Download Explanation of the Memristor`s Hysteresis Loop

6th International Advanced Technologies Symposium (IATS’11), 16-18 May 2011, Elazığ, Turkey
Explanation of Hysteresis Curve of a Fluxdependent Memcapacitor (Memory-capacitor)
Using Taylor Series and Parametric Functions
E. Karakulak1 and R. Mutlu2
1
Vocational School of Technical Sciences, Namik Kemal University, Tekirdag, Turkey, [email protected]
Department of Electronics and Telecommunication Engineering, Namik Kemal University, Tekirdağ, Turkey ,
[email protected]
2
Abstract—Memcapacitor is a two-port circuit element with
a memory. When excited by AC current or voltage, it has a
zero-crossing charge-voltage hysteresis curve. The
memcapacitor hysteresis curve should be taught to
engineering students or engineers, who want to work in this
area. In this work, an easily-comprehensible explanation of
hysteresis curve of flux-dependent memcapacitor is given
using Taylor series and parametric functions.
Keywords—Memcapacitor,
Memcapacitive
Hysteresis Curve, Engineering Education.
fourth section, the memcapacitor frequency behavior is explained.
Voltage-current relationship is also shown by drawing it. The
paper is finished by Conclusion section.
2. MEMCAPACITIVE SYSTEMS AND FLUX-DEPENDENT
MEMCAPACITOR SYSTEMS OR MEMCAPACITORS
Ventra et al have described an nth order voltage-controlled
(voltage dependent) memcapacitive system as
systems,
q(t )  C ( x,V , t )V (t )
1. INTRODUCTION
.
x  f ( x,V , t )
After Dr. Chua claimed that a fundamental circuit element
called memristor must exist in 1971, he and Kang have claimed
that systems called memristive systems with similar properties to
memristors exist and a memristive system should have a zerocrossing current-voltage hysteresis curve in 1976 [1-2]. Almost
37 years later, an HP research team has announced that they
found the missing circuit element [3]. A review on emerging
memristor can be found in [4]. In 2009, Ventra, Pershin and Chua
has written a paper called “Circuit elements with memory” and
claimed that there must be circuit elements inductors and
capacitors with memories beside and with similar properties to
memristors and called them meminductor and memcapacitor [5].
Memcapacitor’s terminal equation and charge-voltage hysteresis
curve are also given by them [5]. Although meminductor has not
been realized in nano dimensions, research on ionic and solidstate memcapacitors are exist in literature [6-7]. Some papers on
memcapacitor are about its modeling and its usage in chaos
circuits [8-12].
In [13], the explanation and drawing of a memristor’s
hysteresis curve is done using Taylor series and parametric
functions. Memcapacitor charge-voltage curve has also a zerocrossing hysteresis curve under ac excitation [5]. To the best of
our knowledge, there is no method in literature given for
explanation and drawing of a memcapacitor’s hysteresis curve.
Such a method would be useful for the best-usage and teaching of
this circuit element. The method used in [13] is easily applicable
to memcapacitor, too. In this paper, the same method is used for
memcapacitor. This paper is arranged in the following way. . In
the second section, general desription os a memcapacitive system
is given and flux-dependent memcapacitor is introduced. In the
third section, using Taylor series expansion, the memcapacitor
hysteresis curve is investigated, explained and drawn. In the
(1)
(2)
Where q(t) is the charge on the capacitor at time t, V(t) is the
corresponding voltage, x is a vector representing n internal state
variables and C is the memcapacitance (short for memory
capacitance), which depends on the state of the system.
A commonly known example for a memcapacitive system is a
variable capacitor or a variac used for frequency tuning in old
radios. In addition, they have defined a subclass of the
memcapacitive systems with memcapacitance being only a
function of memcapacitor flux, and called them voltage-controlled
memcapacitors, when Eqs. (1) and (2) reduce to
 t

q(t )  C   Vc( )d V (t ) ,
 

(3)
q  t   C    t  V  t 
(4)
or

Where  t is the memcapacitor flux, which is the integral of
memcapacitor voltage by respect to time, V(t) is the memcapacitor
C ( ) is the flux-dependent memcapacitor
voltage and
memcapacitance.
Memcapacitor symbol is shown in Figure 1 and its pinched chargevoltage hysteresis curve obtained by simulation is given by Ventra
et al shown in Figure 2 [4].
419
E. Karakulak, R. Mutlu
where
o is average flux
and calculated as
T
1 e
o   V  t  dt
Te 0
Figure 1: Memcapacitor Symbol [4].
(11)
If voltage equation is submitted in (11),
 t   
If
 t 
and
V t 
Vm

(12)
cos t   o
are submitted in (4), the memcapacitor
charge,

V


q  t   C0  K  0  m cos t    Vm sin t 




(13)
KVm2
sin  2t 
2
(14)
q(t )   Co  K 0 Vm sin t  
Figure 2: Memcapacitor’s Pinched Charge-voltage Hysteresis
Curve given in [4].
If electrical angle is designated as
voltage and charge are
3. EXPLANATION OF MEMCAPACITOR’S PINCHED
HYSTERESIS CURVE USING TAYLOR SERIES AND
PARAMETRIC FUNCTIONS AND DRAWING OF FLUXDEPENDENT MEMCAPACITOR HYSTERISIS CURVE
  t ,
the memcapacitor
V    Vm sin  
(15)
q    K1sin    K2 sin  2 
(16)
and
As indicated before, a method is needed for explanation and
drawing of a memcapacitor’s pinched hysteresis curve. If
memcapacitor memcapacitance is expanded into Taylor series,
C k (o ) ( - o ) k
q
k!
k 1

