); 2) introducing the series compensation capacitor to match the coil resonant with the MMRA resonant frequency (i.e., C c = 53.5 µF); and 3) setting the compensation capacitor to the predicted value (i.e., C c = 31.0µF).
In each scenario, we experimentally measure the FPRW and FPV of the system and compare them to the analytical predictions from ( 3) and ( 4).As shown in Fig. 4(a), the FPRW of the system in all three scenarios remains unchanged.The slight difference comes from the parasitic resistance intrinsic to the ceramic capacitors.The optimal driving frequency for maximizing FPRW is experimentally measured at 472.0 Hz in all scenarios.Fig. 4(b) presents the simultaneously measured FPV for the three scenarios.We find that the correct compensation capacitance yields an FPV whose resonance is aligned with the FPRW resonance.The FPV on resonance is also 3.2 times higher than that of the system using the coil alone, i.e., without any compensation capacitor and is 1.62 times higher than when the coil and MMRA have the same resonant frequency.The experimental and analytical results confirm that this efficiency optimization method helps maximize the FPV of the system at the optimal driving frequency without affecting the FPRW of the system.From Fig. 4, we can also see that there is a very good agreement between the analytical model ( 3) and ( 4) and the experimental data for both the FPRW and FPV.

IV. CONCLUSION
In this work, we demonstrate an optimization method that simultaneously maximizes the voltage efficiency and power efficiency of an MMT through the simple addition of a compensation capacitor to the drive coil.Our analytical model exhibits excellent agreement with experimental results.In practical applications, increasing the magnetic moment of an MMT is often necessary to enhance the signal strength or extend the communication distance.One way to achieve this is to use a larger magnet for a single-rotor MMT, while another approach is to employ multiple smaller rotors in a multirotor MMT, which offers advantages such as higher power efficiency and higher resonant frequency [5].In Appendix A, we describe in detail how to leverage this optimization method with multirotor MMTs.Importantly, our numerical analysis with multirotor MMTs, presented in Appendix B, suggests that  The dashed orange lines present the simulated FPV of the system with the coil alone (F P V coil , scenario 1 from the main manuscript).In contrast, the solid orange lines present the simulated FPV of the system with the capacitor-based efficiency optimization applied for each individual case (F P V opt , scenario 3 from the main manuscript).The vertical black dashed lines indicate the optimal driving frequencies and the improvement factors (measured as F P V opt /F P V coil ) are indicated in the plots.The results show that significantly higher factors of improvement are possible than in the single-rotor MMT case.
the improvement factors can be significantly higher than we have shown in our single-rotor experiments.These findings highlight the potential benefits of this efficiency optimization method and suggest promising directions for future research.

APPENDIX A IMPROVING EFFICIENCY OF MULTIROTOR MMTS USING COMPENSATION CAPACITORS
Consider a multirotor MMT consisting of a linear chain of rotors and stators driven by a drive coil [5], [6], [14], as shown in Fig. 5.The differential equations that describe an N -rotor MMT system can be written as where L c , R c , and C c are the inductance, resistance, and capacitance of the coil, respectively.The vector = [θ 1 , θ 2 , . . ., θ N ] T represents the instantaneous angle of each rotor.The vector = [ 1 , 2 , . . ., N ] T represents the coupling where i is the coupling coefficient between the ith rotor and the drive coil.The inertia matrix J is an N × N diagonal matrix written as J = diag(J 1 , J 2 , . . ., J N ) where J i is the moment of inertia of the ith rotor.The damping matrix B can be expanded into two major contributions B = B sus + B eddy .B sus is an N × N diagonal matrix written as J = diag(β 1 , β 2 , . . ., β N ) where β i is the damping coefficient of the ith rotor from its suspension.B eddy is an N × N matrix that represents the eddy current loss coefficient whose value can be calculated from the model developed in [14].The stiffness matrix K can be expanded into three major contributions K = K sus + K r + K s .K sus is a diagonal matrix written as K sus = diag(κ 1 , κ 2 , . . ., κ N ) where κ i represents the restoring stiffness on the ith rotor from its suspension.K r and K s are N × N matrices that represent the restoring stiffness generated by rotors and stators, respectively, and their values can be calculated from the model developed in [14].The mechanically synchronized mode (i.e., the in-phase mode) is particularly well suited for MMT applications [5], [6] since it has the highest resonant frequency, produces the largest magnetic signal, and couples best to the drive coil (the other modes tend to be magnetically dark) [14].The oscillation mode shape for the in-phase mode of the multirotor MMT can be evaluated from the eigenvector of the matrix J −1 K .The differential equations can be rewritten as where θ e is the effective oscillation angle such that = θ e .We can further simplify (A2) into this form where e = T = T is the effective coupling coefficient, J e = T J is the effective moment of inertia, β e = T B is the effective damping coefficient, and κ e = T K is the effective stiffness of the multirotor MMT.The magnetic field at the receiver along the x direction can be written as where A I is the current coefficient and A θ e is the effective angle coefficient.Under the dipole approximation and the small angle approximation, the effective angle coefficient can be evaluated as where Therefore, using (A3) and (A4), we can follow the same procedure and apply the efficiency optimization method described in the main manuscript on multirotor MMTs.

