Memristors are fresh electronic devices, a device that can be used and give a great advantage in many applications such as memory, logic, neuromorphic systems and so on. A computing machine theoretical account of the memristor would be a utile tool to analysis circuit behaviour to assist in develops application of this memristor as inactive circuit component via simulation. In this paper, we incorporate a memristor SPICE for planing memristor circuit which is more concentrating on non-linear theoretical account and parallel circuits. SPICE theoretical account would be appropriate manner to depict existent device operation. We integrating the memristor with assorted window maps that have been proposed in non additive ion impetus memristor devices. In look intoing and qualifying the physical electronic and behavioural belongingss of memristor devices, the circuit analysis of the proposed memristor theoretical accounts are so been studied. The simulation end product should hold a current-voltage hysteresis curve, which looks like bow tie. The loops map the exchanging behaviour of the device. Then, we come out with a simple parallel circuit which in this instance we construct a simple planimeter op-amp and differentiator op-amp circuit and do comparing between memristor implemented circuit and normal circuit. The research verifies the proposed memristor theoretical account, the possibilities of implementing memristor theoretical account and the advantage implementing the memristor in parallel circuit.

## Keywords – memristor, SPICE theoretical account, non linear, window maps, parallel circuit.

## Introduction

Memristor is the contraction of memory resistance which is a inactive device that provides a functional relation between charge and flux. It is a two-terminal circuit component in which the flux between the two terminuss is a map of the sum of electric charge that has passed through the device [ 1 ] . A memristor is said to be charge-controlled if the relation between flux and charge is expressed as a map of electric charge and it is said to be flux-controlled if the relation between flux and charge is expressed as a map of the flux linkage [ 2 ] .

In 1971, Leon Chua proposed that there should be a 4th cardinal passive circuit component to make a mathematical relationship between electric charge and magnetic flux which he called the memristor which is short for memory resistance [ 2 ] .

The current is defined as the clip derived function of the charge. The electromotive force is defined as the clip derived function of the flux harmonizing the Faraday jurisprudence. A resistance is defined by the relationship between electromotive force and current dv=Rdi, the capacitance is defined by the relationship between charge and electromotive force dq=Cdv, the inductance is defined by the relationship between flux and current dI†=Ldi. The 4th cardinal circuit component completes the symmetricalness of the relation between charge and magnetic flux dI†=Mdq. Table 1 show the relationship between the cardinal circuit component.

Basic two terminal devices

Equation

Relationship between cardinal circuit component

Resistor, R

dv=Rdi

V and I

Capacitor, C

dq=Cdv

V and Q

Inductor, L

dI†=Ldi

I and I†

Memristor, M

dI†=Mdq

Q and I†

Table 1: The four cardinal component ( resistance, capacitance, inductance and memristor ) .

In 2008, Stanley Williams and his squad at Hewlett Packard had succesfully fabricated the first memristor in physical device signifier which is a long delay from Leon Chua find in 1971 [ 3 ] . Memristance is a belongings of memristor. When the charge flows in one way through a circuit, the oppositions of the memristor addition. The opposition decreases when the charge flows in the opposite way in the circuit. If the applied electromotive force is turned off, therefore halting the flow of charge and the memristor remembers the last opposition that it had [ 1 ] .

In HP memristor theoretical account, to make a memristor, they used a really thin movie of Ti dioxide ( TiO2 ) . The thin movie is so sandwiched between the two Pt ( Pt ) contacts. One side of TiO2 is doped with O vacancies denoted as TiO2-x which x is normally 0.05. The O vacancies are positively charged ion and do it conductive, therefore it behaves as a semiconducting material. Another side of the TiO2 junction is undoped. The undoped part has insulating belongingss. The device established by HP is shown in Figure 1 [ 3 ] .

Figure 1: Memristor theoretical account adapt from [ 3 ] .

When a positive electromotive force is applied, the positively charged O vacancies in the doped TiO2-x bed are repelled and traveling them towards to the undoped TiO2 bed. When the boundary between the two stuffs moves, the per centum of the carry oning TiO2-x bed is increase. Therefore, the conduction of the whole device additions.

When a negative electromotive force is applied, the positively charged O vacancies are attracted and drawing them out of TiO2 bed. This increases the sum of insulating TiO2, therefore increasing the electric resistance of the whole device. When the electromotive force is turned off, the O vacancies do non travel. The boundary between the two Ti dioxide beds is frozen. This is how the memristor remembers the electromotive force last applied [ 1 ] .

