Journal of the Franklin Institute 336 (1999) 1035}1047
Brief communication
A nonlinear resistor and nonlinear inductor
using a nonlinear capacitor
Emanuel Gluskin *
Department of Electrical and Computer Engineering, Ben-Gurion University of the Negev, P.O. Box 653,
84105 Beer-Sheva, Israel
The Center for Technological Education, Holon, Israel
Received 6 July 1998; received in revised form 15 July 1999
Abstract
Ferroelectric capacitors whose rated voltage is basically de"ned by the requirement of a fair
linearity of the (generally nonlinear) capacitors' voltage}charge characteristic, not by the
breakdown voltage, which can be much higher, are widely used today. It is noted, and
considered with some details, that the use of such a capacitor in a simple switched-capacitive
circuit which imitates a resistance enables a nonlinear frequency-controllable resistance to be
obtained, and that the use of such a capacitor as a load for a gyrator circuit enables a nonlinear
inductor to be obtained. The topic is interesting from the basic circuit theory aspect, in view of
the fact that ferroelectric ceramic capacitors are widely used today, and because of the need to
"nd ways for integral implementation of di!erent nonlinear elements. 1999 The Franklin
Institute. Published by Elsevier Science Ltd. All rights reserved.
1. Introduction
Ferroelectric ceramic capacitors (see Ref. [1] and the catalogs of e.g. AVX) are
widely used today, in particular in circuits where high capacitance precision is not
required (rough "ltration, etc.). These capacitors have already found also some
nontrivial applications as nonlinear elements. They are thus used in electronic starters
for #uorescent lamps [2] and in snubbers for power electronic switches [3}5].
Experimentation with circuits containing such capacitors is also very interesting (e.g.
* Correspondence address: Department of Electrical and Computer Engineering, Ben-Gurion University
of the Negev, P.O. Box 653, 84105 Beer-Sheva, Israel. Fax: #972-7-6472949.
E-mail address:gluskin@ee.bgu.ac.il (E. Gluskin)
0016-0032/99/$ - see front matter 1999 The Franklin Institute. Published by Elsevier Science Ltd. All rights reserved.
PII: S 0 0 1 6 - 0 0 3 2 ( 9 9 ) 0 0 0 2 9 - 0
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E. Gluskin / Journal of the Franklin Institute 336 (1999) 1035}1047
Ref. [6]). The present work is intended to suggest two more applications of these
nonlinear capacitors, which, from the positions of circuit theory, seem to be most
straightforward and even logically necessary, but in some details require a deep
investigation. In these applications, which may be interesting for nonlinear modeling,
nonlinear signal treatment, etc., the capacitors are used to create a nonlinear (frequency controllable) resistance, or a nonlinear inductance. Though we are using
circuits whose linear versions are well known, the nonlinearity raises some nontrivial
points, in particular regarding the frequency range where the circuits can be employed.
In order to use the nonlinear features of a ferroelectric capacitor we require the
voltage actually obtained on the capacitor to be higher than its rated voltage, stressing
that for such capacitors the rated voltage is limited by the requirement that the
voltage}current characteristic of the capacitor be fairly linear, and that this voltage is
usually much lower (sometimes up to 10 times) [7] than the electrical breakdown
voltage.
We shall characterize the ferroelectric capacitor via its nonlinear voltage}charge
characteristic t (q) which usually is close to a third or a "fth degree polynomial with
positive coe$cients:
1
t (q)" q#aq
C
(1)
or
1
t (q)" q#aq#bq.
(2)
C
C , a, b'0, C is the linearized capacitance near q"0. These characteristics pos
sess a monotonically increasing derivative, and the property t (q)&q/C as qP0.
Both of these features are analytically important for the estimation of the frequency
range in Section 3 and Appendix.
In some places we shall use the characteristic t (q) phenomenologically, and in
some places we shall use the analytical form (1)}(2). Using the analytical form, we shall
sometimes assume the nonlinearity to be strong, and sometimes weak. This is clearly
stated in each case.
Our main targets are:
1. to show that the nonlinear resistor is simply obtained from the nonlinear capacitor,
using a switched capacitor circuit (SCC), with the nonlinearity of the resistor being
directly de"ned by that of the capacitor,
2. to investigate the frequency range of the operation of this circuit,
3. to develop the work formulae for the further case of the realization of a nonlinear
inductor, using a gyrator.
