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Memristor-Capacitor Based Startup Circuit for Voltage Reference Generators

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This paper presents the design of Memristorcapacitor based startup circuit. Memristor is a novel device and has many advantages over conventional CMOS devices such as no leakage current and is easy to manufacture. In this work the switching characteristics of memristor is utilized. First the
theoretical equations describing the switching behavior of memristor are derived. To prove the switching capabilities of Memristor, a startup circuit based on series combination of Memristor-capacitor is proposed. This circuit is compared with the reference circuit (which utilizes resistor in place of memristor) and the previously reported MOSFET based startup circuits. Comparison of different circuits was done to validate the results. Simulation results shows that memristor based circuit attains on (I = 2.25 mA) to off state (I = 10 μA) in 2.8 ns while the MOSFET based startup circuits takes (I = 1 mA) to off state (I = 10 μA) in 55.56 ns. However no significant difference in switching time was observed when compared with resistance based startup circuit. The benefit comes in terms of area because much larger die area is required for manufacturing of resistance in comparison to fabrication of memristor.

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Memristor-Capacitor Based Startup Circuit for Voltage Reference Generators

  1. 1. Memristor-Capacitor Based Startup Circuit for Voltage Reference Generators Mangal Das Electronics & Communication Engineering ABES Engineering College Ghaziabad, India mangalforyou@gmail.com Sonal Singhal Electrical Engineering Shiv Nadar University Gautam buddh Nagar, India sonal.singhal@snu.edu.in Abstract— This paper presents the design of Memristor-capacitor based startup circuit. Memristor is a novel device and has many advantages over conventional CMOS devices such as no leakage current and is easy to manufacture. In this work the switching characteristics of memristor is utilized. First the theoretical equations describing the switching behavior of memristor are derived. To prove the switching capabilities of Memristor, a startup circuit based on series combination of Memristor-capacitor is proposed. This circuit is compared with the reference circuit (which utilizes resistor in place of memristor) and the previously reported MOSFET based startup circuits. Comparison of different circuits was done to validate the results. Simulation results shows that memristor based circuit attains on (I = 2.25 mA) to off state (I = 10 μA) in 2.8 ns while the MOSFET based startup circuits takes (I = 1 mA) to off state (I = 10 μA) in 55.56 ns. However no significant difference in switching time was observed when compared with resistance based startup circuit. The benefit comes in terms of area because much larger die area is required for manufacturing of resistance in comparison to fabrication of memristor. Keywords— Startup circuits; Memristors; Voltage Reference generator; Switching circuits component. I. INTRODUCTION This Voltage reference generators are used for the generation of process and temperature independent supply voltages. Self Biased Circuits like voltage reference generator have degenerate bias point, which does not allow these circuits to start itself [1]. It necessitates the use of startup circuit in these circuits [2],[3]. Startup circuit isolates itself electrically from reference generator circuit after giving rise to initial conditions. Conventionally, startup circuits are realized by capacitors and MOSFETs and are characterized in terms of speed and area. These startup circuits are slow because of large time constant of these circuit. The reason of such large time constant is the large resistances (in hundreds of kilo-ohm) between drain and source terminals of MOSFETs. These circuits also consume more area due to presence of capacitors and MOSFETs of longer channel length than other MOSFETs present in the reference generator circuit. As an alternative and new approach, memristor based startup circuits is devised. Memristor is a passive two port element with variable resistance [5]. L. Chua and S. Kang have given theoretical frame work for describing the memrisitve system [6]. In 2008 HP (Hewlett-Packard) has announced that they synthesized a memrisitve system based on TiO2. Since then, numerous models of TiO2 memristor have been reported in literature. All of these models use the physical model put forward by D. B. Strukov [7]. Since then many articles have been published on the physical and electrical properties of memristor. However no report has investigated the switching behaviour of memristor. In this paper we have investigated the switching behaviour of memristor under constant DC bias and suggested a memristor-capacitor based