Thesis Report: “Exploring Memristor Topologies”

Author: Serdar Benderli Adviser: Todd A. Wey School: Lafayette College

Page 2 of 59

Abstract

Memristors may play a crucial role in next generation computational paradigms. By

having access to such a device, circuits with new functions and new architectures of

classic functions can be designed. The purpose of this thesis is to explore a number of

novel topologies that make use of memristive behavior. However, due to the current

unavailability of off-the-shelf memristors and lack of software simulation support for

such devices, creating a behavioral model of a memristor is necessary. The first half of

this thesis deals with the creation of a behavioral model. The second part uses this

model to simulate and analyze three different memristor topologies, demonstrating

potential of memristors and the robustness of the model. By having a behavioral model

of these devices, researchers and designers can simulate circuits which involve

memristors easily and efficiently, accelerating memristor research.

Page 3 of 59

Table of Contents

Abstract ......................................................................................................................................................2

Acknowledgements .................................................................................................................................4

I. Introduction to Memristors..........................................................................................................5

II. Thin-Film TiO2 Memristors ...........................................................................................................8

III. Potential Applications of Memristors .....................................................................................10

IV. Current Problems in Memristor Research.............................................................................11

V. Behavioral Modeling of Thin-Film TiO2 Memristors ...........................................................13

VI. Topologies .....................................................................................................................................23

1. Memristor Switch ................................................................................................................24

2. Memristor AM Modulator ...................................................................................................32

3. Q-Factor Controller for 2ND Order Band-Pass Filter ......................................................49

VII. Conclusions...................................................................................................................................57

Bibliography ............................................................................................................................................59

Page 4 of 59

Acknowledgements

I would like to acknowledge and extend my heartfelt gratitude to the following persons who have helped me complete this thesis:

My adviser, Prof. Todd A. Wey, for all his hard work, encouragement, and support throughout the year. He was always there to help during the countless problems I ran into. He has endured many hours trying to fix my silly syntax errors and he never yelled. His enthusiasm has only gone up throughout the year, and this has always kept me wanting to deliver more than I otherwise would have. Without him, this thesis would not be possible.

Prof. William D. Jemison, for all his important input throughout the year, for accepting to be a part of my thesis committee, and for his comments and corrections on the final draft of this thesis.

Prof. Jeffrey O. Pfaffmann, for accepting to be a part of my thesis committee and for his valuable input on the final draft of this thesis

All Electrical and Computer Engineering faculty members and Staff

…and to my family and Raluca for their constant support and encouragement.

Page 5 of 59

I. Introduction to Memristors

The term ‘Memristor’ was first coined by Prof. Leon Chua in a 1971 paper [1] as a short

for ‘memory resistor’. In traditional circuit theory, the resistor provides the relationship

between current and voltage, the capacitor provides the relationship between voltage

and charge, the inductor provides the relationship between current and flux, dtdqi = , and

dtdv φ

= , but no relationship exists between flux and charge. In order to have a

conceptual symmetry of relationships between four fundamental electrical properties,

Chua postulated that a fourth circuit element should exist which would provide this

missing relationship.

Figure 1 - Fundamental Electrical Properties and Defining Relationships

Image Courtesy of: J. J. Yang/HP Labs

Page 6 of 59

Her argued that if flux is a function of charge, )(qF=φ , then when both sides of this

equation are differentiated, we get:

( ) ( )d q F q dqdt dt dtφ

= (1)

where ( )F qdt

can be re-written as a function of q(t), M(q(t)). Chua defined this as the

memristance. If we replace the derivative of flux with voltage, and the derivative of

charge with current, we get a more familiar equation:

( ) )()()( titqMtv = (2)

As can be seen, memristance has the same units as resistance, ‘ohms’. It is important

to note that the memristor behaves like a resistor at any given time t. However, its

resistance depends on the history of the current that has passed through it.

The basic function )(qF=φ can be re-written in the inverse form, 1( )q F ϕ−= given that it

is invertible. Then, by taking the same steps as before, we can arrive at the equation:

( ) ( ( )) ( )i t W q t v t= , where ( ( ))W q t can be defined as the memductance.

After laying out the basics, Chua moves on to talk about active-circuits he designed by

using transistors and other elements, to behave like a memristor. However, no one was

able to find a physical manifestation of a memristor to serve as a passive device for

over 30 years, until HP Labs did in 2008. They reported their findings in a paper

published in Nature [2]. After HP Labs’ findings, there have been numerous other

reports of devices that show memristor-like behaviors. Since it is out of the scope of this

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paper, they will not be talked about in detail. Please refer to the bibliography for a list of

such devices [3, 4, and 5]. The author has chosen to work with TiO2 memristors

because they are better documented than the other reported realizations are and they

show potential of being available commercially in the near future.

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II. Thin-Film TiO2 Memristors

While working on thin-film TiO2, HP Labs researchers noticed hysteresis in the i-v

curves [2]. This implied that the resistance of the thin-film device depended on the

history of current that had passed through it, essentially resulting in memory.

Figure 2 – An Array of 17 TiO2 Memristors (Left) and 2-dimensional representation of a thin-film TiO2 device (Right)

Images Courtesy of: HP Labs

The memristive behavior of thin-film TiO2 is a direct manifestation of the shift in the

boundary between oxygen-rich (undoped) and oxygen-depleted (doped) regions when

an electrical field is applied [2]. The depleted region acts as if it was doped with +2

charge carriers and thus has a much lower resistance than the undoped region. When

an electrical field is applied, the carriers drift with mobility (µv), changing the widths (w)

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of the low-resistance (RON) and high-resistance (ROFF) layers, effectively changing the

material’s overall resistance. Moreover, this drift is bounded by the length of the material

(D).

The equations governing the width of and the voltage across a TiO2 memristor are as

presented in [2]:

)()( tiD

Rdt

tdw ONvμ= (3)

)()( tqD

Rtw ON

vμ= (4)

)())(1()()( tiDtwR

DtwRtv mOFFONm ×⎟

⎠⎞

⎜⎝⎛ −+= (5)

As can be seen, the memristance of the device is simply the weighted average of the

two regions’ resistances, where the weights are the proportion of the region widths to

the overall device length D. A very important property of these devices can be seen

from (3): when there is no current passing through, dw(t) is 0. This means that the

device retains its memory even when the power is turned off.

