The document summarizes experiments on piezoelectric actuators that observe zero creep in certain locations of the hysteresis loop. Specifically: 1) Relaxation experiments using an offset sinusoidal voltage found relaxation increased hysteresis loop tilt but did not reduce hysteresis. Zero creep was observed at hysteresis loop extremum. 2) Creep experiments varying delay time also increased loop tilt but zero creep locations did not change and coincided with points where loop slope equaled virgin curve slope. 3) Increasing input voltage amplitude did not affect zero creep behavior or virgin curve slope. This characteristic was independent of experiments.

1. Appl. Phys. A 68, 691–697 (1999) / DOI 10.1007/s003399900049
Applied Physics A
Materials
Science & Processing
© Springer-Verlag 1999
Observation of zero creep in piezoelectric actuators
K.R. Koops1,∗ , P.M.L.O. Scholte1 , W.L. de Koning2
1 DelftInstitute of Microelectronics and Submicrontechnology, Faculty of Applied Physics, Delft University of Technology, Lorentzweg 1,
2628 CJ, Delft, The Netherlands
(Fax: +31-15/278-3251)
2 Mathematical System Theory Group, Faculty of Information Technology and Systems, Delft University of Technology, Delft, The Netherlands
Received: 15 February 1999/Accepted: 16 February 1999/Published online: 28 April 1999
Abstract. Piezoelectric actuators are frequently used nowa- is the main source of non-linear and hysteretic behavior of
days in a wide variety of positioning devices. Although very the translation device we have developed a mathematical de-
suitable for small displacements in the range of nm to several scription for the piezo hysteresis, adapted from the field of
hundreds of µm, the actuators always suffer from hysteresis ferromagnetic hysteresis, to be used in a model-based dig-
and creep between the input voltage and resulting displace- ital controller. In previous studies it has been shown that
ments. In scanning applications, the input voltage is often this model provides a satisfactory description for both mag-
used as an indicator of the induced displacement. This pro- netic [1] and piezoelectric hysteresis [2] in the limit of small
cedure can result in a large position error depending on the input signals, i.e. far from the region of saturation. How-
amount of hysteresis and creep. In order to describe and con- ever, relaxation effects are not incorporated in this model. In
trol hysteretic systems various models for hysteresis have order to be able to augment the model, sufficient informa-
been published but little is known about relaxation and creep tion about relaxation effects in piezoelectric materials has to
in piezo materials. In this paper we present detailed studies be obtained. Furthermore, the experimental conditions in the
of the hysteretic behavior and piezo relaxation and creep. We study of ferromagnetic materials can be devised such that the
have identified certain locations on the hysteresis loop that so-called anhysteretic curve is formed [3, 4]. As the name im-
exhibit zero creep. From this observation, a more fundamen- plies, this curve in the B–H plane does not exhibit hysteresis
tal relation between the amount of creep and the local slope (although still non-linear). The experimental conditions in-
of the hysteresis loop and the virgin curve is presented. This volve the application of a relaxation signal in the form of an
observation could be useful in both open-loop and closed- ac magnetic field once a static field is applied. An equiva-
loop position control, since it allows quantification of the lent curve for piezoelectric materials would offer substantial
creep. Futhermore, the experimentally observed relation be- improvement of the positioning capabilities since only the
tween the creep and the hysteresis suggests a reduction of remaining non-linearity has to be compensated. This paper
the creep for non-hysteretic transfers. First measurements on will describe the experimental set-up and measurement pro-
a system with reduced hysteresis support this hypothesis. cedures to perform relaxation measurements in piezoelectric
materials. The results of these measurements have induced
PACS: 77; 07.79.v; 62.20.hg a more specific study of piezo creep, a phenomenon that can
be considered as piezo relaxation in the absence of an ex-
ternally applied relaxation signal. The results from our creep
In our effort to design and construct a sub-angstrom reso- experiments have, finally, led to the study of creep in a system
lution positioning device for scanning probe applications, we with reduced hysteresis.