## Methodology

Our purpose in this research is to supply a simulation plan adequately simulates and can be used as a circuit component in design work. To pattern the electrical features of the memristor, SPICE would be appropriate manner to depict existent device operation [ 4 ] . Furthermore, utilizing the theoretical account as a sub-circuit can extremely vouch a sensible high flexibleness and scalability characteristics [ 5 ] . We use LTSPICE to make a memristor theoretical account and design new symbol of the memristor circuit for the simulation because LTSPICE is much easier to manage compared to others. On the other manus, LTSPICE is a freeware and it will give a great advantage to the pupils in making research for this freshly devices. We use SPICE theoretical account that been adapt from [ 6 ] and we made some accommodation so we can utilize it for several window maps that has been proposed for non additive ion impetus theoretical account.

The SPICE theoretical account is created based on the mathematical theoretical account of the HP Labs memristor. After the memristor has been modeled, we foremost studied the difference between proposed memristor and so we will get down design and implement the memristor with an parallel circuit. We besides investigate and made a comparing between the memristor circuit with parallel circuit to see the difference and analyze the behaviour of the circuits.

## Model Of The Memristor from HP Labs

In the theoretical account of a memristor presented here, there is a thin semiconducting material movie that has two parts, one with a high concentration of dopant that behaves like a low opposition called RON and the other with a low dopant concentration with higher opposition called ROFF [ 3 ] . The movie is sandwiched between two metal contacts as in figure 1.

The entire opposition of the memristor, RMEM, is a amount of the oppositions of the doped and undoped parts, tungsten is the breadth of the doped part and D is the entire length of the TiO2 bed. ROFF and RON will be the bound values of the memristor opposition for w=0 and w=D. The ratio of the two oppositions is normally given as 102 – 103.

( 1 )

( 2 )

From the ohm ‘s jurisprudence relation between the memristor electromotive forces and current, we get

( 3 )

Then, we insert ( 1 ) into ( 3 ) . The electromotive force V ( T ) across the device will travel the boundary between the two parts doing the charged dopants to float. So, there is a drift ion mobility Aµv in the device. The alteration of the boundary is denoted as in ( 5 ) .

( 4 )

( 5 )

To acquire x ( T ) , we so integrates the right side of equation ( 5 ) which so yields the undermentioned expression

( 6 )

By infixing equation ( 6 ) into equation ( 4 ) and since normally RON & lt ; & lt ; ROFF, we so obtained the memristance of this system

( 7 )

Where Aµv is the mean impetus speed and has the units cm2/sV, D is the thickness of titanium-dioxide movie ROFF and RON are on-state and off-state oppositions and Q ( T ) is the entire charge go throughing through the memristor device.

## Non Linear Ion Drift Model

Even a little electromotive force across the nanodevices will bring forth a big electric field [ 7 ] . This doing the ion boundary place will travel in a unquestionably non-linear. Nonlinear dopant impetus adds nonlinear window map degree Fahrenheit ( ten ) to the province equation. The window map decreases as the province variables drift velocity approaches the boundaries until it reaches zero when making either boundaries [ 8 ] . The velocities of the motion of the boundary between the doped and undoped parts are depending on several factors.

( 8 )

Where Aµv is the dopant mobility. The velocity of the boundary between the doped and undoped parts decreases bit by bit to zero at the movie edges [ 1 ] . We simulate the nonlinear ion impetus memristor theoretical account with these window map to see the difference and the issue that been faced by them.

## Window Function

Window map is a map of the province variable. Window map forces the bounds of the device and to add nonlinear behavior near to these bounds. In other words, it creates the boundary for the memristor. Any effectual window map should therefore carry through the undermentioned conditions [ 8 ] :

Take into history the boundary conditions at the top and bottom electrodes of the device ;

Be capable of enforcing nonlinear impetus over the full active nucleus of the device ;

Provide linkage between the additive and nonlinear dopant impetus theoretical accounts ;

Be scalable, intending a scope of fmax ( ten ) can be obtained such that 0 a‰¤ fmax ( x ) a‰¤ 1 ;

Use a constitutional control parametric quantity for seting the theoretical account.

