This document proposes an approach to increase the integration rate of elements in a hybrid comparator circuit by doping a heterostructure. It involves:
1) Considering a heterostructure with epitaxial layers and doping specific sections via diffusion or ion implantation to manufacture field-effect transistors for the circuit.
2) Analyzing the non-linear dynamics of mass transport during annealing of the dopants and radiation defects generated via a system of equations modeling their spatial and temporal distributions.
3) Optimizing the annealing of dopants and defects to decrease the dimensions of the circuit elements and increase their integration density within the heterostructure.
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In this paper, we introduce an approach to increase integration rate of elements of a switched-
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or radiation defects should be optimized.
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heterostructure with specific configuration by diffusion or ion implantation. The doping finished by optimized
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of radiation defects (after implantation of ions of dopant) for optimization of the above annealing
have been done by using recently introduced analytical approach. The approach gives a possibility
to analyze mass and heat transports in a heterostructure without crosslinking of solutions on interfaces
between layers of the heterostructure with account nonlinearity of these transports and variation in time of
their parameters.
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heterostructure should be doped by diffusion and/or ion implantation and optimization of annealing of dopant and/or radiation defects. We analyzed redistribution of dopant with account redistribution of radiation
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with specific configuration. After that several areas of the epitaxial layer have been doped by diffusion or
ion implantation with optimized annealing of dopant and /or radiation defects. At the same time we introduce
an approach of modification of energy band diagram by additional doping of channel of the transistors.
We also consider an analytical approach to model and optimize technological process.
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ion implantation with optimized annealing of dopant and /or radiation defects. At the same time we introduce an approach of modification of energy band diagram by additional doping of channel of the transistors. We also consider an analytical approach to model and optimize technological process.
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which include into their system). In this situation one can also obtain increasing of homogeneity
of dopant in doped area. In this paper we consider manufacturing a field-effect heterotransistor without pn-
junction. Optimization of technological process with using inhomogeneity of heterostructure give us
possibility to manufacture the transistors as more compact.
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or ion implantation. Annealing of dopant and/or radiation defects should be optimized.
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In this paper we introduce an approach to increase integration rate of field-effect heterotransistors in
the framework of a bootstrap switch. In the framework of the approach we consider a heterostructure
with special configuration. Several specific areas of the heterostructure should be doped by diffusion
or ion implantation. Annealing of dopant and/or radiation defects should be optimized.
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In this paper we introduce an approach to increase integration rate of field-effect heterotransistors in
the framework of a bootstrap switch. In the framework of the approach we consider a heterostructure
with special configuration. Several specific areas of the heterostructure should be doped by diffusion
or ion implantation. Annealing of dopant and/or radiation defects should be optimized.
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dimensions. Framework the approach it should be manufactured a heterostructure with specific configuration.
Farther it is necessary to dope certain areas of the hetero structure by diffusion or by ion implantation.
After finishing of the doping process the dopant and/or radiation defects should be annealed. We consider
an approach of optimization of dopant and/or radiation defects for manufacturing more compact double base
heterotransistors.
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In this paper, we introduce an approach to optimize manufacturing of an operational amplifier circuit based on field-effect transistors. Main aims of the optimization are (i) decreasing dimensions of elements of the considered operational amplifier and (ii) increasing of performance and reliability of the considered field-effect transistors. Dimensions of considered field-effect transistors will be decreased due to manufacture of these transistors framework heterostructure with specific structure, doping of required areas of the heterostructure by diffusion or ion implantation, and optimization of annealing of dopant and/or radiation defects. Performance and reliability of the above field-effect transistors could be increased by optimization of annealing of dopant and/or radiation defects and using inhomogeneity of properties of heterostructure. Choosing of inhomogeneity of properties of heterostructure leads to increasing of compactness of distribution of concentration of dopant. At the same time, one can obtain increasing of homogeneity of the above concentration. In this paper, we also introduce an analytical approach for prognosis of technological process of manufacturing of the considered operational amplifier. The approach gives a possibility to take into account variation of parameters of processes in space and at the same time in space. At the same time, one can take into account nonlinearity of the considered processes.
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due to decreasing of their dimensions. The considered approach based on doping of required areas of
heterostructure with specific configuration by diffusion or ion implantation. The doping finished by optimized
annealing of dopant and/or radiation defects. Analysis of redistribution of dopant with account redistribution
of radiation defects (after implantation of ions of dopant) for optimization of the above annealing
have been done by using recently introduced analytical approach. The approach gives a possibility
to analyze mass and heat transports in a heterostructure without crosslinking of solutions on interfaces
between layers of the heterostructure with account nonlinearity of these transports and variation in time of
their parameters.
An Approach to Optimize Regimes of Manufacturing of Complementary Horizontal ...ijrap
In this paper we consider nonlinear model to describe manufacturing complementary horizontal field-effect heterotransistor. Based on analytical solution of the considered boundary problems some recommendations have been formulated to optimize technological processes.
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heterotransistor. Based on analytical solution of the considered boundary problems some recommendations
have been formulated to optimize technological processes.
Optimization of Technological Process to Decrease Dimensions of Circuits XOR,...ijfcstjournal
The paper describes an approach of increasing of integration rate of elements of integrated circuits. The
approach has been illustrated by example of manufacturing of a circuit XOR. Framework the approach one
should manufacture a heterostructure with specific configuration. After that several special areas of the
heterostructure should be doped by diffusion and/or ion implantation and optimization of annealing of dopant and/or radiation defects. We analyzed redistribution of dopant with account redistribution of radiation
defects to formulate recommendations to decrease dimensions of integrated circuits by using analytical
approaches of modeling of technological process.
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approach based on manufacturing a heterostructure, which consist of a substrate and an epitaxial layer
with specific configuration. After that several areas of the epitaxial layer have been doped by diffusion or
ion implantation with optimized annealing of dopant and /or radiation defects. At the same time we introduce
an approach of modification of energy band diagram by additional doping of channel of the transistors.
We also consider an analytical approach to model and optimize technological process.
MODIFICATION OF DOPANT CONCENTRATION PROFILE IN A FIELD-EFFECT HETEROTRANSIST...msejjournal
In this paper we consider an approach of manufacturing more compact field-effect heterotransistors. The
approach based on manufacturing a heterostructure, which consist of a substrate and an epitaxial layer
with specific configuration. After that several areas of the epitaxial layer have been doped by diffusion or
ion implantation with optimized annealing of dopant and /or radiation defects. At the same time we introduce an approach of modification of energy band diagram by additional doping of channel of the transistors. We also consider an analytical approach to model and optimize technological process.
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in a multilayer structure by diffusion or ion implantation under condition of optimization of dopant and/or
radiation defects leads to increasing of sharpness of p-n-junctions (both single p-n-junctions and p-njunctions,
which include into their system). In this situation one can also obtain increasing of homogeneity
of dopant in doped area. In this paper we consider manufacturing a field-effect heterotransistor without pn-
junction. Optimization of technological process with using inhomogeneity of heterostructure give us
possibility to manufacture the transistors as more compact.
On Decreasing of Dimensions of Field-Effect Heterotransistors in Logical CMOP...BRNSS Publication Hub
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This is a presentation by Dada Robert in a Your Skill Boost masterclass organised by the Excellence Foundation for South Sudan (EFSS) on Saturday, the 25th and Sunday, the 26th of May 2024.
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Synthetic Fiber Construction in lab .pptxPavel ( NSTU)
Synthetic fiber production is a fascinating and complex field that blends chemistry, engineering, and environmental science. By understanding these aspects, students can gain a comprehensive view of synthetic fiber production, its impact on society and the environment, and the potential for future innovations. Synthetic fibers play a crucial role in modern society, impacting various aspects of daily life, industry, and the environment. ynthetic fibers are integral to modern life, offering a range of benefits from cost-effectiveness and versatility to innovative applications and performance characteristics. While they pose environmental challenges, ongoing research and development aim to create more sustainable and eco-friendly alternatives. Understanding the importance of synthetic fibers helps in appreciating their role in the economy, industry, and daily life, while also emphasizing the need for sustainable practices and innovation.
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Cambridge International AS A Level Biology Coursebook - EBook (MaryFosbery J...
01_AJMS_195_19_RA.pdf
1. www.ajms.com 1
ISSN 2581-3463
RESEARCH ARTICLE
An Approach to Analyze Non-linear Dynamics of Mass Transport during
Manufacturing of a Hybrid Comparator Circuit: On Increasing of Integration
Rate of Elements of This Circuit
E. L. Pankratov1,2
1
Department of Mathematical and Natural Sciences, Nizhny Novgorod State University, 23 Gagarin Avenue,
Nizhny Novgorod, 603950, Russia, 2
Department of Higher Mathematics, Nizhny Novgorod State Technical
University, 24 Minin Street, Nizhny Novgorod, 603950, Russia
Received: 01-04-2019; Revised: 27-05-2019; Accepted: 01-07-2019
ABSTRACT
In this paper, we introduce an approach to increase integration rate of elements of a hybrid comparator
with the first dynamic amplifying stage and the second quasi-dynamic latching stage. Framework
the approach, we consider a heterostructure with special configuration. Several specific areas of the
heterostructure should be doped by diffusion or ion implantation. Annealing of dopant and/or radiation
defects should be optimized.