It is also important to note that this device does not provide a direct link between flux

and charge. It is simply changing its resistance as a reaction to the current passing

through it.1

1 This was the source of some criticism to the HP findings, noting that it may not be correct to call such a device

‘memristor’ since it is not exactly the same as Chua’s memristor [3, 4].

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III. Potential Applications of Memristors

The first commercial application for memristors will most likely be in the digital domain

where they are used as switches. This is also the first topology that will be considered in

this thesis. HP Labs predicts that such devices could reach the markets within the next

five years. They already reported building a hybrid chip that contains both transistors

and memristors using current manufacturing technology [6].

However, it is likely that memristors will offer their most important advantages in the

analog domain when used as non-volatile analog memory units. For the first time,

researchers and designers would have simple access to non-volatile analog memory

which opens up numerous possibilities for new computational paradigms in the fields of

signal processing, control systems, neural modeling, and neural networks. Very simple

learning circuits that involve a simple RLC circuit and a memristor that model the

learning behavior in amoeba are already being studied [7]. Especially in the emerging

field of neuromorphic engineering, memristors may be used to create simpler and more

efficient neuron models than the currently used transistor architectures [8].

More applications of memristors are likely to emerge as more people learn to design

with them. However, there are still a number of other problems that impede memristor

research.

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IV. Current Problems in Memristor Research

The biggest problem in memristor research is that there are no off-the-shelf memristor

parts to build physical circuits. This problem will likely stay unresolved for at least a few

more years. Until then, researchers and designers who do not have access to nano-

scale manufacturing facilities must rely on a theoretical approach. There are two ways

one can approach theoretical memristor research: in a purely mathematical way where

memristors equations are used along with Kirchhoff’s current and voltage laws to solve

the circuits, or by using circuit simulators such as SPICE. The mathematical method is

not widely used because even though such models may be sufficient for simple circuits,

they cannot be used to solve complex circuits easily and efficiently. Thus, for complex

solvers, solvers such as SPICE have become an industry standard in designing,

simulating, and verifying circuits.

Most circuit solvers have built-in mathematical models for basic circuit elements

(primitives) such as resistors, diodes, and transistors. Furthermore, behavioral models

for more complex elements, such as op-amps, can be created by using these primitives.

Behavioral models are desirable since they capture the behavioral essence of devices

rather than providing an exact mathematical description. Once a behavioral model of a

device is made, it can be used in any circuit topology and thus makes simulation of

different circuits very easy and efficient.

However, no such model for memristors existed at the time the author started his

research. Thus, it was necessary to build a behavioral model for memristors in order to

study potentially useful memristor topologies. The model should be of great relevance to

anyone who decides to study memristors using circuit simulators because it enables

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them to easily and efficiently design, simulate, and verify any circuit design that involves

memristors without having to construct and solve complicated mathematical equations.

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V. Behavioral Modeling of Thin-Film TiO2 Memristors2

The first thing to note about thin-film TiO2 is that equations (3), (4) and (5) are only valid

when w is within the boundaries of the device. Once w reaches either end of the device,

it can move no further, and the overall memristance of the device must remain constant

(clipped) until current in the opposite direction is applied. This dynamic is difficult to

model behaviorally. Thus, the modeling process should be broken into two stages for

simplicity. First, an unconstrained model is built where boundary conditions are not

imposed, and then it is expanded to include the boundary conditions.

The unconstrained model is based on the following memristor relationships:

)()( tiD

Rdt

tdw ONvμ= (3)

)()( tqD

Rtw ON

vμ= (4)

)())(1()()( tiDtwR

DtwRtv mOFFONm ×⎟

⎠⎞

⎜⎝⎛ −+= (5)

2 SPICE language was used to create the behavioral model in this paper. However, other than a few

language-specific rules, the behavioral basics of the model are language-independent and can be

implemented in any circuit simulator.

Page 14 of 59

ROFF

gminCWIw

w(t)

+

v(t)

-

F(w,i(t))

i(t)1

2

Figure 3 shows the proposed 2-terminal model for unconstrained ‘w’, that incorporates

these relationships3:

Figure 3 – Unconstrained Model

3 The values for the parameters µv, D, RON, and ROFF were taken directly from [2] and are 10-10 cm2s-1V-1,

10nm, 100Ω, and 16KΩ respectively and are the same in all circuits and figures throughout the paper.

IW multiplies i(t) by D

RONvμ .

This forms dt

tdw )(

A 1F cap integrates Iw, resulting in w(t) as voltage across CW.

F(w,i(t)) multiplies i(t) by D

RRtw OFFON −)( .

Voltage across F and ROFF gives voltage across memristor.

Page 15 of 59

From (3), dtdw is formed by multiplying the current i(t) by

DRON

vμ , which is the gain of the

current-controlled-current-source Iw. To get (4), both sides of (3) need to be integrated.

This is achieved by a 1F capacitor that integrates a current value of dtdw to give a voltage

of w(t) across4. This capacitance voltage value is then passed to a voltage-controlled-

voltage-source F(w,i(t)) where it is multiplied with the current i(t) and a partial

memristance to get (5). This partial memristance does not include the ROFF term in order

to satisfy SPICE voltage loop criteria when the memristor is driven by a voltage source.

The partial memristance then is given by D

RR OFFON − and the overall voltage across the

memristor is simply the sum of the voltage across ROFF and the voltage across the

dependent source. This completes the unconstrained circuit model and it fully describes

a TiO2 memristor when w(t) is not constrained.

In order to impose the boundary conditions, w(t) must be clipped when it tries to go over

D or below 0. Figure 4 shows the proposed clipping circuit that is used to expand the

unconstrained model described above:

4 A large resistor gmin is placed in parallel with the capacitor in order to satisfy SPICE convergence criteria

Page 16 of 59

Figure 4 - Proposed Clipping Circuit

Four comparators are used in order to ensure that w(t) does not go beyond its limits.

These comparators are ideal and are created by using voltage controlled voltage

sources. Two pairs of comparators clip w(t) at its top and bottom boundaries. When the

external bias is positive and w(t) is equal to the upper boundary, a switch is closed that

connects the CW of the unconstrained circuit to a voltage source equal to D, thereby

effectively clipping w(t) at D. Conversely, when the voltage source is negative and w(t)

is equal to 0, a switch is closed that connects the CW to ground, thereby effectively

clipping w(t) at 0. The clips are always released when the current bias polarity changes

in order to allow w(t) to move away from the boundaries.

