Analysis of Transmission Characteristics on an MSK-Based Rotating Magnet-Based Mechanical Antenna

A rotating magnet-based mechanical antenna (RMBMA) is a new promising paradigm which can signicantly reduce both the size and the power consumption of the super-low frequency electromagnetic transmitter. To further reveal the eects of the rotational motion on transmission, this paper investigates the performance analysis of MSK (Minimum Shift Keying)-based RMBMA for the rst time. Initially, based on the framework of MSK-based RMBMA, both the information loading of RMBMA and the control strategies of the motor are proposed, such that the impacts of the step response induced by the motor on the information loading are characterized. en, the transmission capacity of the MSK-based RMBMA is derived in the closed form. e results indicate that increasing output torque or decreasing inertia load can enhance the transmission capacity. Finally, numerical simulations using a realistic system model demonstrate the validity of the proposed performance analysis. e simulation results show that when the inertia is less than 0.0128 kg∙m and the symbol rate is less than 4 bit/s, the bit error rate is less than 10%, thereby improving the transmission capacity. e proposed comprehensive design principles of RMBMA provide guidance for system design and practical implementation.


Introduction
Super-low frequency (SLF, 30-300 Hz) waves have the characteristics of long propagation distance and good seawater penetration, which have great potential for applications in the eld of underwater communication [1][2][3][4]. However, the low-frequency electromagnetic transmitting antennas of existing technologies are all electrically small antennas (ESAs), which rely on the oscillating current in the conductor to excite electromagnetic waves. ere is a Chu-Harrington limit [5][6][7][8][9], making its transmitting system to have features of bulky size, low radiation e ciency, high transmitting power, and energy consumption, which restricts the development of low-frequency electromagnetic communication in related elds.
To solve the above problems, the US Defense Advanced Research Projects Agency (DARPA) put forward the concept of the mechanical antenna (MA) in December 2016 and launched the AMEBA (a mechanically based antenna) project in August 2017 [10]. It is a new low-frequency electromagnetic transmitting technology that directly excites electromagnetic waves through the mechanical movement of special materials (such as electrets or permanent magnets) [11][12][13][14]. Bene ting from the mature application of rare Earth permanent magnet materials [15] and rotating drive technology [16] in various elds, the rotating magnet-based MA (RMBMA) attracts domestic and foreign attention. RMBMA can generate a static magnetic eld directly without consuming energy, and there is no high-reactance problem and additional loss. us, RMBMA, connected with the motion excitation and control technology of low loss and low damping, can achieve low-frequency electromagnetic waves and information loading at the cost of little energy. It is expected to break through the Chu-Harrington limit of traditional low-frequency ESA [17]. e information loading of RMBMA is realized in the mechanical drive link which means the loaded information is transformed into the control signal of the drive motor. In recent years, the information loading method of RMBMA has been studied. In [18], the modulation of magnetic field amplitude is introduced, which changes the magnetoresistance of the shielding mechanism near the permanent magnet. But the increase of shielding weakens the magnetic field intensity, which is not conducive to the improvement of transmission distance. In [19], the RMBMA technology scheme and information loading method for FSK (Frequency Shift Keying) and MSK (Minimum Shift Keying) are proposed. A prototype is developed to verify the feasibility of FSK modulation, but no quantitative analysis of the transmission performance is performed. In [20], a method based on continuous frequency FSK is introduced, which minimizes the additional torque for the motor to maintain the required bit rate, ensuring that the additional torque is limited to a finite concentration. e driving requirement is reduced, which contributes to the reduction of transmitter power consumption. In summary, the current research on the information loading method of RMBMA is mainly focused on the principle verification of the information modulation method. However, there are few investigations into rotating drives, control performance, and transmission characteristics.
erefore, this paper aims to reveal the influence relationship between the steady-state and transient performance of the rotating servo system and the transmission characteristics of RMBMA. Based on [19], the MSK-based information loading and driving control strategy of RMBMA is introduced. e transmission characteristic analytical method is proposed, and the transceiver link simulation model is established.

