# A Multichannel, High-Bandwidth Wirelane Receiver for D2D Interconnects
## 3 Receiver Circuits
### Mini-Pll
#### 3.3.5 PI  
(1) Method  
PI is the key module in this design, and the linearity of its rotation will directly affect the overall performance of the receiver. In the design, the monotonicity and linearity of PI are the main concerns. Its linearity can be represented by a proportional function. \[\varphi_{out}=k_{PI}n\quad 0\leq n\leq N,\;0\leq\varphi_{out}\leq 2\pi \tag{1}\]  
\[V_{out}=A_{1}\sin(\omega t)+A_{2}\sin(\omega t+\varphi_{d}) \tag{2}\]  
Figure 14: Design of VCO. Figure 13: VCO control circuit. Figure 15: Frequency and KVCO from VCO. (**a**) is for 10 GHz clock, (**b**) is for 8 GHz clock, (**c**) is for 6 GHz clock. \[\varphi_{out}=\arctan(\frac{A_{2}\sin\varphi_{d}}{A_{1}+A_{2}\cos\varphi_{d}}) \tag{3}\]  
\(k_{PI}\) is the gain of PI, and \(n\) is the control code. Equation (1) shows that, when \(n\) increases from 0 to \(N\), the output phase increases from 0 to \(2\pi\). If \(k_{PI}\) remains unchanged, the relationship between \(\varphi_{out}\) and \(n\) is monotonic linear. However, the input and output of the actual PI are nonlinear sinusoidal Equation (2), and the phase and amplitude of the output are determined by \(A_{1}\), \(A_{2}\) and \(\varphi_{d}\) in Equation (3). When the traditional PI takes \(\varphi_{d}\) to take \(90^{\circ}\), 0 to \(2\pi\) are divided into four quadrants. Each quadrant is interpolated separately, which leads to a decrease in the linearity of the phase interpolator. In this work, since the designed minimum phase-locked loop generates 8 clocks with equal phases, PI is interpolated between \(45^{\circ}\) and \(0^{\circ}\), which greatly increases the linearity. When \(\varphi_{d}\) takes \(45^{\circ}\), Equations (4) and (5) are obtained. \[\varphi_{out}=\arctan(\frac{\frac{\sqrt{2}}{2}A_{2}}{A_{1}+\frac{\sqrt{2}}{2} A_{2}}) \tag{4}\]  
\[V_{out}=\sqrt{{A_{1}}^{2}+{A_{2}}^{2}+\sqrt{2}A_{1}A_{2}}\sin(\omega t+\varphi_{ out}) \tag{5}\]  
Due to the symmetry of the quadrants, the model is simulated for linearity in the first quadrant and compared with a traditional PI. Figure 16 shown the linearity analysis and comparison of PI. Figure 16 is the output phase with the thermometer code changing when \(\varphi_{d}=45^{\circ}\). This model has an inflection point at \(\varphi_{d}=22.5^{\circ}\), and the trend of the curve approximates the ideal linearity. The traditional phase interpolator (\(\varphi_{d}=90^{\circ}\)) has an inflection point at \(45^{\circ}\) in Figure 16. Through the fitted curve as shown in Figure 16, phase interpolator (\(\varphi_{d}=45^{\circ}\)) is closer to the ideal value. Therefore, the method of eight equal phases is much better than the traditional PI. (2) Circuit  
Figure 17 shows the structure of a traditional equivalent current source PI. The size of the input transistors M1, M2, M3, and M4 are all the same. The loads R1 and R2 are equal to R. The input signal is two pairs of quadrature differential signals \(v_{IP}\) and \(v_{QP}\), \(v_{IN}\) and \(v_{QN}\), (\(0^{\circ}\), \(90^{\circ}\), \(180^{\circ}\), and \(270^{\circ}\)). PI interpolates the phases of these two pairs of clocks, and a recovered clock with a phase between them can be obtained. The phase of the recovered clock can be adjusted by changing the tail currents of these two differential pairs. When \(\varphi_{d}=45^{\circ}\), the input clocks are eight equal-phase clocks (\(v_{0}=0^{\circ}\), \(v_{1}=45^{\circ}\), \(v_{2}=90^{\circ}\), \(v_{3}=135^{\circ}\), \(v_{4}=180^{\circ}\), \(v_{5}=225^{\circ}\), \(v_{6}=270^{\circ}\), \(v_{7}=315^{\circ}\)), as shown in Figure 17.  
\[V_{out}=A_{1}\sin(\omega t)+A_{2}\sin(\omega t+\varphi_{d}) \tag{2}\]0 Linearity analysis and comparison of PL. (**a**) is the output phase with the thermometer code changing when \(\varphi_{d}=45^{\circ}\), (**b**) is the output phase with the thermometer code changing when \(\varphi_{d}=90^{\circ}\), (**c**) is the fitted curve of (**a**,**b**) and a ideal curve. As shown in the Figure 18, the circuit is the overall circuit design of PI.