Key words: Hybrid comparator, increasing integration rate of field-effect transistors, optimization of
manufacturing
INTRODUCTION
An actual and intensively solving problem of solid-state electronics is increasing of integration rate of the
elements of integrated circuits (p-n-junctions, their systems).[1-8]
Increasing of the integration rate leads
to the necessity to decrease their dimensions are using several approaches. They are widely using laser
and microwave types of annealing of infused dopants. These types of annealing are also widely using
for annealing of radiation defects, generated during ion implantation.[9-17]
Using the approaches give a
possibility to increase integration rate of the elements of integrated circuits through inhomogeneity of
technological parameters due to generating inhomogeneous distribution of temperature. In this situation,
one can obtain decreasing dimensions of elements of integrated circuits[18,19]
with account Arrhenius
law.[1,3]
Another approach to manufacture elements of integrated circuits with smaller dimensions is
doping of heterostructure by diffusion or ion implantation.[1-3]
However, in this case, optimization of
dopant and/or radiation defects is required.[18]
In this paper, we consider a heterostructure. The heterostructure consists of a substrate and several
epitaxial layers. Some sections have been manufactured in the epitaxial layers. Further, we consider
doping of these sections by diffusion or ion implantation. The doping gives a possibility to manufacture
field-effect transistors framework a hybrid comparator circuit with the first dynamic amplifying stage
and the second quasi-dynamic latching stage so as it is shown in Figure 1. The manufacturing gives a
possibility to increase density of elements of the operational amplifier circuit.[4]
After the considered
doping, dopant and/or radiation defects should be annealed. Framework the paper, we analyzed
dynamics of redistribution of dopant and/or radiation defects during their annealing. We introduce
an approach to decrease the dimensions of the element. However, it is necessary to complicate
technological process.
Address for correspondence:
E. L. Pankratov
E-mail: elp2004@mail.ru
2. Pankratov: On increasing of integration rate of elements of hybrid comparator circuit
AJMS/Jul-Sep-2019/Vol 3/Issue 3 2
METHOD OF SOLUTION
In this section, we determine spatiotemporal distributions of the concentrations of infused and implanted
dopants. To determine these distributions, we calculate appropriate solutions of the second Fick’s law.[1,3,18,19]
C x y z t
t
, , ,
( ) =
( )
+
( )
+
x
D
C x y z t
x y
D
C x y z t
y z
D
C
C C C
, , , , , , x
x y z t
z
, , ,
( )
(1)
Boundary and initial conditions for the equations are
∂ ( )
∂
=
=
C x y z t
x x
, , ,
0
0 ,
∂ ( )
∂
=
=
C x y z t
x x Lx
, , ,
0,
∂ ( )
∂
=
=
C x y z t
y y
, , ,
0
0 ,
∂ ( )
∂
=
=
C x y z t
y x Ly
, , ,
0 ,
∂ ( )
∂
=
=
C x y z t
z z
, , ,
0
0,
∂ ( )
∂
=
=
C x y z t
z x Lz
, , ,
0, C (x,y,z,0)= (x,y,z) (2)
The function C(x,y,z,t) describes the spatiotemporal distribution of the concentration of dopant; T is the
temperature of annealing; DС
is the dopant diffusion coefficient. Value of dopant diffusion coefficient
could be changed with changing mat erials of heterostructure, with changing temperature of materials
(including annealing), with changing concentrations of dopant and radiation defects. We approximate
dependences of dopant diffusion coefficient on parameters by the following relation with account results
in Kozlivsky,[20]
Gotra,[21]
Vinetskiy and Kholodar,[22]
Faheyet al.[23]
Figure 1: The considered hybrid comparator[4]
3. Pankratov: On increasing of integration rate of elements of hybrid comparator circuit
AJMS/Jul-Sep-2019/Vol 3/Issue 3 3
D D x y z T
C x y z t
P x y z T
V x y z t
C L
= ( ) +
( )
( )
+
(
, , ,
, , ,
, , ,
, , ,
1 1 1
ξ ς
γ
γ
)
)
+
( )
( )
V
V x y z t
V
* *
, , ,
ς2
2
2 (3)
Here, the function DL
(x,y,z,T) describes the spatial (in heterostructure) and temperature (due toArrhenius
law) dependences of diffusion coefficient of dopant. The function P (x,y,z,T) describes the limit of
solubility of dopant. Parameter γ ∈[1,3]
describes average quantity of charged defects interacted with atom
of dopant.[20]
The function V(x,y,z,t) describes the spatiotemporal distribution of the concentration of
radiation vacancies. Parameter V* describes the equilibrium distribution of concentration of vacancies.
The considered concentration dependence of dopant diffusion coefficient has been described in details in
Kozlivsky.[20]
It should be noted that using diffusion type of doping did not generation radiation defects.
In this situation, z1
=z2
=0. We determine spatiotemporal distributions of the concentrations of radiation
defects by solving the following system of equations.[21,22]
∂ ( )
∂
=
∂
∂
( )
∂ ( )
∂
+
∂
∂
I x y z t
t x
D x y z T
I x y z t
x y
D x y z
I I
, , ,
, , ,
, , ,
, , ,
,
, , ,
T
I x y z t
y
( )
∂ ( )
∂
+
∂
∂
( )
∂ ( )
∂
− ( ) ( )
z
D x y z T
I x y z t
z
k x y z T I x y z t
I I V
, , ,
, , ,
, , , , , ,
, V
V x y z t
, , ,
( )
( ) ( )
2
, , , , , , ,
I I
k x y z T I x y z t
− (4)
∂ ( )
∂
=
∂
∂
( )
∂ ( )
∂
+
∂
∂
V x y z t
t x
D x y z T
V x y z t
x y
D x y z
V V
, , ,
, , ,
, , ,
, , ,
,
, , ,
T
V x y z t
y
( )
∂ ( )
∂
+
∂
∂
( )
∂ ( )
∂
− ( ) ( )
z
D x y z T
V x y z t
z
k x y z T I x y z t
V I V
, , ,
, , ,
, , , , , ,
, V
V x y z t
, , ,
( )
+ ( ) ( )
k x y z T V x y z t
V V
, , , , , , ,
2
Boundary and initial conditions for these equations are
∂ ( )
∂
=
=
ρ x y z t
x x
, , ,
0
0,
∂ ( )
∂
=
=
ρ x y z t
x x Lx
, , ,
0,
∂ ( )
∂
=
=
ρ x y z t
y y
, , ,
0
0,
∂ ( )
∂
=
=
ρ x y z t
y y Ly
, , ,
0,
∂ ( )
∂
=
=
r x y z t
z z
, , ,
0
0,
∂ ( )
∂
=
=
r x y z t
z z Lz
, , ,
0 , ρ (x,y,z,0)=fρ
(x,y,z) (5)
Here, r=I,V. The function I(x,y,z,t) describes the spatiotemporal distribution of the concentration of
radiation interstitials; Dr
(x,y,z,T) is the diffusion coefficients of point radiation defects; terms V2
(x,y,z,t)
and I2
(x,y,z,t) correspond to generation divacancies and di-interstitials; kI,V
(x,y,z,T) is the parameter of
recombination of point radiation defects; kI,I
(x,y,z,T) and kV,V
(x,y,z,T) are the parameters of the generation
of simplest complexes of point radiation defects.
Further, we determine distributions in space and time of concentrations of divacancies FV
(x,y,z,t) and
di-interstitials FI
(x,y,z,t) by solving the following system of equations.[21,22]
∂ ( )
∂
=
∂
∂
( )
∂ ( )
∂
+
∂
∂
Φ Φ
Φ Φ
I
I
I
I
x y z t
t x
D x y z T
x y z t
x y
D x
, , ,
, , ,
, , ,
,
, , ,
, , ,
y z T
x y z t
y
I
( )
∂ ( )
∂
Φ
4. Pankratov: On increasing of integration rate of elements of hybrid comparator circuit
AJMS/Jul-Sep-2019/Vol 3/Issue 3 4
+
∂
∂
( )
∂ ( )
∂
+ ( )
z
D x y z T
x y z t
z
k x y z T I x y z
I
I
I I
Φ
Φ
, , ,
, , ,
, , , , , ,
,
2
t
t k x y z T I x y z t
I
( )− ( ) ( )
, , , , , , (6)
∂ ( )
∂
=
∂
∂
( )
∂ ( )
∂
+
∂
∂
Φ Φ
Φ Φ
V
V
V
V
x y z t
t x
D x y z T
x y z t
x y
D x
, , ,
, , ,
, , ,
,
, , ,
, , ,
y z T
x y z t
y
V
( )
∂ ( )
∂
Φ
+
∂
∂
( )
∂ ( )
∂
+ ( )
z
D x y z T
x y z t
z
k x y z T V x y z
V
V
V V
Φ
Φ
, , ,
, , ,
, , , , , ,
,
2
t
t k x y z T V x y z t
V
( )− ( ) ( )
, , , , , ,
Boundary and initial conditions for these equations are
∂ ( )
∂
=
=
Φρ x y z t
x x
, , ,
0
0,
∂ ( )
∂
=
=
Φr x y z t
x x Lx
, , ,
0 ,
∂ ( )
∂
=
=
Φr x y z t
y y
, , ,
0
0 ,
∂ ( )
∂
=
=
Φr x y z t
y y Ly
, , ,
0 ,
∂ ( )
∂
=
=
Φρ x y z t
z z
, , ,
0
0,
∂ ( )
∂
=
=
Φr x y z t
z z Lz
, , ,
0,
FI
(x,y,z,0)=fFI
(x,y,z), FV
(x,y,z,0)=fFV
(x,y,z) (7)
Here, DFr
(x,y,z,T) is the diffusion coefficients of the above complexes of radiation defects; kI
(x,y,z,T) and
kV
(x,y,z,T) are the parameters of decay of these complexes.
