## 4 Design of Roll Angle Detection Module
### Error Analysis of Tracking Loops  
For a pure PLL, the dominant sources of phase errors are phase jitter and dynamic stress error. The signal-to-noise ratio at the input of PLL is \(\text{SNR}_{i}\), with the bandwidth of \(B_{i}\). For there is no data modulation on the roll modulation signals, the square of the RMS phase jitter \(\overline{\sigma}_{i}^{2}\) at the input of PLL can be represented as [25]:  
\[\overline{\sigma}_{i}^{2}=1/(2 SNR_{i}) \tag{30}\]  
The RMS phase jitter \(\overline{\sigma}_{0}^{2}\) at the output of PLL is given by:  
\[\overline{\sigma}_{0}^{2}=(1/SNR_{i})\cdot(B_{\text{PLL}}/B_{i}) \tag{31}\]  
Comparing (30) and (31), the SNR at the output of PLL can be defined as:  
\[\text{SNR}_{0}=\text{SNR}_{i}\cdot B_{i}/(2B_{\text{PLL}}) \tag{32}\] \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Rotational Speed/(r/s)** & **FLL Noise Bandwidth** & **PLL Noise Bandwidth** & **Damping** & **Integration Time** \\
**Speed/(r/s)** & \(B_{\text{FLL}}\)/(Hz)** & \(B_{\text{PLL}}\)/(Hz) & **Ratio \(\xi\)** & \(T_{roll}\)/(ms)** \\ \hline \(3\leq f_{r}<4\) & 0.3 & 0.3 & 0.3 & 333 \\ \(4\leq f_{r}<10\) & 0.3 & 0.3 & 0.3 & 250 \\ \(10\leq f_{r}\leq 40\) & 0.3 & 0.5 & 0.5 & 100 \\ \(40<f_{r}<300\) & 0.5 & 1 & 0.5 & 50 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Desirable parameters of FLL-assisted PLL. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Rotational Speed/(r/s)** & **Settling Time/(s)** & \begin{tabular}{c} **3 dB Attenuation** \\ **Frequency/(Hz)** \\ \end{tabular}  & **Overshoot** & **Pull-In Range/(Hz)** \\ \hline \(3\leq f_{r}<4\) & 18.76 & 1.04 & 1.45 & \(\pm 1.5\) \\ \(4\leq f_{r}<10\) & 18.76 & 1.04 & 1.45 & \(\pm 2\) \\ \(10\leq f_{r}\leq 40\) & 4.38 & 2.35 & 1.30 & \(\pm 5\) \\ \(40<f_{r}<300\) & 2.3 & 4.70 & 1.21 & \(\pm 10\) \\For most cases, the bandwidth of PLL \(B_{\rm PLL}\) is much narrower than \(B_{i}\), so we can conclude from (32) that the PLL can improve the SNR at its output. Experimental results have shown that for second-order loops, when \(\rm SNR_{o}\geq 4\) (6 dB), a stable operation is generally possible [25]. For example, in this paper, the sampling rate of roll modulation signals is 1000 Hz and the bandwidth of it is approximately \(B_{i}=3\) Hz. When \(B_{\rm PLL}\) is set to 0.5 Hz, the SNR of roll modulation signals must be \(\rm SNR_{i}\geq 1.33\) (1.25 dB), so that a stable operation of PLL is possible. For a second-order PLL, the dynamic stress error can be expressed as [22]:  
\[\theta_{d}=\,1/\omega_{np}^{2}\cdot\left(d^{2}\alpha/dt^{2}\right) \tag{33}\]  
where \(d^{2}\alpha/dt^{2}\) is the phase acceleration of the roll modulation signals (\(\rm deg/s^{2}\)). So, the overall error of PLL is given by:  
\[\sigma_{\rm PLL}=\overline{\sigma}_{0}+\theta_{d} \tag{34}\]  
Figure 17 shows the theoretical predictions of the phase jitter at different SNRs of input signals. The bandwidth of input signal is 3 Hz, and the bandwidth of PLL is set to \(B_{\rm PLL}=0.5\) Hz. As we can see from Figure 17, if the phase jitter of PLL is required to be within 10 degrees, the SNR of the roll modulation signals must be larger than 5.5 (7.4 dB) to ensure the tracking accuracy. For FLL, the dominant sources of frequency errors show frequency jitter, due to thermal noise and dynamic stress error. The thermal noise of FLL in Hz is given by:  
\[\sigma_{\rm fFLL}=\frac{1}{2\pi T_{roll}}\sqrt{\frac{4 FB_{\rm PLL}}{C/N_{0}}(1 +\frac{1}{T_{roll}C/N_{0}})} \tag{35}\]  
where \(F=1\) at a high carrier-to-noise ratio (\(C/N_{0}\)), \(F=2\) when working near the threshold. The relationship between \(C/N_{0}\) and SNR of input signals is \(C/N_{0}=\rm SNR\cdot B_{i}\). For a first-order FLL, the dynamic stress error in Hz can be expressed as:  
\[v_{d}=\,1/(360\cdot\omega_{nf}^{2})\cdot(d^{2}\alpha/dt^{2}) \tag{36}\]  
According to the rule of thumb for tracking threshold, the 3-sigma jitter of FLL shall not exceed one fourth of the pull-in range of FLL discriminator.