Page 17 of 59

The following results can be compared to those reported by HP Labs in [2] to verify that

the circuit models the behavior of the reported thin-film TiO2 memristors correctly. Figure

5 shows voltage and current vs. time and the hysteretic characteristic of the i-v curve

when w(t) is within boundaries. Another important characteristic of these devices can be

seen when the frequency of the input bias is increased. Due to the low mobility of the

positive carriers, the change in w(t) is slow and cannot keep up with increasing bias

frequencies. When the frequency is high enough, the change in w(t) will be

unobservable, essentially biasing the memristor at the current memristance and

practically turning it into a regular resistor. This is reported in [2] with the collapse of the

Lissajous i-v curve, which can also be seen in Figure 5 for a tenfold increase in the

frequency of the input bias from f0 to 10f0.

vm(t)

im(t)

w(t)/D

i-v at f0

i-v at 10f0

Figure 5 - Voltage & Current vs. Time and i-v characteristics for w within boundaries

Page 18 of 59

Figure 6 shows the results in the case where w(t) is clipped:

Figure 6 - Voltage & w(t)/D vs. Time and i-v characteristics when w is clipped

As seen in Figure 6, w(t) is clipped as soon as it tries to go beyond 0 or D and is kept

there until the polarity of the input bias voltage is reversed.

Further nonlinear effects that are observed in the actual devices, such as non-linear drift

mentioned in [2], can also be added to the model. We modeled this nonlinear drift in two

different ways. The first is to replicate a window function such as the one mentioned in

[2] via additional polynomial dependent sources. The second is to increase the

capacitance Cw near the boundaries via additional switches and capacitors thereby

modeling the nonlinear behavior at the boundaries as a piecewise nonlinear

capacitance.

Page 19 of 59

Figure 7 shows the extra capacitance circuit:

Figure 7 ‐ Non‐linear Drift via Added Capacitance

When w(t) is above a chosen value wHIGH and below the a chosen value wLOW,

capacitances C1 and C2 are added in parallel with CW. When w(t) is within these bounds,

the caps are kept at the voltages wHIGH and wLOW for preserve continuity when extra

capacitances are added.

Page 20 of 59

The i-v characteristics in both cases are qualitatively similar. However, the windowing

method is preferable for cases where a simple windowing function can be used to

model the nonlinearity. For TiO2 memristors, a simple window function can approximate

the observed non-linearities as mentioned in [2]. We chose the following window

function that is comparable to the one presented in [2]:

( )2

)()(4D

twDtw − (6)

Imposing this windowing on (3), we get a new equation that governs change in w:

( ) )()()(4)(2 ti

DR

DtwDtw

dttdw ON

vμ−

= (7)

The factor of four is added in order to scale the maximum value of mobility, µV, which

occurs at w=0.5D, to its maximum value when the windowing is not included. This can

be seen in Figure 8 where the dashed line is the mobility without the factor of four in the

window function and the continuous line is the mobility with the factor of four:

Page 21 of 59

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

x 10-8

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

w (nm)

mob

ility

(uv)

Figure 8 ‐ Windowing Rescaling

Figure 9-10 shows the resulting i-v characteristics of both the approaches:

Figure 9 - Voltage & w(t)/D vs. Time and i-v characteristics when a windowing function is used

Page 22 of 59

Figure 10 ‐ Voltage & w(t)/D vs. Time and i‐v characteristics when extra capacitors used

From these results, we can conclude that the model shows excellent agreement with

the results presented in [2]. The robustness of this model will be further reinforced in the

following section as the memristor will be used in three different topologies.

Furthermore, the work in this section provides a behavioral modeling framework for

similar memristive devices.

Page 23 of 59

VI. Topologies

Three topologies will be considered in this section.

The first topology is the memristor switch, where the memristor is used as an on/off

switch. This topology is mainly useful when the memristor is used in a NVRAM

application. One of the key drawbacks of such NVRAM devices would be the slow

switching speed of the memristors due to the low speed of the carriers in the TiO2

memristors. This problem will be analyzed further and comparisons will be made to

current transistor switch technology to see the extent of this drawback.

The second topology is the Memristor AM Modulator. In this topology, the memristor is

used in a simple inverting op-amp configuration. A low frequency signal modulates a

carrier signal by changing the memristance, thereby changing the overall gain of the

opamp. This effectively modulates the carrier signal. Further analysis will be made in

this section to evaluate the performance of this topology.

The third topology is the Q-factor Controller for a 2ND order filter. In this topology, the

memristor’s initial value of ‘w’ is used to set the Q-factor of a 2ND order Band-Pass filter.

Page 24 of 59

1. Memristor Switch

In this topology, the memristor is used alone with an independent current source as

shown in Figure 11:

Figure 11 – Memristor Switching Circuit

The current source is used to vary the memristance between its maximum and the

minimum values as fast as possible thereby switching the memristor on and off.

Since this speed depends on the size of the current, it is constrained by the material

properties of thin-film TiO2. Unfortunately, no such data on the material’s current limits is

currently available. Thus, an exact limit on the speed is not provided in this section.

However, an analysis of the size of the current for a desired speed is provided.

Due to the effects of windowing near the boundaries, this topology should be analyzed

separately for the windowed and the non-windowed memristors. We start the analysis

with the non-windowed case since it is simpler. Suppose we wanted the memristor to

switch in 10usec. From )()( tiD

Rdt

tdw ONvμ= , we can see that for a dw(t) of D, RON=100Ω

and D=10nm, we would need a current of i(t)=10A, to get a dt, which we can call the

Page 25 of 59

switching time ts to be 10usec. ts can also be written in terms of initial w, w0, and final w,

wf, for constant current IM:

MON

V

fsM

ONV

ID

Rww

ttID

Rwtwμ

μ 00)(

−=→+=

It can be seen that for the values we chose for w0=0, and wf=D, this equation also

results in ts=10usec.

