Journal of Computer and Communications, 2013, 1, 5-10
Published Online November 2013 (http://www.scirp.org/journal/jcc)
http://dx.doi.org/10.4236/jcc.2013.16002
Open Access JCC
5
The Realization and Working Conditions of Memristor
Based on Multisim*
Dehua Song1, Xiang Ren1, Mengfei Lv1, Mengmeng Li1, Haiyan g Zho u1, Yunxiao Zu2
1School of Information and Telecommunication Engineering, Beijing University of Posts and Telecommunications, Beijing, China;
2School of Electronic Engineering, Beijing University of Posts and Telecommunications, Beijing, China.
Received July 2013
ABSTRACT
An equivalent circuit is realized using Multisim software by transforming a kind of circuit element according to Map-
ping principle and circuit theory. The effects of every parameter on the equivalent circuit are analyzed and the working
conditions of the equivalent circuit are concluded by simulation.
Keywords: Memristor; Multisim; Equivalent Circuit; Realization; Working Conditions
1. Introduction
Four researchers from HP laboratory successfully manu-
factured memristor based on the metal and metal oxide in
Nano-scale in 2008 and found the mathematical model of
memristor [1]. The memristor was being focused since
then [2-11]. However, analysing the circuit containing
memristor is very difficult because the voltage-current
relation (VCR) of memristor is a multivalued function.
Though researchers usually use a linearization approach
to analyse the circuit, it is also difficult and inaccurate. A
mutator converting other circuit components into me-
mristors is built in this paper according to the mapping
principle and circuit theory b ased on Multisim in order to
analyse circuit containing memristor easily and accu-
rately.
Memristor is an element which provides a functional
relation between flux linkage
ψ
and charge q, and the
basic model is as follows.
ddMq ψ =
(1)
where M is the memristance of the memristor.
2. Circuit of Mutator
Mutator converts VCR of an element into Ψ-q r elation to
simulate memristor. The circuit of mutator is shown in
Figure 1.
The Realization of Memristor Based on Multisim
Assigning hm = 0.4, km = 1000, L = 1 mH, C = 1 mF, R =
1 Ω, I = 1 A. The u-t relation of memristor is shown in
Figure 2 by simulation in Multisim. The longitudinal
axis scale is 500 V/Div. It can be seen that the curve is a
straight line paralleled with axis t, so the memristor is
equivalent to a linear resistor.
Memristor with fixed M is converted by linear resistor,
so no research value. Only M is variable can memristor
has memory function, so a nonlinear component as the
converted component is necessary. The diode can be
converted a memristor with variable M. Because the
VCR cannot be shown directly in Multisim, a 1 Ω resis-
tor is in series in the circuit. In this way, the current of
the circuit can be detected because the voltage and the
current of the 1 Ω resistor are numerically equal. The
channel A of the oscilloscope is used to detect the vol-
tage of memristor while the channel B is used to detect
the voltage of 1 Ω resistor, then the VCR can be obtained
using the A/B mode.
The VCR of the equivalent memristor can be obtained
when the sine alternating current i = sin(2πƒt)A, ƒ = 1
kHz is added on the circuit and let hm = 0.4, km = 1000, L
= 1 mH, C = 1 mF . The u - t relatio n is shown in F ig u r e 3 .
The horizontal axis scale is 500 us/Div and the longitu-
dinal axis scale is 500 mV/Div. The VCR of memristor is
shown in Figure 4. Both the horizontal axis scale and the
longitudinal axis scale are 500 mV/Div. It can be seen
that the VCR of the equivalent memristor is an inclined
“8” and very close to the VCR in literature [2]. So the
circuit can be as the equivalent memristor with little er-
ror.
The circuit of converting the diode into memristor in
*Project supported by the research in novation fund for college
students
of Beijing University of posts and telecommunications.

The Realization and Working Conditions of Memristor Based on Multisim
Open Access JCC
6
+-
-+
++ --
u
1
i
c
i
c
i
2
u
L
u
L
u
2
i
1
u
L
/h
m
L
ic/kmC
i
2
Figure 1. Circuit of mutator.
Figure 2. The u-t relation.
Figure 3. The u-t relation.
Figure 4. The VCR of the equivalent memristor.
Multisim is shown in Figure 5.