Being motivated by the natural questions, arising from circuit theory, this communication cannot touch the technological di$culties associated with the desirable
(though not immediately necessary) integrated implementation of the circuits. It is
E. Gluskin / Journal of the Franklin Institute 336 (1999) 1035}1047
1037
worthwhile, however, to point to the theoretical investigations, e.g., in the works
[8}11], where nonlinearity of a capacitor is obtained by means of switching linear
capacitors at certain voltage values. The circuit methodology of these works, which is
certainly implementable in integral technology, is close, regarding the basic idea, to
our application in Section 2.
The analytical nonlinear characteristic may be realized in a piecewise-linear form
using voltage-level-dependent switching of linear capacitors, but this is complicated,
and thus there is the motivation to attempt to incorporate ferroelectric-layer technology into integrated circuits for nonlinear operations.
Similar comments relate to comparison of Kuboshima and Saito [12] with our
results in Section 4 here. From the point of view of integral implementation, the
advantage of the realization of nonlinear inductors using capacitors, can be even more
important.
2. The realization of nonlinear resistance
Fig. 1 shows a switched capacitor circuit (SCC) which is well known in its linear
version. In the circuit the switches S and S are operated in turn, `f a times per
second, "rst charging the capacitor `Ca to the voltage < (actually very close to that)
via S and then discharging it via S . Since for a linear capacitor the average current,
taken from the voltage source, is fC<, the linear &&resistance'' which `seesa the source is
</( fC<)"1/fC.
For the nonlinear capacitor we obtain in the switched circuit the average current
taken from the source as
i(<)"f ) q(<),
(3)
where q( ) ) is the inverse function with respect to t ( ) ). Eq. (3) presents the nonlinear
conductivity characteristic of the switched circuit, created by the nonlinear capacitive
Fig. 1. The switched capacitive circuit. The capacitor which is periodically charged via S and discharged
via S takes a certain average current from the voltage source, which evokes the respective resistance. For
a linear capacitor we obtain a linear resistor, and for a nonlinear capacitor a nonlinear one. In the latter case
it is more suitable to consider the resistivity (or conductivity) characteristic t(i) (or i(t)).
1038
E. Gluskin / Journal of the Franklin Institute 336 (1999) 1035}1047
circuit. Inverting Eq. (3) and considering (for any more general application) the
battery voltage as a voltage variable t, we obtain, using the function t (q),
t(i)"t
i
f
(4)
which is the nonlinear resistivity characteristic of the SCC.
Changing &&f '' we make only scaling changes; this does not in#uence the general
form of the characteristics i(t) and t(i).
Nonlinear resistors are often used in electrical modeling of physical processes, e.g.
chaotic processes. Usually, nonlinear resistors are created there using ampli"ers.
Despite some circuit complexity, the use of ampli"ers provides more #exibility than is
present here in the creation of di!erent forms of the nonlinear voltage}current
chracteristic t (i). However, here too we have some #exibility, since, using series or
parallel (or mixed) connection of the nonlinear capacitors, we can change the total
t (q) characteristic, as is explained in Ref. [2].
3. A limit for the frequency range in which the modeled resistor can be operated
The simplicity of the inclusion of the nonlinear case in the scheme of the SCC
resistor modeling is somewhat deceptive. Since `ia in Eqs. (3) and (4) is an average
current, the resistor, either linear or nonlinear, realized by the switched circuit is not
a real resistor, and in order to apply the SCC as a resistor in an enlarged circuit, where
any desirable process in which the SCC is involved occurs, we have to "nd a limit for
the frequency range for the process, for which the ripple of the instantaneous current
function could be ignored. For the nonlinear case the issue is not quite analytically
trivial. Besides the quantitative side, there is a qualitatively interesing distinction
between the nonlinear and linear cases. For the strong nonlinearity the discharging
process is dominant concerning the time durations of the transient processes that limit
the frequency range. This is because the duration of the discharge process is mostly at
the low voltages where the di!erential capacitance (equal to C ) is relatively large,
causing a slower process, while for the charging process, where in the most essential
time interval the voltage is high, the di!erential capacitance is relatively small, and the
process quicker.
Denoting the typical frequency of the `externala process as f , we have to require,
in view of the problem of the ripple, that
f ;f.
This is certainly necessary if we wish to deal with the averaged current.
By itself `f a must satisfy the obvious inequality
1
f;
q #q
(5)
E. Gluskin / Journal of the Franklin Institute 336 (1999) 1035}1047
1039
with q
related to the two basic processes in the SCC, i.e. q is the characteristic
period of the charging, and q is the characteristic period of the discharging of the
capacitor.