A more interesting result can be seen in the power consumption which can be found by

the formula P=IM2ROFF. For a constant IM=10A and ROFF=16KΩ.we find that the peak

power consumption by the device is 1.6MW,. This is indeed a huge amount, and it is

proportional to the square of the current. We can integrate the P=IM2M(t) to get the total

energy required for a switching with constant current:

0

2 2

0 0

2

0

0 020

0

( )( ) ( )(1 ) ( )

( ) ( )

( )2

ts tsON OFF

s M ON OFF M OFF

tsON OFF V ON

M OFF M

tsf fON OFF s V ON

M OFF Ms s

R Rw t w tE I R R dt I R w t dtD D D

R R RI R w I t dtD D

w w w wR R t RI R t w t ID t D t

μ

μ

−⎛ ⎞⎛ ⎞= + − =⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

−⎛ ⎞= +⎜ ⎟⎝ ⎠

− −⎡ ⎤−⎛ ⎞= + + ⇐ =⎢ ⎥⎜ ⎟⎝ ⎠⎣ ⎦

∫ ∫

∫

0 0

2f fON OFF

s M OFFV ON

w w w wR RE I R DD Rμ

+ −⎛ ⎞⎛ ⎞−= + ⎜ ⎟⎜ ⎟

⎝ ⎠⎝ ⎠ (8)

For the chosen values of wf=D and w0=0, this equation can be reduced to a simpler

form:

2 1 12

OFFs M

V ON

RDE IRμ

⎡ ⎤= +⎢ ⎥

⎣ ⎦ (9)

Page 26 of 59

For the chosen values of ROFF and RON this equation gives us 8.05J as the energy

required for a single switching of the memristor.

We can manipulate (9) to write the energy in terms of ts rather than IM:

22 2 21 1 1 11 ( ) 12 2

OFF OFFs s s s

V ON V ON V ON ON s

R RD D DE t E tR R R R tμ μ μ

⎛ ⎞ ⎛ ⎞ ⎛ ⎞= + ⇒ = +⎜ ⎟ ⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠ ⎝ ⎠

This equation can be used to plot ES against ts to show graphically the energy required

for a desired switching time:

10-9

10-8

10-7

10-6

10-510

0

101

102

103

104

105

X: 1e-005Y: 8.05

ts (sec)

Ener

gy (J

oule

s)

Figure 12 ‐ Energy vs. Switching Time for Non‐windowed Memristor

Page 27 of 59

We can also find the average power required for a switching the memristor in ts

seconds. PAVG can be defined as PAVG=Es/ts. For ts=10usec, PAVG= 805KW.

Figure 13 shows the SPICE simulation with a 10A current passing through a non-

windowed memristor for 10usec:

Figure 13 – SPICE Simulation Showing a Non‐Windowed Memristor Switching

The memristor in this case switches in 10usec as predicted. At this rate then, the

memristor can be switched with a frequency of 100KHz. For a current of 100A, the

memristor switches in 10usec, 1000KHz. When the power curve from Figure 13 is put

into MATLAB and integrated analytically, we get the predicted energy requirement,

8.049J.

TIME

0s 0.5ms 1.0ms0A

5A

10A

0Ω

10K Ω

20K Ω 0W

1.0MW

2.0MW

CURREN

T M

EMRISTA

NCE

POWER

Page 28 of 59

The windowed case is slightly more complicated to analyze mathematically due to the

resulting integrals. Rearranging (7), we get the following for constant current IM:

( ) )()()(4)(2 ti

DR

DtwDtw

dttdw ON

vμ−

=

( ) dtID

R

DtwDtw

tdwM

ONvμ=−

2

)()(4

)(

Rearranging and integrating both sides gives us:

tID

RKDw

wDM

ONvμ+=⎟⎟

⎠

⎞⎜⎜⎝

⎛⎟⎠⎞

⎜⎝⎛

+− 0ln4

where K0 is an integration constant resulting from the initial value of w0.

Rearranging this, we get:

⎟⎠

⎞⎜⎝

⎛ +

+−=tI

DR

KD M

ONV

eDwtwμ0

4

)()(

Finally, we can write:

AteKDKtw+

= '0

'0)( (10)

where MON

V IDRA 24μ−= and '

0K is another constant resulting from the exponentiation. By

setting t=0, we can find '0K in terms of w(0):

0

0'0'

0

'0

0 1)0(

wDwK

KDKww

−=⇒

+==

Page 29 of 59

We can then use (10) to find tsw:

Aw

wDK

teK

DKw f

f

swAtf

⎟⎟⎠

⎞⎜⎜⎝

⎛ −

=⇒+

=

'0

'0

'0

ln

It is important to note that as we choose wf to be closer to D, tsw goes to infinity. This

makes sense since if w is at the boundary, then dw/dt is 0, regardless of the current

passing through. We can’t choose wf to be equal to D, but we can choose it to be really

close. For wf=0.999D, the above equation gives us tsw=34.53usec. We can integrate the

P=IM2M(t) to get the total energy required for a switching for constant current:

From (10), we can also find the energy Esw as a function of tsw:

' '2 2 0 0

' '0 0

( ) ( )(1 ) (1 )sw M ON OFF M ON OFFAt At

K Kw t w tE I R R dt I R R dtD D K e K e

⎛ ⎞⎛ ⎞= + − = + −⎜ ⎟⎜ ⎟ + +⎝ ⎠ ⎝ ⎠∫ ∫

( ) ( ) ( )2 '0

1 ln Atsw M ON OFF OFFE I R R t K e R t

A⎡ ⎤⎛ ⎞= − − + +⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦

(11)

For the chosen values of ROFF and RON this equation gives us 27.8J as the energy

required for a single switching of the memristor.

We can evaluate the integral in (11) from 0 to tsw and rearrange it to write the energy in

terms of tsw rather than IM:

Page 30 of 59

( ) ( ) ( ) ( )

'0

2

2

2 ' '0 0

ln4

4

1 1( ) ln ln 1

f

f ONM v M

ONv sw

Atswsw sw M ON OFF sw OFF sw

D wK

w RI A IR DtD

E t I R R t K e K R tA A

μμ

⎛ ⎞−⎜ ⎟⎜ ⎟⎝ ⎠= ⇔ = −−

⎡ ⎤⎛ ⎞⎛ ⎞= − − + + + +⎢ ⎥⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠⎣ ⎦

This equation can be used to plot Esw against tsw to show graphically the energy required

for a desired switching time:

10-9

10-8

10-7

10-6

10-5

10-410

0

101

102

103

104

105

106

X: 3.346e-005Y: 28.69

tsw (sec)

Ener

gy (J

oule

s)

Figure 14 – Energy vs. switching time tsw for Windowed Memristor

We can also find the average power required for a switching the memristor in ts

seconds. PAVGw can be defined as PAVGw=Esw/tsw. For tsw=34.53usec, PAVGw= 805KW.