3. The Working Conditions of the
Equivalent Circuit
3.1. The Influences of hm and km
Put
m
m
uk
i hq
ψ =

=

(2)
into the VCR of the diode
set
u
U
iI=
, Equation (3) can be
gotten.
s
e
m
t
k
U
m
hq I
ψ
=
(3)
Figure 5. The simulation circuit in Multisim .
Simplifying Equation (3)
ts
ln
mm
k hq
UI
ψ
=
(4)
Making differential on t in both sides of Equation (4).
s
ts
dd
dd
mm
m
kIh q
UthqI t
ψ
=
(5)
Put
dd
,
dd
q
ui
tt
ψ
= =
into Equation(4).
t
1
m
kui
Uq
=
(6)
Put
0
d
t
q it=
∫
into Equation (6).
t
d
t
m
Ui
ukit
−∞
=
∫
(7)
Analyzing Equation(7) and can found that with the
same current, the bigger hm, the smaller the voltage and
the resistance of the equivalent circuit. Assigning Is = 0.1
uA, Ut = 26 mV, i = sin(2πƒt)A, ƒ = 1 kHz, hm = 1, then
the u-t relation for different km in MATLAB simulation is
shown in Figure 6.
It can be seen from Figure 6 that the voltage peak va-
ries with d ifferent km. The bigger km, the smaller the vol-
tage peak will be. Although the changing rate of the vol-
tage is variable, the u - t relations fo r different km have not
intersected except the origin. Because the value of the
current source is the same at same moment, the bigger
the voltage, the bigger the resistance of the equivalent
circuit, and at different moment the resistance is bigger
for smaller km.
The simulation waveform is the same with that of the-
oretical analysis. For the same current source, the smaller
km, the bigger the voltage peak. Assigning L = 1 mH, C =
1 mF and i = sin(2πƒt)A, ƒ = 1 kHz. The VCR gradually

The Realization and Working Conditions of Memristor Based on Multisim
Open Access JCC
7
Figure 6. The u-t relation for different km.
approaches to a straight line and closes to i axis when km
increasing, while the shape of the u-t curve does not
change but the voltage peak decreasing. Assigning km =
5000, hm = 0.4, and the VCR of the equivalent circuit is
shown in Figure 7, in which both the horizontal axis
scale and the longitudinal axis scale are 500 mV/Div.
Increasing km until the oscilloscope can not display, how-
ever the shape of VCR still remains the same. Decreasing
km, the VCR of the equivalent circuit gradually ap-
proaches to a straight line and closes to u axis. When km
changes very little, the shape of u-t curve does not
change but the voltage peak increasing. When km is
smaller than 10-7, the u-t curve of the equivalent circuit
approaches to the sinusoid and shows the properties of
linear resistor rather than memristor with changing me-
mrisistance. Assigning km = 10−7，hm = 0.4, and the u-t
relation of the equivalent circuit is shown in Figure 8,
and in which the horizontal axis scale is 500 us/Div and
the longitudinal axis scale is 500 MV/D iv. It can be seen
that the curve is very close to sinusoid. Because some
elements of the equivalent circuit are active elements, the
circuit takes some time to get stable, the stability of the
circuit will not be changed when changing km, that is the
VCR curves will never overlap. In a word, when hm = 0.4,
the error of the circuit is small if km is between 10 and
104.
The impact of hm on the equivalent circuit is different
from that of km. For the same current source, the time the
voltage in the u-t curve reaching zero is the same for dif-
ferent hm, and that is right for the voltage reaching the
peak. hm has no impact on the time the voltage reaching
the peak. The simulation waveform in Multisim is dif-
ferent from that of the theoretic analysis. For the same
current source, the bigger hm, the bigger the voltage peak
will be. That is mainly because the VCR of the diode is
only approximately exponential. Assuming
()u fi=
(8)
Put Equation (2) into Equation (8), do differential both
sides of Equation (8) and simplify it.
0
'(d )
t
mm
m
h
uf hiti
k
=∫
(9)
Figure 7. The VCR when km = 5000.
Figure 8. The u-t relation when km = 10−7.
It can be seen that if VCR is not standard exponential
relationship, the equivalent circuit will be affected by hm.
