# LADRC-Based Magnetic Field Measurement Method for a Nuclear Magnetic Resonance Rotation Sensor
###### Abstract  
Magnetic field measurement is fundamental to nuclear magnetic resonance rotation sensors (NMRRS). A phase-locked loop (PLL)-based measurement with two nuclear isotopes is commonly applied to observe the magnetic field. However, the phase-loop and frequency-loop of the nuclear isotopes cannot be optimized simultaneously by a PLL-based method. In this paper, an approach based on a linear active disturbance rejection controller (LADRC) is proposed for synchronous phase-loop control of the two nuclear isotopes. Meanwhile, the frequencies of the nuclear isotopes are observed by linear extended state observers (LESOs). The phase and frequency loops can be decoupled and optimized with the proposed method. An experimental NMRRS prototype used for verification is built. The effectiveness and the feasibility of the proposed method are validated with the experimental results.  
atomic spin gyroscope; active disturbance rejection controller; magnetic field measurement; synchronous control +
Footnote †: journal: Applied sciences  
set of resonant nuclei isotopes, the literature [17] makes it possible to measure the magnetic field. A variant with four frequencies available for measurement is mentioned in [18]; however, it is not commonly applied due to the stringent requirement for the two steady, equal, and opposite magnetic fluxes. Thus, the two-isotope (\({}^{129}\)Xe and \({}^{131}\)Xe) solution of magnetic field measurement is mainly considered in this paper.  
Traditionally, the observed precessional rate is assumed to be equal to the resonance frequency in previous research. These assumptions ignored the transient process of the observer. To make an expected differential rotation measurement, the transient processes of two nuclear isotopes need be synchronous, which is determined by the synchronous performance of the phase-locked loop (PLL). Moreover, observing the frequencies of the two isotopes makes it possible to measure the applied magnetic field. A monotone decreasing step response of the observed frequency can simplify the design of the feedback controller of the magnetic field. Both phase and frequency loops are determined by the parameters of the controllers. It turns out to be a tradeoff, which means that the performance of both loops cannot be optimized at the same time.  
A LADRC is a robust, model-independent approach developed from the classic PID control strategy [19; 20]. One of the most significant features of a LADRC is the linear extended state observer (LESO), which is designed to observe the unknown disturbance without establishing an accurate plant model. In [21; 22; 23], a linear ESO is used to enhance control dynamics of the PLL of the power system, which finds us a new way to overcome the weakness of the traditional method that appears in the NMRRS.  
Therefore, this study makes a major contribution to research on magnetic field measurement by demonstrating a LADRC-based method. The two nuclear spin oscillators are driven synchronously by the LADRC. Two LESOs are designed to observe the Larmor precessional rate of both isotopes. The manipulations of the phase and frequency loops are decoupled, and the optimized performance can be achieved by proper parameter tuning.  
The rest of this paper is organized as follows. The conventional magnetic-field measurement method based on the PLL is briefly introduced in Section 2. Meanwhile, the introduction of the experimental NMRRS and the magnetic field measurement errors are analyzed in this section. Section 3 is concerned with the methodology used for this study. Finally, the experimental setup is introduced in Section 4, and the experimental results are illustrated in Section 5. The main conclusions are drawn in Section 6.