We calculate distributions of the concentrations of point radiation defects in space and time by recently
elaborated approach.[18]
The approach based on transformation of approximations of diffusion coefficients
in the following form: Dr
(x,y,z,T)=D0r
[1+er
gr
(x,y,z,T)], where D0r
is the average values of diffusion
coefficients, 0≤er
1, |gr
(x,y,z,T)|≤1, r=I,V. We also used analogous transformation of approximations of
parameters of recombination of point defects and parameters of generation of their complexes:
kI,V
(x,y,z,T)=k0I,V
[1+eI,V
gI,V
(x,y,z,T)], kI,I
(x,y,z,T)=k0I,I
[1+eI,I
gI,I
(x,y,z,T)] and kV,V
(x,y,z,T)=k0V,V
[1+eV,V
gV,V
(x,y,z,T)], where k0r1,r2
is the their average values, 0≤eI,V
1, 0≤eI,I
1, 0≤eV,V
1, |gI,V
(x,y,z,T)|≤1,
|gI,I
(x,y,z,T)|≤1, |gV,V
(x,y,z,T)|≤1. Let us introduce the following dimensionless variables:
( ) ( ) *
, , , , , ,
I x y z t I x y z t I
=
, ( )
, , ,
V x y z t
= ( )
V x y z t V
, , , *
,ω = L k D D
I V I V
2
0 0 0
, ,Ωρ ρ ρ
= L k D D
I V
2
0 0 0
, ,
2
0 0
I V
D D t L
= , c = x/Lx
, h = y/Ly
, f = z/Lz
. The introduction leads to transformation of Equations (4)
and conditions (5) to the following form.
∂ ( )
∂
=
∂
∂
+ ( )
∂ ( )
I D
D D
g T
I
I
I V
I I
χ η ϕ ϑ
ϑ χ
ε χ η ϕ
χ η ϕ ϑ
, , ,
, , ,
, , ,
0
0 0
1
∂
∂
+
∂
∂
+ ( )
{
χ η
ε χ η ϕ
1 I I
g T
, , ,
×
∂ ( )
∂
+
∂
∂
+ ( )
I D
D D
D
D D
g T
I
I V
I
I V
I I
χ η ϕ ϑ
η ϕ
ε χ η ϕ
, , ,
, , ,
0
0 0
0
0 0
1
∂ ( )
∂
− ( )
I
I
χ η ϕ ϑ
ϕ
χ η ϕ ϑ
, , ,
, , ,
× + ( )
( )− + ( )
ω ε χ η ϕ χ η ϕ ϑ ε χ η ϕ
1 1
I V I V I I I I I
g T V g T
, , , ,
, , , , , , , , ,
Ω
( )
I 2
χ η ϕ ϑ
, , , (8)
∂ ( )
∂
=
∂
∂
+ ( )
∂ ( )
V D
D D
g T
V
V
I V
V V
χ η ϕ ϑ
ϑ χ
ε χ η ϕ
χ η ϕ ϑ
, , ,
, , ,
, , ,
0
0 0
1
∂
∂
+
∂
∂
+ ( )
{
χ η
ε χ η ϕ
1 V V
g T
, , ,
×
∂ ( )
∂
+
∂
∂
+ ( )
V D
D D
D
D D
g T
V
I V
V
I V
V V
χ η ϕ ϑ
η ϕ
ε χ η ϕ
, , ,
, , ,
0
0 0
0
0 0
1
∂ ( )
∂
− ( )
V
I
χ η ϕ ϑ
ϕ
χ η ϕ ϑ
, , ,
, , ,
5. Pankratov: On increasing of integration rate of elements of hybrid comparator circuit
AJMS/Jul-Sep-2019/Vol 3/Issue 3 5
× + ( )
( )− + ( )
ω ε χ η ϕ χ η ϕ ϑ ε χ η ϕ
1 1
I V I V V V V V V
g T V g T
, , , ,
, , , , , , , , ,
Ω
( )
V 2
χ η ϕ ϑ
, , ,
∂ ( )
∂
=
=
ρ χ η ϕ ϑ
χ χ
, , ,
0
0,
∂ ( )
∂
=
=
ρ χ η ϕ ϑ
χ χ
, , ,
1
0,
∂ ( )
∂
=
=
ρ χ η ϕ ϑ
η η
, , ,
0
0,
∂ ( )
∂
=
=
ρ χ η ϕ ϑ
η η
, , ,
1
0,
∂ ( )
∂
=
=
ρ χ η ϕ ϑ
ϕ ϕ
, , ,
0
0 ,
∂ ( )
∂
=
=
ρ χ η ϕ ϑ
ϕ ϕ
, , ,
1
0 ,
ρ χ η ϕ ϑ
χ η ϕ ϑ
ρ
ρ
, , ,
, , ,
*
( ) =
( )
f
(9)
We determine solutions of Equations (8) with conditions (9) framework recently introduced approach,[18]
i.e. as the power series.
ρ χ η ϕ ϑ ε ω ρ χ η ϕ ϑ
ρ ρ
, , , , , ,
( ) = ( )
=
∞
=
∞
=
∞
∑
∑
∑ i j k
ijk
k
j
i
Ω
0
0
0
(10)
Substitution of the series (10) into Equations (8) and condition (9) gives us possibility to obtain equations
for initial order approximations of the concentration of point defects
I000 χ η ϕ ϑ
, , ,
( )and
V000 χ η ϕ ϑ
, , ,
( )
and corrections for them
Iijk χ η ϕ ϑ
, , ,
( ) and
Vijk χ η ϕ ϑ
, , ,
( ), i ≥1, j ≥1, k ≥1. The equations are presented
in the Appendix. Solutions of the equations could be obtained by standard Fourier approach.[24,25]
The
solutions are presented in the Appendix.
Now, we calculate distributions of the concentrations of simplest complexes of point radiation defects in
space and time. To determine the distributions, we transform approximations of diffusion coefficients in
the following form: DFr
(x,y,z,T)=D0Fr
[1+eFr
gFr
(x,y,z,T)], where D0Fr
is the average values of diffusion
coefficients. In this situation, the Equation (6) could be written as
∂ ( )
∂
=
∂
∂
+ ( )
∂ ( )
∂
Φ Φ
Φ Φ Φ
I
I I I
I
x y z t
t
D
x
g x y z T
x y z t
x
, , ,
, , ,
, , ,
0 1 e
+ ( ) ( )
k x y z T I x y z t
I I
, , , , , , ,
2
+
∂
∂
+ ( )
∂ ( )
∂
+
∂
∂
D
y
g x y z T
x y z t
y
D
z
I I I
I
I
0 0
1 1
Φ Φ Φ Φ
Φ
e , , ,
, , ,
+
+ ( )
∂ ( )
∂
eΦ Φ
Φ
I I
I
g x y z T
x y z t
z
, , ,
, , ,
( ) ( )
, , , , , ,
I
k x y z T I x y z t
−
∂
∂
∂
∂
ε
∂
∂
Φ Φ
Φ Φ Φ
V
V V V
V
x y z t
t
D
x
g x y z T
x y z t
x
, , ,
, , ,
, , ,
( )
= + ( )
( )
0 1
+ ( ) ( )
k x y z T I x y z t
I I
, , , , , , ,
2
+ + ( )
( )
+
D
y
g x y z T
x y z t
y
D
z
V V V
V
V
0 0
1 1
Φ Φ Φ Φ
Φ
∂
∂
ε
∂
∂
∂
∂
, , ,
, , ,
+
+ ( )
( )
ε
∂
∂
Φ Φ
Φ
V V
V
g x y z T
x y z t
z
, , ,
, , ,
− ( ) ( )
k x y z T I x y z t
I , , , , , ,
Farther, we determine solutions of above equations as the following power series.
Φ Φ
Φ
ρ ρ ρ
ε
x y z t x y z t
i
i
i
, , , , , ,
( ) = ( )
=
∞
∑
0
(11)
Now, we used the series (11) into Equation (6) and appropriate boundary and initial conditions. The using
gives the possibility to obtain equations for initial order approximations of concentrations of complexes
of defects Fr0
(x,y,z,t), corrections for them Fri
(x,y,z,t) (for them i≥1), and boundary and initial conditions
for them. We remove equations and conditions to the Appendix. Solutions of the equations have been
calculated by standard approaches[24,25]
and presented in the Appendix.
6. Pankratov: On increasing of integration rate of elements of hybrid comparator circuit
AJMS/Jul-Sep-2019/Vol 3/Issue 3 6
Now, we calculate distribution of the concentration of dopant in space and time using the approach, which
was used for analysis of radiation defects. To use the approach, we consider the following transformation
of approximation of dopant diffusion coefficient: DL
(x,y,z,T)=D0L
[1+eL
gL
(x,y,z,T)], where D0L
is the
average value of dopant diffusion coefficient, 0≤eL
1, |gL
(x,y,z,T)|≤1. Farther, we consider solution of
Equation (1) as the following series:
C x y z t C x y z t
L
i j
ij
j
i
, , , , , ,
( ) = ( )
=
∞
=
∞
∑
∑ε ξ
1
0
.
Using the relation into Equations (1) and condition (2) leads to obtaining equations for the functions
Cij
(x,y,z,t) (i≥1, j≥1), boundary and initial conditions for them. The equations are presented in the
Appendix. Solutions of the equations have been calculated by standard approaches.[24,25]
The solutions
are presented in the Appendix.
We analyzed distributions of concentrations of dopant and radiation defects in space and time analytically
using the second-order approximations on all parameters, which have been used in appropriate series.
Usually, the second-order approximations are enough good approximations to make qualitative analysis
and to obtain quantitative results. All analytical results have been checked by numerical simulation.
