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DOI: 10.14738/tnc.86.9699
Publication Date: 08th February 2021
URL: http://dx.doi.org/10.14738/tnc.86.9699
VOLUME 8, NO. 6
ISSN: 2054 -7420
SOCIETY FOR SCIENCE AND EDUCATION
UNITED KINGDOM
TRANSACTIONS ON TNC NETWORKS AND COMMUNICATIONS
Characteristics of Nanopositioning Electroelastic
Digital-to-Analog Converter for Communication Systems
Sergey Mikhailovich Afonin
National Research University of Electronic Technology (MIET), Moscow, Russia;
eduems@mail.ru
ABSTRACT
The characteristics of the nanopositioning electroelastic digital-to-analog converter for communication
systems are examined. In the static and dynamic regimesthis characteristics are received. The static strain
and control characteristics of the nanopositioning electroelastic digital-to-analog converter are obtained.
The transfer function of the nanopositioning electroelastic digital-to-analog converter is received.
Keywords: Nanopositioning electroelastic digital-to-analog converter; Static and dynamic characteristics;
Deformation; Piezoconverter; Strain and control characteristics; Transfer function.
1 Introduction
The nanopositioning electroelastic converter on the piezoelectric effect or the electrostriction effect
solves task of precise adjustment for nanotechnology, adaptive optics, antennas, communication systems
[1−8]. The tasks of providing accuracy and speed of the control system using the nanopositioning
electroelastic digital-to-analog converter with the coded control are topical. These tasks can be solved by
determining the static and dynamic characteristics of the nanopositioning electroelastic converter and its
static strain and control characteristics [7, 8]. The application of this actuator with the multilayer sectional
electroelastic converter using the coded control makes it possible to efficiently use the electromechanical
digital-to-analog conversion for the nanodisplacement.
The nanopositioning electroelastic digital-to-analog converter converter for communication systems is
used in the form a piezoelectric or electrostriction converter. In the scanning probe microscopy and the
gene manipulator the multilayer piezoconverter is applied for the scanner and the manipulator [4–8]. The
strain and control characteristics of the electroelastic digital-to-analog converter for communication
systems are obtained.
2 Static characteristics
Let us consider the characteristics of the nanopositioning electroelastic digital-to-analog converter for
communication systems at the transverse, shear, longitudinal piezoeffects. The piezolayers in the
nanopositioning electroelastic digital-to-analog converter are connected in parallel in each section of the
converter, and the control voltage is supplied to each section via the multiplexer. The number of
piezolayers in the section is equal to the power of 2. To calculate the deformation of the nanopositioning
electroelastic digital-to-analog converter, considering the specific features of its strain and control
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Sergey Mikhailovich Afonin; Characteristics of Nanopositioning Electroelastic Digital-to-Analog Converter for
Communication Systems, Transactions on Networks and Communications, Volume 8 No. 6, Dec (2020); pp: 35-44
URL:http://dx.doi.org/10.14738/tnc.86.9699 36
characteristics is necessary [4-15]. The electroelastic digital-to-analog converter consists of N sections
with all n layers.
For the electroelastic digital-to-analog converter at the transverse piezoeffect the equation of the
electroelasticity [11−34] has the form
1 31 3 11T1 S d E s E = + , (1)
where ` S1 , d31 , E3 , E s11 , T1 are the relative deformation along the axis 1, the transverse piezomodule, the
electric field strength along the axis 3, the elastic compliance with E = const , the mechanical tension along
the axis 1.
The displacement of first section [11, 14] on Figure 1 the electroelastic digital-to-analog converter at the
transverse piezoeffect has the form
∆l
1 = (d31l
1 δ)U , (2)
where 1
l is length 1 section δ is the thickness
From (2) the displacement of k section the electroelastic digital-to-analog converter for transverse
piezoeffect at F = 0 has the following form
l (d l ) U l (d lk )U k k
k ∆ = δ = ∆ = δ − −
1 31
1 1
31 1 2 2 . (3)
At F = 0 from (3) the displacement of the electroelastic digital-to-analog converter for transverse
piezoeffect is obtained in the form
( )
= ∆ ∑
∆ = δ ∑ =
−
=
− N
k
k
k
N
k
k l d l ak U l a
1
1
1 1
1
31 1 2 2 , (4)
where ak is the binary code for k section on Figure 1.