C( )  C (o )  
(5)
Where K1=Co.Vm and
C (o ) ( - o )1
q
1!
(6)
Renaming
C
K,
q
(7)
memcapacitor memcapacitance can be written as a linear function
similar to the Hp memristor memristance. Then, its
memcapacitance is given as,
C     C0  K   t 
(8)
To obtain its hysteresis curve, a periodic signal must be applied to
the memcapacitor. If a sinusoidal signal is applied,
V (t )  Vm sin t 
t
dq  
 K1cos    K 2 2cos  2 
d
(17)
dV  
 Vm cos  
d
(18)
and
(9)
Memcapacitor flux is then obtained as
  t    V  t  dt  o
.
These equations are parametric equations depending on  as
parameter.
A careful look at Eqs. (15) and (16) shows that both
parametric equations are zero when   0 . This can easily be
interpreted as charge-voltage hysteresis curve goes through the
origin. Both the memcapacitor charge and the memcapacitor
voltage have a period of 2  radian by respect to electrical angle.
In addition to that, the functions are odd-functions and the curve is
symmetric by respect to origin. Therefore, if the curve is drawn for
the range of [0,  ] and its symmery is taken by respect to origin,
the full curve can be obtained.
If the derivatives of the memcapacitor charge and the
memcapacitor voltage parametric functions by respect to the
electrical angle,  , are taken,
If the second or more order terms are ignored, the
memcapacitance can be approximated as
C( )  C (o ) 
KVm2
K2  
2
are obtained. Then, the tangent of the hysteresis curve is
(10)
m
0
420
dq  
d
dv 
d

K1cos    2 K 2cos  2 
Vm cos  
(19)
Explanation of Hysteresis Curve of a Flux-dependent Memcapacitor (Memory-capacitor)….
The points where the tangent of the curve is zero are the
extremum points. They can be found by finding the electrical
angle which makes (19) equal to zero. Remembering that
cos  2   2cos 2    1
(20)
Making (19) zero and plugging (20) into (19);
0  2cos 2   
K1
cos    1
2K2
(21)
This equation has two roots and they are equal to
Figure 4: The memcapacitor’s hysteresis curve drawn for Vm=0.25
V, K1=0.75, and K2=0.25.
2
1 
 K1 1  2 K1 

 8,

8K 2 4  K 2 
(22)
4. MEMCAPACITOR’S FREQUENCY RESPONSE
and
The method also allows us to investigate the effect of the
frequency. If Eq. (14) is used,
2
 K1 1  2 K1 

