## 3 Measurement Errors of PLLs
### Stand-Alone PLLs  
The signal transfer relationship of the stand-alone loop in a Laplace domain is shown in Figure 3. The tracking loop includes a discriminator, a low-pass filter \(F(s)\), while a numerically controlled oscillator (NCO) \(K_{o}/S\). \(\theta_{i}(s)\) is the input signal, and \(\theta_{0}(s))\) is the output signal. \(\omega_{\varphi}(s)\) is the thermal noise in the tracking loops. \(\theta_{clk\_error}(s)\) is the local oscillator error, added to \(\theta_{i}(s)\) by down conversion. In addition, the dynamic stress also leads to tracking error. Therefore, the error sources of the carrier phase include thermal noise \(\sigma_{PLL}\), vibration-induced oscillator jitter \(\sigma_{r\
u}\), Allan variance-induced oscillator jitter \(\sigma_{rA}\), and dynamic stress \(\theta_{e}\). Since all error sources are independent, the mean squared error of total errors can be written as follows:  
\[\sigma_{PLL}=\sqrt{\sigma_{tPLL}^{2}+\sigma_{r\
u}^{2}+\sigma_{rA}^{2}}+ \theta_{e}/3 \tag{1}\]  
Taking the second-order PLL as an example, the models of the error sources can be expanded as follows [26]:  
\[\sigma_{tPLL}=\frac{180}{\pi}\sqrt{\frac{B_{n}}{C/N_{0}}.\left(1+\frac{1}{2T_ {coh}\cdot C/N_{0}}\right)}(^{\circ}) \tag{2}\]  
\[\sigma_{r\
u}=\frac{180}{\pi}\sqrt{\frac{\pi^{2}{f_{0}}^{2}{K_{g}}^{2}{G_{g}} }{2.67B_{n}}}(^{\circ}) \tag{3}\]  
Figure 3: Error transfer model of stand-alone phase-locked loop (PLL). Figure 2: Power spectrum of acceleration data from 2011 Honshu earthquake. \[\sigma_{rA}=\frac{180}{\pi}\sqrt{2\pi^{2}{f_{0}}^{2}\cdot(\frac{\pi^{2}h_{-2}}{ \sqrt{2}(1.89B_{n})}+\frac{\pi h_{-1}}{4(1.89B_{n})^{2}}+\frac{h_{0}}{4\sqrt{2}(1.89B_{n})})}(^{\circ}) \tag{4}\]  
\[\theta_{e}=\frac{\Delta\Re}{(1.89B_{n})^{2}}(^{\circ}) \tag{5}\]  
where \(B_{n}\) is the noise bandwidth, \(C/N_{0}\) is the carrier-to-noise ratio (CN0), \(T_{coh}\) is the coherent integration time, \(f_{0}\) is the carrier frequency, \(K_{g}\) is oscillator sensitivity, \(G_{g}\) is the single-sided vibration spectral density, \(h_{i}\) is a constant, and \(\Delta\Re\) is the frequency ramp-up amplitude. The Costas PLL is insensitive to phase inversion of 180\({}^{\circ}\). When \(\sigma_{PLL}\) is larger than the threshold 15\({}^{\circ}\), the PLL loses lock. Based on above error model, the impacts of GNSS signal strength and the bandwidth of tracking loops on tracking error can be analyzed under strong earthquake motion. An oven-controlled crystal oscillator (OCXO) is used as our receiver clock, with parameters that \(h_{-2}=2.51\times 10^{-22},h_{-1}=2.51\times 10^{-23},h_{0}=2.51\times 10^{-26}\). When the signal CN0 is 40 dB-Hz, the movement acceleration is 2 g, and the coherent integration time of PLL is set as 1 ms, Figure 4 shows tracking errors as a function of the PLL bandwidth. The tracking error caused by thermal noise increases with the increase of the bandwidth. Since the signal with a CN0 of 40 dB-Hz is a strong signal, the tracking error increases slowly with bandwidth increasing. Tracking errors caused by the oscillator and dynamic stress decrease with the increase of the bandwidth. Because the tracking error caused by dynamic stress changes significantly larger than others, it is necessary to increase the bandwidth to reduce the total tracking error appropriately. To ensure the total loop tracking error less than the threshold, the bandwidth must be wider than 17 Hz. When the movement acceleration maintains 2 g, the PLL bandwidth is 20 Hz, and the coherent integration time is 1 ms, Figure 5 shows tracking errors as a function of signal CN0. As is seen in Figure 5, the errors caused by dynamic stress and oscillator are in dependent of GNSS signal strength. With the signal CN0 reducing, the tracking error caused by thermal noise increases greatly. When the CN0 is less than 34 dB-Hz, the total phase error exceeds the tracking threshold. Hence, when the GNSS signal is weak under dynamics, the bandwidth should be compromised.