This document presents a memristor-capacitor based startup circuit for voltage reference generators. It derives theoretical equations describing memristor switching behavior and proposes a startup circuit using a series combination of memristor and capacitor. Simulation results show the memristor-based circuit achieves an on to off state transition in 2.8 ns, much faster than a conventional MOSFET-based startup circuit which takes 55.56 ns. While no significant speed difference is seen compared to a resistor-based startup circuit, the memristor circuit offers area savings due to memristors' smaller size compared to resistors.

1. Memristor-Capacitor Based Startup Circuit for
Voltage Reference Generators
Mangal Das
Electronics & Communication Engineering
ABES Engineering College
Ghaziabad, India
mangalforyou@gmail.com
Sonal Singhal
Electrical Engineering
Shiv Nadar University
Gautam buddh Nagar, India
sonal.singhal@snu.edu.in
Abstract— This paper presents the design of Memristor-capacitor
based startup circuit. Memristor is a novel device and
has many advantages over conventional CMOS devices such as
no leakage current and is easy to manufacture. In this work the
switching characteristics of memristor is utilized. First the
theoretical equations describing the switching behavior of
memristor are derived. To prove the switching capabilities of
Memristor, a startup circuit based on series combination of
Memristor-capacitor is proposed. This circuit is compared with
the reference circuit (which utilizes resistor in place of
memristor) and the previously reported MOSFET based startup
circuits. Comparison of different circuits was done to validate the
results. Simulation results shows that memristor based circuit
attains on (I = 2.25 mA) to off state (I = 10 μA) in 2.8 ns while the
MOSFET based startup circuits takes (I = 1 mA) to off state (I =
10 μA) in 55.56 ns. However no significant difference in switching
time was observed when compared with resistance based startup
circuit. The benefit comes in terms of area because much larger
die area is required for manufacturing of resistance in
comparison to fabrication of memristor.
Keywords— Startup circuits; Memristors; Voltage Reference
generator; Switching circuits component.
I. INTRODUCTION
This Voltage reference generators are used for the
generation of process and temperature independent supply
voltages. Self Biased Circuits like voltage reference generator
have degenerate bias point, which does not allow these circuits
to start itself [1]. It necessitates the use of startup circuit in
these circuits [2],[3]. Startup circuit isolates itself electrically
from reference generator circuit after giving rise to initial
conditions.
Conventionally, startup circuits are realized by capacitors
and MOSFETs and are characterized in terms of speed and
area. These startup circuits are slow because of large time
constant of these circuit. The reason of such large time constant
is the large resistances (in hundreds of kilo-ohm) between drain
and source terminals of MOSFETs. These circuits also
consume more area due to presence of capacitors and
MOSFETs of longer channel length than other MOSFETs
present in the reference generator circuit.
As an alternative and new approach, memristor based
startup circuits is devised. Memristor is a passive two port
element with variable resistance [5]. L. Chua and S. Kang have
given theoretical frame work for describing the memrisitve
system [6]. In 2008 HP (Hewlett-Packard) has announced that
they synthesized a memrisitve system based on TiO2. Since
then, numerous models of TiO2 memristor have been reported
in literature. All of these models use the physical model put
forward by D. B. Strukov [7]. Since then many articles have
been published on the physical and electrical properties of
memristor. However no report has investigated the switching
behaviour of memristor.
In this paper we have investigated the switching behaviour
of memristor under constant DC bias and suggested a
memristor-capacitor based startup circuit for voltage reference
generators. Spice model of memristor suggested by Z. Biolek is
used for simulation [4].
This paper is organized as follows. Section II describes the
electrical properties of memristor in terms of physical
parameters. Section III describes switching behaviour of
memristor. It is studied under initial and final conditions.
Section IV presents the application of switching characteristics
of memristor in a startup circuit. A new startup circuit has been
proposed which utilizes the MOSFET and MEMRISTOR.
Section V presents simulation results for the suggested circuit.
Simulation has been done on SPICE.
II. ELECTRICAL AND PHYSICAL PARAMETERS OF
MEMRISTOR
The basic equation of current controlled memrisitve system
is given below [6]:
v=R(w,i)i (1)
dw =f(w,i)
(2)
dt
where w is the state variable of physical dimension of
Memristor, R(w,i) is memresistance, v is the voltage across
memresistance, and i is the current through memristance.