We thus have to require that
1
f ;
.
q #q
(6)
For the case of the linear capacitor it is conventional to take
q "q "RC,
where R is the resistance (assumed to be linear) of the switches, which we assume to be
similar, in the conductive state. Then Eq. (6) turns into
1
.
f ;
2RC
In order to treat the nonlinear case, we introduce a speci"c parameter k using which
we shall de"ne the charging and the discharging periods, now denoted as q and q ,
I
I
respectively, noting that these parameters should be applied to the consideration of
the linear case also.
Considering the charge-function of the capacitor q(t) during the charging process,
we shall connect k and q by the equality
I
q(q )"kq , 0(k(1,
(7)
I
where q
is the asymptotic `"nala charge on the capacitor, de"ned by the equality
(8)
t (q )"<
and where we assume that at the beginning of the charging process q(0)+0. The
system of equations (A.1) in the appendix presents a precise de"nition of q .
I
Thus, q(q ) is required to be a certain part, characterized by k, of the q . The
I
value of k is dictated by an engineering reason. Such a reason need not require,
of course, k to equal 1/e even in the linear case where the process is exponential,
and the more so in the nonlinear case. Engineering requirements should dictate k
to be rather close to 1, in order to e!ectively use the battery voltage, and actually,
in the derivation of the expression 1/fC for the modeled linear resistance we meant
such a k.
For simplicity we de"ne the discharging time constant using the same k, i.e.
q(q )"(1!k)q ,
(9)
I
where q(t) is now the solution of the discharge problem described in the appendix by
system of equations (A.3), which presents a precise de"nition of q .
I
In these concepts, we "nd in the appendix analytical expressions for q
, using the
I
relevant circuits' equations, and we "nd from these expressions some `'a-type
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E. Gluskin / Journal of the Franklin Institute 336 (1999) 1035}1047
inequality for q #q , which corresponds, according to Eq. (6), to the `(a-type
I
I
inequality for f . The "nal result for kP1 is as follows:
1
f ;
.
R[(dt /dq"
)\#(dt /dq" )\] ln(1!k)
O
OO
For model (1), in the case of strongly nonlinearity, this is
1
f ;
RC ln(1!k)
(kP1).
4. Realization of a nonlinear inductance
Nonlinear capacitors may also be used for the creation of nonlinear inductors,
using the well-known gyrator circuit, as is shown in Fig. 2. The equations of the
gyrator (i is directed as is accepted in the theory of 2-ports) are [14]
1
t " (!i ),
G
i "Gt .
Because of the capacitive load, t "t and
dq
!i " ,
dt
(10)
(11)
(12)
where `qa is the capacitor's charge.
Using, sequentially, Eqs. (11), (12) and (10), we obtain
di
dt
dt (q)
"G "G dt
dt
dt
Fig. 2. The ideal gyrator loaded by the nonlinear capacitor. This circuit imitates a nonlinear inductor, or
a nonlinear #ux-current dependence.
E. Gluskin / Journal of the Franklin Institute 336 (1999) 1035}1047
"G
1041
dt dq
dt
"G (!i )
dq dt
dq
"G
dt
t ,
dq i.e.
dq di
.
t "G\
dt dt
Since, further,
(13)
dq(t ) dq(t ) dq(G\i )
"
"
dt
dt
d(G\i )
dq(G\i )
,
"G
di
we obtain from Eq. (13)
t "G\
dq(G\i ) di
di
dt
t "G\
dq(G\i )
dt
or
(14)
which has to be compared with the usual inductor described by the equation
dt(i)
t"
dt
or
t"¸(i)
di
dt
, ¸" ,
dt
di
where t is the magnetic #ux multiplied by the number of turns of the winding of the
inductor.
We have here the correspondence (actually equality, in the modeling sense, after
replacement of i by i)
G\q(G\i ) t(i)
(15)
or
dq(G\i )
¸(i)
(16)
di
depending on whether one prefers to use the nonlinear function ¸(i), or directly
t(i).
G\
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E. Gluskin / Journal of the Franklin Institute 336 (1999) 1035}1047
Thus the modeled inductor's characteristic is de"ned by q(t )(q( ) )).
For the linear case where q(t )"Ct we obtain from Eq. (16) the well-known result
¸"G\C.
For the nonlinear case, we consider (1) and, assuming that in the work region the
nonlinearity is weak, i.e.
1
aq; q,
C
we rewrite Eq. (1) as
q"C t !bt
with
b"aC .
This turns Eqs. (15) and (16), respectively, into
((i)"C G\(i!aG\C i)
and
¸(i)"C G\(1!3G\aC i)
with the relatively small nonlinear terms.
The current-inversion symmetry properties, ((!i)"!((i) and ¸(!i)"¸(i), obtained due to the features of the realistic characteristic t (q), are typical for also the
(either linear or nonlinear) ((i) and ¸(i) which would be obtained in a usual inductor
which employs magnetic #ux.