Figure 1. Nanopositioning electroelastic digital-to-analog converter at transverse piezoeffect
The strain characteristic of the electroelastic digital-to-analog converter has the form
( ) ( ) = const, = const ∆ = ∆ k k U a l F l U ,a ,F . (5)
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Transactions on Networks and Communications; Volume 8, No. 6, Dec 2020
Copyright © Society for Science and Education, United Kingdom 37
After transformations the strain characteristic of the electroelastic digital-to-analog converter has the
following form
( ) max 1 F Fmax ∆l = ∆l − , (6)
where max ∆l , Fmax are maximum the displacement at F = 0 and are maximum the force at ∆l = 0 .
The strain characteristic of the electroelastic digital-to-analog converter for transverse piezoeffect has the
following form
( ) 11 0 1
1 l d31 l
1 a 2 U s Fl S E N
k
k
k −
∆ = δ ∑
=
− , (7)
therefore,
( ) E N
k
k l d l ak U F C11 1
1
31 1 2 −
∆ = δ ∑
=
− , (8)
where C S (s l) E E
11= 0 11 is the rigidity of the electroelastic digital-to-analog converter for transverse
piezoeffect
For the strain characteristic (6) of the electroelastic digital-to-analog converter at transverse piezoeffect
the maximum displacement and the maximum force have the form
( )
= ∆ ∑
∆ = δ ∑ =
−
=
− N
k
k
k
N
k
k
k l d l a U l a
1
1
1 1
1
max 31 1 2 2 , (9)
F (d l ) a (S s l)U E N
k
k
k 0 11 1
1
max 31 1 2
= δ ∑
=
− , (10)
therefore,
F (d l ) a C U
E N
k
k
k 11 1
1
max 31 1 2
= δ ∑
=
− . (11)
For the electroelastic digital-to-analog converter from ceramic PZT at 31 d = 2.5∙10-10 m/V, l
1 δ = 1, U =
60 V the displacement of first section is obtained 1 ∆l = 15 nm.
Control characteristic of the electroelastic digital-to-analog converter at transverse piezoeffect and elastic
load has the form
( )
( )
= ∆ ∑
∑
+
δ ∆ = =
−
=
− N
k
k
k
N
k
k
E k
e
a U l a
C C
d l l 1
1
1 1
1
11
31 1 2 2
1 , (12)
therefore,
( )
( )
U
C C
d l l E
e 11
31 1
1 1+
δ ∆ = , (13)
where Ce is the rigidity of the load.
For the electroelastic digital-to-analog converter at the shear piezoeffect the equation of the
electroelasticity [11−34] has the form
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Sergey Mikhailovich Afonin; Characteristics of Nanopositioning Electroelastic Digital-to-Analog Converter for
Communication Systems, Transactions on Networks and Communications, Volume 8 No. 6, Dec (2020); pp: 35-44
URL:http://dx.doi.org/10.14738/tnc.86.9699 38
5 15 1 55T5 S d E s E = + , (14)
where ` S5 , d15 , E1 , E s55 , T5 are the relative deformation along the axis 5, the shear piezomodule, the
electric field strength along the axis 1, the elastic compliance with E = const , the mechanical tension along
the axis 5. The strain characteristic (6) of electroelastic digital-to-analog converter at shear piezoeffect
has the following form
( )
= ∆ ∑
∆ = δ ∑ =
−
=
− N
k
k
k
N
k
k l d l ak U l a
1
1
1 1
1
max 15 1 2 2 , (15)
F (d l ) a (S s l)U E N
k
k
k 0 55 1
1
max 15 1 2
= δ ∑
=
− , (16)
therefore,
F (d l ) a C U
E N
k
k
k 55 1
1
max 15 1 2
= δ ∑
=
− , (17)
∆l
1 = (d15 l
1 δ)U , (18)
where C S (s l) E E
55= 0 55 is the rigidity of electroelastic digital-to-analog converter for shear piezoeffect. For
electroelastic digital-to-analog converter from ceramic PZT at 15 d = 5∙10-10 m/V, l
1 δ = 1, U = 60 V the
displacement of first section is received 1 ∆l = 30 nm.