Fig 1 shows the memristor model in terms of it’s physical
parameters. The relation between the physical and electrical
behaviour of memristor are given by following equations [7]:

2. v(t)= R w(t) +R 1- w(t) i(t)
⎛ ⎛ ⎞ ⎞
⎜ ⎜ ⎟ ⎟ ⎝ on D off
⎝ D
⎠ ⎠
(3)
dw(t) =μ R on
i(t)
dt v
D
(4)
Where w(t) is the width of doped TiO2-x at any time “t”, D is
the semiconductor thickness, μv is dopant mobility, v(t) is the
voltage across memristor, i(t) is the current through
memristor, on R is on-state resistance of memristor, off R is
off state resistance of memristor.
w
2 1
III. SWITCHING CHARACTERISTICS OF MEMRISTOR
This section presents the theoretical framework for
analyzing the switching behavior of memristor. Switching
behavior can be analyzed from equation (3) and (4). From
these equations it is seen that the memristor can be made to
work as switch as at time t = 0 and at t → ∞. At these two
extreme time instants the value of
w(t) (0,1)
D
∈ which give
rise to two values of v(t) at t = 0 and t → ∞. The memristor
switch makes the transition from ON to OFF or OFF to ON
state in accordance with the polarity applied across the
memristor.
In order to analyze the behaviour of TiO2 based memristor
resistance with respect to time we have used the equations
given by Joglekar and Wang [8],[9]:
2 2
mem 0 d R (t)=R-2kR φ(t) (5)
where k = μv Ron/D2, μv is the dopant drift mobility,
t0
φ(t)=∫ v(τ)dτ is the flux at time ‘t’, d off on R =R -R is the
difference of boundary resistances and 0 0 R =R(x ) is the
initial resistance at t = 0. The value of resistance at any time
‘t’ mem on off R ∈(R ,R ) .
Radwan has extended the theoretical framework given by
Joglekar and Wang [8],[9] to investigate the memristance
resistance under DC excitation [10]. The equation is
reproduced here
2 2
mem 0 d dc R (t)=R-2kR V t (6)
From equation (3), (4), (5), (6) we can identify the switching
activities for two extreme cases
(i) At t = 0, equation (6) and (3) can be written as
R (0)=R = R w(0) +R 1- w(0)
⎛ ⎛ ⎞ ⎞
⎜ ⎜ ⎟ ⎟
⎝ ⎝ ⎠ ⎠
mem 0 on off
D D
(7)
(ii) At time t = ∞ (long times under bias) resistance of
memresistor will tend to Ron or Roff depending on the
polarity of applied voltage.
Strokov proposed that in hard switching case (large voltage
excursions or long times under bias) there appears to be
clearly defined threshold voltage however effect is actually
dynamical [7].
Saturation time is the time taken by memristor to reach it’s
one of the final values either Ron or Roff.
Saturation time of memristor is given as [7]:
2
off
sat
D R
v on dc
t =
2μ R V
(8)
IV. STARTUP CIRCUIT BASED ON MEMRISTOR CAPACITOR
COMBINATION
A startup circuit based on memristor and capacitor is
proposed. It is designed keeping in view on the capability
of memristor to work as a time dependent resistive switch
(Refer to equation (5)).
Fig 2 shows the circuit diagram of proposed startup
circuit utilizing MOSFET and memristor-capacitor
combination. In the proposed circuit voltage at point A and
B depends on the type of MOSFET.
For the proposed circuit, VGS of MOSFET is written as:
GS mem mem V (t)=R (t) I (t) (9)
Where Rmem(t) is given by equation (5), Imem(t) is the
current flowing through the memristor at any time. Multu
has given the relation for Imem(t) with time and is given by
[11] :
-(t-tsat )/
I (t)= (-V (t )+V )e
c sat DC
mem
R
τ
(10)
Where Vc(t) is the voltage at capacitor at time ‘t’, τ =
Rmem(t)C is the time constant of the memristor-capacitor
circuit, VDC is the applied Voltage.
From equation (10), the current behavior for time t = 0 and t
= ∞ is analyzed (equation (11) and (12) respectively)
tsat /
I (0)= (-V (t )+V )e
c sat DC
mem
R
τ
(11)
mem I (∞)=0 (12)
Therefore at time t = 0 and at t→∞, VGS (gate to source
voltage) of MOSFET can be written as (equation (13) and
(14) respectively)
tsat /
c sat DC
GS mem
(-V (t )+V )e
V (0)=R (0)
R
τ
(13)
GS V (∞)=0 (14)
i v
Fig. 1.Memristor model (Shaded area shows TiO2-x region remaining
part is.TiO2).

3. V. SIMULATION RESULTS
In order to investigate the resistance behavior of memristor
with respect to input voltage, transient analysis is carried out.
Biolek suggested a window function [4] and the same is used
in the simulations:
f(x)=1-(x-stp(-i))2p (15)
Where p is a positive integer, i is the memristor current, and
≥ ⎧⎨
⎩ ≤
1 proi 0
stp(i)=
0 proi 0
(16)
Fig 3 shows the transient analysis of memristor resistance
which exhibits Ron=100, p=10, Rinit=11kΩ, Roff =16kΩ, D =10
nm for different step inputs (1 V, 2 V, 3 V, 4 V).