The known possibility of creating the gyrator as an integrated-circuit module,
associated with the ease of integrated-circuit implementation of the layer-structured
capacitor, suggests that the nonlinear inductor can be integrated into a module.
5. Conclusions
Considering the fact that nonlinear ferroelectric capacitors are easily available
today, and considering the ease in integrated-circuit implementation of a capacitor,
we have suggested realizations of a nonlinear resistor/conductor and a nonlinear
inductor by means of the nonlinear capacitor employed in either an SCC or a gyrator.
The characteristic of each of the elements modelled is directly given by the capacitor's
voltage}charge characteristic.
These results are strongly di!erent from, for instance, the results of Gregorian and
Temes [13] related to nonlinear capacitance inaccuracies in SCC, associated with the
physics of MOS implementation. (The results of Gregorian and Temes [13] are
relevant to the precision of the formulae derived in Ref. [8].)
The switched character of the circuit which models the nonlinear resistor suggests
a limit for the operational frequency of such a resistor. In the derivation of the limit we
E. Gluskin / Journal of the Franklin Institute 336 (1999) 1035}1047
1043
have assumed that the switches S and S (see Fig. 1) can be characterized, in the
conductive state, by a linear resistance.
In some detailed estimations we used a polynomial model for t (q). For the model
used, the complete charging and discharging processes would require, in principle, an
in"nite time, as in the linear case when we have exponential damping processes. The
maximal and the minimal charges which may be obtained in the capacitor in the
switched circuit are de"ned using a speci"c parameter (k) and the asymptotic charge
value de"ned by Eq. (8). The associated time constants of the charging and discharging processes are then precisely de"ned by Eqs. (A.1) and (A.3). Formulae (3)}(4) and
(15)}(16) are correct for any unique t (q). For correctness of formula (A.6) it is
su$cient to require, additionally, t (q) to be monotonically increasing. Asymptotic
formulae for the important case of k very close to 1 (kP1) are also derived.
A simple qualitative di!erence between the nonlinear and linear cases is that the
nonlinearity causes (Section 3) the charging and discharging processes to have
di!erent durations, which is because during the charging process the nonlinear
element is, most of the time, under higher voltage. In the present case, the nonlinearity
is such that this causes the charging process be much quicker. This is a good
heuristical point for understanding nonlinear circuits, and it allows one to simplify
estimation of the frequency range for operation of a strongly nonlinear switched
circuit.
Among the analytical points, related to the switched circuit, which can be suggested
for further investigation, there is the interesting and di$cult question of the meaning
of the `;a inequality for f when the nonlinear modeled resistor is in a circuit
involved in a chaotic process. A problem which can appear here, is considered in
Ref. [15].
The possibility of the realization of a nonlinear inductor by means of the capacitor
and gyrator, suggested in section, is also relevant to the topic of the integrated
implementation of nonlinear (e.g. oscillatory) circuits.
The technological di$culties of the implementation cannot be discussed in this
Communication. However, "nding new elements for integrated implementation is
a timely topic. It is argued in Ref. [16] that integral implementation of nonlinear
elements, which allows us to create multi-element nonlinear circuits, will be promising
for the electronic modeling of statistical processes.
Regarding the realistic t (q) characteristic of the ferroelectric capacitor, it has to be
commented that such a characteristic usually possesses [1] some weak hysteresis,
which we have not considered here. However:
1. As the investigation of Gluskin [2] shows, ignoring the hysteresis may still give
good agreement with experiment, even when the capacitor has, as it can in the
speci"c application of Gluskin [2], a noticeable hysteresis.
2. As in the case of magnetic hysteresis, the dielectric hysteresis is associated with
power losses inside the mass of the material. Thus the strength of the hysteresis of
t (q) is associated with the volume of the capacitor's dielectric layer, and the thinner
the layer, the weaker the hysteresis. For integrated-circuit capacitor performance,
when the layers are very thin, we can expect the hysteresis to be negligible.
1044
E. Gluskin / Journal of the Franklin Institute 336 (1999) 1035}1047
It is hoped that the possibilities noted will contribute to electronic modeling,
"ltering, and other developments, and one can expect that this, together with the ease,
today, of "nding ferroelectric capacitors, will lead to novel device applications,
and*last but not least*will encourage the manufacturers to give a speci"cation and
guarantee for di!erent important uses of the capacitors in the nonlinear region of its
characteristic t (q).
Acknowledgements
The author is grateful to the anonymous referees for their valuable comments.
