## 2 Principle
\begin{aligned} V_{NCOI}(t)&=A_{NCO}\cos(\omega_{NCO} t+\psi_{NCO})\\ V_{NCOQ}(t)&=A_{NCO}\sin(\omega_{NCO}t+\psi_{NCO}) \end{aligned}  \tag{15}\]  
The NCO in-phase and quadrature signals are multiplied, integrated, and averaged with the input signals to obtain \(V_{INTI}(t)\) and \(V_{INTQ}(t)\), respectively. \(V_{INTI}(t)\) and \(V_{INTQ}(t)\) can be expressed as Equation (16). Subsequently, interference terms such as \(\omega_{het}+\omega_{NCO}\) in the multiplication result are filtered out, and the \(\omega_{het}-\omega_{NCO}\) containing the phase difference are retained. This process is analogous to a low-pass filter in a conventional phase-locked loop. \[ \begin{aligned} V_{INTI}(t)&=\frac{A_{S}A_{NCO}}{2} \sin[(\omega_{het}-\omega_{NCO})t+(\psi_{i}-\psi_{NCO})]\approx\frac{A_{S}A_{NCO }}{2}\sin(\psi_{i}-\psi_{NCO})\\ V_{INTQ}(t)&=\frac{A_{S}A_{NCO}}{2}\cos[(\omega_{het }-\omega_{NCO})t+(\psi_{i}-\psi_{NCO})]\approx\frac{A_{S}A_{NCO}}{2}\cos(\psi_{ i}-\psi_{NCO})\end{aligned}  \tag{16}\]  
After completing the integration for the mean value, the output signals of the in-phase and quadrature branches perform the _atan_ operation to directly solve for the phase difference information \(\psi_{i}-\psi_{NCO}\). The phase difference between the input signal and the NCO signal at this point is obtained and used as the input signal of the loop filter, and the NCO frequency word is adjusted accordingly. The phase difference solving process is shown in Equation (17). \[arc\tan\left(\frac{\psi_{INTI}(t)}{V_{INTQ}(t)}\right)=\psi_{i}-\psi_{NCO}\ \ \ \ (i=1,2,3,4) \tag{17}\]  
After this, by constantly combining the current phase difference \(\psi_{i}-\psi_{NCO}\) to the NCO frequency \(\omega_{NCO}\) feedback adjustment. This process gradually leads to the convergence of the phase difference to 0. Following the completion of phase tracking, the NCO index serves as the current moment of the phase information. This is then utilized in the subsequent phase angle conversion step, thereby facilitating the high accuracy measurement of the angle. In a phase-locked loop, the input signal \(V_{S}(t)\) will contain additive Gaussian white noise \(n\)(t) with mean 0 and variance \(\sigma^{2}\), and the phase jitter variance \(\sigma^{2}\)\({}_{\psi n}\) generated by this noise can be expressed as Equation (18), where \(SNR\) denotes the input signal-to-noise ratio, and \(P_{sig}\) and \(P_{n}\) denote the signal and noise power, respectively. \[\sigma^{2}_{\psi n}=\frac{1}{2 SNR}=\frac{P_{n}}{2P_{sig}} \tag{18}\]  
Gaussian white noise \(n(t)\) will introduce phase noise containing systematic and random errors to the phase-locked loop. In this case, the systematic error is accompanied by the phase higher-order dynamic stress of the input signal, and the random error is generated by the noise factor. Combining the one-sided power spectral density \(N_{0}\) of the noise \(n(t)\) and the input signal power \(P_{in}\) of the phase-locked loop, the conversion of the phase jitter variance \(\delta^{2}\)\({}_{\psi n}\) generated by the random error to the equivalent loop bandwidth \(B_{n}\) can be expressed as Equation (19) [23]. It can be seen that the increase in the loop bandwidth \(B_{n}\) will lead to a decline in the loop phase locking performance, thereby affecting the angle measurement performance. The main reason is that excessive adjustment of the loop leads to the introduction of random errors. Meanwhile, if \(B_{n}\) is too small, it will also make it difficult for the loop to correctly detect the signal phase. The main reason is that the adjustment of the loop being too small leads to the introduction of systematic errors. In this part, further analysis is carried out in combination with the relevant data in the experimental system. \[\delta^{2}_{\psi n}=\frac{N_{0}}{2P_{in}}\int\limits_{0}^{+\infty}|H(j2\pi f )|^{2}df=\frac{N_{0}B_{n}}{2P_{in}} \tag{19}\]  
Furthermore, the system noise sources and \(SNR\) are also influenced by the power of the local oscillation laser beam \(P_{LO}\).