DISCUSSION
In this section, we analyzed spatiotemporal distributions of the concentrations of dopants. Figure 2
shows typical spatial distributions of the concentrations of dopants in neighborhood of interfaces of
heterostructures. We calculate these distributions of the concentrations of dopants under the following
condition: Value of dopant diffusion coefficient in doped area is larger than the value of dopant diffusion
coefficient in nearest areas. In this situation, one can find increasing of compactness of field-effect
transistors with increasing of homogeneity of distribution of the concentration of dopant at 1 time.
Changing relation between values of dopant diffusion coefficients leads to opposite result [Figure 3].
It should be noted that framework the considered approach one should optimize annealing of dopant and/
or radiation defects. To do the optimization, we used recently introduced criterion.[26-34]
The optimization
based on approximation real distribution by step-wise function y (x,y,z) [Figure 4]. Farther, the required
values of optimal annealing time have been calculated by minimization the following mean squared
error.
Figure 2: (a) Dependences of concentration of dopant, infused in heterostructure from Figure 1, on coordinate in direction,
which is perpendicular to interface between epitaxial layer substrate. Difference between the values of dopant diffusion
coefficient in layers of heterostructure increases with increasing of the number of curves. Value of dopant diffusion
coefficient in the epitaxial layer is larger than the value of dopant diffusion coefficient in the substrate. (b) Dependences of
the concentration of dopant, implanted in heterostructure from Figure 1, on coordinate in direction, which is perpendicular
to interface between epitaxial layer substrate. Difference between the values of dopant diffusion coefficient in layers of
heterostructure increases with increasing of the number of curves. Value of dopant diffusion coefficient in the epitaxial layer
is larger than the value of dopant diffusion coefficient in the substrate. Curve 1 corresponds to homogeneous sample and
annealing time Θ = 0.0048 (Lx
2
+Ly
2
+Lz
2
)/D0
. Curve 2 corresponds to homogeneous sample and annealing time Θ = 0.0057
(Lx
2
+Ly
2
+Lz
2
)/D0
. Curves 3 and 4 correspond to heterostructure from Figure 1; annealing times Θ = 0.0048 (Lx
2
+Ly
2
+Lz
2
)/D0
and Θ = 0.0057 (Lx
2
+Ly
2
+Lz
2
)/D0
, respectively
a b
7. Pankratov: On increasing of integration rate of elements of hybrid comparator circuit
AJMS/Jul-Sep-2019/Vol 3/Issue 3 7
Figure 3: (a) Distributions of the concentration of dopant infused in average section of epitaxial layer of heterostructure from
Figure 1 in direction parallel to interface between epitaxial layer and substrate of heterostructure. Difference between the
values of dopant diffusion coefficients increases with increasing of the number of curves. Value of dopant diffusion coefficient
in this section is smaller than the value of dopant diffusion coefficient in nearest sections. (b) Calculated distributions of
implanted dopant in epitaxial layers of heterostructure. Solid lines are spatial distributions of implanted dopant in system
of two epitaxial layers. Dashed lines are spatial distributions of implanted dopant in one epitaxial layer. Annealing time
increases with increasing of number of curves
Figure 5: (a) Dimensionless optimal annealing time of infused dopant as a function of several parameters. Curve 1 describes
the dependence of the annealing time on the relation a/L and x=g= 0 for equal to each other values of dopant diffusion
coefficient in all parts of heterostructure. Curve 2 describes the dependence of the annealing time on value of parameter e for
a/L=1/2 and x=g=0. Curve 3 describes the dependence of the annealing time on value of parameter x for a/L=1/2 and e =g=0.
Curve 4 describes the dependence of the annealing time on value of parameter g for a/L=1/2 and e=x=0. (b) Dimensionless
optimal annealing time of implanted dopant as a function of several parameters. Curve 1 describes the dependence of the
annealing time on the relation a/L and x=g=0 for equal to each other values of dopant diffusion coefficient in all parts of
heterostructure. Curve 2 describes the dependence of the annealing time on value of parameter e for a/L=1/2 and x=g=0.
Curve 3 describes the dependence of the annealing time on value of parameter x for a/L=1/2 and e =g = 0. Curve 4 describes
the dependence of the annealing time on value of parameter g for a/L=1/2 and e = x = 0
a b
Figure 4: (a) Distributions of the concentration of infused dopant in depth of heterostructure from Figure 1 for different values
of annealing time (curves 2-4) and idealized step-wise approximation (curve 1). Increasing of the number of curve corresponds
to increasing of annealing time. (b) Distributions of the concentration of implanted dopant in depth of heterostructure from
Figure 1 for different values of annealing time (curves 2-4) and idealized step-wise approximation (curve 1). Increasing of
the number of curve corresponds to increasing of annealing time
a b
a b
8. Pankratov: On increasing of integration rate of elements of hybrid comparator circuit
AJMS/Jul-Sep-2019/Vol 3/Issue 3 8
( ) ( )
0 0 0
1
, , , , ,
y
x z
L
L L
x y z
U C x y z x y z d z d y d x
L L L
= Θ −
∫ ∫ ∫ (12)
We show optimal values of annealing time as functions of parameters in Figure 5. It is known that
standard step of manufactured ion-doped structures is annealing of radiation defects. In the ideal case,
after finishing the annealing dopant achieves interface between the layers of heterostructure. If the dopant
has no enough time to achieve the interface, it is practicably to anneal the dopant additionally. Figure 5b
shows the described dependences of optimal values of additional annealing time for the same parameters
as for Figure 5a. Necessity to anneal radiation defects leads to smaller values of optimal annealing of
implanted dopant in comparison with optimal annealing time of infused dopant.
CONCLUSIONS
In this paper, we introduce an approach to increase the integration rate of element of a hybrid comparator
with the first dynamic amplifying stage and the second quasi-dynamic latching stage. The approach
gives us possibility to decrease area of the elements with smaller increasing of the element’s thickness.
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heterotransistors. Multidiscip Model Mater Struct 2016;12:578-604.
