## 4 Design of Roll Angle Detection Module
### Design of Tracking Loops  
Figure 9 shows a block diagram of the proposed roll angle detection module. The FLL-assisted PLL based Costas loop contributes the most to the module. FLL-assisted PLL combines the advantages of both tracking loops, which means it can track the phases of input signals precisely under high dynamic stress [24]. A second-order PLL with first-order FLL assisted filter is selected, because both loops are insensitive to constant velocity, while the roll angle measurements here focus on a constant rotational speed with tiny signal dynamic stress to the roll modulation signals. The loop filter is as shown in Figure 10, and there are four parameters that need to be determined. They are damping ratio \(\xi\), the natural frequency of PLL \(\omega_{np}\), the natural frequency of FLL \(\omega_{nf}\), and the integration time \(T_{roll}\). \(\Omega_{e}\) is the frequency error, and \(\Phi_{e}\) is the phase error between the roll modulation signals and local replicas. Figure 8: The relationship between squaring loss and the signal-to-noise ratio (SNR) of correlation results. In the stable tracking state, PLL plays a more important role than FLL, because it has lower noise bandwidth and makes a more accurate estimate of the phases of input signals. Figure 11 shows the block diagram of a linear PLL in S-domain. According to Figures 10 and 11, the transfer function of the PLL's loop filter and the second-order PLL is given by:  
\[F_{p}(s)=2\xi\omega_{np}+\frac{\omega_{np}^{2}}{s} \tag{22}\]  
\[H_{p}(s)=\frac{F_{p}(s)}{s+F_{p}(s)}=\frac{2\xi\omega_{np}s+\omega_{np}^{2}}{s ^{2}+2\xi\omega_{np}s+\omega_{np}^{2}} \tag{23}\]  
respectively. The noise bandwidth \(B_{\rm PLL}\) of PLL can be calculated from \(\xi\) and \(\omega_{np}\) by:  
\[B_{\rm PLL}=\int\limits_{0}^{\infty}\bigl{|}H_{p}(j2\pi f)\bigr{|}^{2}df=\frac {\omega_{np}}{2}\biggl{(}\xi+\frac{1}{4\xi}\biggr{)} \tag{24}\]  
Figure 11: Block diagram of the PLL in S-domain. Figure 10: Block diagram of a second-order Phase-Locked Loop (PLL) with first-order Frequency-Locked Loop (FLL) assisted filter. Figure 9: Block diagram of the proposed roll angle detection module. The amplitude-frequency responses and step responses of PLL are simulated with different parameters, which are shown in Figure 12.  
As seen from Figure 12a,b, the damping ratio \(\xi\) and noise bandwidth \(B_{\text{PLL}}\) both determine the cut-off, settling the time and overshoot characteristics of PLL. The larger the damping ratio and the noise bandwidth is, the smaller the settling time and overshoot would be, and the cut-off frequency of PLL could also be increased, leading to a reduction of tracking accuracy. While the frequency of the input signal is between 3 to 300 Hz, the 3 dB attenuation frequency of PLL has to be less than 3 Hz to guarantee the filtering performance when the PLL is used to track the input signals with minimum frequency. In that case, the parameters of PLL must be \(B_{\text{PLL}}\leq 0.5\) Hz, \(\xi\leq 0.7\). The transfer function and noise bandwidth \(B_{\text{FLL}}\) of first-order FL can be expressed as follows:  
\[H_{f}(s)=\frac{F_{f}(s)}{s+F_{f}(s)}=\frac{\omega_{nf}}{s+\omega_{nf}} \tag{25}\]  
\[B_{\text{FLL}}=\int\limits_{0}^{\infty}\Bigl{|}H_{f}(j2\pi f)\Bigr{|}^{2}df= \frac{\omega_{nf}}{4} \tag{26}\]  
The amplitude-frequency responses and step responses of FLL with different parameters are shown in \[F_{p}(s)=2\xi\omega_{np}+\frac{\omega_{np}^{2}}{s} \tag{22}\]0  
As shown in Figure 13a,b, the responses pattern of FLL with different parameters is similar to that of PLL. The larger the bandwidth is, the greater the 3 dB attenuation frequency and the shorter the  
\[F_{p}(s)=2\xi\omega_{np}+\frac{\omega_{np}^{2}}{s} \tag{22}\]1 Amplitude–frequency (**a**) and step (**b**) responses of the second-order PLL. \[F_{p}(s)=2\xi\omega_{np}+\frac{\omega_{np}^{2}}{s} \tag{22}\]2 Amplitude–frequency (**a**) and step (**b**) responses of the first-order FLL. settling time would be.