There are several window maps that have been proposed for non-linear theoretical account boulder clay day of the month which are by Strukov, Joglekar and, Biolek, and Prodromakis. Strukov proposed the following window map [ 3 ] .

( 9 )

However, as we can see in the figure 2, this window map deficiencies of flexibleness.

Figure 2: Plot of Strukov window map.

Another window map was proposed by Joglekar [ 4 ] , which has a control parametric quantity P which is a positive whole number. The intent of holding a control parametric quantity as an advocate is to integrate scalability and flexibleness in window map degree Fahrenheit ( ten ) that describes the dopant dynamicss.

( 10 )

Figure 3 displays a graphical representation of the window map described by Joglekar for assorted Ps parametric quantity ( p=1, 5 and 10 ) . This control parametric quantity controls the one-dimensionality of the theoretical account, where it becomes more additive as P additions. This window map ensures zero impetus at the boundaries. From the aforethought graph, we noticed that the maximal degree Fahrenheit ( x ) value is occurs at the centre of the device and nothing is obtained at two boundaries. However, a important liability of this theoretical account lies in the fact that if w hits any of the boundaries ( w = 0 or w = D ) the province of the device can non be farther adjusted. This will be from now on termed as the terminal province job.

Figure 3: Plot of Joglekar window map for p=1, 5 and 10.

Then, Biolek proposed another window map that allows the memristor to come back from the terminal province job.

( 11 )

The reversed prejudice is now should travel back the province variable after it reaches either boundary. This characteristic is described by a current dependent measure map, STP ( I ) , which is a portion of a new window map degree Fahrenheit ( ten ) that behaves otherwise in each electromotive force bias way.

( 12 )

Figure 4: Plot of Biolek window map for p=1, 5 and 10.

Figure 4 displays a graphical representation of the window map described by Biolek for assorted Ps parametric quantity ( p=1, 5 and 10 ) . When ten starts at 0, we noticed that the map equal to 1. As ten addition nearing D, the map approaches 0. Once the current contrary the way, the map instantly exchange to 1. As ten lessening back to 0, the map besides decreases to 0. Biolek window map eliminates convergence issues at the devices boundaries.

The last window map for non-linear theoretical account is proposed by Prodromakis [ 8 ] .

( 13 )

Figure 5 displays a graphical representation of the window map described by Prodromakis for assorted Ps parametric quantity ( p=1, 5 and 10 ) . As we can see, it allows the window map to scale upwards which implies that fmax ( ten ) can take any value within 0 & lt ; fmax ( x ) & lt ; 1. Besides noticed that when p=1, it become indistinguishable to Strukov theoretical account. In add-on, P can take any positive existent figure leting a greater extent of flexibleness. The boundary issues are besides resolved with the window map returning a zero-value at the active bi-layer borders.

Figure 5: Plot of Prodromakis window map for p=1, 5 and 10.

## SPICE Model of Memristor

Figure 6: Stucture of the SPICE theoretical account from [ 6 ] .

In the above circuit in figure 6, VMEM is the input electromotive force and Imem is modeled to be the current through the memristor. The flux is calculated by incorporating the electromotive force VMEM and the charge is calculated by incorporating the current IMEM.

Figure 7: Resistive port of the memristor theoretical account.

As we can see in figure 7, the circuit is really referred to entire resistance RMEM. RMEM ( x ) = ROFF -xa?†R where a?†R= ROFF-RON. ROFF is the resistance in series electromotive force beginning whose terminal electromotive force is controlled by the expression -xa?†R.

Figure 8: Differential equation mold of the memristor.

Figure 8 shows the differential equation mold of the memrsitor. It consist a portion of the electromotive force controlled beginning xa?†R and the differential equation from equation ( 6 ) which serves as an planimeter of the measures on the right side of the province equation ( 6 ) which is to acquire the value of normalize x. EMEM is the electromotive force beginning whose terminal electromotive force is controlled harmonizing to the expression -xa?†R. GX is a current beginning whose current is controlled harmonizing to the equation IMEMf ( V ( x ) ) where V ( x ) is the electromotive force across the capacitance Cx and it theoretical accounts the normalized breadth ten of the doped bed. F ( V ( x ) ) is the window map, K is AµvRON/D2 and x0 is the initial electromotive force of the capacitance. [ 6 ] .

