# LADRC-Based Magnetic Field Measurement Method for a Nuclear Magnetic Resonance Rotation Sensor
## 2 Dual Nuclear Magnetic-Field Measurement Method
### Dynamic Model of the PLL Controlled Spin  
The dynamics of the noble gas spin ensemble is the basis of this method. The motion of the magnetization vector of the Xe nuclei is described by Bloch equations as follows:  
\[ \begin{array}{l}\frac{dM_{x}}{dt}=\gamma\left(B_{z}M_{y}+B_{y}M_{x}\right)- \frac{M_{x}}{T_{2}}\\ \frac{dM_{y}}{dt}=\gamma\left(B_{x}M_{z}+B_{z}M_{x}\right)-\frac{M_{y}}{T_{2} }\\ \frac{dM_{y}}{dt}=\gamma\left(B_{y}M_{x}+B_{x}M_{y}\right)-\frac{M_{x}}{T_{1} }+R_{se}\end{array}  \tag{4}\]  
where \(M_{x}\), \(M_{y}\) and \(M_{z}\) are the three components of the magnetization vector. \(B_{x}\), \(B_{y}\), and \(B_{z}\) are the magnetic field. \(T_{1}\) and \(T_{2}\) are the longitudinal and transverse relaxation times of noble gas. \(\gamma\) is the gyromagnetic ratio. \(R_{se}\) is the relaxation ratio. For convenience, a phasor representation of \(M_{+}\), \(B_{+}\) is defined as,  
\[ \begin{array}{l}M_{+}=M_{x}+iM_{y}=M_{\perp}e^{-i\phi}\\ B_{+}=B_{x}+iB_{y}=B_{\perp}e^{-i(\phi+\frac{\pi}{2})}\end{array}  \tag{5}\]  
Figure 1: Typical schematic diagram of the NMRRS. (**a**) is the typical experimental arrangement of the NMRRS. (**b**) is the PLL-based single nuclear feedback control. Meanwhile, Equation (4) could be rewritten as a phasor representation,  
\[ \begin{array}{l}\frac{dM_{+}}{dt}=-i\gamma B_{z}M_{+}+i\gamma B_{\perp}M_{z}- \frac{M_{\pm}}{T_{2}}\\ \frac{dM_{+}}{dt}=-\gamma B_{+}\times M_{+}+R_{se}-\frac{M_{\pm}}{T_{1}}\end{array}  \tag{6}\]  
A main magnetic field is applied in the z-axis direction to make the nucleon ensemble precession, and a drive magnetic field \(B_{1}\) with the same precession frequency is added in the transverse x-axis. Thus, the evolution equation of the amplitude and phase of the spin magnetic moment of the nucleon can be obtained by introducing Equation (5) into Equation (6),  
\[\frac{dM_{\perp}}{dt}=-\frac{M_{\perp}}{T_{2}}+\Omega_{1}M_{z}\cos(\phi-\theta) \tag{7}\]  
\[\frac{d\phi}{dt}=-\frac{\Omega_{1}M_{z}}{M_{\perp}}sin(\phi-\theta)+\Omega_{0} \tag{8}\]  
where \(\Omega_{z}=\gamma B_{0}\pm\omega_{r}\) and \(\Omega_{1}=\gamma B_{1}\). It can be seen from the above formula that the precession phase dynamics of the noble gas nucleon is a nonlinear process, and its changing rate is determined by the detuning phase and intensity of the driving magnetic field, the amplitude of the z-axis magnetic moment, and the amplitude of the transverse magnetic moment. When the NMR gyro operates, the nuclear spin resonator is in resonance or near detuning state, and the changing rate of \(M_{z}\) is very small, which could be ignored. Therefore, substituting \(\frac{dM_{\perp}}{dt}=0\) into Equation (7), the following can be obtained,  
\[M_{\perp}=T_{2}\Omega_{1}M_{z}\cos(\phi-\theta) \tag{9}\]  
Substituting Equation (6) into Equation (5), it can be derived,  
\[\frac{d\phi}{dt}=-\frac{1}{T_{2}}\tan(\phi-\theta)+\Omega_{z} \tag{10}\]  
Assuming \(\omega_{1}=d\theta/dt\), \(\Gamma=\frac{1}{T_{2}}\), a differential equation about the detune phase \(y\) could be obtained,  
\[\frac{dy}{dt}=-\Gamma\tan y+\Omega_{z}-\omega_{1} \tag{11}\]  
Considering the detuning phase \(y\) is extremely small, Equation (11) can be linearized as  
\[\frac{dy}{dt}=-\Gamma y+\Omega_{z}-\omega_{1} \tag{12}\]  
Thus, the block diagram of the PID-based nuclear spin control system is shown as Figure 2.  
Figure 2: Block diagram of PLL-based nuclear spin control loop. \(D\)(s) is the external disturbance, \(Y\)(s) is the observed detuning phase, \(R\)(s) is the reference phase, and \(U\)(s) is the output of the controller. \(G_{p}\)(s) and \(G_{c}\)(s) is the transfer function of the plant and the controller, respectively.