Implementation of a Second-Order Low-Pass Active ...

Implementation of a Second-Order Low-Pass Active Time-Varying Filter with a symbolic-modelled memristor Arturo Sarmiento-Reyes1 , Luis Hernández-Martínez1 , Jesús Jiménez-León1 , Gerardo Ulises Días Arango1 , Hector Vázquez-Leal2 1 2 INAOE, Electronics Department; Universidad Veracruzana, Faculty of Electronic Instrumentation {jarocho,jesus.leon}@inaoep.mx, [email protected]

Abstract

The Time-Invariant Filter

A time-varying lter is achieved by incorporating a memristor to a linear time-invariant biquad lter topology. The memristor is modelled with a i(t) − u(t) symbolic function. A chief characteristic of LTV lters is obtained, namely a faster settling time.

Figure 1 shows the biquadratic topology that has been used as the time-invariant prototype [1]. where Ho is the DC gain, ωo is the cut-o frequency and Q the quality factor of the lter.

Introduction In modern electronics, adaptability and programmability are some characteristics that are commonly expected in the implementation of analog lters. A smart form of incorporating these characteristics to analog lter design is by generating structures whose parameters change with the time, i.e. time-varying lters. This can be done by linking a time-dependent function to the constitutive branch relationships of those elements in the lter structure that dene the parameters of the transfer function such as gain, quality factor and cut-o frequency. The specic memory-resistance property of the memristor makes this device specially suited for implementing time-varying lters. In this work, this is achieved by using a memristor as the element that controls the parameters of the transfer function. It yields that, the time-dependance of the memristor is reected in transfer function of the lter. The result is a lter that possesses identical frequency-domain response as the original but a faster settling time.

Conclusion A parameter varying low-pass lter has been designed from a time-invariant biquadratic topology. The model of the memristor has been thoroughly analyzed. The frequency response of both lter implementations agree, whereas the transient response of the memristor-based time-varying lter exhibits a faster settling time that its time-invariant counterpart, which proofs the feasibility of the implementing LVT lters using memristors.

References [1] Adel S Sedra and Peter O Brackett. Filter theory and design: active and passive. Matrix Pub, 1978. [2] Miguel Ángel Gutiérrez De Anda, Arturo Sarmiento Reyes, Luis Hernández Martínez, Jacek Piskorowski, and Roman Kaszy«ski. The reduction of the duration of the transient response in a class of continuous-time ltv lters. Circuits and Systems II: Express Briefs, IEEE Transactions on, 56(2):102106, 2009.

The design formulas for the lter are: Ho = Figure 1: Time-invariant Biquad Filter.

ωo =

If the operational ampliers have ideal gains, then the transfer function has the form:

T (s) =

Ho ωo2 ωo 2 s +s Q +ωo2

q

Q=

R2 − Rin

(3)

1 R1 C1 R2 C2

C2 C3

q

(2)

(4)

R1 C1 R2 C2

Herein, it can be noticed that an important com(1) ponent for xing ω and Q is R . o 1

The Memristor Symbolic Model In [3], a circuit simulation-oriented model has been directly derived from the linear drift equation mechanism [4] that governs the functioning of the memristor. The equation (5) constitutes the symbolic memristor model. M = (α − 1)   !2  2 µAp a1 3 3 −  Xo (5Xo2 − 1)(Xo − 1)3 (Xo + 1)3 Ron  2    4 ∆ ω           µA a  p 1 2 2 2 −   X (X − 1) (X + 1) R o o o on   2ω  ∆      −1    [1 + (Xo − 1)α]  + Xo  Ron   (α − 1)     !2   2   µA a p 1 2 3 3 3     Xo (5Xo − 1)(Xo − 1) (Xo + 1) Ron     2 ∆ ω  +  cos(ωt)            µA a p 1   2 2 2 + X (X − 1) (X + 1) R   o o o on 2 ∆ ω    "   #  2   1 µA a   p 1 2 3 3 3 + Xo (5Xo − 1)(Xo − 1) (Xo + 1) Ron cos(2ωt) 2 4 ∆ ω

(a)

(b)

(5)

The parameter values associated to the HP memristor are shown in Table 1. Figure 2: (a) Voltage and Current waveforms; (b) (c)

Parameter

Symbol Value

Mobility of the charges ON-State Resistance Full lenght of the Semiconductor Amplitud of the Current Source Coecient of the window function Ratio of ON-State and OFF-State Resistance

µ Ron ∆ Ap a1 α

10−10 cm2 s−1 V −1 100 Ω 10 nm 40 µA 3 160

Table 1: Parameter Values for the model in (5).

Pinched hysteresis loop; (c) Memristance-Current loop.

Figure 2 summarizes the electrical behavior of the memristor at ω = 1. The memristance spans from Mmin = 6, 448 Ω to M∞ = 14, 410 Ω.

The Time-varying low-pass lter

Figure 3 shows the time-varying version of the low-pass lter where the resistor R1 has been sustituted by a memristor M1 dened by the model given in (5). Figure 4 shows the low-pass magnitude response (top) and the time response (bottom) for both, the time invariant an the time-varying lter.

[3] Arturo Sarmiento-Reyes, Luis HernandezMartinez, Carlos Hernandez Mejia, Gerardo Ulises Diaz Arango, and Hector Vazquez-Leal. A fully symbolic homotopy-based memristor model for applications to circuit simulation. In Circuits and Systems (LASCAS), 2014 IEEE 5th Latin American Symposium on, pages 14. IEEE, 2014.

[4] Omid Kavehei, A Iqbal, YS Kim, Kamran Eshraghian, SF Al-Sarawi, and Derek Abbott. The fourth element: characteristics, modelling and electromagnetic theory of the memristor. In Proceed-

ings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, page

rspa20090553. The Royal Society, 2010.

Information For further information consult the complete article:

Figure 3: The Time-varying low-pass lter.

For the specs (ω = 1, Ho = 10, Q = 5), the values of the elements are: C = C1 = C2 = 1µF

R 1 = M∞ R2 =

C3 =

1 ω 2 C 2 M∞

C Q

q

Rin =

M∞ R2

R2 |Ho |

Figure 4: Magnitude response and Time response for both lters. It can be noticed that the frequency response of the time-varying lter is identical to its time-invariant counterpart. It also demonstrates that the settling time is shorter for the time varying low-pass implementation.