The relation between memristor current and electromotive force is modeled as on the footing of RMEM ( x ) = ROFF -xa?†R where a?†R= ROFF-RON. The electromotive force V ( x ) across the capacitance CX theoretical accounts the normalized breadth ten of the doped bed. The initial province of ten is modeled by the initial electromotive force of the capacitance. The flux is calculated by the time-integral of electromotive force, and the charge is calculated by the time-integral of current.

## Result and Discussion

All theoretical accounts were simulated in LTSpice utilizing SPICE theoretical account that was given in [ 6 ] , we add new nonlinear window maps that was proposed by prodromakis and strukov to the theoretical account and compare all suggested window maps.

## memristor

Figure 9: Memristor circuit.

Figure 9 shows constellation of individual memristor for mensurating the behaviour of memristor theoretical account in LTSPICE with a sine moving ridge input electromotive force of 1.2V with 1Hz frequence. The values for the memristor parametric quantities Aµv, D, RON, ROFF and RINITIAL are 10-10cm2s-1V-1, 10 nanometer, 100ohm, 16kohm and 11Kohm. All theoretical account are utilizing same window map parametric quantity p=10.

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Figure 10: Strukov memristor theoretical account – electromotive force, IMEM, RMEM and normalized ten.

Figure 10 shows the simulation consequence of memristor SPICE theoretical account for Strukov window map of electromotive force, IMEM, RMEM and normalized ten. As we can see, the current of the memristor, IMEM is changing up to about 100AµA for upper limit of 1.2V electromotive force applied. The RMEM for this theoretical account show that the values are in scope of 11kOhm boulder clay 12kohm which means the consequence of the electromotive force applied to the memristor merely give somewhat alterations on the value of the memristor. Noticed that when positive electromotive force is applied, the conduction of the device increases therefore the memristance is lessening. When negative electromotive force is applied, the electric resistance of the device addition therefore the memristance is besides addition. This verifies the memristive system on the device. In normalize x graph, we besides noticed that the normalized ten is at higher province in the beginning. Figure 11 shows the I-V feature of the devices and the relationship between charge and flux. The charge and flux curve curves confirms the good known fact that there is a one-to-one correspondence between them in malice of the 1-4 hysteresis consequence. Strukov memristor shows deficiency of flexibleness of commanding the device.

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Figure 11: Strukov memristor theoretical account – I-V loop hysteresis and relationship of charge and flux.

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Figure 12: Joglekar memristor theoretical account – electromotive force, IMEM, RMEM and normalized ten.

Joglekar window map seems to be assuring as the being of commanding parametric quantity. Figure 12 shows the simulation consequence of memristor SPICE theoretical account for Joglekar window map of electromotive force, IMEM, RMEM and normalized ten. Same as strukov memristor, when positive electromotive force is applied, the conduction of the device increases therefore the memristance is lessening. When negative electromotive force is applied, the electric resistance of the device addition therefore the memristance is besides addition. The current of the memristor, IMEM is changing up to about 300AµA for upper limit of 1.2V electromotive force applied. Joglekar window map give higher current compared to others. It shows that the current in the memristor are much easier to travel. The RMEM are within scope of about 0ohm to 11kohm which give full scope of value for the memristor. Figure 13 show the I-V hysteresis cringle of the devices and the relationship between charge and flux. The exchanging behaviour is much more sensitive on the electromotive force degree than Strukov window map. But, in term of stableness, Joglekar window map can non execute for an arbitrary length of clip. This failure is caused by the convergence issue where when the memristor range w=0 or w=D, the province of the device can non be farther adjusted.

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Figure 13: Joglekar memristor theoretical account – I-V loop hysteresis and relationship of charge and flux.

Biolek window maps are supposed to work out terminal province job as in literature [ 4 ] . It should work out the boundry job of the terminal province. Figure 14 and 15 shows the simulation consequence of memristor SPICE theoretical account for Biolek window map. The current of the memristor, IMEM is changing up to about 220AµA for upper limit of 1.2V electromotive force applied. The RMEM are within scope of about 1kohm to 11kohm. Figure 15 show the I-V hysteresis cringle of the devices and the relationship between charge and flux. From the figures, we observe that the biolek memristor preserve the extremely non-linear device characteristic behaviour. In add-on, Biolek ‘s theoretical account allows for general asymmetric I-V device behaviour mold.