have made extensive study of piezo hysteresis, relaxation, and
creep. The main goal of the current project is to combine
precision mechanics, sensor technology, and digital signal 1 Experimental details
processing into a 3D translation stage with positioning and
tracking capabilities beyond the angstrom level. Straightfor- The measurements were performed on a home-built piezo-
ward applications will be in the field of metrology, nano- actuated STM positioning device [5]. This device is equipped
lithography and atomic manipulation. An operational proto- with capacitive position sensors for both the X and Y trans-
type is currently used to perform pilot experiments in order lation directions and serves as a test bed for the develop-
to increase our understanding of piezo behavior and non- ment of a next-generation sub-angstrom resolution 3D po-
linear control applications. Since the piezoelectric actuator sitioning device. The actuators are 10-µm maximum exten-
sion piezo stacks from Physik Instrumente [6]. In the course
∗ E-mail: of the development of the 3D positioning device we have
r.koops@tn.tudelft.nl

2. 692
integrated the prototype stage in a digital control environ-
0.5
ment consisting of dSPACE [7] signal acquisition, signal 0.4 V
generation, and signal processing modules. The dSPACE 0.4
hardware is programmed from within MATLAB [8] using 0.2 V
0.3
the simulink graphical programming environment. Since the 0V
0.2
hardware containing the capacitive sensors has not been cal-
OUTPUT [V]
ibrated, the gain factor between the input voltage applied to 0.1
the piezo and the resulting output voltage of the capacitive 0
position sensors is only known approximately and is of minor
-0.1
importance for the presented results. We will therefore dis-
play our measurement results by the internal representation -0.2
within the measurement program in units of V for the input -0.3
signal, representing the voltage applied to the piezo and also
-0.4
in units of V of the output signal representing the measured
displacement detected by the sensors. In order to convert the -0.5
displayed values of the input signal to the real voltage on the
-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5
piezo, multiply by 350, and to convert the output signal to the
INPUT [V]
approximate displacement in nm, multiply by 5000.
The test signal to induce relaxation of the piezo for var- Fig. 2. The resulting hysteresis loops for different amplitudes (0 V, 0.2 V,
and 0.4 V) of the relaxation signal
ious offset voltages is schematically shown in Fig. 1 and
consists of a slowly varying sinusoidal offset with relaxation
signals in between. The sinusoid for both the main waveform or the total area enclosed by the loop, has not changed sig-
and the relaxation signal are of sufficiently low frequency nificantly. Therefore, relaxation does not reduce the hysteretic
( 10 Hz) so as not to invoke the system dynamics. The sys- behavior of the piezo and an anhysteretic curve is not ob-
tem dynamics can result in overshoot and oscillatory behavior tained using this type of relaxation in piezoelectric materials.
of the piezo elongation which can significantly change the Note that the observed tilt of the loops upon application of
relaxation behavior of the piezo [9]. In order to reach the the relaxation signal actually means a change in the effect-
relaxed state in a controlled fashion, the amplitude of the re- ive piezo sensitivity. The effective piezo sensitivity, as defined
laxation signal is slowly increased from zero to a maximum by the slope of the line connecting the turning points of the
value and then slowly decreased to zero again. For every off- loop, increases about 35% for the 0.4 V relaxation curve as
set value we have measured the sensor response just before compared to the 0 V curve.
and just after the application of the relaxation signal. The actual amount of relaxation at every point along the
This procedure has been performed for different values loop for the various relaxation amplitudes is plotted in Fig. 3.
of the maximum relaxation amplitude A, i.e. 0 V, 0.2 V, and For comparison, the input sinusoid (solid line, not to scale)
0.4 V, see Fig. 2. The duration T of the relaxation signal was has been added to the graph (Note that the first quarter of the
30 s for all measurements. The curves show the response of first sinusoid corresponds to the virgin curve). The amount
the position sensor as a function of the input voltage. Al- of relaxation is periodic with the same period as the input
though some relaxation is visible, the amount of hysteresis, signal but not in phase with the input signal. Although, the re-
expressed either in terms of the maximum vertical aperture laxation increases with increasing amplitude of the relaxation
signal, the relation is non-linear and has substantial values
0.04
A 0.03
0.02
RELAXATION [V]
0.4 V
INPUT VOLTAGE
0.2 V
0.01
T 0V
0
-0.01
-0.02
-0.03
-0.04
0 20 40 60 80 100 120 140
TIME
Fig. 1. The waveform of the input signal for the relaxation experiments. INDEX
A slowly varying sinusoid is periodically interrupted by an amplitude- Fig. 3. The amount of relaxation at every point along the hysteresis loop for
modulated sinusoid of higher frequency for a duration T with a maximum different amplitudes (0 V, 0.2 V, and 0.4 V) of the relaxation signal. The
amplitude A solid curve represents the input sinusoid (not to scale)

3. 693
0.015
even when the relaxation amplitude is zero. At the extremum
values of the input signal, the amount of relaxation is equal
for all relaxation amplitudes. Furthermore, for values of the 0.01
60 s
input signal just beyond the extremum value, the relaxation
for all three measurements becomes zero. We have associated 10 s
the relaxation in the absence of an active relaxation signal, the 0.005
CREEP [V]
0 V curve in Fig. 3, with piezo creep. Using this interpreta-
0.1 s
tion we observe two distinct locations on the hysteresis loop 0
where the creep in the previous experiment becomes zero.
