# New Nonlinear Second-Order Phase-Locked Loop with Adaptive Bandwidth Regulation
## 3 Simulation and Results  
The eigenfrequency of the VCO is set \(\omega_{o}=2\pi\times 2\times 10^{5}\) rad/s, 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  
Figure 7: Flow chart of the algorithm. \(\alpha\) = 1 \(\times\) 10\({}^{6}\) Hz, and the simulation time is 3 ms. The nonlinear PLL (NPLL) is evaluated under two conditions: NPLL\({}_{1}\): \(y^{*}\) = 0.05, \(m_{l}\) = 1.5, \(m_{h}\) = 5; and NPLL\({}_{2}\): \(y^{*}\) = 0, \(m_{l}\) = 0, \(m_{h}\) = 5. The second situation can be viewed as an extreme condition in which the nonlinear element is a linear amplification structure. Test condition 1: the input angular frequency and the input noises are \(\omega_{o}\) = 2\(\pi\)\(\times\) 2 \(\times\) 10\({}^{5}\) rad/s and \(n_{i}\) = 0.01 \(\times\) sin(1 \(\times\) 10\({}^{4}\)\(\times\) t). The input-output phase error of the new PLL under this condition is shown in Figure 8. It can be seen that NPLL\({}_{1}\) and NPLL\({}_{2}\) can be locked more quickly than the traditional PLL. Given that the NPLL\({}_{2}\) is a linear amplification structure, it does not suppress noises through small \(m_{l}\) in the end section, which determines its higher locking speed than that of NPLL\({}_{1}\). In Figure 8b, the phase error of NPLL\({}_{2}\) fluctuates considerably, which weakens the noise suppression capability. Therefore, the linear amplification structure fails to solve the contraction between tracking speed and accuracy fundamentally. However, the phase error fluctuation of NPLL\({}_{1}\) is similar to that of the traditional PLL, indicating that NPLL\({}_{1}\) suppresses noises successively through the compression of \(m_{l}\) in the end section. Meanwhile, the tracking speeds of NPLL are increased by the amplification effect of \(m_{h}\). This condition realizes a compromise between the tracking speed and accuracy of the traditional PLL. Test condition 2: the input angular frequency of the PLL is \(\omega_{o}\) = 2\(\pi\)\(\times\) 1.99 \(\times\) 10\({}^{5}\) rad/s. Under this condition, the input-output frequency offset of PLL is 1 kHz, and the input noise is \(n_{i}\) = 0.01 \(\times\) sin (1 \(\times\) 10\({}^{4}\)\(\times\) t). The input-output phase error of PLL under this condition is shown in Figure 9.  
Figure 8: Phase error of PLL under no frequency offset: (**a**) phase error of PLL under no frequency offset; (**b**) phase error of PLL after 2.95 ms.  
Figure 9: Phase error of PLL under a frequency offset of 1 KHz: (**a**) phase error of PLL under a frequency offset of 1 kHz; (**b**) phase error of PLL after 2.95 ms. Figure 9 shows that, when the input-output frequency offset is 1 kHz, NPLL\({}_{1}\) and NPLL\({}_{2}\) still can be locked more quickly than the traditional PLL. The phase error fluctuation amplitude of NPLL\({}_{1}\) is equivalent to that of the traditional PLL, indicating the strong adaptation of the new PLL under large acquisition bandwidth. Test condition 3: the input angular frequency of the PLL is \(\omega_{o}=2\pi\times 1.96\times 10^{5}\) rad/s. Under this condition, the input-output frequency offset of the PLL is 4 kHz, and the input noise is \(n_{i}=0.01\times\) sin (\(1\times 10^{4}\times\) t). The input-output phase error of the new PLL under this condition is shown in Figure 10.  
Figure 10 shows that, when the input-output frequency offset is 4 kHz, the traditional PLL develops a cycle-skipping phenomenon. Meanwhile, the NPLL\({}_{1}\) and NPLL\({}_{2}\) still can be locked successfully, indicating the stronger adaption of new PLL to the large acquisition bandwidth than that of traditional PLL. In order to detail what sets the upper boundary limit on the frequency difference, multiple sets of simulations are implemented.