33. Pankratov EL, Bulaeva EA. An approach to increase the integration rate of planar drift heterobipolar transistors. Mater
Sci Semicond Process 2015;34:260-8.
APPENDIX
Equations for the functions
Iijk χ η ϕ ϑ
, , ,
( ) and
Vijk χ η ϕ ϑ
, , ,
( ), i ≥0, j ≥0, k ≥0 and conditions for them
∂ ( )
∂
=
∂ ( )
∂
+
∂
I D
D
I I
I
V
000 0
0
2
000
2
2
000
χ η ϕ ϑ
ϑ
χ η ϕ ϑ
χ
χ η ϕ ϑ
, , , , , , , , ,
(
( )
∂
+
∂ ( )
∂
η
χ η ϕ ϑ
ϕ
2
2
000
2
I , , ,
∂ ( )
∂
=
∂ ( )
∂
+
∂
V D
D
V V
V
I
000 0
0
2
000
2
2
000
χ η ϕ ϑ
ϑ
χ η ϕ ϑ
χ
χ η ϕ ϑ
, , , , , , , , ,
(
( )
∂
+
∂ ( )
∂
η
χ η ϕ ϑ
ϕ
2
2
000
2
V , , ,
;
∂ ( )
∂
=
∂ ( )
∂
+
∂ ( )
∂
I D
D
I I
i I
V
i i
00 0
0
2
00
2
2
00
χ ϑ
ϑ
χ η ϕ ϑ
χ
χ η ϕ ϑ
η
, , , , , , ,
2
2
2
00
2
0
0
+
∂ ( )
∂
+
I D
D
i I
V
χ η ϕ ϑ
ϕ
, , ,
×
∂
∂
( )
∂ ( )
∂
+
∂
∂
( )
∂
−
χ
χ η ϕ
χ η ϕ ϑ
χ η
χ η ϕ
g T
I
g T
I
I
i
I
, , ,
, , ,
, , ,
100 i
i− ( )
∂
+
100 χ η ϕ ϑ
η
, , ,
+
∂
∂
( )
∂ ( )
∂
−
ϕ
χ η ϕ
χ η ϕ ϑ
ϕ
g T
I
I
i
, , ,
, , ,
100
, i≥1,
∂ ( )
∂
=
∂ ( )
∂
+
∂ ( )
∂
V D
D
V V
i V
I
i i
00 0
0
2
00
2
2
00
χ ϑ
ϑ
χ η ϕ ϑ
χ
χ η ϕ ϑ
η
, , , , , , ,
2
2
2
00
2
+
∂ ( )
∂
+
∂
∂
( )
V
g T
i
V
χ η ϕ ϑ
ϕ χ
χ η ϕ
, , ,
, , ,
×
∂ ( )
∂
+
∂
∂
( )
∂
− −
V D
D
D
D
g T
V
i V
I
V
I
V
i
100 0
0
0
0
100
χ η ϕ ϑ
χ η
χ η ϕ
, , ,
, , ,
χ
χ η ϕ ϑ
η ϕ
χ η ϕ
, , ,
, , ,
( )
∂
+
∂
∂
( )
g T
V
×
∂ ( )
∂
−
V D
D
i V
I
100 0
0
χ η ϕ ϑ
ϕ
, , ,
, i≥1,
10. Pankratov: On increasing of integration rate of elements of hybrid comparator circuit
AJMS/Jul-Sep-2019/Vol 3/Issue 3 10
∂ ( )
∂
=
∂ ( )
∂
+
∂
I D
D
I I
I
V
010 0
0
2
010
2
2
010
χ η ϕ ϑ
ϑ
χ η ϕ ϑ
χ
χ η ϕ ϑ
, , , , , , , , ,
(
( )
∂
+
∂ ( )
∂
η
χ η ϕ ϑ
ϕ
2
2
010
2
I , , ,
− + ( )
( ) ( )
1 000 000
ε χ η ϕ χ η ϕ ϑ χ η ϕ ϑ
I V I V
g T I V
, , , , , , , , , , ,
∂ ( )
∂
=
∂ ( )
∂
+
∂
V D
D
V V
V
I
010 0
0
2
010
2
2
010
χ η ϕ ϑ
ϑ
χ η ϕ ϑ
χ
χ η ϕ ϑ
, , , , , , , , ,
(
( )
∂
+
∂ ( )
∂
η
χ η ϕ ϑ
ϕ
2
2
010
2
V , , ,
− + ( )
( ) ( )
1 000 000
ε χ η ϕ χ η ϕ ϑ χ η ϕ ϑ
I V I V
g T I V
, , , , , , , , , , ,
;
∂ ( )
∂
=
∂ ( )
∂
+
∂
I D
D
I I
I
V
020 0
0
2
020
2
2
020
χ η ϕ ϑ
ϑ
χ η ϕ ϑ
χ
χ η ϕ ϑ
, , , , , , , , ,
(
( )
∂
+
∂ ( )
∂
η
χ η ϕ ϑ
ϕ
2
2
020
2
I , , ,
− + ( )
( ) ( )+
1 010 000 000
ε χ η ϕ χ η ϕ ϑ χ η ϕ ϑ
I V I V
g T I V I
, , , , , , , , , , ,
χ
χ η ϕ ϑ χ η ϕ ϑ
, , , , , ,
( ) ( )
V010
∂ ( )
∂
=
∂ ( )
∂
+
∂
V D
D
V V
I
V
020 0
0
2
020
2
2
020
χ η ϕ ϑ
ϑ
χ η ϕ ϑ
χ
χ η ϕ ϑ
, , , , , , , , ,
(
( )
∂
+
∂ ( )
∂
η
χ η ϕ ϑ
ϕ
2
2
020
2
V , , ,
− + ( )
( ) ( )+
1 010 000 000
ε χ η ϕ χ η ϕ ϑ χ η ϕ ϑ
I V I V
g T I V I
, , , , , , , , , , ,
χ
χ η ϕ ϑ χ η ϕ ϑ
, , , , , ,
( ) ( )
V010 ;
∂ ( )
∂
=
∂ ( )
∂
+
∂
I D
D
I I
I
V
001 0
0
2
001
2
2
001
χ η ϕ ϑ
ϑ
χ η ϕ ϑ
χ
χ η ϕ ϑ
, , , , , , , , ,
(
( )
∂
+
∂ ( )
∂
η
χ η ϕ ϑ
ϕ
2
2
001
2
I , , ,
− + ( )
( )
1 000
2
ε χ η ϕ χ η ϕ ϑ
I I I I
g T I
, , , , , , , ,
∂ ( )
∂
=
∂ ( )
∂
+
∂
V D
D
V V
V
I
001 0
0
2
001
2
2
001
χ η ϕ ϑ
ϑ
χ η ϕ ϑ
χ
χ η ϕ ϑ
, , , , , , , , ,
(
( )
∂
+
∂ ( )
∂
η
χ η ϕ ϑ
ϕ
2
2
001
2
V , , ,
− + ( )
( )
1 000
2
ε χ η ϕ χ η ϕ ϑ
I I I I
g T V
, , , , , , , ,
;
∂ ( )
∂
=
∂ ( )
∂
+
∂
I D
D
I I
I
V
110 0
0
2
110
2
2
110
χ η ϕ ϑ
ϑ
χ η ϕ ϑ
χ
χ η ϕ ϑ
, , , , , , , , ,
(
( )
∂
+
∂ ( )
∂
+
η
χ η ϕ ϑ
ϕ
2
2
110
2
0
0
I D
D
I
V
, , ,
×
∂
∂
( )
∂ ( )
∂
+
∂
∂
( )
∂
χ
χ η ϕ
χ η ϕ ϑ
χ η
χ η ϕ
g T
I
g T
I
I I
, , ,
, , ,
, , ,
010 010
0 χ η ϕ ϑ
η ϕ
χ η ϕ
, , ,
, , ,
( )
∂
+
∂
∂
( )
g T
I
×
∂ ( )
∂
− ( ) ( )+
I
I V I
010
100 000
χ η ϕ ϑ
ϕ
χ η ϕ ϑ χ η ϕ ϑ
, , ,
, , , , , , 0
000 100
χ η ϕ ϑ χ η ϕ ϑ
, , , , , ,
( ) ( )
V
11. Pankratov: On increasing of integration rate of elements of hybrid comparator circuit
AJMS/Jul-Sep-2019/Vol 3/Issue 3 11
× + ( )
1 ε χ η ϕ
I I I I
g T
, , , , ,
∂ ( )
∂
=
∂ ( )
∂
+
∂
V D
D
V V
V
I
110 0
0
2
110
2
2
110
χ η ϕ ϑ
ϑ
χ η ϕ ϑ
χ
χ η ϕ ϑ
, , , , , , , , ,
(
( )
∂
+
∂ ( )
∂
η
χ η ϕ ϑ
ϕ
2
2
110
2
V , , ,
+
∂
∂
( )
∂ ( )
∂
+
∂
∂
(
D
D
g T
V
g T
V
I
V V
0
0
010
χ
χ η ϕ
χ η ϕ ϑ
χ η
χ η ϕ
, , ,
, , ,
, , ,
)
)
∂ ( )
∂
V010 χ η ϕ ϑ
η
, , ,
+
∂
∂
( )
∂ ( )
∂
− +
ϕ
χ η ϕ
χ η ϕ ϑ
ϕ
ε χ η
g T
V
g
V V V V V
, , ,
, , ,
, ,
, ,
010
1 ϕ
ϕ,T
( )
× ( ) ( )+ ( ) ( )
V I V I
100 000 000 100
χ η ϕ ϑ χ η ϕ ϑ χ η ϕ ϑ χ η ϕ ϑ
, , , , , , , , , , , ,
;
∂ ( )
∂
=
∂ ( )
∂
+
∂
I D
D
I I
I
V
002 0
0
2
002
2
2
002
χ η ϕ ϑ
ϑ
χ η ϕ ϑ
χ
χ η ϕ ϑ
, , , , , , , , ,
(
( )
∂
+
∂ ( )
∂
η
χ η ϕ ϑ
ϕ
2
2
002
2
I , , ,
− + ( )
( ) ( )
1 001 000
ε χ η ϕ χ η ϕ ϑ χ η ϕ ϑ
I I I I
g T I I
, , , , , , , , , , ,
∂ ( )
∂
=
∂ ( )
∂
+
∂
V D
D
V V
V
I
002 0
0
2
002
2
2
002
χ η ϕ ϑ
ϑ
χ η ϕ ϑ
χ
χ η ϕ ϑ
, , , , , , , , ,
(
( )
∂
+
∂ ( )
∂
η
χ η ϕ ϑ
ϕ
2
2
002
2
V , , ,
− + ( )
( ) ( )
1 001 000
ε χ η ϕ χ η ϕ ϑ χ η ϕ ϑ
V V V V
g E V V
, , , , , , , , , , ,
;
∂ ( )
∂
=
∂ ( )
∂
+
∂
I D
D
I I
I
V
101 0
0
2
101
2
2
101
χ η ϕ ϑ
ϑ
χ η ϕ ϑ
χ
χ η ϕ ϑ
, , , , , , , , ,
(
( )
∂
+
∂ ( )
∂
η
χ η ϕ ϑ
ϕ
2
2
101
2
I , , ,
+
∂
∂
( )
∂ ( )
∂
+
∂
∂
(
D
D
g T
I
g T
I
V
I I
0
0
001
χ
χ η ϕ
χ η ϕ ϑ
χ η
χ η ϕ
, , ,
, , ,
, , ,
)
)
∂ ( )
∂
I001 χ η ϕ ϑ
η
, , ,
+
∂
∂
( )
∂ ( )
∂
− + (
ϕ
χ η ϕ
χ η ϕ ϑ
ϕ
ε χ η ϕ
g T
I
g T
I I I
, , ,
, , ,
, , ,
001
1 )
)
( ) ( )
I V
100 000
χ η ϕ ϑ χ η ϕ ϑ
, , , , , ,
∂ ( )
∂
=
∂ ( )
∂
+
∂
V D
D
V V
V
I
101 0
0
2
101
2
2
101
χ η ϕ ϑ
ϑ
χ η ϕ ϑ
χ
χ η ϕ ϑ
, , , , , , , , ,
(
( )
∂
+
∂ ( )
∂
η
χ η ϕ ϑ
ϕ
2
2
101
2
V , , ,
+
∂
∂
( )
∂ ( )
∂
+
∂
∂
(
D
D
g T
V
g T
V
I
V V
0
0
001
χ
χ η ϕ
χ η ϕ ϑ
χ η
χ η ϕ
, , ,
, , ,
, , ,
)
)
∂ ( )
∂
V001 χ η ϕ ϑ
η
, , ,
+
∂
∂
( )
∂ ( )
∂
− + (
ϕ
χ η ϕ
χ η ϕ ϑ
ϕ
ε χ η ϕ
g T
V
g T
V V V
, , ,
, , ,
, , ,
001
1 )
)
( ) ( )
I V
000 100
χ η ϕ ϑ χ η ϕ ϑ
, , , , , , ;
12. Pankratov: On increasing of integration rate of elements of hybrid comparator circuit
AJMS/Jul-Sep-2019/Vol 3/Issue 3 12
∂ ( )
∂
=
∂ ( )
∂
+
∂
I D
D
I I
I
V
011 0
0
2
011
2
2
011
χ η ϕ ϑ
ϑ
χ η ϕ ϑ
χ
χ η ϕ ϑ
, , , , , , , , ,
(
( )
∂
+
∂ ( )
∂
− ( )
η
χ η ϕ ϑ
ϕ
χ η ϕ ϑ
2
2
011
2 010
I
I
, , ,
, , ,
× + ( )
( )− + ( )
1 1
000
ε χ η ϕ χ η ϕ ϑ ε χ η ϕ
I I I I I V I V
g T I g T
, , , ,
, , , , , , , , ,
( ) ( )
I V
001 000
χ η ϕ ϑ χ η ϕ ϑ
, , , , , ,
∂ ( )
∂
=
∂ ( )
∂
+
∂
V D
D
V V
V
I
011 0
0
2
011
2
2
011
χ η ϕ ϑ
ϑ
χ η ϕ ϑ
χ
χ η ϕ ϑ
, , , , , , , , ,
(
( )
∂
+
∂ ( )
∂
− ( )
η
χ η ϕ ϑ
ϕ
χ η ϕ ϑ
2
2
011
2 010
V
V
, , ,
, , ,
× + ( )
( )− + ( )
1 1
000
ε χ η ϕ χ η ϕ ϑ ε χ η ϕ
V V V V I V I V
g T V g t
, , , ,
, , , , , , , , ,
( ) ( )
I V
000 001
χ η ϕ ϑ χ η ϕ ϑ
, , , , , , ;
∂ ( )
∂
=
=
ρ χ η ϕ ϑ
χ
ijk
x
, , ,
0
0,
∂ ( )
∂
=
=
ρ χ η ϕ ϑ
χ
ijk
x
, , ,
1
0,
∂ ( )
∂
=
=
ρ χ η ϕ ϑ
η η
ijk , , ,
0
0,
∂ ( )
∂
=
=
ρ χ η ϕ ϑ
η η
ijk , , ,
1
0,
∂ ( )
∂
=
=
ρ χ η ϕ ϑ
ϕ ϕ
ijk , , ,
0
0,
∂ ( )
∂
=
=
ρ χ η ϕ ϑ
ϕ ϕ
ijk , , ,
1
0 (i≥0, j≥0, k≥0);
ρ χ η ϕ χ η ϕ ρ
ρ
000 0
, , , , , *
( ) = ( )
f ,
ρ χ η ϕ
ijk , , ,0 0
( ) = (i≥0, j≥0, k≥0).