System Model of RMBMA
RMBMA is a typical electromechanical coupled system, and according to the conversion relationship among electricity, mechanic, and magnetism energy, the system model of RMBMA is established, as shown in Figure 1. e spinning magnet acts as the load of the servo system, overcoming the frictional damping and moment of inertia and rotating with the permanent magnet synchronous motor (PMSM). e induced/radiated electromagnetic field can be equivalently generated by induction/radiation damping R rad . A part of the mechanical energy is converted into radiated electromagnetic energy; hence, the mechanical motion equation of the spinning magnet can be expressed as where T em is electromagnetic torque, R Ω1 is frictional damping, ω is the angular frequency; J m , Ω, and N are the moment of inertia, rotational angular velocity, and pole pairs of the spinning magnet, respectively, and N � ω/Ω. As shown in Figure 2, the induced/radiated electromagnetic energy in the incident direction r is received by the magnetic antenna after superimposing the channel noise in the transmission channel. e magnetic antenna is equal to an induction coil hinging on a time-varying magnetic field whose outputinduced voltage U [21] can be expressed as follows: where N 0 and S are the numbers of turns and the crosssectional area of the coil, respectively, f is the excitation electromagnetic wave frequency, B is the received magnetic field intensity, and θ′ is the angle between the axis of the coil and the r direction (θ' � 0°indicates that the coil axis is horizontal).

Information Loading and Control Strategy
Using MSK e rotational state of the spinning magnet cannot be changed abruptly due to the rotational inertia, and the modulated waveform is required to have constant amplitude and continuous phase to reduce distortion, so MSK is used as the information loading method of RMBMA. For an MSK signal, the k th symbol can be expressed as [22]: where A is the amplitude, f c is the carrier frequency, θ k (t) is the additional phase of the k th symbol, T RB � 1/R B is the symbol time, and R B is the symbol rate. e rotational speed n of the spinning magnet corresponds to the excitation electromagnetic wave frequency f, and the rotation angle Θ (rotor position P) corresponds to the phase θ of the excitation electromagnetic wave. Figure 3 gives the diagram of the rotating speed and rotor position control signals generated from the symbol. Here, n/f � Δn/ Δf � 60N; P is equal to θ and always changes ±π/2 within a symbol time T RB , i.e., linearly increases π/2 when the symbol is 1 and linearly decreases π/2 when the symbol is 0.
As shown in Figure 4, the control loop of PMSM [23] corresponding to the MSK signal is introduced, which mainly consists of the automatic speed regulator (ASR) and the automatic current regulator (ACR). In the rotating speed control, the ASR obtains the given current of the current loop I * q based on the error between the given speed n * and the calculated actual speed n. e ACR adjusts the output voltage of the power converter in real time to control the output torque of the motor based on the vector control strategy with zero direct shaft current [24] and finally achieves a fast and high-precision tracking response to the given speed. e rotor position control is realized by the magnetic coding current [25], and it changes the output voltage of the power converter in real time based on the estimated rotor position information and the input given signal according to the adaptive control strategy [26] so that the rotor tracks the input given signal. us, the information loading for MSK signals can be achieved by implementing real-time control for n and P.