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Figure 14: Biolek memristor theoretical account – electromotive force, IMEM, RMEM and normalized ten.

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Figure 15: Biolek memristor theoretical account – I-V loop hysteresis and relationship of charge and flux.

Prodromakis window maps are besides said to work out the boundry issue. Figure 16 shows the simulation consequence of memristor SPICE theoretical account for Joglekar window map of electromotive force, IMEM, RMEM and normalized ten. The current of the memristor, IMEM is changing up to about about 180AµA for upper limit of 1.2V electromotive force applied. The RMEM are within scope of about 3kohm to 11kohm.

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Figure 16: Prodromakis memristor theoretical account – electromotive force, IMEM, RMEM and normalized ten.

Figure 17 show the I-V hysteresis cringle of the devices and the relationship between charge and flux. The hysteresis cringle is shown to be asymmetrical while the OFF province of the device is extremely non-linear compared with other.

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Figure 17: Prodromakis memristor theoretical account – I-V loop hysteresis and relationship of charge and flux.

In comparing of I-V characteristic hysteresis cringle, as we can see in figure 18, it shows all hysteresis cringles for all proposed window maps. By utilizing same parametric quantity we can see the difference in each theoretical account. Joglekar window map seems to hold a strong memristance compared to others.All theoretical accounts seem to be a good estimate of the measuring of the existent memristor green goodss by HP Labs. But, Prodromakis memristor theoretical account satisfies all the requirements and improves on the defects of bing theoretical accounts.

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Figure 18: I-V Hysteresis Loop for all theoretical accounts.

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Figure 19: Prodromakis I-V Hysteresis Loop when p=1, p=5 and p=10.

Then, we try alter the parametric quantity of P of the theoretical account. In this instance, we use prodromakis memristor theoretical account and alter the value of whole number p=1, p=5 and p=10 to see the difference. As we can see in figure 19, as the value of P is addition, the hysteresis is shriveling. Similar with altering the parametric quantity P, figure 20 besides confirm that the hysteresis psychiatrists at higher frequences.

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Figure 20: Prodromakis I-V Hysteresis Loop when frequence f=1Hz, f=2Hz and f=5Hz.

In term of power dissipation, as we simulates our consequences. We can acquire the value of maximal IMEM for each theoretical account. We can cipher the power by utilizing P=IV equation. Table 2 show the maximal power dissipation for each memristor.

Memristor

theoretical account

Max IMEM, AµA

Power, W

Strukov

100AµA

120AµW

Joglekar

300 AµA

360AµW

Biolek

220AµA

264AµW

Prodromakis

180 AµA

216AµW

Table 2: IMEM and Power dissipation for all at the memristor.

As in table 2, we can see that the Strukov theoretical account spring lowest power which is 120AµW while Joglekar model give much higher power dissipation which is about 360AµW compared to the others. We besides noticed that as the memristor theoretical account is improves, the power become lesser. Prodromakis give rather good power dissipation which is 216AµW as the best Windowss map and theoretical account as boulder clay now.

In implementing memristor with parallel circtuit, we pick a two simple parallel circuit to be tested. Figure 21 shows the SPICE topology of the memristor based planimeter op amplifier with the input electromotive force Vp-p=2.4V from -1.2V to 1.2V and C1=25AµF. Using the memristor theoretical account that we create earlier with assorted types of window maps, we see the difference on the fake consequence on each theoretical account. The values for the memristor parametric quantities are same for all theoretical account with uv, D, RON, ROFF and RINITIAL are 10-10cm2s-1V-1, 10nm, 100ohm, 16kohm and 11Kohm.

memristor planimeter op A

Figure 21: Memristor Implemented Integrator Circuit

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Figure 22: Positive Integrator simulation

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Figure 23: Negative Integrator simulation

Figure 22 and 23 shows the simulation consequence for the enforced memristor planimeter op amplifier for positive input and negative input severally. In this instance, we implement prodromakis memristor to the planimeter circuit. As we know, the planimeter acts like a storage component that produces aA voltageA end product which is relative to the built-in of its input electromotive force with regard to clip. The magnitude of the end product signal is determined by the length of clip a electromotive force is present at its input as theA currentA through theA feedback loopA charges or discharges theA capacitorA as the requiredA negative feedbackA occurs through the capacitance. For positive starting input, we vary the electromotive force from 1.2V down to -1.2V and traveling back to 1.2V over clip. When positive electromotive force are applied in the beginning, the end product electromotive force tend to dispatch and drop from 0V to negative electromotive force and bear downing back to 0V when the input electromotive force are bead to negative electromotive force. The end product for negative electromotive force applied from get downing point give a frailty versa consequence. The charging and discharge are depends the electromotive force applied over clip and the value of the capacitance. We can state the memristor theoretical accounts give rather good consequence for an planimeter.