Solutions of the above equations could be written as
ρ χ η ϕ ϑ χ η ϕ ϑ
ρ ρ
000
1
1 2
, , ,
( ) = + ( ) ( ) ( ) ( )
=
∞
∑
L L
F c c c e
n n
n
,
where F nu nv n w f u v w d wd vd u
n n
ρ ρ
ρ
π π π
= ( ) ( ) ( ) ( )
∫
∫
∫
1
0
1
0
1
0
1
*
cos cos cos , , , cn
(c) = cos (pnc),
e n D D
nI V I
ϑ π ϑ
( ) = −
( )
exp 2 2
0 0 , e n D D
nV I V
ϑ π ϑ
( ) = −
( )
exp 2 2
0 0 ;
I
D
D
nc c c e e s u c
i
I
V
n nI nI n n
00
0
0
2
χ η ϕ ϑ π χ η ϕ ϑ τ
, , ,
( ) = − ( ) ( ) ( ) ( ) −
( ) ( ) v
v
I u v w
u
i
n
( )
∂ ( )
∂
−
=
∞
∫
∫
∫
∫
∑
100
0
1
0
1
0
1
0
1
, , ,τ
ϑ
× ( ) ( ) − ( ) ( ) ( ) ( ) −
c w g u v w T d wd vd u d
D
D
nc c c e e
n I
I
V
n nI nI
, , , τ π χ η ϕ ϑ
2 0
0
τ
τ
ϑ
( ) ( ) ( )
∫
∫
∫
∑
=
∞
c u s v
n n
n 0
1
0
1
0
1
× ( ) ( )
∂ ( )
∂
−
−
∫c w g u v w T
I u v w
v
d wd vd u d
D
D
nc
n I
i I
V
, , ,
, , ,
100
0
1
0
0
2
τ
τ π n
n nI nI
n
c c e e
χ η ϕ ϑ τ
ϑ
( ) ( ) ( ) ( ) −
( )
∫
∑
=
∞
0
1
× ( ) ( ) ( ) ( )
∂ ( )
∂
−
∫
c u c v s w g u v w T
I u v w
w
d wd vd u d
n n n I
i
, , ,
, , ,
100
0
1
0
τ
τ
1
1
0
1
∫
∫ , i≥1,
V
D
D
nc c c e e s u c
i
V
I
n nV nI n n
00
0
0
2
χ η ϕ ϑ π χ η ϕ ϑ τ
, , ,
( ) = − ( ) ( ) ( ) ( ) −
( ) ( ) v
v g u v w T
V
n
( ) ( )
∫
∫
∫
∫
∑
=
∞
, , ,
0
1
0
1
0
1
0
1
ϑ
13. Pankratov: On increasing of integration rate of elements of hybrid comparator circuit
AJMS/Jul-Sep-2019/Vol 3/Issue 3 13
× ( )
∂ ( )
∂
− ( ) ( ) ( ) ( )
−
c w
V u
u
d wd vd u d
D
D
nc c c e e
n
i V
I
n nV n
100 0
0
,τ
τ χ η ϕ ϑ I
I n n
n
c u s v
−
( ) ( ) ( )
∫
∫
∫
∑
=
∞
τ
ϑ
0
1
0
1
0
1
× ( ) ( )
∂ ( )
∂
−
−
∫
2 2
100
0
1
0
0
π
τ
τ π χ
c w g u v w T
V u
v
d wd vd u d
D
D
nc
n V
i V
I
n
, , ,
,
(
( ) ( ) ( ) ( )
=
∞
∑ c c enV
n
η ϕ ϑ
1
× −
( ) ( ) ( ) ( ) ( )
∂ ( )
∂
−
e c u c v s w g u v w T
V u
w
d wd vd u d
nI n n n V
i
τ
τ
τ
, , ,
,
100
0
1
1
0
1
0
1
0
∫
∫
∫
∫
ϑ
, i≥1,
where sn
(c) = sin (p n c);
ρ χ η ϕ ϑ χ η ϕ ϑ τ
ρ ρ
010 2
, , ,
( ) = − ( ) ( ) ( ) ( ) −
( ) ( ) ( )
c c c e e c u c v c w
n n n n n n n n (
( )
∫
∫
∫
∫
∑
=
∞
0
1
0
1
0
1
0
1
ϑ
n
× + ( )
( ) ( )
1 000 000
ε τ τ
I V I V
g u v w T I u v w V u v w d wd vd u
, , , , , , , , , , ,
d
dτ ;
ρ χ η ϕ ϑ χ η ϕ ϑ τ
ρ ρ
020
0
0
2
, , ,
( ) = − ( ) ( ) ( ) ( ) −
( ) ( )
D
D
c c c e e c u c
I
V
n n n n n n n v
v c w
n I V
n
( ) ( ) +
∫
∫
∫
∫
∑
=
∞
1
0
1
0
1
0
1
0
1
ε
ϑ
,
× ( )
( ) ( )+ (
g u v w T I u v w V u v w I u v w
I V
, , , , , , , , , , , , ,
010 000 000
τ τ τ )
) ( )
V u v w d wd vd u d
010 , , ,τ τ ;
ρ χ η ϕ ϑ χ η ϕ ϑ τ
ρ ρ
001 2
, , ,
( ) = − ( ) ( ) ( ) ( ) −
( ) ( ) ( )
c c c e e c u c v c w
n n n n n n n n (
( )
∫
∫
∫
∫
∑
=
∞
0
1
0
1
0
1
0
1
ϑ
n
× + ( )
( )
1 000
2
ε ρ τ τ
ρ ρ ρ ρ
, , , , , , , ,
g u v w T u v w d wd vd u d
;
ρ χ η ϕ ϑ χ η ϕ ϑ τ
ρ ρ
002 2
, , ,
( ) = − ( ) ( ) ( ) ( ) −
( ) ( ) ( )
c c c e e c u c v c w
n n n n n n n n (
( )
∫
∫
∫
∫
∑
=
∞
0
1
0
1
0
1
0
1
ϑ
n
× + ( )
( ) ( )
1 001 000
ε ρ τ ρ τ
ρ ρ ρ ρ
, , , , , , , , , , ,
g u v w T u v w u v w d wd vd u
d
dτ ;
I
D
D
nc c c e e s u
I
V
n n n nI nI n
110
0
0
2
χ η ϕ ϑ π χ η ϕ ϑ τ
, , ,
( ) = − ( ) ( ) ( ) ( ) −
( ) ( ) c
c v c u
n n
n
( ) ( )
∫
∫
∫
∫
∑
=
∞
0
1
0
1
0
1
0
1
ϑ
× ( )
∂ ( )
∂
− ( ) (
−
g u v w T
I u v w
u
d wd vd u d
D
D
nc c
I
i I
V
n n
, , ,
, , ,
100 0
0
2
τ
τ π χ η)
) ( ) ( )
=
∞
∑ c e
n nI
n
ϕ ϑ
1
× −
( ) ( ) ( ) ( ) ( )
∂ ( )
∂
−
e c u s v c u g u v w T
I u v w
v
d wd vd
nI n n n I
i
τ
τ
, , ,
, , ,
100
u
u d
D
D
I
V
τ π
ϑ
0
1
0
1
0
1
0
0
0
2
∫
∫
∫
∫ −
× ( ) −
( ) ( ) ( ) ( ) ( )
∂ ( )
−
n e e c u c v s u g u v w T
I u v w
nI nI n n n I
i
ϑ τ
τ
, , ,
, , ,
100
∂
∂
∫
∫
∫
∫
∑
=
∞
w
d wd vd u d
n
τ
ϑ
0
1
0
1
0
1
0
1
14. Pankratov: On increasing of integration rate of elements of hybrid comparator circuit
AJMS/Jul-Sep-2019/Vol 3/Issue 3 14
× ( ) ( ) ( )− ( ) ( ) ( ) ( ) −
( ) ( ) ( )
c c c c e c c e c u c v c
n n n n nI n n nI n n n
χ η ϕ χ ϑ η ϕ τ
2 v
v I V
n