Simulation Results shown in fig. 3 are in accordance with
equation (8). From equation (8) we can observe that saturation
time ‘tsat’ is inversely proportional to applied DC Voltage with
other parameters constant. Similar trend is obtained in the
simulation studies that as the applied voltage increases the
saturation time decreases. From fig. 3 we can observe that
when applied step input is changed from 1 V, 2 V to 3 V, the
change in saturation time (tsat) is significant whereas when
applied excitation is changed from 3 V to 4 V the change in
‘tsat’ is small. For simulations the power supply of 3 V is used
as it optimizes the power and speed.
Transient behavior of proposed startup circuit here in
referred as PSC is investigated with the step input of 3 V. The
peak value of current is obtained 2.52mA. To compare the
result, similar analysis is done on a resistive startup circuit
(RSC). In this startup circuit (RSC) memristor is replaced by a
resistor. Again to validate result, the startup circuit already
reported by Giustolisi [2], [12] is compared with PSC. The
startup used by Giustolisi and Yu-Hsuan [2], [12] is MOSFET
based startup circuit and is referred as (MSC).
Simulation of PSC was done with PMOS (W=50μ, L=1μ),
Capacitor (C=1ps), Memristor (Rinit = 137.5Ω, Roff = 250Ω,
Ron = 25Ω, p=10, D = 10 nm). RSC (Resistive startup circuit)
uses resistance (250 ohm) in place of memristor. MSC uses
two PMOS (W=20μ,4μ; L=4μ,10μ) and a capacitor (C=1ps).
Fig. 4 shows the transient behavior of the currents of
proposed startup circuit (shown as PSC), MOSFET based
startup circuit used (shown as MSC), and a startup circuit
which uses resistance in place of memristor (shown as RSC).
Table I shows the peak values of current attained by circuits
(MSC, PSC, RSC). The peak current value is more in PSC,
RSC in comparison with MSC because of very low resistance
value in comparison to drain - source terminals of MOSFETs
used in MSC. However it is not good for our circuit because
the peak power dissipation will be high. Table II shows the
time taken by the currents (MSC, PSC, RSC) to drop to the
10%, 63%, 90% of their peak value.
The very fast transition of currents for PSC can be
explained in the following way: Time constant (τ) for an RC-series
circuit is given by τ = RC. Resistance or capacitance is
to be decreased in order to decrease time constant. In the
proposed circuit memristor is biased in such a way that as time
increases the resistance of memristor decreases, thereby
reducing the time constant. With decrease in resistance of
memristor (connected between gate and source) the VGS (gate
to source voltage) of MOSFET will also decrease, thus
pushing MOSFET towards cutoff very rapidly. Combined
Fig. 2.Proposed circuit diagram of startup circuit
Fig.3 .Transient analysis of Resistance (Memristor parameters
Ron=100, p=10, Rinit=11k•, Roff =16kΩ, D=10 nm, under different step
inputs (1 V, 2 V, 3 V, 4 V)).
Fig.4. Comparison of transient behavior of currents

4. effect of these two phenomena causes current to decrease very
rapidly.
From table II, it is seen that PSC and RSC are comparable
in speed. It is also seen that proposed startup circuit is much
faster than MOSFET based startup circuit.
In comparison to resistance based startup circuit (RSC)
the memristor based startup circuit (PSC) will be an
economical choice. It is because RSC consumes more area
than PSC. Length of a memristor typically varies from 30nm
to 3nm. Startup circuit which is suggested in the paper uses a
memristor of length (D) 10nm. This is less even in comparison
with the modern CMOS technology (28 nm) used today. Much
larger die area is required for manufacturing of resistance than
the area required for the manufacturing of memristor.
TABLE I. PEAK VALUE OF CURRENT
PSC RSC MSC
Peak Value of
Current (mA) 2.52 2.51 0.67
TABLE II. TRASIENT BEHAVOIUR OF CIRCUIT
% Drop From Peak
Value
TIME (nsec)
PSC RSC MSC
90 % 2.00 2.00 6.85
63% 2.06 2.09 16.91a
10 % 2.31 2.56 45.372a
a. Not shown in fig 4.
CONCLUSION
In this paper we have suggested a new hybrid circuit which
includes the memristor-capacitor as its components and
capable of working as a startup circuit in voltage reference
generators. Current in proposed startup circuit drops from 2.5
mA to 10 μA (2.8 ns) which is much faster than conventional
CMOS startup circuit. Proposed circuit also saves area as
dimensions and number of components used in the circuit is
less than any conventional startup circuit.
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