This document discusses the implementation of low power integrators and differentiators using memristors. It presents the mathematical models of memristors and describes how memristor-based integrator and differentiator circuits were designed and simulated. The results show that the memristor-based circuits achieve much lower power consumption in the nano-watt range compared to traditional op-amp based implementations, demonstrating the potential of memristors for low power analog circuit applications.

1. Implementation of Low Power Integrator & Differentiator Using
Memristor
1
Joy Chowdhury
PG Student, School of Electronics Engineering
KIIT University, Bhubaneswar
joychowdhury87@yahoo.in
2
J. K. Das, 3
N. K. Rout
2,3
Associate Professor
School of Electronics Engineering
KIIT University, Bhubaneswar
jkdas12@gmail.com, nkrout@kiit.ac.in
Abstract—When the fourth passive circuit
element(Memristor) was discovered in 2008 it had tremendous
potential to replace the MOSFETS and this made its
mathematical modeling very imperative. Various mathematical
device models for the memristor had been proposed earlier such
as the linear ion drift model, non linear ion drift model, tunnel
barrier model, and a recently proposed TEAM model. But the
linear ion drift model continued to be the most simple and
efficient model. Due to its dual capability to be used as a bipolar
resistive switch as well as memory device it has tremendous
potential to harness low power and more compact designs by
replacing or in conjunction to the standard CMOS design styles
giving new hybrid structures.
Keywords- Memristor, Integrator, differentiator, low power
INTRODUCTION
The idea of memristors was first postulated by Leon Chua in
1970. Its invention in 2008 by HP Labs was a major
breakthrough in the world of electronic circuit theory. A
major leap was taken and a new fourth fundamental passive
element was introduced. The basic idea struck Chua from the
symmetry conditions exhibited in Aristotle's THEORY OF
MATTER . The working principle of memristor basically
revolves around the two most basic circuit parameters namely
flux(φ) and charge(q). This new device is very similar to a
resistor except that it is defined in terms of its Memristance
(M) which is given as
where M denotes a functional relationship between magnetic
flux(φ) and electric charge(q).From the mathematical
relations of the memristor it can be comprehended that the
electrical resistance of the memristor is not constant rather it
depends on the previous values of charge that was previously
present and the present value of current. Thus the device is
said to have memory. It could be effectively used for storing
the previous state information or data bits even without using
flip flops.
MATHEMATICAL MODEL
One of the most simple and computationally efficient models
which is prevalent in memristor modeling is the linear ion
drift model. This model assumes the memristor device to be a
combination of series connected resistors with effective
switching possible between them. This concept was supported
in the fabrication process by varying the doping concentration
of the . The stated fact can be seen from the hp lab memristor
model in figure 1.
Figure 1: HP Lab Memristor Model
The device released by Stanley Williams from HP Labs was
made up of thin film titanium dioxide (TiO2) which was
selectively doped. The doping mentioned here was
implemented by incorporating extra oxygen vacancies in the
undoped thin film. The doped region has a resistance Ron
while the undoped area has a resistance Roff. Several
phenomenon are assumed here which include a) ohmic
conductance , b) equal average ion mobility c)linear ion drift
in presence of uniform electric field. The transport of charge
carriers within the thin film titanium dioxide can be explained
from the figure 2.
International Journal of Applied Engineering Research ISSN 0973-4562 Volume 10, Number 9 (2015)
© Research India Publications ::: http://www.ripublication.com
9009