In order to study this effect in more detail, the previous
experiment was repeated but now for A = 0 and for various -0.005
interval times T , which we will refer to as the delay time.
The resulting hysteresis loops for various delay times (0.1,
-0.01
10, and 60 s), Fig. 4, show a similar tilt of the loop for increas-
ing delay times as was observed for increasing relaxation
amplitudes. The effective piezo sensitivity therefore also in- -0.015
0 20 40 60 80 100 120 140
creases as a result of creep (for longer delay times) but the INDEX
effect is smaller, about 12%. Observe that the behavior at the
Fig. 5. The amount of creep for every point along the hysteresis loop. The
beginning of the virgin curve is identical for all creep meas- solid line represents the input sinusoid (not to scale). The delay times are
urements, Fig. 4. Analysis of the same region in the relaxation 0.1, 10, and 60 s, respectively
measurements, Fig. 2, also shows identical behavior.
The amount of creep at every point along the hystere-
2
sis loop is displayed in Fig. 5. Again, the creep is periodic
with the same period as the input signal and out of phase.
In contrast to the relaxation results, Fig. 3, all curves show
1.5
zero creep at the same locations with respect to the input sig-
nal, independent of the delay time. Note that the horizontal
axis in Fig. 5 indicates the index of the points on the hystere-
sis loops and not the time. The time scale for the individual 1
SLOPE
curves varies from about 2 min for the 0.1-s delay curve to 60 s
about 2 h for the 60-s delay curve. The curves in Fig. 4 can 10 s
be interpreted in various ways: the points before and after the 0.1 s
0.5
delay procedure can be looked at separately as points defin-
ing a non-relaxed curve and a relaxed curve, respectively.
Additionally one can define a curve determined by the lines 0
connecting relaxed points to non-relaxed points, i.e. the tran-
sition from a certain location after the delay time to the next
position just before the delay procedure. As such, a slope can
-0.5
be associated at the points defining either curve. 0 20 40 60 80 100 120 140
INDEX
Fig. 6. The curve slope at every point along the hysteresis loop defined by
the relaxed points for various delay times. The solid line represents the
slope at the beginning of the virgin curve
0.5
0.4 In Fig. 6, the slopes at the individual points for the relaxed
0.3
loops are displayed along with a solid line representing the
value of the slope at the beginning of the virgin curve. Com-
0.2
paring Figs. 5 and 6 observe that the locations of the points of
OUTPUT [V]
0.1 zero creep coincide with the locations on the hysteresis loop
0
where the value of the slope of the curve equals the value of
the slope at the beginning of the virgin curve.
-0.1
In order to investigate the influence of the amplitude of the
-0.2 input signal (for fixed delay time) on the behavior of creep,
-0.3
we have measured the piezo response for various amplitudes.