Page 31 of 59

Figure 15 shows a windowed memristor switching on and off consecutively with a

current of 10A with a w0 of 0.001D:

Figure 15 – SPICE Simulation showing a Windowed Memristor Switching

The off value of memristance is 15.984KΩ whereas the on value is 100.8Ω. The

memristor switches on in 34.53usec. We can see the dramatic increase in the amount

of time it takes for the memristor to switch when non-linear drift effects are taken into

account. Peak power consumption is 15.98MW and total energy required for a single

switch is 27.61J. These values are dependent on the value of w0 we choose, the farther

away from the boundaries we pick the initial value, the less energy will be required to

switch it. We can conclude that the model is robust and performs as expected in this

topology. However, these values suggest that such an application would be unfeasible if

indeed the switch is made from a single memristor where its memristance is varied by a

large current source. Perhaps as HP Labs develops their memory chips in the coming

years, more light will be shed on this matter, along with more data for further analysis.

TIME

0s 0.5ms 1.0ms0A

5A

10A

0Ω

10K Ω

20K Ω 0W

1.0MW

2.0MW

CURREN

T M

EMRISTA

NCE

POWER

Page 32 of 59

2. Memristor AM Modulator

In this topology, the memristor is used as the feedback resistor in an inverting gain op-

amp configuration as shown in Figure 16. The purpose of this topology is to modulate a

high-frequency carrier signal vRF, referred to as the carrier, by varying the memristance

which effectively changes the gain of the inverting op-amp. The memristance is varied

by a low-frequency signal vAM, referred to as the signal, which we wish to transmit. The

modulation is achieved by controlling the amplitude of the carrier by changing gain of

the op-amp. As with a typical AM modulator, the carrier is enveloped by the signal and

can be retrieved on the receiver side back by envelope detection. The output of the op-

amp is the combination of the modulated carrier and the signal itself. To retrieve only

the modulated carrier that we wish to transmit to the receiver, a high-pass filter is used.

Due to the inherent integration within the memristor, the signal is passed through a

differentiator in order to keep amplitude proportional over the circuit.

Figure 16 - Proposed Memristor AM Modulator

Page 33 of 59

For this application to work, it is important that the signal be a sufficiently low frequency

and the carrier is sufficiently high frequency. The reason for this constraint lies in the

slow frequency response of the w of the TiO2 memristor due to low carrier speeds. As

mentioned before, the memristor acts like a resistor when the frequency of the applied

signal is high enough because the w(t) cannot keep up. Thus, the effect of the carrier on

the memristance is negligible given that the low frequency signal is of sufficiently high

amplitude and sufficiently low frequency so that it can change the memristance. The

modulation depth depends on how much the gain of the op-amp varies overall.

It is insightful to start with a mathematical derivation of what the output should be so that

we can compare and justify the behavior of the model. The current through the

memristor can be written as:

( ) ( )1

( )1 AMRF

d v ti t v tR dt

⎡ ⎤= ⋅ −⎢ ⎥⎣ ⎦ (12)

Plugging (12) into (3) and integrating both sides, we get the following expression for

w(t):

)()( tiD

Rdt

tdw ONvμ=

( )01

( ) (0) ( ) ( )tV ON

RF AMRw t w v t dt v t

R Dυ ⎡ ⎤= + ⋅ −⎢ ⎥⎣ ⎦∫ (13)

Page 34 of 59

Using this expression for w(t), we can re-write (5) as follows:

( )

( ) ( )

1

2 01 1

( )(0) 1( ) ( )

( )1( ) ( ) ( )

AMM OFF ON OFF RF

tV ON AMON OFF RF AM RF

d v twv t R R R v tD R dt

R d v tR R v t dt v t v tR D R dtυ

⎡ ⎤⎡ ⎤= + ⋅ − ⋅ ⋅ − +⎢ ⎥⎢ ⎥⎣ ⎦ ⎣ ⎦

⎡ ⎤⎡ ⎤− ⋅ − ⋅ ⋅ −⎢ ⎥⎢ ⎥⎣ ⎦ ⎣ ⎦∫ (14)

We also know that vout(t) is:

( )( ) ( )AMopamp M

d v tv t v tdt

= − (15)

Using these equations, we can mathematically solve for w(t) and the op-amp output

voltage. For this analysis, we pick the carrier to be a 1 KHz cosine wave with amplitude

1V, R1 to be 5KΩ, and the signal to be a 1Hz sine wave with amplitude 0.2V.

Figure 17 shows the MATLAB analysis results. The top graph is the w(t) for the

memristor, and the bottom graph is the vout(t) of the op-amp prior to the filtering which is

the combination of the signal and the modulated carrier.

Page 35 of 59

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.5

1x 10

-8

X: 0.7503Y: 9.003e-009

Leng

th (m

)

w(t)

X: 0.2377Y: 1.009e-009

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-10

-5

0

5

10

X: 0.0955Y: 5.78

Am

plitu

de (v

olts

)

vout(t)

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-4

-2

0

2

4

X: 0.2575Y: 2.881

Time (sec)

Am

plitu

de (v

olts

)

vfiltered(t)

X: 0.7505Y: 0.338

Figure 17 ‐ MATLAB Analysis Results

The peak values for w(t) are 9nm and 1nm. If vout(t) is passed through a perfect filter

which eliminates the low-frequency signal component, we get the filtered vfiltered(t) shown

in the bottom graph. The crest peak value of the modulated carrier is 2.88V whereas the

trough is 0.32V. The modulation index can be found by C T

C

v vmv−

= which equals 88% in

this case.

The results of the SPICE simulation for the corresponding circuit are given in Figures

18-20:

Page 36 of 59

Figure 18 – SPICE Simulation of w(t) with Non‐Windowed Memristor

Figure 19 ‐ SPICE Simulation of vout(t) with Non‐Windowed Memristor

Page 37 of 59

Figure 20 ‐ SPICE Simulation of vfiltered(t) with Non‐Windowed Memristor

As can be seen, the memristance is successfully varied at a rate of 1Hz, and the carrier

is successfully modulated. Furthermore, we can see that the effect of the carrier on the

memristance is negligible for the chosen frequencies and amplitudes of the carrier and

the signal. The magnitude of vout(t) is as predicted by the MATLAB analysis. The slight

difference between the predicted maximum peak and the maximum peak of simulated

vfiltered(t) is a side-effect of the non-perfect filtering

Thus, we can conclude that the model behaves according to theory.