The control characteristic of the electroelastic digital-to-analog converter at shear piezoeffect and elastic
load has the form
( )
( )
= ∆ ∑
∑
+
δ ∆ = =
−
=
− N
k
k
k
N
k
k
E k
e
a U l a C C
d l l 1
1
1 1
1
55
15 1 2 2
1 , (19)
( )
( )
U
C C
d l l E
e 55
15 1
1 1+
δ ∆ = . (20)
For the electroelastic digital-to-analog converter at the longitudinal piezoeffect the equation of the
electroelasticity [11−34] has the form
3 33 3 33T3 S d E s E = + , (21)
where ` S3 , d31 , E3 , E s33 , T3 are the relative deformation along the axis 3, the longitudinal piezomodule,
the electric field strength along the axis 3, the elastic compliance with E = const , the mechanical tension
along the axis 3.
For the strain characteristic (6) of the electroelastic digital-to-analog converter at longitudinal piezoeffect
the maximum displacement and the maximum force are obtained in the form
( )
= ∆ ∑
∆ = δ ∑ =
−
=
− N
k
k
k
N
k
k
k l d l a U l a
1
1
1 1
1
max 33 1 2 2 , (22)
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Transactions on Networks and Communications; Volume 8, No. 6, Dec 2020
Copyright © Society for Science and Education, United Kingdom 39
F (d l ) a (S s l)U E N
k
k
k 0 33 1
1
max 33 1 2
= δ ∑
=
− , (23)
therefore,
F (d l ) a C U
E N
k
k
k 33 1
1
max 33 1 2
= δ ∑
=
− , (24)
∆l
1 = (d33 l
1 δ)U , (25)
where C S (s l) E E
33= 0 33 is the rigidity of the electroelastic digital-to-analog converter for longitudinal
piezoeffect. For electroelastic digital-to-analog converter from ceramic PZT at 33 d = 4∙10-10 m/V, l
1 δ = 1,
U = 60 V the displacement of first section is received 1 ∆l = 24 nm.
The control characteristic of the electroelastic digital-to-analog converter at longitudinal piezoeffect and
elastic load has the form
( )
( )
= ∆ ∑
∑
+
δ ∆ = =
−
=
− N
k
k
k
N
k
k
E k
e
a U l a
C C
d l l 1
1
1 1
1
33
33 1 2 2
1 , (26)
( )
( )
U
C C
d l l E
e 33
33 1
1 1+
δ ∆ = , (27)
where C S (s l) E E
33= 0 33 is the rigidity of the digital-to-analog converter for longitudinal piezoeffect.
For the electroelastic digital-to-analog converter the equation in general of the electroelasticity [11−34]
has the form
j
E Si = dmiEm + sij T , (28)
where i S , dmi , Em , E
ij s , Tj are the relative deformation, the coefficient of electro elasticity, the electric
field strength, the elastic compliance with E = const , the mechanical tension, and the indexes i, j, m. For
the strain characteristic (6) of the electroelastic digital-to-analog converter the maximum displacement
and the maximum force are received in the form
l (d l ) a U N
k
k
mi k
∆ = δ ∑
=
−
1
1
max 1 2 , (29)
F (d l ) a U(S (s l)) E
ij
N
k
k
mi k 0 1
1
max 1 2
= δ ∑
=
− , (30)
therefore,
F (d l ) a C U
E
ij
N
k
k
mi k
= δ ∑
=
−
1
1
max 1 2 , (31)
∆l
1 = (dmi l
1 δ)U , (32)
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Sergey Mikhailovich Afonin; Characteristics of Nanopositioning Electroelastic Digital-to-Analog Converter for
Communication Systems, Transactions on Networks and Communications, Volume 8 No. 6, Dec (2020); pp: 35-44
URL:http://dx.doi.org/10.14738/tnc.86.9699 40
where C S (s l) E
ij
E
ij= 0 is the rigidity of the electroelastic digital-to-analog converter.