In Fig. 7 several hysteresis loops for different amplitudes of
-0.4
the input signal are displayed. The amplitude of the input sig-
-0.5 nal was varied from 0.1 V to 0.5 V in steps of 0.1 V. For
clarity only three measurements are shown but the behavior
-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5
for all measurements at the beginning of the virgin curve is
INPUT [V] again identical. The slope at this point therefore is a char-
Fig. 4. The resulting hysteresis loops for various delay times (0.1, 10, and acteristic property of the piezo that is not influenced by the
60 s) experiments we have performed. The amount of creep at indi-

4. 694
0.4 1.2
0.3 1
0.2
0.8
1
OUTPUT [V]
0.1
2
0.6
SLOPE
0
0.4
3
-0.1
0.2
-0.2
0
-0.3
-0.2
-0.4 0 20 40 60 80 100 120 140
-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5
INDEX
INPUT [V]
Fig. 9. The various slopes at each point along the hysteresis loop. The slope
Fig. 7. The resulting hysteresis loops for different values of the amplitude for the curve through the relaxed points (1) follows the non-linearity of the
of the input sinusoid (0.1 V, 0.3 V, and 0.5 V) and a delay time of 10 s hysteresis loop. The slope for the lines connecting relaxed points to non-
relaxed points (2) is nearly constant and equal to the slope at the beginning
of the virgin curve (3)
0.01
0.4
0.5 V
0.3
0.005 0.4 V
0.3 V
CREEP [V]
0.2
0.2 V
0.1 V
OUTPUT [V]
0 0.1
0
-0.005
-0.1
-0.01 -0.2
0 20 40 60 80 100 120 140 -0.3
INDEX
Fig. 8. The amount of creep for every point along the hysteresis loop for -0.4
-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5
different values of the amplitude of the input sinusoid (0.1 V, 0.2 V, 0.3 V,
0.4 V, and 0.5 V). The solid line represents the input signal (not to scale) INPUT [V]
Fig. 10. The trajectory of zero creep (solid line) as defined by the measured
points of zero creep (black dots) superimposed on the hysteresis loops
vidual points along the curve for all measurements is shown
in Fig. 8. Again the creep curves are periodic with the same Note that this trajectory can not be reached in a single step
period as the input signal (solid line in Fig. 8) and have com- but requires a more sophisticated approach procedure. Nev-
mon zero crossings. However, the slope connecting relaxed ertheless, the trajectory could be very useful in positioning
points to adjacent non-relaxed points show a relaxation to- experiments. The slope relaxation and the relation between
wards the value at the beginning of the virgin curve as indi- the locations of zero creep and the local slope suggest a more
cated by the solid line in Fig. 9. This observation is consistent fundamental relation between the creep and the slope of the
with the experimental results presented by [10] in which sev- hysteresis loop.
eral small signal hysteresis loops have been superimposed on In Fig. 11 a creep experiment is displayed for small indi-
a large signal hysteresis loop. The slope for the smaller loops vidual steps and sufficient delay time (10 s) between each step
was observed to be equal for all loops independent of the pos- to fully relax (within the resolution of our measurements) at
ition on the larger hysteresis loop. The zero crossings in the every position. The inset shows the region around the turning
creep values, Fig. 8, again coincide with the locations on the point at the upper right corner. Observe that the direction of
hysteresis loop where the local slope equals the slope at the the creep remains upward even after the direction of the input
beginning of the virgin curve. signal has changed. Since we use a sinusoidal input signal, the
When all locations of zero creep for the individual meas- size of each individual voltage step changes along the curve.
urements are plotted in one graph, Fig. 10, a trajectory of zero In order to compare the creep along the curve we have cal-
creep within the operating space of the piezo can be defined. culated the creep per unit step size for every point along the

5. 695
0.9
0.88
branch is negative and that ∆V for the lower or ascending
0.8
branch is positive. Observe that, although C(∆V ) is propor-
tional to ∆V , the direction of the creep is not changed when
0.876
0.7 0.872
∆V changes sign because dH − dH V =0 also changes sign at
dV dV
0.6 0.868
the turning points. It is only around the locations where the
slope equals dH V =0 that the creep crosses zero and changes
dV
OUTPUT [V]
0.5
0.864
sign. From (1) we can derive that the creep is essentially de-
0.86
0.97 0.975 0.98 0.985 0.99 0.995 1
scribed by the local non-linearity or deviation of the local
0.4
slope as compared to the slope at the beginning of the virgin
0.3 curve.