startup circuit for voltage reference generators. Spice model of memristor suggested by Z. Biolek is used for simulation [4]. This paper is organized as follows. Section II describes the electrical properties of memristor in terms of physical parameters. Section III describes switching behaviour of memristor. It is studied under initial and final conditions. Section IV presents the application of switching characteristics of memristor in a startup circuit. A new startup circuit has been proposed which utilizes the MOSFET and MEMRISTOR. Section V presents simulation results for the suggested circuit. Simulation has been done on SPICE. II. ELECTRICAL AND PHYSICAL PARAMETERS OF MEMRISTOR The basic equation of current controlled memrisitve system is given below [6]: v=R(w,i)i (1) dw =f(w,i) (2) dt where w is the state variable of physical dimension of Memristor, R(w,i) is memresistance, v is the voltage across memresistance, and i is the current through memristance. Fig 1 shows the memristor model in terms of it’s physical parameters. The relation between the physical and electrical behaviour of memristor are given by following equations [7]:
  2. 2. v(t)= R w(t) +R 1- w(t) i(t) ⎛ ⎛ ⎞ ⎞ ⎜ ⎜ ⎟ ⎟ ⎝ on D off ⎝ D ⎠ ⎠ (3) dw(t) =μ R on i(t) dt v D (4) Where w(t) is the width of doped TiO2-x at any time “t”, D is the semiconductor thickness, μv is dopant mobility, v(t) is the voltage across memristor, i(t) is the current through memristor, on R is on-state resistance of memristor, off R is off state resistance of memristor. w 2 1 III. SWITCHING CHARACTERISTICS OF MEMRISTOR This section presents the theoretical framework for analyzing the switching behavior of memristor. Switching behavior can be analyzed from equation (3) and (4). From these equations it is seen that the memristor can be made to work as switch as at time t = 0 and at t → ∞. At these two extreme time instants the value of w(t) (0,1) D ∈ which give rise to two values of v(t) at t = 0 and t → ∞. The memristor switch makes the transition from ON to OFF or OFF to ON state in accordance with the polarity applied across the memristor. In order to analyze the behaviour of TiO2 based memristor resistance with respect to time we have used the equations given by Joglekar and Wang [8],[9]: 2 2 mem 0 d R (t)=R-2kR φ(t) (5) where k = μv Ron/D2, μv is the dopant drift mobility, t0 φ(t)=∫ v(τ)dτ is the flux at time ‘t’, d off on R =R -R is the difference of boundary resistances and 0 0 R =R(x ) is the initial resistance at t = 0. The value of resistance at any time ‘t’ mem on off R ∈(R ,R ) . Radwan has extended the theoretical framework given by Joglekar and Wang [8],[9] to investigate the memristance resistance under DC excitation [10]. The equation is reproduced here 2 2 mem 0 d dc R (t)=R-2kR V t (6) From equation (3), (4), (5), (6) we can identify the switching activities for two extreme cases (i) At t = 0, equation (6) and (3) can be written as R (0)=R = R w(0) +R 1- w(0) ⎛ ⎛ ⎞ ⎞ ⎜ ⎜ ⎟ ⎟ ⎝ ⎝ ⎠ ⎠ mem 0 on off D D (7) (ii) At time t = ∞ (long times under bias) resistance of memresistor will tend to Ron or Roff depending on the polarity of applied voltage. Strokov proposed that in hard switching case (large voltage excursions or long times under bias) there appears to be clearly defined threshold voltage however effect is actually dynamical [7]. Saturation time is the time taken by memristor to reach it’s one of the final values either Ron or Roff. Saturation time of memristor is given as [7]: 2 off sat D R v on dc t = 2μ R V (8) IV. STARTUP CIRCUIT BASED ON MEMRISTOR CAPACITOR COMBINATION A startup circuit based on memristor and capacitor is proposed. It is designed keeping in view on the capability of memristor to work as a time dependent resistive switch (Refer to equation (5)). Fig 2 shows the circuit diagram of proposed startup circuit utilizing MOSFET and memristor-capacitor combination. In the proposed circuit voltage at point A and B depends on the type of MOSFET. For the proposed circuit, VGS of MOSFET is written as: GS mem mem V (t)=R (t) I (t) (9) Where Rmem(t) is given by equation (5), Imem(t) is the current flowing through the memristor at any time. Multu has given the relation for Imem(t) with time and is given by [11] : -(t-tsat )/ I (t)= (-V (t )+V )e c sat DC mem R τ (10) Where Vc(t) is the voltage at capacitor at time ‘t’, τ = Rmem(t)C is the time constant of the memristor-capacitor circuit, VDC is the applied Voltage. From equation (10), the current behavior for time t = 0 and t = ∞ is analyzed (equation (11) and (12) respectively) tsat / I (0)= (-V (t )+V )e c sat DC mem R τ (11) mem I (∞)=0 (12) Therefore at time t = 0 and at t→∞, VGS (gate to source voltage) of MOSFET can be written as (equation (13) and (14) respectively) tsat / c sat DC GS mem (-V (t )+V )e V (0)=R (0) R τ (13) GS V (∞)=0 (14) i v Fig. 1.Memristor model (Shaded area shows TiO2-x region remaining part is.TiO2).