Page 38 of 59

The next step is to look at the performance of this circuit. This can be done by

calculating the Total Harmonic Distortion (THD)5 in the signal after it has been

demodulated. Two main factors contribute to THD:

i. Imperfect components (i.e. op-amp, filter, etc.)

ii. Modulation Depth

Since we used a non-windowed memristor in this first analysis, we would expect the

best-case THD to be very close to 0. This is because for a perfectly filtered vout(t), the

envelope of the filtered output is simply w(t), and there is the only distortion in w(t) is

due to the high-frequency signal. Figure 21 shows the FFT of the w(t):

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000-600

-500

-400

-300

-200

-100

0X: 1Y: 0

Frequency (Hz)

Mag

nitu

de R

elat

ive

to F

unda

men

tal F

requ

ency

(dB

)

X: 1000Y: -147.5

Figure 21 ‐ MATLAB Result of FFT on the Envelope

Y‐axis is Magnitudes relative to fundamental frequency in dB

For a modulation index of 88%, the best-case THD is 6.34e-5%, which is very small as

we expected. We can also plot THD over a range of modulation indices:

5 The THD is calculated to be the sum of the squares of the amplitudes of the harmonic frequencies over the square of the amplitude of the fundamental frequency, in this case, the frequency of the signal, 1 Hz.

Page 39 of 59

0 10 20 30 40 50 60 70 80 90 10010-5

10-4

10-3

10-2

10-1

100

101

X: 88.18Y: 6.338e-005

Modulation Index (%)

THD

(%)

To calculate THD on the SPICE simulation, the output of the op-amp in the SPICE

simulation is passed through an imperfect high-pass filter. The 1st order high-pass filter

is a simple RC filter with a cutoff frequency (fc) of 100Hz. This output of the filter is then

sent into MATLAB where it is passed through a perfect envelope detector. The detected

envelope is the received signal that we wanted to transmit. Figures 22-23 shows the

MATLAB output of the envelope detection and the FFT of the envelope. The THD for

this case is 0.018%. We can conclude that the extra distortion is due to the imperfect

filtering.

Figure 22 ‐ MATLAB Result of Envelope Detection

Page 40 of 59

0 50 100 150 200 250 300 350 400 450 500-700

-600

-500

-400

-300

-200

-100

0X: 1Y: 0

X: 2Y: -86.79

Mag

nitu

de R

elat

ive

to F

unda

men

tal F

requ

ency

(dB

)

Frerquency (Hz)

Figure 23 ‐ MATLAB Result of FFT of the Envelope

Y‐axis is Magnitudes relative to fundamental frequency in dB

A valid question is to ask how the non-linear windowing on the memristor affects the

performance. We would expect the THD to be higher in this case due to the non-

linearity we impose on w(t). To answer this question mathematically, we must re-write

equations (13) and (14) to include the windowing. The other equations from the non-

windowed analysis are still the same. In order to re-write (13), we combine (7) and (12)

and integrate both sides:

( )2

( ) ( )( ) 4 ( )w ww ONv

w t D w tdw t R i tdt D D

μ−

= (7)

( ) ( )1

( )1 AMRF

d v ti t v tR dt

⎡ ⎤= ⋅ −⎢ ⎥⎣ ⎦ (12)

Page 41 of 59

⎟⎟⎠

⎞⎜⎜⎝

⎛−+=

+− ∫ )()(ln4 01

0 tvdttvDR

RKDw

wDAM

t

RFON

vw

w μ

The subscript ‘w’ shows that we are solving for the windowed case. We can now

rearrange and solve for ww(t)

21 0

21 0

4 ( ) ( )'0

4 ( ) ( )'0

( )

1

tON

v RF AM

tON

v RF AM

R v t dt v tR D

w R v t dt v tR D

DK ew t

K e

μ

μ

⎛ ⎞⎜ ⎟−⎜ ⎟⎝ ⎠

⎛ ⎞⎜ ⎟−⎜ ⎟⎝ ⎠

∫=

∫+

(16)

where '0K is 0

0

wD w−

, evaluated the same way as in the memristor switch analysis.

We can then plug (16) into (5) to get vMW(t).

Page 42 of 59

Using these equations, we can analyze how the AM Modulator should behave for the

input signals we chose. We need to make sure that the modulation index for this case is

also 88%, so we can have a valid comparison between the windowed and non-

windowed cases. To achieve this, we need to use a trial-and-error approach to pick the

correct amplitude of vAM(t) that will range the ww(t) between 0.9mm and 0.1mm. We

would expect this amplitude to be higher than the 0.2V we used for the non-windowed

case, due to the slowing carrier speeds in the windowed case. An amplitude of 0.275V

gives us the desired variance of ww(t) and the below figures show the analysis of the

AM Modulator for this analysis:

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.5

1x 10

-8

X: 0.7504Y: 9.002e-009

Leng

th (m

)

w(t)

X: 0.2389Y: 1.002e-009

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-10

-5

0

5

10

X: 0.0825Y: 7.503

Am

plitu

de (v

olts

)

vout(t)

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-4

-2

0

2

4

X: 0.2615Y: 2.882

X: 0.7505Y: 0.3372

Time (sec)

vfiltered(t)

Am

plitu

de (v

olts

)

Figure 24 ‐ MATLAB Analysis Results for Windowed AM Modulator

We know that the best-case THD can be found by looking at the FFT of the envelope,

which is w(t) for a perfectly filtered vout(t). For a modulation index of 88%, the best case

THD is 0.54%. In other words, if we used perfect components, the best THD we could

achieve is 0.54%. Figure 25 shows the FFT of w(t):

Page 43 of 59

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000-400

-350

-300

-250

-200

-150

-100

-50

0X: 1Y: 0

Mag

nitu

de R

elat

ive

to F

unda

men

tal F

requ

ency

(dB

)

Frequency (Hz)

X: 3Y: -51.11

X: 5Y: -97.7

Figure 25 ‐ MATLAB Result of FFT of ww(t)

Y‐axis is Magnitudes relative to fundamental frequency in dB

We can also plot the best-case THD values over a range of modulation indices:

0 10 20 30 40 50 60 70 80 90 10010

-3

10-2

10-1

100

101

X: 29.76Y: 0.001897

Modulation Index (%)

THD

(%)

X: 0.9169Y: 2.454

X: 88.01Y: 0.5419

Figure 26 – THD over a Range of Modulation Indices for Windowed Memristor

Page 44 of 59

As we can see, we would achieve the lowest best-case THD of 1.89e-3% for a

modulation index of %29.76. As modulation index goes down, the amplitude of the high-

frequency carrier rises relative to the amplitude of the low-frequency signal and thus its

effects on w(t) become more significant, increasing THD. At a modulation index of

0.91%, the best-case THD is 2.45%. As modulation index goes up, THD caused by

windowing goes up since w(t) gets closer to the boundaries. At a modulation index of

88%, the best-case THD is 0.54%.