( ) +
∫
∫
∫
∫
∑
=
∞
1
0
1
0
1
0
1
0
1
ε
ϑ
,
× ( )
( ) ( )+ (
g u v w T I u v w V u v w I u v w
I V
, , , , , , , , , , , , ,
100 000 000
τ τ τ )
) ( )
V u v w d wd vd u d
100 , , ,τ τ
V
D
D
nc c c e e s u
V
I
n n n nV nV n
110
0
0
2
χ η ϕ ϑ π χ η ϕ ϑ τ
, , ,
( ) = − ( ) ( ) ( ) ( ) −
( ) ( ) c
c v c u
n n
n
( ) ( )
∫
∫
∫
∫
∑
=
∞
0
1
0
1
0
1
0
1
ϑ
× ( )
∂ ( )
∂
− ( ) (
−
g u v w T
V u v w
u
d wd vd u d
D
D
nc c
V
i V
I
n n
, , ,
, , ,
100 0
0
2
τ
τ π χ η)
) ( ) ( )
=
∞
∑ c e
n nV
n
ϕ ϑ
1
× −
( ) ( ) ( ) ( ) ( )
∂ ( )
∂
−
e c u s v c u g u v w T
V u v w
v
d wd vd
nV n n n V
i
τ
τ
, , ,
, , ,
100
u
u d
D
D
V
I
τ π
ϑ
0
1
0
1
0
1
0
0
0
2
∫
∫
∫
∫ −
× ( ) −
( ) ( ) ( ) ( ) ( )
∂ ( )
−
ne e c u c v s u g u v w T
V u v w
nV nV n n n V
i
ϑ τ
τ
, , ,
, , ,
100
∂
∂
∫
∫
∫
∫
∑
=
∞
w
d wd vd u d
n
τ
ϑ
0
1
0
1
0
1
0
1
× ( ) ( ) ( )− ( ) ( ) ( ) ( ) −
( ) ( ) ( ) +
c c c c e c c e c u c v
n n n n nI n n nV n n
χ η ϕ χ ϑ η ϕ τ
2 1 ε
ε
ϑ
I V I V
n
g u v w T
, , , , ,
( )
∫
∫
∫
∫
∑
=
∞
0
1
0
1
0
1
0
1
× ( ) ( ) ( )+ ( )
c w I u v w V u v w I u v w V u v
n
100 000 000 100
, , , , , , , , , , ,
τ τ τ w
w d wd vd u d
,τ τ
( )
;
I
D
D
nc c c e e s u
I
V
n n n nI nI n
101
0
0
2
χ η ϕ ϑ π χ η ϕ ϑ τ
, , ,
( ) = − ( ) ( ) ( ) ( ) −
( ) ( ) c
c v g u v w T
n I
n
( ) ( )
∫
∫
∫
∫
∑
=
∞
, , ,
0
1
0
1
0
1
0
1
ϑ
× ( )
∂ ( )
∂
− ( ) ( ) ( )
c w
I u v w
u
d wd vd u d
D
D
nc c c e
n
I
V
n n n n
001 0
0
2
, , ,τ
τ π χ η ϕ I
I
n
ϑ
( )
=
∞
∑
1
× ( ) ( ) ( )
∂ ( )
∂
−
∫
∫s v c w g u v w T
I u v w
v
d wd vd u d
D
n n I , , ,
, , ,
001
0
1
0
1
0
2
τ
τ π I
I
V
nI n n n
n
D
ne c c c
0 1
ϑ χ η ϕ
( ) ( ) ( ) ( )
=
∞
∑
× −
( ) ( ) ( ) ( ) ( )
∂ ( )
∂
e c u c v s w g u v w T
I u v w
w
d wd vd u d
nI n n n I
τ
τ
, , ,
, , ,
001
τ
τ χ η ϕ
ϑ
0
1
0
1
0
1
0 1
2
∫
∫
∫
∫ ∑
− ( ) ( ) ( )
=
∞
c c c
n n n
n
× ( ) −
( ) ( ) ( ) ( ) + ( )
e e c u c v c w g u v w T I u
nI nI n n n I V I V
ϑ τ ε
1 100
, , , , , ,
, , , , , ,
v w V u v w d wd vd ud
τ τ τ
ϑ
( ) ( )
∫
∫
∫
∫
000
0
1
0
1
0
1
0
V
D
D
nc c c e e s u
V
I
n n n nV nV n
101
0
0
2
χ η ϕ ϑ π χ η ϕ ϑ τ
, , ,
( ) = − ( ) ( ) ( ) ( ) −
( ) ( ) c
c v g u v w T
n V
n
( ) ( )
∫
∫
∫
∫
∑
=
∞
, , ,
0
1
0
1
0
1
0
1
ϑ
15. Pankratov: On increasing of integration rate of elements of hybrid comparator circuit
AJMS/Jul-Sep-2019/Vol 3/Issue 3 15
× ( )
∂ ( )
∂
− ( ) ( ) ( )
c w
V u v w
u
d wd vd u d
D
D
nc c c e
n
V
I
n n n n
001 0
0
2
, , ,τ
τ π χ η ϕ I
I nV n
n
e c u
ϑ τ
ϑ
( ) −
( ) ( )
∫
∫
∑
=
∞
0
1
0
1
× ( ) ( ) ( )
∂ ( )
∂
−
∫
∫s v c w g u v w T
I u v w
v
d wd vd u d
D
n n I , , ,
, , ,
001
0
1
0
1
0
2
τ
τ π I
I
V
nI n n n
n
D
ne c c c
0 1
ϑ χ η ϕ
( ) ( ) ( ) ( )
=
∞
∑
× −
( ) ( ) ( ) ( ) ( )
∂ ( )
∂
e c u c v s w g u v w T
V u v w
w
d wd vd u d
nV n n n V
τ
τ
, , ,
, , ,
001
τ
τ χ η ϕ
ϑ
0
1
0
1
0
1
0 1
2
∫
∫
∫
∫ ∑
− ( ) ( ) ( )
=
∞
c c c
n n n
n
× ( ) −
( ) ( ) ( ) ( ) + ( )
e e c u c v c w g u v w T I u
nV nV n n n I V I V
ϑ τ ε
1 100
, , , , , ,
, , , , , ,
v w V u v w d wd vd u d
τ τ τ
ϑ
( ) ( )
∫
∫
∫
∫
000
0
1
0
1
0
1
0
;
I c c c e e c u c v c w
n n n nI nI n n n
011 2
χ η ϕ ϑ χ η ϕ ϑ τ
, , ,
( ) = − ( ) ( ) ( ) ( ) −
( ) ( ) ( ) (
( ) ( )
{
∫
∫
∫
∫
∑
=
∞
I u v w
n
000
0
1
0
1
0
1
0
1
, , ,τ
ϑ
× + ( )
( )+ + ( )
1 1
010
ε τ ε
I I I I I V I V
g u v w T I u v w g u v w T
, , , ,
, , , , , , , , ,
( ) ( )}
I u v w V u v w d wd vd ud
001 000
, , , , , ,
τ τ τ
V c c c e e c u c v c w
n n n nV nV n n n
011 2
χ η ϕ ϑ χ η ϕ ϑ τ
, , ,
( ) = − ( ) ( ) ( ) ( ) −
( ) ( ) ( ) (
( ) ( )
{
∫
∫
∫
∫
∑
=
∞
I u v w
n
000
0
1
0
1
0
1
0
1
, , ,τ
ϑ
× + ( )
( )+ + ( )
1 1
010
ε τ ε
I I I I I V I V
g u v w T I u v w g u v w T
, , , ,
, , , , , , , , ,
( ) ( )}
I u v w V u v w d wd vd ud
001 000
, , , , , ,
τ τ τ .
Equations for functions Fi
(x,y,z,t), i≥0 to describe concentrations of simplest complexes of radiation
defects.
