Proposed Memristor-based Loop Passive Filter for Phase Locked Loop

This paper highlights a way to introduce memristive device into loop filter circuits for phase locked loop (PLL) applications. The fundamental behaviors of memristor are carefully examined. In light of this, we proposed two commonest topologies of memristor-based passive loop filter (LP) for PLL applications. The transfer functions of the LPs are obtained, and the amplitude-frequency response, phasefrequency response, and noise-frequency response are presented. Finally, Simulation results demonstrate our analyses, and the memristive elements can be extended to other types of loop filters such as active filter, digital loop filter and sequential filter.


I. INTRODUCTION -MEMRISTOR
The advent of HP labs' 2 memristor has resulted in a large number of proposed memristor-based models and implementation in recent years [1][2][3]. Memristor is regarded as a fourth fundamental passive element whose resistance changes with the magnitude and direction of excitation or input signal (voltage or current). Basically, a memristor has two states -and . We define these states as the weight of memristor which is given in (1).
State variable, ( ) = ( ) ε (0,1) Where ( ) and D are the widths of doped region and total region respectively as shown in Fig. 1.
. Fig. 1. HP memristor structure In this paper, we adopt HP 2 memristor model in our design and simulations. Although memristor was introduced by Leon Chua in 1971 [4], it was not developed until 2008 when a team at HP lab, led by Stanley Williams, manufactured the first memristor. The memristor was fabricated by sandwiching 2 (undoped region) and 2− (doped region) between platinum contacts. The doped region has high carrier or ion concentrations which makes it a low resistance region, , and vice versa for the undoped region, [5,6].
When an electric current passes through the structure shown in Fig. 2, the boundary between the doped and undoped moves in the same direction, thus, the total resistance of memristor is obtained from the series connection of both doped and undoped regions [5].
It could be inferred from (2) that whenever ( ) increases, decreases and vice versa. The two possible extreme cases are ( ) = 0 and ( ) = ; ( ) = 0 1 respectively. Invariably, is clipped to and respectively, until the applied voltage or current changes its direction.

II. MEMRISTOR-BASED LOOP FILTER
The loop filter (LF) in Phase Locked Loop (PLL) is a lowpass filter, which is used to remove the high frequency components from phase detector/charge pump (PD/CP) [7]. LF is an essential block in PLL because it filters out the ripple in the Voltage Controlled Oscillator (VCO) control voltage and ensures a dc-controlled signal to the VCO. Loop filter does not only filter high frequency components but also control the dynamic characteristics capture, lock range, bandwidth and transient response in PLL. Studies have shown that there are majorly four different topologies of loop filter; passive filter, active filter, digital filter, and sequential filter [8]. In this work, we only examined the two commonest passive loop filter topologies in phase locked loop [7,[9][10][11][12] . For our proposed second order loop filter, it is worth mentioning that we replaced the resistors with memristors, M, 1 , and 2 as shown in Fig. 3 and 4. M, memristance, is the inherent property of a charge-controlled memristor. It is similar to resistance and also has the same unit, Ohm (Ω).

III. ANALYSIS OF MEMRISTOR-BASED LOOP FILTER
In order to understand the behaviour characteristics of these filters, it is necessary to identify their transfer functions.
The transfer function of loop filter in Fig. 3 has a unit of impedance (Ω), being the output generated from CP output current [13]. The impedance of the loop filter is denoted as as ( ). Where: Similarly for Fig. 4, which is a type I second order filter, the transfer function is given in (7).   5 shows that the phase difference between voltage and current reduces as frequency increases. This validates that a memristor behaves similar to a resistor at high frequency. The frequency responses of our proposed filters are shown in Fig. 9 and 11. From Fig. 9, we obtained a cut-off frequency of 158 Krad/s for the proposed analog loop filter which makes it suitable for high frequency applications such as PLL. Similarly, Fig. 11 shows the frequency response of our proposed filters. To ensure stable tracking in phase locked loop, we have proposed filter that has equal number of poles and zeros.
It is quite important to analyse the noise-rejection ability of our proposed loop filter. LF should be able to filter noise to ensure stable controlled input voltage to VCO. SPICE simulation of noise analyses are shown in Fig. 12 and 13. The result shows that our proposed filters do not only remove high frequency component from PD/CP output, but also filter noise interference, thus improving signal-to-noise ratio (SNR). From Fig. 13 , it is clearly shown that: = * (9)

V. CONCLUSION AND FUTURE WORK
This work started with understanding the behavior of memristor to sinusoidal voltage excitation at varying frequencies. Following this, we examined the two commonest low pass filter topologies that have been used in phase locked loop. We obtained their transfer functions and plotted their Bode plots with Octave and SPICE respectively. In our analyses, we showed that the frequency responses of our proposed filter are consistent with the existing resitancebased loop filter. However, our proposed memristor-based loop filters are not only CMOS compatible (crossbar architecture), but also consume less power and reduced area occupied on the chip without compromising the functionalities of filters. Aside the fact that memristor is a suitable choice for nanoelectronic systems and applications, the advent of soluble memristor provides environmentally friendly electronic devices, most especially it prevents valuable information from falling into the hands of enemy. In addition, memristor offers a large resistance value to keep PLL stable, and occupies small area on the chip.
The proposed memristor-based loop filter, together with memristive emulator and MOSCAP , can be designed using EDA tools in current CMOS technology and simulated.