Effect of Spinning Magnet Parameters on Amplitude.
e physical model of the spinning magnet is equated to a rotating magnetic dipole and can be regarded as two orthogonal vibrating magnetic dipoles with the same frequency in the same plane [27]. e time-varying electromagnetic field component generated by a spinning magnet in a uniform lossy medium can be expressed as where μ 0 is the magnetic permeability, r is the propagation distance, m 0 � VB r /μ 0 is the magnetic moment of the magnet (V is the volume of the magnet, and B r is the remanence), r, θ, φ are the unit vectors in the direction of r, θ, and φ, respectively, c � β − jα is the wave number of the lossy medium, and α and β are the attenuation and phase shift factors, respectively.
where ε and σ are the dielectric constant and conductivity, respectively. Assuming that the magnetic antenna is placed in the r direction and the angle θ′ with the r direction is 0°, the magnetic field intensity of the hinge of the induction coil can be expressed as To simplify the analysis, assuming that the permanent magnet is a cylinder and the material is NdFeB (ρ � 7800 kg/ m 3 is the density), its outer diameter and height are R and h, respectively, the rotational inertia J m � MR 2 /2, M � ρV is the mass of the magnet, and V � πR 2 h is the volume of the magnet. When B 0 is constant, the aspect ratio λ � h/R (the structure of the magnet) is changed by changing J m when the receiving distance is certain. e rotational inertia J m � 0.0032 kg·m 2 (V � 500 cm 3 and λ � 2.4) is taken as the ideal value [28,29] in this paper.

Effect of Rotational Motion on Frequency.
Due to the rotational inertia of both the spinning magnet and the rotor, abrupt speed variation is not allowed. As shown in  International Journal of Antennas and Propagation Figure 5, the MSK modulation signal in turn appears as the speed step change. erefore, the steady-state and dynamic performance of the servo system determines the transmission characteristics of RMBMA. Assuming N � 1, n c /f c � Δn/Δf � 60 and Ω � ω � (π/30)n, particularly, for MSK, Δn � 15/T RB , Δf � 1/(4T RB ). It can be seen that the given speed increases suddenly from 60 (f 0 − 1/(4T RB )) to 60 (f 0 + 1/(4T RB )) at t 1 , and it decreases suddenly from 60 (f 0 + 1/(4T RB )) to 60 (f 0 − 1/(4T RB )) at t 1 + T RB . e corresponding rotation angular velocity is Ω 1 ∼Ω 2 at t 1 ∼t 1 + Δt r . For the drive characteristics of PMSM, T em is constant [30], since T em ≫ (R Ω + R rad )Ω, dΩ/dt is approximately a constant from equation (1). Hence, the transient process of the speed step change can be regarded as a linear variation, and the times reaching stable performance are Δt r and Δt f , respectively (ignoring the oscillation process).
Neglect the higher-order oscillation process, the angular velocity whose step-change response is approximated as a linear rise and decay process in Figure 5 can be expressed as Since n/f � Δn/Δf � 60, frequency has the same variation as angular velocity. en, neglecting the effects of R Ω1 and R rad in (1), for the rise and decay processes in Figure 5, we have where ΔΩ�(π/30)•Δn, T em ′ is the braking torque.T em ′ � 0 indicates that PMSM drives without a brake control mode, and Δt f ≫ Δt r . To reduce the drive loss and avoid an excessive difference between Δt f and Δt r , the feedback braking control strategy [31] is adopted. e rotor mechanical energy is converted into electrical energy and fed back to the DC bus through boosting, which can make T em and T em ′ approximately equal while recovering the braking energy.

Effect of Rotational Motion on Phase.
e amount of change in the additional phase is equal to the actual rotational angular velocity of the spinning magnet minus the integral amount of the unmodulated rotational angular velocity. For the step response process shown in Figure 5, supposing the initial phase at t 1 is 0, θ k (t), then the additional phase difference Δθ k (t) can be expressed as It can be seen that the amount of change of θ k (t) at the rise and decay process is 0 from (9), which means θ k (t) behaves as a period of Δt r or Δt f time delay at the rise and decay process, and the change of phase within a T RB is no longer ±π/2. As shown in Figure 6, if Δt r ≠ Δt f , the phase error increases continuously with the accumulation of time, and if the control strategy is adopted, i.e., Δt r � Δt f , the phase error generated in the previous rise/decay process will be compensated in the next rise/decay process, and there is no longer a cumulative error.  International Journal of Antennas and Propagation