0.2
2 Reduction of hysteresis and creep
0.1
0
The experimental observation (1) would suggest that the
creep can be minimised if the deviations of the slope of the
-0.1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
transfer function H can be minimised. In other words, the
creep can be minimised if the linearity of the transfer can
INPUT [V]
be optimised. This can be achieved by, for example, reduc-
Fig. 11. Creep experiment for the virgin curve and a small section of the sta- ing the hysteresis in the system. This hypothesis can be tested
tionary loop. The inset, with the details near the turning point, shows that
the direction of the creep is not determined by the direction of the input on piezoelectric systems that exhibit no hysteresis, as for
signal example quartz. However, quartz is a very insensitive piezo-
electric material and requires large voltages to produce a sig-
nificant displacement. Furthermore, our set-up does not allow
curve, see Fig. 12. In the same figure, the difference between an easy exchange of the actuators. Another way to reduce the
the local slope at each individual location along the curve and hysteresis in ceramic piezoelectric actuators can be realised
the slope at the beginning of the virgin curve is displayed. The by using charge control instead of voltage control. Since pre-
striking equivalence between the two data sets suggest the fol- vious papers [11–13] have reported on a reduced hysteretic
lowing relation between the local slope and the total amount behavior when the piezo is driven with a charge instead of
of creep: when driven with a voltage, a test set-up has been realized to
assess the possibilities of charge control. A voltage-to-charge
dH dH converter has been realized using a single opamp [12–14]
C (∆V ) = − ∆V , (1) as shown in Fig. 13a. Although the maximum output volt-
dV dV V =0 age swing is restricted to the range of the supply voltage
where C(∆V ) is the total amount of creep, H a pair of func- (+/– 15 V) it was verified that sufficient hysteresis could be
tions describing the ascending and descending branches of observed in the case of voltage control. The FET input opamp
the hysteresis loop, dH the local slope, dH V =0 the slope at TL071 has been selected for its relatively low bias current
dV dV
the beginning of the virgin curve, and ∆V the step size of (Ib = 200 pA) and availability. Since all measurements were
the input signal. Note that ∆V for the upper or descending performed in the quasi-static regime, the dynamic behavior of
the opamp is of little concern. The external capacitor was se-
lected for low leakage. Analysis of the circuit yields, for an
ideal opamp and an ideal external capacitor:
10
U+ = U− and Ib = 0 ,
8
so
6
Uin = UC ,
CREEP/STEPSIZE
4
and
2
0
Q ext = UC × Cext = Uin × Cext .
-2
Uin +15 V Uin +15 V
-4 + +
Uout Uout
-6 - -
-15 V Piezo -15 V
-8 Ib Uc
0 20 40 60 80 100 120 140
INDEX Cext
a b Piezo
Fig. 12. The amount of creep per unit step size for every point along the
curve (dots) is equal to the difference between the local slope and the slope
at the beginning of the virgin curve (solid line) Fig. 13a,b. Electronic circuits for charge control (a) and voltage control (b)

6. 696
Since
Q ext = Q piezo ,
we have
Q piezo = Uin × Cext , (2)
that is, the charge on the piezo is proportional to the input
voltage with a conversion factor determined by the value of
the external capacitor, irrespective of the piezo capacitance.
In order to exclude any influence from the opamp in
comparing the charge control measurement and the voltage
control measurement the same opamp has been used in a cir-
cuit for voltage control, see Fig. 13b. Following similar argu-
ments, analysis of the circuit yields
Uin = U+ = U− = Uout , (3)
so the voltage across the piezo equals the input voltage. Fig. 15. Creep at individual locations along the hysteresisloops for charge
In Fig. 14 the voltage and charge measurements are dis- control and voltage control. The triangular input signal (solid line) is not to
played in the same graph. In contrast to the previous experi- scale
ments, the waveform of the input signal was triangular in
order to make the voltage steps equidistant. For clarity, the 0.08
curves have been separated by an artificial offset. Since the
transfer between the input voltage and the resulting elonga- 0.06
tion of the piezo is different for the two electronic circuits
(2),(3), we have adjusted the input voltage for both measure- 0.04
ments so as to obtain approximately the same piezo response.
CREEP/STEPSIZE
Additionally, the input voltage used in the charge control ex- 0.02
periment has been scaled in Fig. 14 to the values used in the
voltage control experiment. This way we are able to compare 0
the creep for voltage and charge control experiments. Because
of the simplicity of the electronic circuit we used to perform -0.02
charge control, significant leakage currents were present re-
-0.04
sulting in drift in the charge control measurement. The charge
control curve in Fig. 14 has been corrected for linear drift.