Assigning L = 1 mH, C = 1 mF, i = sin(2πƒt)A and ƒ =
1 kHz. When hm is not largely changed, only the voltage
peak in the u-t curve increasing, while the shape of the
curve does not changing. When hm is bigger than 104, the
shape of the u-t curve will obviously change, the voltage
peak is not being constant, the voltage near the peak va-
ries very quickly and the maximum positive value and
the maximum negative value is not equal. Assigning km =
1000, hm = 104, the u-t curve of the equivalent circuit is
shown in Figure 9. The scales of the horizontal axis and
the longitudinal axis are 500 us/Div and 5 V/Div respec-
tively. It can be seen that the u-t relation is no longer
periodic.
In the same way, increasing hm, the u-i curve of the
equivalent circuit changes a lot, from a convex and obli-
que “8” changes to a concave and oblique “8”, which is
quite different from the u-i curve in literature [1]. Figure
10 is the u-i curve when km = 1000, hm = 104. Both the
scales of the horizontal axis and the longitudinal axis are
1 V/Div. It can be seen that the curve obviously twists
and different from the curve in Figure 4. Figure 11 is
the amplified curve partially by changing the scale of the
horizontal axis to 10 V/Div and the longitudinal axis to
50 mV/Div. It can b e seen that the u-i curve is composed
of many intersected curves. So when hm = 104, the equiv-
alent circuit is no longer stable and changes in every pe-
riod of the current source, which is also why the voltage
peak in Figure 10 is a fixed value.
Not only the voltage peak and the u-t relation, but also
the stability of circuit will be affected by increasing hm.
The bigger hm, the less stable the circuit. The u-i curve of
the equivalent circuit gets flatter, approaches a straight
line and closer to the horizontal axis when decreasing hm.
Assigning km = 1000, hm = 0.1, the simulated VCR is

The Realization and Working Conditions of Memristor Based on Multisim
Open Access JCC
8
Figure 9. The u-t curve when hm = 104.
Figure 10. The u-i curve when hm = 104.
Figure 11. The amplified u-i curve when hm = 104.
shown in Figure 12. Both the horizontal axis scale and
the longitudinal axis scale are 500 mV/Div. The u-t curve
is shown in Figure 13, the scales of the horizontal axis
and the longitudinal axis are 1 ms/Div and 200 mV/Div
respectively. It can be seen that the voltage peak changes
less than that of Figure 3. When hm is not largely
changed, only the voltage peak of the u-t curve decreases,
the shape of the curve doesn’t change much and the
curve gets smoother. When hm is smaller than 0.01, the
u-t curve significantly changes and closes to the normal
sinusoid. Assigning km = 1000, hm = 0.1, the u-t curve is
shown in Figure 14, the horizontal axis scale is 500
us/Div and the longitudinal axis scale is 50 mV/Div. It
can be seen that the curve is very close to normal sinu-
soid, while the u-i curve of the equivalent circuit is close
to a straight line, the circuit shows the characteristic of
linear resistor rather than memristor with changing me-
mrisistance. The stability of the circuit will not be af-
fected when hm decreases. So when km = 1000, hm = 0.01
~ 104, the error of the equivalent circuit is small.
3.2. The Influences of C and L
It can be seen from Equation (7) that the capacitor and
inductor have no influences on the equivalent circuit. The
Figure 12. The u-i curve wh en hm = 0.1.
Figure 13. The u-t curve wh en hm = 0.1.
Figure 14. The u-t curve wh en hm = 0.01.
simulation result is the same with that of the theoretical
analysis. Although the capacitance and inductance in the
equivalent circuit are adjustable, the curves of VCR and
u-t don’t change with the capacitance and inductance.
When the inductance is in a certain range, assigning i =
sin(2πƒt)A, ƒ = 1 kHz, hm = 0.4, km = 1000, changing the
capacitance or inductance respectively, the curves of
VCR and u-t are totally the same with that of Figure 3
and Figure 4, which shows that the capacitor and induc-
tor have no influences on the equivalent circuit. However,
when the inductance is bigger than 1 mH, there will be
errors in simulation. Though simulation can be continued
by cutting down the peak value of current source, the u-i
curve is close to a straight line and the u-t curve is close
to sinusoid because of the too small current. So the cir-
cuit shows the characteristic of linear r esistor and cannot
represent memristor. Such simulation circumstances
won’t happen when changing the capacitance.
