## 3 Steady-State Error Modeling and Analysis
### Steady-State Error Modeling of Normal Second-Order PLLs  
A conservative rule of thumb for tracking threshold is that the three-sigma jitter must not exceed one-fourth of the phase pull-in range of the PLL discriminator [1]. When the PLL two-quadrant arctangent discriminator is used, it has a phase pull in the range of 180\({}^{\circ}\). Previous research shows that the dominant sources of phase error in a normal PLL are phase jitter and dynamic stress error; while the phase jitter is the root sum squared (RSS) of every source of uncorrelated phase error, such as thermal noise and oscillator noise. The dynamic stress error is a three-sigma effect and is additive to the phase jitter [1]. Therefore, the one-sigma rule threshold for the PLL for the two-quadrant arctangent discriminator is:  
\[\sigma_{PLL}=\sqrt{\sigma_{PLL}^{2}+\sigma_{r}^{2}+\sigma_{r,t}^{2}}+\theta_{e }\ /\ 3\leq 15\big{(}{{}^{\circ}}\big{)} \tag{3}\]where \(\sigma_{PLL}\) is one-sigma tracking error of the normal PLL in degrees. \(\sigma_{PLL}\) is one-sigma thermal noise in degrees. \(\sigma_{{}_{\pi\
u}}\) is one-sigma vibration-induced oscillator jitter in degrees. \(\sigma_{{}_{\pi d}}\) is Allan variance-induced oscillator jitter in degrees. \(\theta_{e}\) is dynamic stress error in the PLL.  
In the second-order PLL, the typical relationship of the noise bandwidth \(B_{n}\) (Hz) and the natural radian frequency \(\omega_{n}\) (rad/s) is \(B_{n}=0.53\omega_{n}\)[1]. Therefore, the thermal noise jitter for the second-order PLL is computed as follows [1]:  
\[\sigma_{{}_{\pi PLL}}=\frac{360}{2\pi}\sqrt{\frac{B_{n}}{C\,/\,N_{0}}\bigg{[}1 +\frac{1}{2T_{coh}\cdot C\,/\,N_{0}}\bigg{]}}\big{(}^{\circ}\big{)} \tag{4}\]  
where \(C\,/\,N_{0}\) is the carrier-to-noise power expressed as a ratio (Hz) and \(T_{coh}\) is coherent integration time (seconds).  
The vibration-induced oscillator phase noise is a complex analysis problem. Assuming that all occurring vibrations are equally distributed across the entire frequency range, the vibration-induced oscillator phase jitter for the second-order PLL is written as follows [20]:  
\[\sigma_{{}_{\pi\
u}}=180\sqrt{\frac{f_{0}^{2}K_{g}^{2}G_{g}}{2.67B_{n}}}\, \big{(}^{\circ}\big{)} \tag{5}\]  
where \(K_{g}\) is the g-sensitivity of the oscillator, \(G_{g}\) is the single-sided spectral density of vibration and \(f_{0}\) is carrier frequency.  
The Allan deviation phase noise is caused by the drift of the receiver oscillator, which is determined by the oscillator's material and craft. The second-order PLL jitter due to Allan deviation phase noise is written as follows [20]:  
\[\sigma_{{}_{\pi d}}=180\sqrt{2f_{0}^{2}\bigg{[}\frac{\pi^{2}h_{{}_{2}}}{\sqrt{ 2(1.89B_{n})}^{3}}+\frac{\pi h_{{}_{-1}}}{4(1.89B_{n})^{2}}+\frac{h_{0}}{4 \sqrt{2}  
\[\sigma_{PLL}=\frac{180}{\pi}\left\{\begin{array}{l}\frac{B_{n}}{C \,/\,N_{0}}(1+\frac{1}{2T_{coh}\cdot C\,/\,N_{0}})+\frac{\pi^{2}\,f_{0}^{2}K_{g }^{2}G_{g}}{2.67\,B_{n}}\\ +2\pi^{2}f_{0}^{2}\big{(}\frac{\pi^{2}h_{-2}}{\sqrt{2}(1.89\,B_{n})^{3}}+\frac{ \pi h_{-1}}{4(1.89\,B_{n})^{2}}+\frac{h_{0}}{4\sqrt{2}(1.89\,B_{n})}\big{)}+ \frac{\Delta\Re}{3(1.89\,B_{n})^{2}}\big{(}^{\circ}\big{)}\end{array}\right. \tag{8}\]  
As we know the running conditions of the receiver, including GNSS signal strength and the vehicle dynamics, the optimal bandwidth of the normal PLL can be calculated based on the error model in Equation (8), as described in [21]. Using the idea of the optimal bandwidth of the normal PLL, a tool for the bandwidth selection of the INS-aided PLLs can be proposed. Therefore, if the INS errors in the INS-aided PLLs are modeled in detail, an error model of the INS-aided PLLs, like Equation (8), could be built, which could truly reflect the effect of the INS error on the tracking loop physically and be used for the parameter optimization of the INS-aided PLLs.