To see the results of the SPICE simulation, we can simply plug in the windowed

memristor model into the circuit. In order to make a valid comparison between the two

circuits, we must make sure that we are comparing two scenarios where the modulation

indices are the same. So, we need to set the amplitude of the signal to 0.25V to achieve

a modulation index of %88, as found in the previous section.

Figure 27 ‐ SPICE Simulation of w(t) with Windowed Memristor

Page 45 of 59

Figure 28 ‐ SPICE Simulation of vout(t) with Windowed Memristor

Figure 29 ‐ SPICE Simulation of vfiltered(t) with Windowed Memristor

Page 46 of 59

As can be seen, the memristance is successfully varied at a rate of 1Hz, however the

non-linear effects are apparent. The magnitude of vout(t) is as predicted by the MATLAB

analysis. The slight difference between the predicted maximum peak and the maximum

peak of simulated vfiltered(t) is a side-effect of the non-perfect filtering.

To calculate THD on the SPICE simulation, the output of the op-amp in the SPICE

simulation is passed through an imperfect high-pass filter. The 1st order high-pass filter

is a simple RC filter with a cutoff frequency (fc) of 100Hz. This output of the filter is then

sent into MATLAB where it is passed through a perfect envelope detector. The detected

envelope is the received signal that we wanted to transmit. Figures 30-31 shows the

MATLAB output of the envelope detection and the FFT of the envelope:

Figure 30 ‐ MATLAB Result of Envelope Detection for Windowed Memristor

Page 47 of 59

0 50 100 150 200 250 300 350 400 450 500-700

-600

-500

-400

-300

-200

-100

0X: 1Y: 0

X: 3Y: -50.47

X: 5Y: -96.74

Frequency (Hz)

Mag

nitu

de R

elat

ive

to F

unda

men

tal F

requ

ency

(dB

)

Figure 31 ‐ MATLAB Result of FFT of the Envelope

Y‐axis is Magnitudes relative to fundamental frequency in dB

The THD in this case is 0.69%, slightly higher than the predicted value of 0.54% due to

imperfect filtering.

We can repeat this process for the two other data points in Figure 26 to show that the

circuit shows the THD characteristics predicted mathematically. A total of three points is

shown in Figure 32, superimposed on the mathematically predicted curve:

Page 48 of 59

0 10 20 30 40 50 60 70 80 90 10010

-3

10-2

10-1

100

101

Modulation Index (%)

THD

(%)

Figure 32 – SPICE Simulation Results of 3 THD values and Modulation Indices

As we can see, the simulation results largely agree with the predicted values. The

offsets are due to the imperfect filtering.

In this section, a mathematical framework was developed to analyze a memristor AM

Modulator. It was seen that the results of SPICE simulations showed excellent

agreement with the theoretically predicted values. It can be concluded that the AM

Modulator behaves according to the mathematical theory both for the non-windowed

and the windowed cases. Furthermore, the robustness of the memristor model is shown

through the analysis and simulations in this section.

Page 49 of 59

3. Q-Factor Controller for 2ND Order Band-Pass Filter

In this topology, the memristor is used in a 2ND order cascaded Sallen-Key filter

topology to set a desired Q-factor. By setting an initial value for the memristor, we can

set the Q-factor of the filter to a desired value. By varying the Q-factor, we can change

the frequency tolerance of the filter around the center frequency w0. The general

transfer function for a 2nd order band-pass filter is:

2 200

( ) KsH ss s

Qω ω

=+ +

This transfer function can be realized in a cascaded topology using integrator blocks as

shown in Figure 33:

Out1

1

Integrator1

1s

Integrator

1s

Gain4

1/Q

Gain3

-1

Gain2

K

Gain1

w0

Gain

w0

In11

Figure 33 – 2nd order Cascaded Sallen-Key Filter

Page 50 of 59

In this cascaded configuration, the gain of the feedback op-amp is the coefficient 1/Q.

Thus, a memristor can be placed here in order to adjust the Q-factor by setting it to an

initial value. As long as the input signal is of a high enough frequency, the memristance

would remain unaffected, and the filter would be stable at the desired Q-factor.

Two different filters are shown in Figure 34. Both are centered on f0 = 10KHz (ω0 =

62,832Rad/sec). One has a Q-factor of 10, and the other has a Q-factor of 1. In order to

achieve unity gain at f0, the filter with Q-factor of 10 needs K=0.1, and the filter with Q-

factor of 1 needs K=1.

Q-Factor = 10

Frequency (Hz)

Q-Factor=1

Frequency (Hz)

103

104

105-40

-35

-30

-25

-20

-15

-10

-5

0

Mag

nitu

de (d

B)

102

103

104

105

106-50

-40

-30

-20

-10

0

Mag

nitu

de (d

B)

Figure 34 ‐ MATLAB Analysis of the Transfer Function

Page 51 of 59

The circuit that realizes the filter topology in Figure 33 is shown in Figure 35:

Figure 35 ‐ SPICE Circuit for Q‐Factor Controller

The ω0 of this circuit is 1/RC, which is 62,832Rad/sec. An inverting amplifier is placed at

the output of the second integrator for two reasons: To provide the negative feedback at

the input of the first integrator, and to provide the positive feedback at the input of the

second integrator since the memristor gain circuit also an inverting amplifier. The gain K

is realized by an inverting gain amplifier where varying RK to changes the gain of the

amplifier and inverts back the already inverted output of the second integrator, thus

providing the correct polarity at the output.

