Memristor-based reactance-less oscillator M. Affan Zidan, H. Omran, A.G. Radwan and K.N. Salama The first reactance-less oscillator is introduced. By using a memristor, the oscillator can be fully implemented on-chip without the need for any capacitors or inductors, which results in an area-efficient fully integrated solution. The concept of operation of the proposed oscillator is explained and detailed mathematical analysis is introduced. Closedform expressions for the oscillation frequency and oscillation conditions are derived. Finally, the derived equations are verified with circuit simulations showing excellent agreement.
Introduction: The memristor, the fourth basic circuit element, was first proposed by Chua in 1971 [1], but it was not until 2008 that a passive physical implementation was realised [2]. Since then, the memristor has attracted researchers from multiple disciplines with its unique characteristics and has shown potential importance in many fields, including non-volatile memory, reconfigurable logic and circuit design [3– 5]. In this Letter, we introduce a reactance-less memristor-based oscillator. The increase/decrease of memristor resistance according to the applied voltage resembles the charging/discharging of a reactive element. The inherent delay in the memristor response is exploited to realise the oscillator function. The ‘resistance-storage’ property of the memristor eliminates the need for an energy-storing reactive element, i.e. capacitor or inductor. It should be noted that even ring oscillators are formed of delay stages, which depend on the charging/discharging of intrinsic and extrinsic capacitances. Using the mathematical model of the HP memristor in [6], the memristor resistance as a function of time is given by R2m (t) = R2o + 2k ′
t Vm (t)d t
(1)
o
where Rm is the memristor resistance, Ro is the initial resistance of the memristor, Vm is the voltage across the memristor, and k′ ¼ mvRon (Roff 2 Ron)/d 2, where mv is the dopant drift mobility, Ron and Roff are the minimum and maximum memristor resistances, respectively, and d is the length of the device. The dopant drift mobility is the physical limit that determines the response time of the memristor. For current values of mv , the proposed oscillator is suitable for low-frequency applications. However, the introduced concept is general and can be extended to higher frequencies given that technology advancement improves the response speed. Low-frequency oscillators find usage in biomedical applications and embedded systems [7, 8]. However, off-chip components are usually used because large capacitances are required [7]. In [8], a novel technique was used to implement the oscillator on-chip, but the capacitor consumed 77.8% of the total chip area. By using a memristor, the implementation proposed in this Letter eliminates the need for capacitors or inductors, allowing a fully integrated implementation in a very small area.
Vn
M
Rm Rm + Ra
(2)
The memristor is connected in a polarity such that Rm increases for positive Vi and decreases for negative Vi. The threshold voltages Vp and Vn should be selected such that Vn , 0 , Vp. A simple implementation of F(Vi) using two comparators and an AND gate is shown in Fig. 1, while other implementations are also possible. Using Fig. 2, the operation of the oscillator can be traced assuming that we start at ‘a’: a b: At ‘a’ a positive voltage is applied to the memristor since Vo ¼ Voh. The memristor resistance will increase, and so will Vi until the operating point reaches ‘b’. b c: At ‘b’ the value of Vi will just pass Vp , thus Vo will switch to Vol , and the operating point will jump to ‘c’. c d: At ‘c’ a negative voltage is applied to the memristor. The memristor resistance will decrease, and so will |Vi| until the operating point reaches ‘d’. d a: At ‘d’ the value of Vi will just pass Vn , thus Vo will switch to Voh , and the operating point will jump to ‘a’. The circuit will oscillate independent of the initial memristor resistance (Ro). If the initial point is at Vi , Vn , the memristor resistance will decrease until reaching ‘d’. If it starts at 0 , Vi , Vp , the memristor resistance will increase until reaching ‘b’. In both cases, the circuit will start oscillation from ‘b’ or ‘d’ and will follow the path described previously. The cases Vn , Vi , 0 and Vi . Vp will not happen since Vi and Vo must have the same polarity. Oscillation condition: Based on (2), there is a single equivalent memristor resistance for a given value of Vi. The resistances at Vi ¼ Vp and Vi ¼ Vn are given by Rmp = Ra
Vp Vn , Rmn = Ra Voh − Vp Vol − Vn
(3)
Rmp and Rmn have to be selected such that Ron , Rmn , Rmp , Roff. Based on the circuit tracing given in the preceding Section, the oscillation will occur if Vp and Vn are selected such that Vp − Vn
Voh .0 Vol
(4)