Effect of Rotational Motion on Transmission Performance.
For MSK signals with bandwidth B w � 1/T RB , assuming Δt � Δt r � Δt f , the transmission performance is analyzed in terms of transmission capacity (B W R B ), which can be expressed according to (7) as With K SNR is inversely correlated with SNR, which reflects signal distortion of about Δt/T RB . e signal distortion is generated by the frequency step response and phase fixation error introduced by rotational motion, and the smaller Δt or K SNR is, the higher the SNR value is. When K SNR is small, the signal distortion is mainly outside the band, which has a little effect on the in-band signal and is not easy to occur symbol error; when K SNR > 1/π, the phase error generated in the previous rise/decay process cannot be compensated in the next rise/decay process, and the phase accumulation error occurs, which affects the in-band signal and is bound to occur bit error; when K SNR is small and remains unchanged, the transmission capacity of RMBMA (B w R B ) is proportional to T em /J m , that is, the transmission capacity can be improved by increasing the output torque or decreasing the inertia.

Transmitter and Transceiver Link Modelling Based on MSK.
e transmission characteristics of RMBMA are determined by two aspects: the output field intensity and the rotating servo performance of the rotating permanent magnet. Corresponding to the signal amplitude of receiving distance, and the transient response process of frequency and phase, respectively. For channel noises are white Gaussian noises, the transmitter and receiver link modes are established as shown in Figure 7. e transmitter link model offers five modules: MSK signal generation, magnetic field distribution model [27], structural design and stress field analytical model [28], electromagnetic field analytical model of PMSM [32], and servo control model. According to the correspondence shown in Figure 3, the MSK signal generation module converts the input symbol data into the given speed. As required for receiving distance and field intensity, the magnetic moment m 0 is calculated by the magnetic field distribution model, then the spinning magnet structure and size are determined by the structural design and stress field analytical model, and the rotational inertia of the spinning magnet is obtained. With the PMSM parameters, the PMSM electromagnetic field analytical model calculates its output electromagnetic torque T em . e servo control model adjusts the actual speed with reference to the given speed. Finally, the simulation waveform B(t) of RMBMA output magnetic field strength is obtained by the magnetic field intensity simulation of the spinning magnet. e receiver link converts the electromagnetic wave B(t) output from the transmitting link into the induced voltage U(t) of the induction coil after superimposing a certain amount of noise power on it. From (2), U(t) is proportional to f(t), so no coherent demodulation of B′(t) can be achieved by extracting the envelope of U(t). e received chip data are obtained after envelope detection, shaping filtering, and normalization, and finally, the bit error rate (BER) is analyzed.

Simulation
Results. MATLAB is used for the simulation of the proposed structure. e PMSM parameters are cited from [28] with the rated speed at 60,000 rpm, T em � 0.8 N•m. Based on the above transmitter and transceiver link model, MSK modulation is adopted, and the transmission characteristics of RMBMA are simulated and analyzed by adjusting the rotational servo control parameters and the rotational inertia J m of the spinning magnet. Taking f c � 300 Hz, Δf � 1 Hz, R B � 4 bit/s, and J m � 0.0032 kg•m 2 , a feedback braking control strategy is used, and its effect is adjusted by ACR. e speed step response of the spinning magnet corresponding to different feedback braking effects is shown in Figure 8. e fall time (Δt f1 � 45.70 ms) for speed from 18060 rpm to 17940 rpm is much larger than the rise time (Δt r1 � 9.72 ms) for speed from 17940 rpm to 18060 rpm when the feedback braking effect is poor, and Δt r2 and Δt f2 are equal to 9.72 ms when the feedback braking control effect is good.  Removing the speed rise time of the PMSM, the phase change is observed in a steady state. e simulated comparison of the phase and spectrum change corresponding to different feedback braking effects is shown in Figure 9. From Figure 9(a), it can be seen that when the feedback braking effect is poor (Δt r <Δt f ), the cumulative error of the phase keeps increasing with time. When the feedback braking control effect is good (Δt r = Δt f ), the phase error is only shown as a fixation error in the decay process, and there is no more cumulative error. From Figure 9(b), From Figure 9(b), it can be seen that whether Δtr and Δtf are equal has little effect on the in-band signal spectrum. But harmonic components is appeared in f c ±kΔf (k is odd and k>1) when Δt r <Δt f (green), causing out-band distortion.
When the magnetic moment is constant, changing the rotational inertia J m affects the structure of the magnet.