-0.06
Compared to the voltage controlled piezo, the hysteresis in
-0.08
0 10 20 30 40 50 60 70
0.02
INDEX
Fig. 16. Comparison between the creep per unit step size (solid line) and
0.015
the difference between the local slope and the slope at the beginning of
CHARGE CONTROL the virgin curve (dotted line) for the charge control measurement. Although
0.01 the data contain more noise as compared to previous results, the correlation
between the two curves suggests that equation (1) is still valid
OUTPUT [V]
0.005
the charge control experiment has been reduced to about 1/3
0
of the value measured in the voltage control experiment.
VOLTAGE CONTROL
-0.005
The creep at individual points for both experiments is dis-
played in Fig. 15 along with the triangular input signal. The
-0.01
amount of creep in the charge control measurement is sig-
nificantly reduced as compared to the voltage control meas-
-0.015
urement. Furthermore, the phase of the creep in the charge
control measurement is almost inverted as compared to the
-0.02 phase of the creep in the voltage control signal.
-0.02 -0.015 -0.01 -0.005 0 0.005 0.01 0.015 0.02
Figure 16 furthermore indicates that the amount of creep
INPUT [V]
for the charge control measurement is also determined by (1).
Fig. 14. Creep experiments in a system with reduced hysteresis for a charge- More detailed inspection of the charge control measurement
controlled piezo compared to a voltage-controlled piezo. In order to com-
pare the two measurements the values of the input voltage for charge reveals a residual non-linearity, that suggests that even for
control have been scaled to the values used in the voltage control experi- zero hysteresis, the transfer will still be non-linear and the
ment creep will not completely vanish.

7. 697
3 Conclusions piezoelectric actuators. Finally, the assumption imposed by
the experimentally obtained relation between the creep and
The relaxation mechanism in piezoelectric materials does not the local slope, that the creep should reduce when the linear-
result in an anhysteretic curve and also does not reduce the ity is improved has been verified on a system with reduced
hysteresis. The effective sensitivity of the piezo is increased hysteresis.
upon application of relaxation signals (up to 35%) and as a re-
sult of creep (up to 12%). This effect may be important in Acknowledgements. This research is supported by the Technology Founda-
tion (STW).
the calibration procedures of (open loop) piezo materials. Es-
pecially in scanning probe applications, a distinction can be
made between a fast scanning direction and a slow scanning References
direction resulting in two different times scales for the scan- 1. B. Coleman, M. Hodgdon: Int. J. Eng. Sci. 24(6), 897 (1986)
ning directions. Even when identical piezoelectric actuators 2. H.J. Adriaens, W.L. de Koning, R. Banning: Feedback-linearization
are used, the effective piezo sensitivity will be different for control of a piezo-actuated positioning mechanism, Submitted to the
each direction. 1999 European Control Conference, 1998b
Several points on the hysteresis loop exhibit zero relax- 3. R. Bozorth: Ferromagnetism (Van Nostrand, New York 1951)
4. D.C. Jiles, D.L. Atherton: J. Magn. Magn. Mater. 61, 48 (1986)
ation and zero creep. Within the operating voltage of the 5. A. Holman, C. Laman, P. Scholte, W. Heerens, F. Tuinstra: Rev. Sci.
piezo, the points of zero creep form a trajectory of zero creep Instrum. 67, 2274 (1996)
that may be useful in positioning strategies. The slope at these 6. Physik Instrumente (PI) UHV compatible piezo stack, type P-171.00,
points equals the slope at the begining of the virgin curve. 10 micrometer nominal expansion at −1000 V
7. dSPACE (digital signal processing and control engineering) GmbH,
Once the virgin curve of the piezo is characterised, the pos- Germany
itions of zero creep can be obtained in closed-loop control 8. MATLAB, the MathWorks Inc., USA, www.mathworks.com
systems by measuring the local slope. In contrast to com- 9. K. Koops: Tech. rep., Delft University of Technology, 1998
mon belief, the direction of the creep is not primarily deter- 10. A. Holman: PhD Thesis, Delft University of Technology, 1996
mined by the direction of the input signal but solely by the 11. R.H. Comstock: United States patent 4263527
12. K. Takata: United States patent 4841191
shape of the hysteresis loop. From the observation of points 13. C.V. Newcomb, L. Flinn: Electron. Lett. 18(11), 442 (1982)
of zero creep we have found a more general relation between 14. A.M. John, E. Garcia: J. Guidance, Control and Dynamics 20(3), 479
the amount of creep and the slope of the hysteresis loop in (1997)