Appendix A. Derivation of the inequality for fop
Since the charging and the discharging processes are parts of the general periodic
process in the SCC, and the capacitor's voltage is a continuous function, the initial
and "nal values for the particular processes must be adjusted.
For the charging we have (see Fig. 1) for q(t):
R
dq
#t (q)"<,
dt
q(0)"(1!k)q ;
q(q )"kq .
I
Separating variables we obtain
(17)
IO dq
.
(18)
<!t (q)
\IO It is easy to show that since (dt /dq)(q )O0 for models (1)}(2), when taking in the
upper limit of the integration k"1 we would obtain in view of Eq. (8) a logarithmic
integral singularity at the upper limit, which would correspond to the in"nite period
of the complete (up to <) charging of the capacitor in the nonlinear case, as it is in the
linear case. Contrary to that, a weak change in the lower limit of the integration
relatively weakly in#uences q .
I
For the discharging we have
q "R
I
R
dq
#t (q)"0
dt
q(0)"kq ,
q(q )"(1!k)q .
I
Integrating and inverting the limits of the integration, we obtain
q "R
I
IO dq
.
t (q)
\IO (19)
(20)
E. Gluskin / Journal of the Franklin Institute 336 (1999) 1035}1047
1045
The value of q is "nite because of the positive lower border of the integration
I
(since t (q)&q, as qP0, taking k"1, or the lower border as zero, we would have
a logarithmic singularity here also), and in this case the precision in the upper limit
does not very strongly in#uence the time constant.
From Eqs. (18) and (20)
IO dq
.
(21)
t
(q)[<!
t (q)]
\IO For k very close to 1, in the evaluation of the integral large terms of the type
ln(1!k) appear. Assuming that k is not so close to 1 that these terms would be
dominant, we can suggest a very simple estimation.
In this estimation we use that since t (q) and <!t (q) are essentially positive
inside the interval of the integration, the known inequality for positive numbers,
q #q "R<
I
I
(ab) (a#b)
can be applied. This gives for the denominator in the integrand
1
<
t (q)[<!t (q)]) (t (q)#<!t (q))"
4
4
while the equality is obtained only for t (q)"</2. This gives, in view of Eq. (21) the
inequality
4R IO 4R
q #q '
dq" (2k!1)q .
<
<
\IO In view of Eq. (6) we obtain for f
<
f ;
.
4R(2k!1)q
For the linear case q "C<, and we have
1
.
f ;
4(2k!1)RC
(22)
For the nonlinear characteristic (1), assuming (consider Eq. (8) for the characteristic)
that aq <C\q , i.e. that aq +<, and thus q +(</a), we obtain from
Eq. (22)
<a
f ;
.
4R(2k!1)
For k very close to 1, more precisely estimating the integral in Eq. (21) we would
obtain in (22) and in what follows it an expression of order ln(1!k) in the denominators, instead of 2k!1. This gives, of course, a more precise (lower) upper limit for f ,
for such a k. For this estimation it is suitable to refer directly to Eqs. (18) and (20),
replacing the integrand there by the function which is asymptotically precise in the
1046
E. Gluskin / Journal of the Franklin Institute 336 (1999) 1035}1047
dominant part of the interval of the integration, where the integrand is large. This
gives asymptotically precise expressions, as kP1, for both q and q .
I
I
For Eq. (18) the dominant interval is near q , and using the continuity of the
derivatives of dt /dq and that t(q )"< we write
dt
dt
<!t (q)+<! <! (q!q ) " (q!q ).
dq dq O
O
This gives
\
dt
ln"q!q "
!ln"q!q "
q &R
OIO O\IO I I
dq O
\ 1!k
dt
"R
ln
dq k
O
\
dt
ln(1!k).
&R
dq O
For Eq. (20), where the dominant interval is near q+0, we write (see Eqs. (1) and
(2))
[
"
"
]
1
t (q)+ q
C
and obtain similarly
q & RC ln(1!k).
I I
We observe that for the strong nonlinearity in Eq. (1) when aq <C\q , i.e.
aq <C\,
\
dt
1
C
"
; (C
dq
3aq
3
OO and thus for kP1, q ;q .
I
I
In general, for kP1 the inequality for f is
1
f ;
.
R[(dt /dq"
)\#(dt /dq" )\]ln(1!k)
O
OO
For model (1), in the case of strongly nonlinearity, this is
1
f ;
(kP1),
RC ln(1!k)
since we can ignore the small "nal di!erential capacitance (dt/dq" )\"
OO
1/3aq "3a\<\ in this case.
E. Gluskin / Journal of the Franklin Institute 336 (1999) 1035}1047
1047
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