Oscillation frequency: The time required by the circuit to change its state from ‘a’ to ‘b’ is determined by the time required for the memristor to change its resistance from Rmn to Rmp. The derivative of (1) with respect to time is given by Rm dRm = k ′ Vi (t)dt
(5)
+ –
By solving the integration, the time of the positive half cycle is
F(Vi ) Vo
–
Vi
Vi (t) = Vo (t)
By substituting (2) into (5) then integrating, the equation can be written as Rmp TH 1 dt = ′ (Rm + R)dRm (6) k Voh Rmn 0
+
Vp
Ra
Proposed circuit: The proposed circuit is based on a voltage divider between a resistor and a memristor, and a feedback function (F(Vi)) as shown in Figs. 1 and 2. The voltage on the memristor is given by
TH =
R2mp − R2mn + 2R(Rmp − Rmn ) 2k ′ Voh
Fig. 1 Proposed memristor-based reactance-less oscillator
(7)
Similarly, the time of the negative half cycle is Vo
c V'n Vol
d
Voh
a Vn
V'p
TL = b Vp
Vi
R2mn − R2mp + 2R(Rmn − Rmp ) 2k ′ Vol
(8)
The duty cycle expression can be derived in a simple form by substituting (3) into (7) and (8):
Fig. 2 Transfer function of F (Vi) showing transition between different operating points
ELECTRONICS LETTERS 27th October 2011 Vol. 47 No. 22
D=
|Vol | Voh − Vol
(9)
Using the same substitutions and after simplification, the frequency of oscillation is given by fo =
2Dk ′ Voh (Voh − Vp )2 (Vn − Vol )2 + Vn Voh )(2Voh Vol − Vol Vp − Voh Vn )
R2a (Vp Vol
(10)
Circuit simulation: As the access to the experimental realisation of the memristor is very limited, researchers resort to SPICE and behavioural models of the HP memristor [9], or emulate the memristor model using active circuitry [3]. The proposed oscillator was simulated using both SPICE and Verilog-A models, both giving similar results. We chose Ron , Roff , d and mv to be 100 V, 38 kV, 10 nm and 10210 cm2 s21 V21, respectively. Fig. 3 shows the transient simulation results. The memristor resistance oscillates between Rmn and Rmp , which defines the location of the operating points. By substituting the circuit parameters into (3) and (10): Rmn ¼ 3 kV, Rmp ¼ 9 kV and fo ¼ 3.51 Hz, which shows an excellent match to the simulation results. For further verification of the mathematical analysis presented, the oscillation frequency was tuned by sweeping Ra from 2 to 12 kV. The maximum error was 2.13% and 0.1% for SPICE and Verilog-A simulations, respectively. Rm
R, W (×103)
10.0
6.0
2.0 Vp
V, V
1.0
Vn
Vi
0.5 0 –0.5
Vo
V, V
1.5 0.5 –0.5 –1.5 0
0.25
0.50
0.75 time, s
1.00
1.25
which finds applications in biomedical and embedded systems. Mathematical analysis for the new oscillator is presented and verified using circuit simulation.
1.50
Fig. 3 Transient simulation results for Rm (upper), Vi (middle), and Vo (lower), for Ra ¼3 kV, Voh ¼ 1 V, Vol ¼ 21 V, Vp ¼ 0.75 V and Vn ¼ 20.5 V
# The Institution of Engineering and Technology 2011 24 August 2011 doi: 10.1049/el.2011.2700 One or more of the Figures in this Letter are available in colour online. M. Affan Zidan, H. Omran, A.G. Radwan and K.N. Salama (Electrical Engineering Program, King Abdullah University of Science and Technology (KAUST), Thuwal 23955-6900, Saudi Arabia) E-mail:
[email protected] A.G. Radwan: Also with the Department of Engineering Mathematics, Cairo University, Cairo, Egypt References 1 Chua, L.: ‘Memristor – the missing circuit element’, IEEE Trans. Circuit Theory, 1971, 18, (5), pp. 507–519 2 Strukov, D.B., Snider, G.S., and Stewart, D.R.: ‘The missing memristor found’, Nature, 2008, 435, pp. 80–83 3 Pershin, Y.V., and Di Ventra, M.: ‘Memristive circuits simulate memcapacitors and meminductors’, Electron. Lett., 2010, 46, (7), pp. 517– 518 4 Witrisal, K.: ‘Memristor-based stored-reference receiver – the UWB solution?’, Electron. Lett., 2009, 45, (14), pp. 713–714 5 Talukdar, A., Radwan, A.G., and Salama, K.N.: ‘Generalized model for memristor-based Wien family oscillators’, Microelectron. J., 2011, 42, (9), pp. 1032–1038 6 Radwan, A.G., Zidan, M.A., and Salama, K.N.: ‘HP memristor mathematical model for periodic signals and DC’. IEEE Int. Midwest Symp. on Circuits and Systems, (MWSCAS’10), Seattle, WA, USA, 2010, pp. 861 –864 7 Elwakil, A.S., and Ozoguz, S.: ‘A low frequency oscillator structure’. European Conf. on Circuit Theory and Design, (ECCTD’09), Antalya, Turkey, 2009, pp. 588–590 8 Hwang, C., Bibyk, S., Ismail, M., and Lohiser, B.: ‘A very low frequency, micropower, low voltage CMOS oscillator for noncardiac pacemakers’, IEEE Trans. Circuits Syst. I, Fundam. Theory Appl., 1995, 42, (11), pp. 962 –966 9 Biolek, Z., Biolek, D., and Biolkova, V.: ‘SPICE model of memristor with nonlinear dopant drift’, Radioengineering, 2009, 18, (2), pp. 210– 214
Conclusions: A memristor-based oscillator without using any capacitors or inductors is presented. The introduced reactance-less oscillator enables an area-efficient implementation for low-frequency oscillators,
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