# New Nonlinear Second-Order Phase-Locked Loop with Adaptive Bandwidth Regulation
## 2 Materials and Methods
### Design of the Nonlinear Element Parameters  
A few classical simulations are implemented to find the relationship between the performance of the nonlinear second-order PLL and the parameters of the nonlinear element. Moreover, on the basis of the relationship obtained from the simulation results, an algorithm is proposed to solve the optimal parameters of the nonlinear element \(N\). The simulations are carried out as follows: a specific parameter of the nonlinear element is chosen to be three typical values while the other two parameters remain unchanged. First, the parameter \(m_{l}\) selects as 1, 4, and 7 respectively, while \(y^{\star}\) is 0.05 and \(m_{h}\) is 4. The simulation result is presented in Figure 4. Then, the parameter \(m_{h}\) is chosen to be 1, 4, and 7 respectively, while the other two remains unchanged, the corresponding value of \(y^{\star}\) selects as 0 and \(m_{l}\) as 1. And, the simulation result is shown in Figure 5. The other simulation parameters of the loop are: the eigenfrequency of the VCO is set \(\omega_{o}=2\pi\times 2\times 10^{5}\) rad/s, the input angular frequency of the PLL is \(\omega_{i}=2\pi\times 1.99\times 10^{5}\) rad/s, the input noise is \(n_{i}=0.01\times\sin(1\times 10^{4}\times\mathrm{t})\), and the gain of PD is \(K_{d}=1\) V. The LP parameters are \(\tau_{1}=500\times 10^{-6}\) and \(\tau_{2}=50\times 10^{-6}\). The sensitivity and sampling frequency of VCO are \(K_{o}=1.3\times 10^{5}\) rad/(s V) and \(f_{s}=1\times 10^{6}\) Hz, respectively. The cut-off frequency of the LPF is \(\alpha=1\times 10^{6}\) Hz, and the simulation time is 3 ms.  
The three parameters are mutually constrained to jointly determine the comprehensive performance of the loop. On the basis of the analytical and simulation results, the fast locking performance is determined by the tracking speed and the noise reduction ability can be determined by the standard deviation of the frequency-phase error. Therefore, a new algorithm is proposed to solve the optimal parameters of the nonlinear element \(N\). The final object function is constructed, which uses the minimum product of the locking time \(t^{*}\), namely the phase error when \(t\geq t^{*}\) is less than \(30^{\circ}\) and the standard deviation of the phase error as a target solution, as shown as in Equation (13).  
\[G=min(t^{*}(y*,m_{l},m_{h})\times std(phase\_error(y*,m_{l},m_{h}))), \tag{13}\]  
The process of algorithm is as follows:  
Step 1: In the numerical simulation process, dynamically select \(y^{*}\) according to the output of the loop filter.  
Step 2: Determine the dynamic range of the two segments slopes of the nonlinear element. In this paper, the settings of the dynamic range are \(m_{h}\in(1,10)\) and \(m_{l}\in(0,0.9)\).  
Step 3: Calculate \(t^{*}\) when the loop is locked, namely when \(phase\_error(t\geq t^{*})\leq 30^{\circ}\), then calculate the standard deviation of the phase error.  
Step 4: Calculate the object function based on the dynamic parameters of the nonlinear element.  
Step 5: Repeat steps 1 through 4 and solve the object function, \(G=min(t^{*}(y^{*},m_{l},m_{h})\times std(phase\_error(y^{*},m_{l},m_{h})))\).  
The specific flow chart of the algorithm is shown in Figure 7.  
With the above proposed algorithm, the calculated result of the object function in every iteration is stored in a three-dimensional matrix. In addition, the minimum value of the three-dimensional matrix corresponds to the optimal solution of the three parameters considering the comprehensive performance of the loop. Finally, with the parameters of the loop in this paper, the optimal parameters of the nonlinear element solved by the proposed algorithm are \(y^{*}=0.05\), \(m_{h}=5\), and \(m_{l}=1.5\) respectively. The performance test results of the nonlinear second-order PLL are presented in Section 3.