# New Nonlinear Second-Order Phase-Locked Loop with Adaptive Bandwidth Regulation
## 2 Materials and Methods  
### Mathematical Model of the Traditional Second-Order PLL  
Phase-locked loop is a closed feedback system used to synchronize the frequency and phase of the output signal into the input signal. Figure 1a illustrates the structure of the traditional second-order PLL. It is mainly composed of three parts [24]: (1) phase discriminator (PD), (2) loop filter (LP), and (3) voltage-controlled oscillator (VCO) (Figure 1a). The three components are connected in a series. In addition, the output of the VCO is feedback connected to the other input port of the phase discriminator. Figure 1b presents the corresponding model of the second-order PLL. The phase discriminator is modeled as a multiplier. The loop filter is modeled as the corresponding transfer function, which is an ideal integrator. And, the voltage-controlled oscillator is modeled as an integrator.  
The oscillating frequency of the VCO is controlled by the output, namely the control voltage, of the loop filter.  
Supposing the input signal of the PLL is  
\[U_{i}(t)=A_{i}\sin(\omega_{i}t+\phi_{i}(t))+n_{i}(t), \tag{1}\]  
where \(A_{i}\) is the amplitude of input signal, \(\omega_{i}\) is the angular frequency of input signal, \(\phi_{i}\) is the initial phase of input signal, and \(n_{i}(t)\) is the sinusoidal input noise, i.e., tone interference, the amplitude of output signal of the PLL is  
\[U_{o}(t)=A_{o}\cos(\omega_{o}t+\phi_{o}(t)+\phi_{n}(t)), \tag{2}\]  
where \(A_{o}\) is the amplitude of input signal, \(\omega_{o}\) is the angular frequency of output signal, \(\phi_{o}\) is the phase of output signal, and \(\phi_{n}\) is the phase noise of the output signal.  
The state equations of traditional second-order PLL are constructed in accordance with Figure 1. They are expressed as  
\[U_{d}(t)=K_{d}[\sin(\omega_{i}t+\phi_{i}(t))+n_{i}(t)]\times\cos(\omega_{o}t+ K_{o}\int U_{c}(t)+\phi_{n}(t)), \tag{3}\]  
\[\overset{\bullet}{x}(t)=u_{d}(t), \tag{4}\]  
\[U_{c}(t)=(\frac{1}{\tau_{1}})x(t)+(\frac{\tau_{2}}{\tau_{1}})\overset{\bullet }{x}(t). \tag{5}\]  
In the state Equation (3), the PD in the PLL is modeled into a multiplying unit, and its gain is \(K_{d}\). Sensitivity of the VCO is \(K_{o}\). The amplitudes of the input and output signals are normalized into 1. \(K\) is the loop gain, and \(K=1/2\times K_{d}\times K_{o}\). \(\tau_{1}\) and \(\tau_{2}\) are time constants of the ideal integral filter. \(U_{c}(t)\) is the control voltage of VCO.  
In the state equations, the output signal phase of the traditional second-order PLL is completely determined by the output signal of the LP. Therefore, LP design is the key in the design of traditional PLL. After the LP design is completed, the bandwidth of traditional PLL is fixed. Given the large frequency offset of input and output signals and even dynamic frequency changes in the input signal (i.e., synchronous receiving highly dynamic GPS signals), the large loop bandwidth is required to increase the acquisition bandwidth to guarantee a satisfactory tracking speed. However, large loop bandwidth weakens noise suppression ability and reduces tracking accuracy, thereby influencing receiving and demodulation of GPS signals. However, if small loop bandwidth is used to suppress noises, then the requirements on tracking speed of large bandwidth cannot be met. Hence,  
Figure 1: (**a**) Basic structure of the second-order phase-locked loop (PLL). (**b**) Model of the second-order PLL.  
an irreconcilable contradiction exists between the tracking speed and accuracy of the traditional PLL. To solve such contradiction, a new PLL coupled with a nonlinear element is proposed in Section 2.2.