APPENDIX B NUMERICAL ANALYSIS FOR MULTIROTOR MMTS
Let us consider a multirotor MMT with N identical rotors and suspension systems, and the same resonant frequency ω r es as the single-rotor MMT, i.e., assuming i as the rotor index we have J i = J r , β i = β r , and κ i = κ r for all individual rotors.As a reminder, ω r es is the resonance for the transfer function H θ/I (ω), and it is only set by the mechanics in our model.For simplicity we can also assume the N -rotor MMT has an uniform mode shape, i.e., is an identity matrix, and thus J e = N J r , β e = Nβ r , and κ e = N κ r .Here it is important to emphasize that in a real multirotor system the rotor participation is generally nonuniform, and the synchronized oscillation mode of a multirotor MMT typically has a much higher frequency ω r es than the individual rotors [14].For the sake of simplicity and insight development, we will choose to ignore these subtleties, but details for interested readers can be found in [14].
Based on the above assumptions, we find that as the number of rotors N increases, the coil size will need to increase to accommodate the size of the rotor array.Since both coil size (and thus the coil inductance and resistance) and magnetic moment increase by a factor of N , the coupling coefficient increases by a factor of N 2 as it linearly depends on both the coil size and the magnetic moment [5], i.e., e = N 2 r .If we assume this rapid scaling on the coupling coefficient and maintain the above assumptions on individual rotors, it can be shown that there will be a more substantial improvement in the FPV of the MMT when a compensation capacitor is used.Since the analytical arguments for this conclusion are not easily presented, we instead conducted numerical simulations for MMTs with varying numbers of rotors N = 1, 5, 10, 20.The results of this numerical analysis are presented in Fig. 6.This analysis confirms that, for larger numbers of rotors, the improvement in FPV can be significantly higher than what we have reported in single-rotor MMT experiments conducted in the main manuscript.These findings highlight the potential benefits of this efficiency optimization method for multirotor systems.

Fig. 1 .
Fig. 1.Operational principle of an MMT.(a) Schematic of a single-rotor MMT.The drive coil is supplied with an alternating voltage V (t), which induces an alternating current I (t) flowing through the coil.The drive coil generates an alternating magnetic field B c (t) in the x direction.The rotor and stators in the MMRA have uniform magnetization, M r and M s , respectively, pointing in the ŷ direction at rest, which is orthogonal to B c (t) to maximize the driving torque on the rotor.During operation, stators remain static, and the rotor oscillates about its longitudinal axis with a time-varying angular position θ (t) with amplitude θ m .The switch is used to implement the ON-OFF keying modulation.(b) Schematic showing the relative position of the MMT and the receiver (scalar magnetic field sensor).The MMT generates a magnetic field at any point in space, ⃗B, which varies with the angular position of the rotor magnet.The scalar magnetic field sensor is placed in the x direction to measure the projected magnetic field B meas (t) along the x direction.

Fig. 3 .
Fig. 3. (a) Schematic of the single-rotor MMT.The single-rotor MMT uses a cylindrical neodymium magnet (NdFeB) as the rotor, two cuboidal NdFeB with a square cross section as stators, and two copper beryllium (CuBe) wires as the suspension system.The drive coil is a 100-turn coil made of AWG 18 enameled copper wire and the capacitors are nonpolarized ceramic capacitors.(b) Photograph of the experimental setup.The single-rotor MMT consists of a 3-D-printed frame and two 3-D-printed wheels to adjust the tension on the CuBe wires.We use a flux-gate magnetometer (Texas Instruments DRV425EVM) as the scalar magnetic field sensor to pick up the magnetic field signal as illustrated in Fig. 1(b).

Fig. 4 .
Fig. 4. Analytical (solid lines) and experimental (dots) results of (a) FPRW and (b) FPV of the system at different driving frequencies.Red color represents scenario 1 (no capacitor in the experiment), blue color represents scenario 2 (C c = 54.1 µF in the experiment), and green color represents scenario 3 (C c = 29.0µF in the experiment).The vertical black dashed line marks the optimal driving frequency, which is ω opt = 472.0Hz in all three scenarios.FPVs of three scenarios at the optimal driving frequency are 41.8, 82.7, and 133.8 µT/V, respectively, as indicated in (b).

Fig. 5 .
Fig. 5. Operational principle of a multirotor MMT.(a) Schematic of a multirotor MMT.The rotors and stators in the MMT have uniform magnetization, ⃗ M r i and ⃗ M s , respectively, pointing in the ŷ direction at rest, which is orthogonal to ⃗ B c (t) to maximize the drive torque on the rotor.(b) Schematic showing the relative position of the MMT and the receiver (scalar magnetic field sensor).

Fig. 6 .
Fig.6.Numerical results for MMTs with varying numbers of rotors (N = 1, 5, 10, and 20) at different driving frequencies.The solid blue lines represent the simulated FPRW of the MMTs.The dashed orange lines present the simulated FPV of the system with the coil alone (F P V coil , scenario 1 from the main manuscript).In contrast, the solid orange lines present the simulated FPV of the system with the capacitor-based efficiency optimization applied for each individual case (F P V opt , scenario 3 from the main manuscript).The vertical black dashed lines indicate the optimal driving frequencies and the improvement factors (measured as F P V opt /F P V coil ) are indicated in the plots.The results show that significantly higher factors of improvement are possible than in the single-rotor MMT case.