3.3. The Influences of the Source’s Peak and
Frequency
It can b e seen fr om Equation (7) that the current source’s
peak has impact on the VCR and u-t relation of the
equivalent circuit. Put i = Isin(2πƒt) into Equation (7)
and simplify it.
m
2 sin(2 )
cos(2) 1
t
Uf ft
ukf
ππ
π
=−
(10)
It can be seen from Equation (10) that the voltage has
no relationship with the current peak. When increasing
the current peak, the horizontal peak increases, but the
vertical peak doesn’t change. So the bigger the current

The Realization and Working Conditions of Memristor Based on Multisim
Open Access JCC
9
peak, the flatter the u-i curve, and closer to the horizontal
axis.
Assigning i = Isin(2πƒt)A, ƒ = 1 kHz, hm = 0.4, km =
1000, L = 1 mH, C = 1 mF. When I = 10 A the u-i curve
is shown in Figure 15, the scales of the horizontal axis
and the longitudinal axis are 5 V/Div. It can be seen that
the curve is very close to the horizontal axis. When the
current peak I is bigger than 50 A, the stability of the
equivalent circuit weakens and the u-t curve loses peri-
odicity. Whe n I = 50 A the u-i curve is shown in Figure
16, the scales of the horizontal axis and the longitudinal
axis are 2 V/Div. It can be seen that the equivalent circuit
is very unstable, many curves lap over, and the analysis
result of the circuit is not accurate. When decreasing the
current, the u-i curve is close to a straight line. However,
the slope k of the curve keeps constant when changing
the current value. This is mainly because that the current
value is so small that the resistance of the equivalent cir-
cuit varies less and can be approximately regarded as a
linear resistor in a short period. For example, when I =
0.01 A the u-i curve is shown in Figure 17. The scales of
the horizontal axis and the longitudinal axis are 10
mV/Div. It can be seen that the u-i curve is very close to
a straight line. So the circuit shows the characteristic of
linear resistor and the characteristic of memristor is not
obvious. However, the stability of circuit will not be af-
fected when decreasing the current. In a word, the circuit
is not stable if the current is very big, and the circuit will
show the characteristic of linear resistor if the current is
over-small. So the equivalent circuit in Figure 1 can si-
mulate the memristor only when the effective value of
the current is 0.01 ~ 50 A.
Figure 15. The VCR when I = 10 A.
Figure 16. The u-i curve wh en I = 50 A.
Figure 17. The u-i curve wh en I = 0.01 A.
It can be seen from Equation (10) that the frequency of
the current source has impact on the VCR and u-t curve
of the equivalent circuit. Assigning i = sin(2πƒt)A, hm =
0.4, km = 1000, L = 1 mH, C = 1 mF. Increasing the fre-
quency of current source in a certain range, the u-t curve
doesn’t change obviously, and the node of the oblique “8”
of the u-i curve offsets upward but not so obvio us ly.
Assigning f = 100 kHz, the u-i curve of the equivalent
circuit is shown in Figure 18. The scales of the horizon-
tal axis and the longitudinal axis are 500 mV/Div. It can
be seen that the u-i curve is no longer an oblique “8” but
an oval, the circuit is unstable, and the curve is composed
of many curves. The u-t curve of the equivalent circuit is
shown in Figure 19. The scales of the horizontal axis and
the longitudinal axis are 10 us/Div and 500 mV/Div re-
spectively. It can be seen that the curve is close to sinu-
soid, the voltage peak is bigger in the first cycle of the
current source, and then the voltage peak gets to stable,
so the stability of circuit gets less. When decreasing the
frequency of the current source, the u-i curve gets flatter
and closes to the horizontal axis, and the characteristic of
the equivalent circuit approaches to the linear resistor.
The u-t curve has not obviously change. Assigning f =
100 Hz, the u-i curve of the equivalent circuit is shown in
Figure 20. The scales of the horizontal axis and the lon-
gitudinal axis are 500 mV/Div. It can be seen that the
curve is very similar to Figure 16. In a word, if the fre-
quency is over high, the u-i curve of the equivalent cir-
cuit is obviously distorted and the circuit is uns table, and
Figure 18. The u-i curve wh en ƒ = 100 kHz.
