# New Nonlinear Second-Order Phase-Locked Loop with Adaptive Bandwidth Regulation
## 2 Materials and Methods
### New PLL Coupled with a Nonlinear Element  
To solve the contradiction between the tracking speed and accuracy of traditional PLL, a nonlinear element is coupled into the traditional second-order PLL based on the nonlinear control idea. The proposed new PLL conduct adaptive regulation of output of LP to balance the tracking speed and accuracy of the PLL on the basis of the controlling effect of nonlinear element. A large \(U_{c}(t)\), which is caused by the large frequency-phase error, usually happens in the initial acquisition stage of the PLL control process. Therefore, the nonlinear element should enhance the control voltage of the VCO to reduce the frequency-phase error more rapidly. The frequency-phase error can gradually decrease, which causes a smaller \(U_{c}(t)\), due to the feedback control effect. At this time, the nonlinear element must then decrease \(U_{c}(t)\) as well as the input noise to improve the tracking accuracy in the tracking stage. Thus, a nonlinear element which consists of two piecewise linear segments is adopted to achieve the above described function. The new designed PLL structure is shown in Figure 2.  
The proposed new PLL adds one nonlinear element N after the LP, and this nonlinear element cooperates with one low-pass filter (LPF) to control the output of the VCO. The mathematical expression of the nonlinear element N is designed as  
\[N(U_{c}(t))=\left\{ \begin{array}{ll}m_{h}U_{c}(t)+(m_{h}-m_{l})\times y^{*},&U_{c}(t)\leq-y^{*}\\ m_{l}U_{c}(t),&-y^{*}\leq U_{c}(t)\leq y^{*}\\ m_{h}\times U_{c}(t)-(m_{h}-m_{l})\times y^{*},&y^{*}\leq U_{c}(t)\end{array} \right. \tag{6}\]  
where \(y^{*}\) is the turning point of nonlinear element. \(m_{l}\) and \(m_{h}\) are the slopes of two sections. The graph of the nonlinear element N is shown in Figure 3. The output of the nonlinear element is adaptively adjusted according to the control voltage of the loop. To suppress the input noise, the larger slope of the nonlinear element should not utilize a value too large. Moreover, a smaller value of the other slope is adopted to further suppress the noise. And, the inflection point is selected based on the output of the loop filter. Figure 3: Schematic of three-stage nonlinear element. Figure 2: Basic structure of the new nonlinear second-order PLL. Large input-output phase error and high controlling voltage of the VCO occur when the system starts to requisition, and the oscillation frequency of the VCO is adjusted quickly through the amplification of \(m_{h}\) to reduce the input-output phase error. When the PLL runs for a certain period, the input-output phase error is decreased as a response to the controlling effect of large \(m_{h}\) in the new PLL. In this case, residual noises in the controlling voltage can be suppressed by the compression effect of a smaller slope \(m_{l}\), and the accurate tracking of PLL can be realized. The relationships between the three parameters and the PLL performance can be summarized as follows: the PLL shows high tracking speed at small \(y^{\star}\) and high \(m_{h}\) and has strong noise suppression ability at high \(y^{\star}\) and low \(m_{l}\). The nonlinear element is coupled to perform adaptive regulation on the loop noise bandwidth according to the input-output phase error. When a large phase error occurs, the proposed PLL reduces this phase error quickly by using a large loop bandwidth with high \(m_{h}\) and small \(y^{\star}\). When the phase error is decreased by the loop controlling effect, the method suppresses noises through the small bandwidth with small \(m_{l}\) and large \(y^{\star}\) to realize accurate fast tracking.