International Journal of Antennas and Propagation
Taking f c � 300 Hz, Δf � 1 Hz, and R B � 4 bit/s, the feedback braking control strategy is adopted and keeps a good braking effect so that Δt r ≈ Δt f . e simulation results of the phase and spectrum with different inertias are shown in Figure 10. It can be seen that when J m is small, Δt r and Δt f are much smaller than T RB , the fixation error of phase is small, and there is no accumulated error, which has a little effect on the in-band signal spectrum and mainly introduces out-band distortion; when J m is large, the error of phase increases with Δt r and Δt f getting increased, and the introduced distortion increases accordingly. Especially when Δt r and Δt f are nearing or over T RB , the error of phase is no longer the previous fixed value, and the accumulated error is generated instead. e in-band distortion is introduced, and the error bit is generated. e same as that revealed by (11) and (12), the transmission capacity of RMBMA is inversely proportional to J m .
For Δf � 1/(4T RB ) and T RB � 1/R B , changing the symbol rate R B affects Δf. Taking f c � 300 Hz and J m � 0.0032 kg•m 2 , the feedback braking control strategy is adopted and keeps a good braking effect so that Δt r ≈ Δt f . e simulation results of the phase and spectrum for R B � 2 bit/s(Δf � 0.5 Hz), R B � 4 bit/s(Δf � 1 Hz), and R B � 8 bit/s(Δf � 2 Hz) are shown in Figures 11 and 12. It can be seen that as R B increases, Δt r and Δt f increase, leading to an increase in the fixed error of the phase. When R B is small, the increase in the fixation error of the phase has a little effect on the in-band signal spectrum; when R B increases, T RB decreases, and the decrease of the out-band signal power increases as R B . Especially when Δt r and Δt f are nearing or over T RB , the fixation error of the phase has an effect on the in-band signal spectrum, and the error bit is generated, which is the same as that revealed by equation (12).

International Journal of Antennas and Propagation
According to the transmitter and transceiver link mode introduced in Figure 7, the BER is analyzed with different symbol rates and inertias. Having f c � 300 Hz, the simulation results of the BER with no coherent demodulation are shown in Figure 13. In the light of Figure 13(a), R B � 4 bit/s (Δf � 1 Hz), when J m ≥ 0.0128 kg·m 2 , and Δt r and Δt f are nearing or over T RB . When the symbol data change, the bit error easily appears, of which the BER is greater than 10 −1 . Moreover, Δt r and Δt f decrease with the decrease of J m , resulting in the decline of the BER for MSK signals. From

Conclusions
In this paper, the information loading and driving control strategy for MSK modulation-based RMBMA is introduced, and the transient response analytical method of transmission characteristics is proposed. e transmission capacity of RMBMA is proportional to T em /J m for MSK signals. e greater the output torque and the less the inertia load are, the higher the permitted transmission capacity is for a certain SNR. e transmission characteristics of RMBMA with MSK modulation are simulated and analyzed. e frequency step change fits the speed step change of the spinning magnet. However, there is a transient transition time of the speed step change, corresponding to the stable time of the frequency step change and the phase error, due to the spinning magnet inertia. It has a direct influence on signal quality and receiving BER. In practical applications, transmission performance can be enhanced by designing PMSM with big enough torque. To reduce harmonic distortion, a suitable control strategy should be adopted to ensure the rise/decay process times Δt r and Δt f of the step change are equal and much smaller than T RB . In order to reduce the BER and improve the transmission capacity, the symbol rate should be less than 4 bit/s, and the inertia should be less than 0.0128 kg•m 2 .

Data Availability
e data used to support the findings of this study are included within the article.

Conflicts of Interest
e authors declare that they have no conflicts of interest.