Figure 19. The u-t curve wh en ƒ = 100 kHz.
Figure 20. The u-i curve wh en ƒ = 100 Hz.

The Realization and Working Conditions of Memristor Based on Multisim
Open Access JCC
10
if the frequency is over small, the circuit will show the
characteristic of linear resistor. Thus the equivalent cir-
cuit in Figure 1 can simulate the memorize only when
the frequency of the current source is 0.1 ~ 10 kHz.
4. Conclusion
A mutator is built based on Multisim as an equivalent
circuit of memristor in this paper. The parameters which
impact on the equivalent circuit are analyzed, and the
working conditions of the equivalent cir cuit of memristor
are given. The errors of the equivalent circuit are smaller
when km is 10 ~ 104 and hm is 0.01 ~ 104, the inductor and
capacitor have no effects on the equivalent circuit when
L < 1 mH, the frequency should be ƒ < 10 kHz, and the
current should be I < 50 A. Which means the circuit can
not be use d i n hi gh freque nc y circui t .
5. Acknowledgements
The work is supported by the Research Innovation Fund
for College Students of Beijing University of Posts and
Telecommunications.
REFERENCES
[1] D. B. Strukov, G. S. Snider, D. R. Stewart and R.
S.Williams, “The Missing Memristor Found,” Nature,
Vol. 453, 2008, pp. 80-83.
[2] X. Zhang, Y. Z. Zhou, Q. Bi, X. H. Yang and Y. X. Zu,
“The Mathematical Model and Properties of Memristor
with Border Constraint,” Acta Physica Sinica, Vol. 59,
No. 9, 2010, pp. 6669-6676.
[3] Q. H. Wang and W. P. Song, “Realization of Memristor
by T wo-Port Mutator,” Journal of Electrical & Electronic
Education, Vol. 33, No. 3, 2011, pp. 56-57.
[4] M. Itoh and L. O. Chua, “Memristor Oscillators,” Inter-
national Journal of Bifurcation and Chaos, Vol. 18, No.
11, 2008, pp. 3183-3206.
http://dx.doi.org/10.1142/S0218127408022354
[5] Y. Zhang, X. L. Zhang and J. B. Yu, “Approximated
SPICE Model for Emristor,” International Conference on
Communications, Circuits and Systems, 2009, pp. 928-
931. http://dx.doi.org/10.1109/ICCCAS.2009.5250371
[6] D. P. Wang, Z. H. Hu, X. Yu and J. B. Yu, “A PWL
Model of Memristor and Its Application Example,” In-
ternational Conference on Communications, Circuits and
Systems, 2009, pp. 932-934,
http://dx.doi.org/10.1109/ICCCAS.2009.5250372
[7] Klaus Witrisal, “A Memristor-Based Multicarrier UWB
Receiver,” Proceedings of IEEE International Conference
on Ultra-Wideband, IEEE Press, Sep. 2009, pp. 678-683,
http://dx.doi.org/10.1109/ICUWB.2009.5288703
[8] D. Varghese and G. Gandhi, “Memristor Based High
Linear Range Differentia l Pair, ” International Conference
on Communications, Circuits and Systems, Jul. 2009, pp.
935-938.
http://dx.doi.org/10.1109/ICCCAS.2009.5250373
[9] A. Delgado, “Input-Output Linearization of Memristive
Systems,” Proceedings of IEEE Nanotechnology Mate-
rials and Devices Conference, IEEE Press, July. 2009, pp.
154-157.
http://dx.doi.org/10.1109/NMDC.2009.5167523
[10] M. Di Ventura, Y. V. Pershin and L. O. Chua, “Circuit
Elements with Memory: Memristors, Memcapacitors, and
Meminductors,” Proceedi ngs of IEEE, Vol. 97, Oct. 2009,
pp. 1717-1724.
http://dx.doi.org/10.1109/JPROC.2009.2021077
[11] L. O. Chua, “Memristor—The Missing Circuit Element,”
IEEE Transactions on Circuit Theory, Vol. 18, Sep. 1971,
pp. 507-519.
http://dx.doi.org/10.1109/TCT.1971.1083337