Q-factor of this circuit is one over the gain of the inverting opamp circuit with the

memristor. Thus, to achieve the Q-factors of 10 and 1, the gain of the opamp would

need to be 0.1 and 1 respectively. To achieve this gain with R2=10KΩ, the memristor

would need to be initialized at 1KΩ and 10 KΩ respectively. To find the w0 that will give

us the required memristances, we use the memristor equation:

0 0 0 001 1 100 16000 1 0.94ON OFF

w w w wM R R K w DD D D D

⎛ ⎞ ⎛ ⎞= + − ⇒ Ω = + − ⇒ =⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

0 0 0 001 10 100 16000 1 0.38ON OFF

w w w wM R R K w DD D D D

⎛ ⎞ ⎛ ⎞= + − ⇒ Ω = + − ⇒ =⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

Page 52 of 59

The input signal to both of these filters would need to be at a frequency that is outside

the band-pass region. We chose to feed in a sine wave at 100Hz frequency with

amplitude of 1V. To get the frequency response of these two circuits, we do an AC

sweep on them. The results are shown in Figures 36-37:

Figure 36 – Frequency Response with Q‐Factor=10, K=0.1

Figure 37 – Frequency Response with Q‐Factor=1, K=1

Page 53 of 59

We can conclude that the frequency response of the filter circuit is in full agreement with

the mathematically predicted response. We have two degrees of freedom in setting the

Q-factor of this circuit:

1- We can vary the w0 between 0 and D, we can achieve any memristance

between 100Ω and 16000Ω.

2- We can choose R2 to be any desired value.

Thus, we can conclude that we can design this circuit to have any desired Q-factor.

A valid question to ask is “how does this circuit behave in real-time?” We can simulate

the circuit in real-time and see the result. We would expect ‘w’ to be largely unaffected

since the input signal is outside the band-pass region and is filtered sufficiently before it

effects the memristor. Figures 38-40 shows the results of a transient analysis on a filter

with Q-factor of 10. Figure X shows ‘w’ of the memristor as a fraction of D, Figure X

shows the voltage at the input of the memristor, and Figure X shows the output of the

filter:

Page 54 of 59

Figure 38 – SPICE Transient Analysis of w(t) with Q‐Factor=10

Figure 39 ‐ SPICE Transient Analysis of voltage at Memristor Node With Q‐Factor=10

Page 55 of 59

Figure 40 ‐ SPICE Transient Analysis of Output Voltage Q‐Factor=10

As we predicted, the ‘w’ is largely unaffected by the input signal since it is largely

filtered out before it reaches the input node of the memristor as can be seen from Figure

39. Furthermore, as can be seen from Figure 40, the input voltage is filtered out at the

output by a factor of 0.001 is -60dB, as predicted from Figures 34 and 36 for an input

frequency of 100Hz.

However, it can also be seen that aside from sinusoidal change, there is a slow but

steady downward drift on ‘w’. Although this downward drift is negligible in the short-run,

in the long-run, this filter would not perform as it was first designed. In fact, as ‘w’

decreases, the Q-Factor would increase, which would widen the band-pass region. The

filter would then let through more of the input signal, which in turn would deteriorate ‘w’

even more, making the filter more unstable as time goes on. Thus, a correcting

Page 56 of 59

feedback circuit would be necessary in order to keep the ‘w’ stable. Such a circuit is

possible, although it was not analyzed in this thesis.

In this section, a mathematical framework was developed to analyze a 2nd order Band-

Pass filter with controllable Q-factor. It was seen that the results of SPICE simulations

showed excellent agreement with the theoretically predicted values. It can be concluded

that the Q-factor controller circuit behaves according to the mathematical theory.

Furthermore, the robustness of the memristor model is shown and reinforced by the

analysis and simulations in this section.

Page 57 of 59

VII. Conclusions

Memristors may play a very important role in the future of electronics. It could help us

keep up with Moore’s Law in the digital domain, as applications that use memristors as

switches hit the markets within the next five years. However, their most important role

will most likely be in the analog domain. By having access to non-volatile analog

memory, we can now design new circuits that can do new things. Already, there are

patents of applications that use memristors in the fields of signal processing, neural

networks, and control systems6. Furthermore, by being able to model neurons without

the use of complicated transistor topologies, it may become feasible to built large-scale

neuromorphic computers and take advantage of vast parallel-computation, much like a

human brain does.

To speed up memristor research and achieve these breakthroughs, the problems

standing in the way of memristor research must be solved. One of these problems is the

lack of software simulation support for these devices. This problem was addressed in

the first half of this thesis by creating a behavioral model of TiO2 memristors that can be

used to simulate circuits that involve memristors. The results show that the model

behaves appropriately and the behavior of the devices reported in [2] are modeled

correctly. It is the author’s hope that the model described in this paper is used for easily

and efficiently simulating circuits that involve TiO2 memristors, and that it serves as a

framework for the behavioral modeling of other reported memristive devices, in order to

speed up the research and development of memristor electronics.

6 U.S. Patent 7,302,513; U.S. Patent 7,359,888; U.S. Patent Application 11/976927

Page 58 of 59

The second half of the thesis used this behavioral model to design, simulate, and

analyze three circuit topologies: Memristor Switch, Memristor AM Modulator and Q-

Factor Controller for 2nd order Band-Pass Filter.

For the Memristor Switch, it was concluded that the memristor switch uses unfeasibly

high amounts of energy in order to switch on and off both in the non-windowed case

and the windowed case. Further research and explanations by HP Labs should shed

light on how memristors can be feasibly used as digital switches.

For the AM Modulator, a circuit topology was proposed that uses the memristor in a

simple inverting op-amp configuration to modulate a high-frequency carrier. This circuit

was analyzed mathematically and SPICE simulation results were compared to theory

both for the non-windowed and the windowed memristor. It was concluded that the

model behaves according to theory in both cases. Furthermore, performance analysis

was done by using looking at THD values. It was concluded that the non-windowed

memristor model creates close to zero distortion whereas the windowed memristor

introduces distortion as modulation index increases and w(t) approaches the

boundaries. In both cases, THD went up as the input signal amplitude decreased,

increasing the distortion caused by the high-frequency carrier.

For the Q-Factor controller, the memristor was used in the feedback gain section of a

cascaded 2nd order Sallen-Key band-pass filter, again in an inverting op-amp

configuration. The Q-Factor was controlled by varying the initial condition on the w(t) of

the memristor.

It was concluded overall that the memristor model created by the author is robust and

can be used to simulate new circuit topologies easily and efficiently.

Page 59 of 59

Bibliography

1- Leon, O. C., “Memristor - The Missing Circuit Element”, IEEE Transactions on Circuit Theory,” pp. 507-519, 1971.

2- Strukov D. B., Snider G. S., Stewart D. R., and Williams R. S., “The missing memristor found,” Nature , pp. 80-83, 2008.

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