Figure 24: Memristor Implemented Differentiator Circuit.

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esultdifferentiatordifferentiator.png

Figure 25: Saw tooth input of Memristor implemented differentiator op-amp simulation.

Figure 25 shows the simulation consequence for the enforced memristor discriminator op amplifier. We are utilizing prodromakis memristor theoretical account for the memristor execution. As we know, for discriminator op-amp, the magnitude of its end product is determined by the rate at which the electromotive force is applied to its input alterations. The faster the input electromotive force alterations, the greater the end product electromotive force becomes. If a proverb tooth input signal is applied to the input of the discriminator op-amp a square wave signal will be produced. As we can see in figure 25, the simulation shows a rather good consequence for discriminator. We besides noticed some spikes at the end product electromotive force. Each spikes occurs merely occurs the brief minute the proverb tooth is altering from one degree to the following. The electromotive force spikes represent a impermanent end product electromotive force.

## Decision

As a decision to this research is that it could convey a new visible radiation of familiarisation in the integrating of memristive constituents in any sorts of electronic devices that are at nanoscale. It is utile to hold a computing machine theoretical account of the memristor as a tool for the analysis of the behaviour of the circuits in developing application of this memristor as inactive circuit component via simulation. SPICE theoretical account will decidedly assist us to carry on interesting simulation experiments and can be of great importance for such a research in future while the memristor are still difficult to manufacture to analyze the behaviour of the circuit. Different theoretical accounts with strong behaviour and ground give a batch of benefits in development intent to make the possibilities of the execution in an incorporate circuit. The possibilities for execution of the memristor with parallel circuit are broad unfastened.

## Appendix

.SUBCKT memristor plus minus PARAMS:

+ Ron=100 Roff=16K Rinit=11K D=10N uv=10F p=10

## ***********************************************

* DIFFERENTIAL EQUATION MODELING *

## ***********************************************

Gx 0 x value= { I ( Emem ) *uv*Ron/D**2*f ( V ( x ) , P ) }

Cx x 0 1 IC= { ( Roff-Rinit ) / ( Roff-Ron ) }

Raux x 0 1T

* RESISTIVE PORT OF THE MEMRISTOR *

## ***********************************************

Emem plus aux value= { -I ( Emem ) *V ( x ) * ( Roff-Ron ) }

Roff aux minus { Roff }

## ***********************************************

*Flux computation*

## ***********************************************

Eflux flux 0 value= { SDT ( V ( plus, minus ) ) }

## ***********************************************

*Charge computation*

## ***********************************************

Echarge charge 0 value= { SDT ( I ( Emem ) ) }

## ***********************************************

* WINDOW FUNCTIONS

* FOR NONLINEAR DRIFT MODELING *

## ***********************************************

*proposed by joglekar

; .func degree Fahrenheit ( x, p ) = { 1- ( 2*x-1 ) ** ( 2*p ) }

*proposed by biolek

; .func degree Fahrenheit ( x, one ) = { 1- ( x-stp ( -i ) ) ** ( 2*p ) }

*proposed by prodromakis

; .func degree Fahrenheit ( x, p ) = { 1- ( ( ( x-0.5 ) **2 ) +0.75 ) **p }

*proposed by strukov

.func degree Fahrenheit ( x, p ) = { x-x*2 }

.ENDS memristor

## Recognition

This paper participates in the IEEE Student Conference Research & A ; Development SCORED 2012. The writer would wish to thank Dr. Wan Fazlida Hanim bte Abdullah for being supervisor in this concluding twelvemonth undertaking. The writer besides would wish to thank Universiti Teknologi MARA for funding the research work through the Excellence Fund Grant 600-RMI/ST/DANA 5/3/RIF ( 360/2012 ) .