2. Figure 2 : Transport of oxygen vacancies in TiO2 thin film.
The relation between the derivative of the state variable and
voltage is stated as
where Ron is the resistance when w(t)= D and Roff is the
resistance when w(t)= 0.
To aid the linear ion drift model a window function is used to
limit the movement of charge carriers within the device
bounds. Similar to signal processing applications a window
function is multiplied to the derivative of the state variable to
nullify it. The ideal choice is rectangular window function.
But due to the absence of any non linearity consideration
several modifications to this window has been presented in .
By far the best performance is demonstrated by the
Prodromakis window which also ensure scalability of the
window function by introducing a scalar multiplicative factor.
APPLICATIONS
The memristor serves as a bipolar resistive switch with
history mechanisms. It can be used to substitute conventional
electronic switches which are made of bipolar junction
transistors or Mosfets. Sometimes they can be used as a
substitute for resistors. They find extensive applications in
neuromorphic systems and recently with the development of
memcapacitance and meminductance they are a potential
component of high frequency RF circuits. The most widely
prevalent and commercial application of memristor is non
volatile Random Access Memory which has already been
developed by HP Labs. Crossbar latch memory is another
widespread application which is expected to hit the market in
the next two years.
Minimal power consumption and very less die area are two of
the most lucrative properties of the memristor which can be
extensively dominated. In this paper we have tried to
implement memristor based circuits for analog computation
and instrumentation. The circuits by marked reduction in
power consumption and hence better compactness and greater
reliability. Also the fabrication of memristors is compatible
with the existing CMOS technology.
ANALOG CIRCUITS
The Integrator and Differentiator:
In this paper the analysis covers the integrator and the
differentiator. This two circuits are widely used for linear
wave shaping, analog computation, design of communication
systems(transmitters and receivers) , data converters , spectral
shaping and even data encryption.
The integrating amplifier more commonly known as the
integrator is a circuit in which gives the output waveform as
the integral of the applied input voltage. Such as circuit can be
achieved by using an opamp based inverting amplifier
configuration in which a capacitor replaces the resistor in the
feedback path.
Figure 3 : Integrator circuit diagram with input and output waveform.
From figure3 it is seen that since the inverting input is at
virtual ground
(1)
(2)
By applying Kirchhoff's current law at the inverting input we
get
(3)
therefore,
(4)
or , (5)
where ) is the constant of integration and its value
is proportional to the value of output voltage at time t=0.
International Journal of Applied Engineering Research ISSN 0973-4562 Volume 10, Number 9 (2015)
© Research India Publications ::: http://www.ripublication.com
9010

3. The integrator is very widely used in analog computers,
analog -to- digital converters (ADC) and signal - waveshaping
circuits.
The differentiating amplifier or the differentiator as
implied from the name performs the mathematical operation
of differentiation. The output waveform is given as the
derivative of the applied input signal. This differentiator is
constructed by from the basic inverting amplifier by
interchanging the input resistor by a capacitor and the usual
resistor in the feedback path.
Figure 4 : Differentiator circuit
Since the inverting input is at virtual ground,
(6)
(7)
Then applying KCL at the inverting input we obtain,
(7)
therefore ,
(8)
or, (9)
Thus the output voltage is RC(time constant) times the
negative instantaneous rate of change of the input voltage Vin
with time. A triangular waveform at the input produces a
square wave at the output.
The differentiator is very widely used in waveshaping circuits
for detecting high-frequency components in the input signal,
also used as a 'rate-of-change' detector in FM modulators.
RESULTS
The opamp based circuits were simulated in cadence 180nm
technology and transient analysis was run.
Then the normal circuits were suitably modified to form the
memristor based integrator and differentiator circuits. The on
resistance of the memristor has to be adjusted in the modeling
code to suit the RC time constant values. The transient
analysis was run and the transient power was calculated for all
the configurations. The power values are tabulated below.
Figure 5: Output of normal integrator for square wave input
Figure 6 : Transient Power Analysis of normal integrator
Figure 7: Output response of memristor based integrator
Figure 8: Transient power Analysis of Memristor based integrator
Figure 9: Differentiator output for triangular wave input
International Journal of Applied Engineering Research ISSN 0973-4562 Volume 10, Number 9 (2015)
© Research India Publications ::: http://www.ripublication.com
9011

4. Figure 10: Transient Power analysis for normal differentiator
Figure 11: Output response of memristor based differentiator for triangular
wave input
Figure 12: Transient Power analysis of memristor based differentiator.
From the analysis of the above results we that in the
memristor based circuits we obtain a greater power reduction
which is in the nano watts range as compared to the normal
opamp - resistor based circuits. The comparison in power is
shown in the table below. The value of Ron in all cases is
taken to be 500 ohms.
Table 1 : Power comparison of the implemented designs
Design Technology Device
Length
Transient Power
(watts)
Integrator CMOS
OPAMP
based
L=
180nm
×
Integrator Memristor
based
D= 3nm 49.492 × 10-9
Differentiator CMOS
OPAMP
based
L=
180nm
15.786 × 10-6
Differentiator Memristor
based
D=3nm 16.48 × 10-9
CONCLUSION
Thus we can see that in case of the circuits implemented by
memristors the power dissipation is very minimal and the
leakage power is almost negligible as compared to the
existing VLSI design styles. Also due to its extremely small
device length it further enhances design integration and
miniaturization.
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International Journal of Applied Engineering Research ISSN 0973-4562 Volume 10, Number 9 (2015)
© Research India Publications ::: http://www.ripublication.com
9012