At elastic load in the static regime from (28) the control characteristic of the electroelastic digital-to- analog converter with N sections is obtained in the following form
( )
( ) a U l с
C C
d l l k U N
k
k
E k
e ij
mi
a 1 1
1 1 2
1 = ∆
∑
+
δ ∆ = = =
− (33)
( )
( )
∑
+
δ = =
− N
k
k
E k
e ij
mi
a a
C C
d l k 1
1 1 2
1 , ( )
( ) E
e ij
mi
C C
d l U l
+
δ ∆ = 1
1
1 , (34)
= ∑
=
− N
k
k
k с a
1
1 2 ,
∆ = ∆ ∑
=
− N
k
k
k l l a
1
1
1 2 , (35)
where a k is the transfer coefficient of the digital-to-analog converter, 1 ∆l is the deformation of the first
section of the converter, c is the decimal code. In the work the static strain and control characteristics of
the nanopositioning electroelastic digital-to-analog converter are obtained.
3 Dynamic characteristics
The transfer function [14, 32] of the nanopositioning electroelastic digital-to-analog converter is obtained.
At elastic-inertial load at m << M on Figure 2 the transfer function W (p) for the lumped parameters of
the nanopositioning electroelastic digital-to-analog converter has the form
( ) ( )
( ) 2 1 2 2 + ξ + = Ξ = T p T p
k
U p
p W p
ij ij ij
a , (36)
( )
( )
∑
+
δ = =
− N
k
k
E k
e ij
mi
a a
C C
d l k 1
1 1 2
1 , (37)
( ) E Tij = M Ce + Cij , ( )
ξ = α + E
e ij
E E
ij l Cij 3c M C C 2 , (38)
where m,M are the masses of the actuator and the load, Ξ(p), U(p) are the Laplace transforms of the
displacement and the voltage, p is the operator, a k , Tij , ij ξ are the transfer coefficient, the time
constant, the coefficient attenuation of the electroelastic digital-to-analog converter.
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Transactions on Networks and Communications; Volume 8, No. 6, Dec 2020
Copyright © Society for Science and Education, United Kingdom 41
Figure 2. Nanopositioning electroelastic digital-to-analog converter at transverse piezoeffect and elastic-inertial load
At elastic-inertial load in the dynamic regime from (36) the transient response ξ(t) of the displacement
the nanopositioning electroelastic digital-to-analog converter is obtained in the form
( ) ( ) ( )
ξ −
ξ = ξ = ξ − β + φ − ξ ij
ij
ij ij ij
T
t
t h t t e 2
0 0 1 sin 1 , (39)
( ) ( ) ij
ij
ij ij ij
T
t
h t t e
ξ −
= − β + φ − ξ2 1 sin 1 , (40)
0 U0 k = a ξ , ij ij T ij
2 β = 1− ξ ,
φ = − ξ ξ ij ij ij
2 arctg 1 ,
where t is the time, 0 ξ , U0 are the amplitudes of the displacement and the voltage.
The transfer function and the transient response of the nanopositioning electroelastic digital-to-analog
converter at elastic-inertial load are received for the communications systems.
4 Conclusion
The characteristics in general of the nanopositioning electroelastic digital-to-analog converter for the
static and dynamic regimes are obtained. The strain and control characteristics in general of the
electroelastic digital-to-analog converter for the communications systems are determined.
The transfer function and the transient response of the nanopositioning electroelastic digital-to-analog
converter are received for the communications systems. The strain and control characteristics of the
electroelastic digital-to-analog converter for the transverse, shift, longitudinal piezoelectric effects are
determined.
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Sergey Mikhailovich Afonin; Characteristics of Nanopositioning Electroelastic Digital-to-Analog Converter for
Communication Systems, Transactions on Networks and Communications, Volume 8 No. 6, Dec (2020); pp: 35-44
URL:http://dx.doi.org/10.14738/tnc.86.9699 42
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