  3. 3. V. SIMULATION RESULTS In order to investigate the resistance behavior of memristor with respect to input voltage, transient analysis is carried out. Biolek suggested a window function [4] and the same is used in the simulations: f(x)=1-(x-stp(-i))2p (15) Where p is a positive integer, i is the memristor current, and ≥ ⎧⎨ ⎩ ≤ 1 proi 0 stp(i)= 0 proi 0 (16) Fig 3 shows the transient analysis of memristor resistance which exhibits Ron=100, p=10, Rinit=11kΩ, Roff =16kΩ, D =10 nm for different step inputs (1 V, 2 V, 3 V, 4 V). Simulation Results shown in fig. 3 are in accordance with equation (8). From equation (8) we can observe that saturation time ‘tsat’ is inversely proportional to applied DC Voltage with other parameters constant. Similar trend is obtained in the simulation studies that as the applied voltage increases the saturation time decreases. From fig. 3 we can observe that when applied step input is changed from 1 V, 2 V to 3 V, the change in saturation time (tsat) is significant whereas when applied excitation is changed from 3 V to 4 V the change in ‘tsat’ is small. For simulations the power supply of 3 V is used as it optimizes the power and speed. Transient behavior of proposed startup circuit here in referred as PSC is investigated with the step input of 3 V. The peak value of current is obtained 2.52mA. To compare the result, similar analysis is done on a resistive startup circuit (RSC). In this startup circuit (RSC) memristor is replaced by a resistor. Again to validate result, the startup circuit already reported by Giustolisi [2], [12] is compared with PSC. The startup used by Giustolisi and Yu-Hsuan [2], [12] is MOSFET based startup circuit and is referred as (MSC). Simulation of PSC was done with PMOS (W=50μ, L=1μ), Capacitor (C=1ps), Memristor (Rinit = 137.5Ω, Roff = 250Ω, Ron = 25Ω, p=10, D = 10 nm). RSC (Resistive startup circuit) uses resistance (250 ohm) in place of memristor. MSC uses two PMOS (W=20μ,4μ; L=4μ,10μ) and a capacitor (C=1ps). Fig. 4 shows the transient behavior of the currents of proposed startup circuit (shown as PSC), MOSFET based startup circuit used (shown as MSC), and a startup circuit which uses resistance in place of memristor (shown as RSC). Table I shows the peak values of current attained by circuits (MSC, PSC, RSC). The peak current value is more in PSC, RSC in comparison with MSC because of very low resistance value in comparison to drain - source terminals of MOSFETs used in MSC. However it is not good for our circuit because the peak power dissipation will be high. Table II shows the time taken by the currents (MSC, PSC, RSC) to drop to the 10%, 63%, 90% of their peak value. The very fast transition of currents for PSC can be explained in the following way: Time constant (τ) for an RC-series circuit is given by τ = RC. Resistance or capacitance is to be decreased in order to decrease time constant. In the proposed circuit memristor is biased in such a way that as time increases the resistance of memristor decreases, thereby reducing the time constant. With decrease in resistance of memristor (connected between gate and source) the VGS (gate to source voltage) of MOSFET will also decrease, thus pushing MOSFET towards cutoff very rapidly. Combined Fig. 2.Proposed circuit diagram of startup circuit Fig.3 .Transient analysis of Resistance (Memristor parameters Ron=100, p=10, Rinit=11k•, Roff =16kΩ, D=10 nm, under different step inputs (1 V, 2 V, 3 V, 4 V)). Fig.4. Comparison of transient behavior of currents