 8,
2 

8K 2 4  K 2 
(23)
q(t )   C0  K 0 Vm sin t  
The hysteresis curve can be drawn using these values. The
specific values of the parametric functions within the range of
[0,  ] needed to draw the hysteresis curve is given in Table 1.
Using its values, the hysteresis curve of the parametric equations
of (15) and (16) can be drawn for an alternance and, by taking its
symmetry by respect to origin, the hysteresis curve for a full
period can be obtained as shown in Figure 3.
Also, the memcapacitor’s hysteresis curve is drawn for
hypothetical values of Vm=0.25 V, K1=0.75, and K2=0.25 and
shown in Figure 4.
KVm2
sin  2t 
2
(24)
If the electrical angular speed,  , or the electrical frequency,
increases so much, Eq. (24) turns into
q(t)  C  0 Vm sin t  .
(25)
This is the terminal equation of a linear time-invariant
capacitor. Eq. (25) means that a memcapacitor starts behaving as a
linear time-invariant capacitor. Its hyesteresis curve ges narrower
by increasing frequency as shown in Figure 5. At high frequencies,
its curve is almost same as that of a linear time-invariant
capacitor’s with a capacitance value of C0  .
Table 1: The specific values of the parametric functions within
the range of [0,  ].
Figure 5: The memcapacitor charge-voltage curve for several
frequencies.
To see the memcapacitor’s current-voltage curve, the
memcapacitor current is calculated by taking the derivative of the
memcapacitor charge given by Eq. (24);
dq(t ) ,
dt
i  CO  KO  cos(t )  KVm2 cos(2t ) .
(26)
i  K1 cos( )  K 2 2 cos(2 ) .
(28)
i(t ) 
Figure 3: Memcapacitor Hysteresis Curve Drawn by Using
Parameteric Functions.
421
(27)
E. Karakulak, R. Mutlu
Using Eqs. (9) and (28), the memcapacitor’s current-voltage
curve under ac excitation is simulated and shown in Figure 6. It is
different than that of a linear time-invariant capacitor. The curve
would be an ellipse for LTI capacitor. However, when the
frequency of memcapacitor voltage source increases, the currentvoltage curve turns into an ellipse as that of a LTI capacitor as
shown in Figure 7 and 8. The reason for this is that while the
frequency increases, the fundamental current increases and the
second harmonic current stays the same. This is also an important
property of a memcapacitor to teach since the current-voltage
curve of a LTI capacitor is commonly taught in circuit and/or
circuit laboratory courses.
5. CONCLUSION
Memcapacitor’s hysteresis curve for small signals is explained
easily using Taylor series and parametric functions. The chargevoltage hysteresis curve shows that it has zero-crossing property.
Also this method is able to explain the frequency behavior of fluxdependent memcapacitor. At high frequencies, the narrowed
hysteresis curve looks no different than a linear capacitor’s since it
behaves almost as a linear capacitor. The parametric memcapacitor
hysteresis curve is quite similar to the one given in [5]. Also, the
current-voltage curve of a memcapacitor is drawn and it is shown
that it is different than that of an LTI capacitor. However, it turns
into an ellipse as that of an LTI capacitor as frequency increases.
The circuit elements with memory such as memristor,
memcapacitor, and meminductor should be taught to today’s
students, the future’s engineers. New methods are needed for
teaching them easily and effectively. This method given in this
paper can be used to teach students on memcapacitor hysteresis
curve for this purpose.
Memcapacitor Current (V)
0.5
0
REFERENCES
-0.5
-1
-0.5
0
Memcapacitor Voltage (V)
[1] L. O. Chua, ”Memristor - The Missing Circuit Element,” IEEE Trans.
Circuit Theory, vol. 18, pp. 507-519, 1971.
[2] L. O. Chua and S. M. Kang, ”Memristive devices and systems,”
Proc.IEEE, vol. 64, pp. 209-223, 1976.
[3] D. B. Strukov, G. S. Snider, D. R. Stewart, and R. S. Williams, ”The
missing memristor found,” Nature (London), vol. 453, pp. 80-83, 2008.
[4] The fourth element: characteristics, modeling and electromagnetic
theory of the memristor, O. Kavehei, A. Iqbal, Y.S. Kim, K. Esraghian,
S.F. AL-Sarawi and D. Abbott, Proc. R. Soc. A, publishes
[5] M. Di Ventra ,Yu. V. Pershin and L. O. Chua “Circuit Elements With
Memory: Memristors, memcapacitors and meminductors” Proc. IEEE, vol.
97, pp. 1717–1724, 2009.
[6] J Martinez, M Di Ventra. Pershin, Yu. V., “Solid-state memcapacitive
system with negative and diverging capacitance”, Physical Review B,
adsabs.harvard.edu, 2010.
[7] Matt Krems, Yuriy V. Pershin and Massimiliano Di Ventra, “Ionic
Memcapacitive Effects in Nanopores” Nano letters, ACS Publications,
2010.
[8] D.Biolek, Z. Biolek and V. Biolkova “SPICE Modeling of Memristive,
Memcapacitative and Meminductive Systems”, European Conference on
Circuit Theory and Design, ECCTD 2009, page(s):249–252, 23-27 Aug.
2009.
[9] Blaise Mouttet “ A Memadmittance Systems Model for Thin Film
Memory Materials”, Arxiv.
[10] Pershin, Y.V.Di Ventra, M.; “Memristive circuits simulate
memcapacitors and meminductors” Electronicsletters, Volume: 46, Issue:
7, page(s): 517– 518, 2010.
[11] Zhiheng Hu, Yingxiang Li, Li Jia, and Juebang Yu, “ Chaos in a
charge-controlled memcapacitor circuit”, International Conference on
Communications, Circuits and Systems (ICCCAS), Chengdu, pages: 828–
831, July 2010.
[12] Zhiheng Hu, Yingxiang Li, Li Jia, and Juebang Yu, ,“Chaotic
oscillator based on voltage-controlled memcapacitor”, International
Conference on Communications, Circuits and Systems (ICCCAS),
pages: 824-827, July 2010.
[13] Reşat MUTLU, “Taylor Serisi ve Kutupsal Fonksiyonlar Kullanarak
Memristorün (Hafızalı Direncin) Histeresis Eğrisinin Açıklanması”, 3. Ileri
Muhendislik Teknolojileri Sempozyumu, 29-30 Mayis 2010, Cankaya
Universitesi, Ankara.
0.5
Figure 6: The Memcapacitor Current vs. The Memcapacitor
Voltage for K1=0.625, K2=-0.125, Vm=0.5 V and ω=1 rad/s.
3
2
Memcapacitor Current (V)
1
0
-1
-2
-3
-4
-0.5
0
Memcapacitor Voltage (V)
0.5
Figure 7: The Memcapacitor Current vs. The Memcapacitor
Voltage for K1=1.25, K2=-0.125, Vm=0.5 V and ω=5 rad/s.
40
30
Memcapacitor Current (V)
20
10
0
-10
-20
-30
-40
-0.5
0
Memcapacitor Voltage (V)
0.5
Figure 8: The Memcapacitor Current vs. The Memcapacitor
Voltage for K1=5, K2=-0.125,
Vm=0.5 V and ω=50 rad/s.
422