  4. 4. effect of these two phenomena causes current to decrease very rapidly. From table II, it is seen that PSC and RSC are comparable in speed. It is also seen that proposed startup circuit is much faster than MOSFET based startup circuit. In comparison to resistance based startup circuit (RSC) the memristor based startup circuit (PSC) will be an economical choice. It is because RSC consumes more area than PSC. Length of a memristor typically varies from 30nm to 3nm. Startup circuit which is suggested in the paper uses a memristor of length (D) 10nm. This is less even in comparison with the modern CMOS technology (28 nm) used today. Much larger die area is required for manufacturing of resistance than the area required for the manufacturing of memristor. TABLE I. PEAK VALUE OF CURRENT PSC RSC MSC Peak Value of Current (mA) 2.52 2.51 0.67 TABLE II. TRASIENT BEHAVOIUR OF CIRCUIT % Drop From Peak Value TIME (nsec) PSC RSC MSC 90 % 2.00 2.00 6.85 63% 2.06 2.09 16.91a 10 % 2.31 2.56 45.372a a. Not shown in fig 4. CONCLUSION In this paper we have suggested a new hybrid circuit which includes the memristor-capacitor as its components and capable of working as a startup circuit in voltage reference generators. Current in proposed startup circuit drops from 2.5 mA to 10 μA (2.8 ns) which is much faster than conventional CMOS startup circuit. Proposed circuit also saves area as dimensions and number of components used in the circuit is less than any conventional startup circuit. REFERENCES [1] R.Jacob Baker, “Voltage References,” in CMOS Circuit Design , Layout And Simulation,Ed. New Delhi: Wiley India (P) Ltd., 2005, pp. 613-651. [2] G. Giustolisi, G. Palumbo, M. Criscione, and F. Cutrì, “A low-voltage low-power voltage reference based on subthreshold MOSFETs,” IEEE J. Solid-State Circuits, vol. 38, no. 1, pp. 151–154, Jan. 2003. [3] A. Bendali and Y. Audet, “A 1-V CMOS current reference with temperature and process compensation,” IEEE Trans. on Circuits and Systems I, vol. 54, pp. 1424-1429, 2007. [4] Z. Biolek, D. Biolek, and V. Biolkov´a, “SPICE model of memristor with nonlinear dopant drift,” Radioengineering, vol. 18, no. 2, pp. 210– 214,2009. [5] L. Chua, “Memristor: The missing circuit element,” IEEE Trans. Circuit Theory, vol. 18, no. 5, pp. 507–519, Sep. 1971. [6] L. Chua and S. Kang, “Memristive devices and systems,” Proc. IEEE, vol. 64, no. 2, pp. 209–223, Feb. 1976. [7] D. B. Strukov, G. S. Snider, D. R. Stewart, and S. R. Williams, “The missing memristor found,” Nature, vol. 453, no. 7191, pp. 80–83, May 2008. [8] Y. N. Joglekar and S. J. Wolf, “The elusive memristor: Properties of basic electrical circuits,” Eur. J. Phys., vol. 30, no. 4, pp. 661–675, Jul. 2009. [9] F. Y. Wang, “Memristor for intrductory physics,” Physics. class-ph, pp. 1–4, 2008. [10] A. G. Radwan, M. A. Zidan, K. N. Salama,“HP Memristor Mathematical Model for Periodic Signals and DC,” IEEE International Midwest Symposium on Circuits and Systems (MWSCAS’10), pp. 861– 864, 2010. [11] Resat MUTLU,“Solution of TiO2 memristor-capacitor series circuit excited by a constant voltage source and its application to calculate operation frequency of a programmable TiO2 memristor-capacitor relaxation oscillator,” Turkish Journal of Electrical Engineering & Computer Sciences, in press. [12] Yu-Hsuan Chiang and Shen-Iuan Liu, “A Submicrowatt 1.1-MHz CMOS Relaxation Oscillator With Tempreture Compensation,” IEEE Transactions on Circuits and Systems II, vol.99,pp.1-5,Oct. 2013.

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