∂
∂
∂
∂
∂
∂
∂
Φ Φ Φ
Φ
I
I
I I
x y z t
t
D
x y z t
x
x y z t
y
0
0
2
0
2
2
0
2
2
, , , , , , , , ,
( )
=
( )
+
( )
+
Φ
ΦI x y z t
z
0
2
, , ,
( )
∂
+ ( ) ( )− ( ) ( )
k x y z T I x y z t k x y z T I x y z t
I I I
, , , , , , , , , , , , ,
2
∂
∂
∂
∂
∂
∂
∂
Φ Φ Φ
Φ
V
V
V V
x y z t
t
D
x y z t
x
x y z t
y
0
0
2
0
2
2
0
2
2
, , , , , , , , ,
( )
=
( )
+
( )
+
Φ
ΦV x y z t
z
0
2
, , ,
( )
∂
+ ( ) ( )− ( ) ( )
k x y z T V x y z t k x y z T V x y z t
V V V
, , , , , , , , , , , , ,
2
;
∂
∂
∂
∂
∂
∂
∂
Φ Φ Φ
Φ
I i
I
I i I i
x y z t
t
D
x y z t
x
x y z t
y
, , , , , , , , ,
( )
=
( )
+
( )
+
0
2
2
2
2
2
Φ
ΦI i x y z t
z
, , ,
( )
∂ 2
+ ( )
( )
+ (
−
D
x
g x y z T
x y z t
x y
g x y z T
I I
I i
I
0
1
Φ Φ Φ
Φ
, , ,
, , ,
, , , )
)
( )
−
ΦI i x y z t
y
1 , , ,
16. Pankratov: On increasing of integration rate of elements of hybrid comparator circuit
AJMS/Jul-Sep-2019/Vol 3/Issue 3 16
+ ( )
( )
−
∂
∂
∂
∂
z
g x y z T
x y z t
z
I
I i
Φ
Φ
, , ,
, , ,
1
, i≥1,
∂
∂
∂
∂
∂
∂
∂
Φ Φ Φ
Φ
V i
V
V i V i
x y z t
t
D
x y z t
x
x y z t
y
, , , , , , , , ,
( )
=
( )
+
( )
+
0
2
2
2
2
2
Φ
ΦV i x y z t
z
, , ,
( )
∂ 2
+ ( )
( )
+ (
−
D
x
g x y z T
x y z t
x y
g x y z T
V V
V i
V
0
1
Φ Φ Φ
Φ
∂
∂
∂
∂
∂
∂
, , ,
, , ,
, , , )
)
( )
−
∂
∂
ΦV i x y z t
y
1 , , ,
+ ( )
( )
−
∂
∂
∂
∂
z
g x y z T
x y z t
z
V
V i
Φ
Φ
, , ,
, , ,
1
, i≥1;
Boundary and initial conditions for the functions take the form
∂ ( )
∂
=
=
Φρ i
x
x y z t
x
, , ,
0
0,
∂ ( )
∂
=
=
Φρ i
x L
x y z t
x
x
, , ,
0,
∂ ( )
∂
=
=
Φρ i
y
x y z t
y
, , ,
0
0,
∂ ( )
∂
=
=
Φri
y L
x y z t
y
y
, , ,
0 ,
∂ ( )
∂
=
=
Φρ i
z
x y z t
z
, , ,
0
0,
∂ ( )
∂
=
=
Φρ i
z L
x y z t
z
z
, , ,
0, i≥0; Fr
0(x,y,z,0)=fFr
(x,y,z),
Fri
(x,y,z,0)=0, i≥1.
Solutions of the above equations could be written as
Φ Φ Φ
r r r
0
1
1 2
x y z t
L L L L L L
F c x c y c z e t
x y z x y z
n n n n n
n
, , ,
( ) = + ( ) ( ) ( ) ( )
=
∞
∑
∑ ∑
+ ( ) ( ) ( )
=
∞
2
1
L
n c x c y c z
n n n
n
× ( ) −
( ) ( ) ( ) ( ) ( ) ( )
e t e c u c v c w k u v w T I u v w
n n n n n I I
Φ Φ
ρ ρ
τ τ
, , , , , , ,
2
0
L
L
L
L
t z
y
x
∫
∫
∫
∫ 0
0
0
− ( ) ( )
k u v w T I u v w d wd vd u d
I , , , , , ,τ τ ,
where
F c u c v c w f u v w d wd vd u
n n n n
L
L
L z
y
x
Φ Φ
ρ ρ
= ( ) ( ) ( ) ( )
∫
∫
∫ , ,
0
0
0
, e t n D t L L L
n x y z
Φ Φ
ρ ρ
π
( ) = − + +
( )
− − −
exp 2 2
0
2 2 2
,
cn
(x) = cos (p n x/Lx
);
Φ Φ Φ
ρ
π
τ
ρ ρ
i
x y z
n n n n n n
x y z t
L L L
nc x c y c z e t e s u
, , ,
( ) = − ( ) ( ) ( ) ( ) −
( ) (
2
2 )
) ( ) ( )
∫
∫
∫
∫
∑
=
∞
c v g u v w T
n
L
L
L
t
n
z
y
x
Φρ
, , ,
0
0
0
0
1
× ( )
( )
− ( ) ( ) ( )
−
c w
u v w
u
d wd vd u d
L L L
nc x c y c z
n
I i
x y z
n n n
∂ τ
∂
τ
π
ρ
Φ 1
2
2
, , ,
e
e t e
n n
t
n
Φ Φ
ρ ρ
τ
( ) −
( )
∫
∑
=
∞
0
1
17. Pankratov: On increasing of integration rate of elements of hybrid comparator circuit
AJMS/Jul-Sep-2019/Vol 3/Issue 3 17
× −
( ) ( ) ( ) ( ) ( )
( )
−
e c u s v c w g u v w T
u v w
v
d wd v
n n n n
I i
Φ Φ
Φ
ρ ρ
ρ
τ
∂ τ
∂
, , ,
, , ,
1
d
d u d
L L L
n
L
L
L
t
x y z n
z
y
x
τ
π
0
0
0
0
2
1
2
∫
∫
∫
∫ ∑
−
=
∞
× ( ) −
( ) ( ) ( ) ( )
( )
−
e t e c u c v s w
u v w
w
g u v w
n n n n n
I i
Φ Φ Φ
Φ
ρ ρ
ρ
ρ
τ
∂ τ
∂
1 , , ,
, , ,T
T d wd vd u d
L
L
L
t z
y
x
( )
∫
∫
∫
∫ τ
0
0
0
0
× ( ) ( ) ( )
c x c y c z
n n n , i≥1,
where sn
(x) = sin (p n x/Lx
).
Equations for the functions Cij
(x,y,z,t) (i≥0, j≥0), boundary and initial conditions could be written as
∂ ( )
∂
=
∂ ( )
∂
+
∂ ( )
∂
+
C x y z t
t
D
C x y z t
x
D
C x y z t
y
L L
00
0
2
00
2 0
2
00
2
, , , , , , , , ,
D
D
C x y z t
z
L
0
2
00
2
∂ ( )
∂
, , ,
;
∂ ( )
∂
=
∂ ( )
∂
+
∂ ( )
∂
+
∂
C x y z t
t
D
C x y z t
x
C x y z t
y
C
i
L
i i
0
0
2
0
2
2
0
2
2
, , , , , , , , , i
i x y z t
z
0
2
, , ,
( )
∂
+
∂
∂
( )
∂ ( )
∂
+
∂
∂
(
−
D
x
g x y z T
C x y z t
x
D
y
g x y z T
L L
i
L L
0
10
0
, , ,
, , ,
, , , )
)
∂ ( )
∂
−
C x y z t
y
i 10 , , ,
+
∂
∂
( )
∂ ( )
∂
−
D
z
g x y z T
C x y z t
z
L L
i
0
10
, , ,
, , ,
, i≥1;
∂ ( )
∂
=
∂ ( )
∂
+
∂ ( )
∂
+
C x y z t
t
D
C x y z t
x
D
C x y z t
y
L L
01
0
2
01
2 0
2
01
2
, , , , , , , , ,
D
D
C x y z t
z
L
0
2
01
2
∂ ( )
∂
, , ,
+
∂
∂
( )
( )
∂ ( )
∂
+
∂
∂
D
x
C x y z t
P x y z T
C x y z t
x
D
L L
0
00 00
0
γ
γ
, , ,
, , ,
, , ,
y
y
C x y z t
P x y z T
C x y z t
y
00 00
γ
γ
, , ,
, , ,
, , ,
( )
( )
∂ ( )
∂
+
∂
∂
( )
( )
∂ ( )
∂
D
z
C x y z t
P x y z T
C x y z t
z
L
0
00 00
γ
γ
, , ,
, , ,
, , ,
;
∂ ( )
∂
=
∂ ( )
∂
+
∂ ( )
∂
+
C x y z t
t
D
C x y z t
x
D
C x y z t
y
L L
02
0
2
02
2 0
2
02
2
, , , , , , , , ,
D
D
C x y z t
z
L
0
2
02
2
∂ ( )
∂
, , ,
+
∂
∂
( )
( )
( )
∂ ( )
−
D
x
C x y z t
C x y z t
P x y z T
C x y z t
L
0 01
00
1
00
, , ,
, , ,
, , ,
, , ,
γ
γ
∂
∂
+
∂
∂
( )
( )
( )
−
x y
C x y z t
C x y z t
P x y z T
01
00
1
, , ,
, , ,
, , ,
γ
γ
×
∂ ( )
∂
+
∂
∂
( )
( )
−
C x y z t
y z
C x y z t
C x y z t
P x y z
00
01
00
1
, , ,
, , ,
, , ,
, , ,
γ
γ
T
T
C x y z t
z
( )
∂ ( )
∂
00 , , ,
×
∂ ( )
∂
+
∂
∂
( )
( )
−
C x y z t
y z
C x y z t
C x y z t
P x y z
00
01
00
1
, , ,
, , ,
, , ,
, , ,
γ
γ
T
T
C x y z t
z
D
x
C x y z t
P x y z
L
( )
∂ ( )
∂
+
∂
∂
( )
00
0
00
, , , , , ,
, , ,
γ
γ
T
T
( )
18. Pankratov: On increasing of integration rate of elements of hybrid comparator circuit
AJMS/Jul-Sep-2019/Vol 3/Issue 3 18
×
∂ ( )
∂
+
∂
∂
( )
( )
∂
C x y z t
x y
C x y z t
P x y z T
C x y z t
01 00 01
, , , , , ,
, , ,
, , ,
γ
γ
(
( )
∂
+
∂
∂
( )
( )
∂ ( )
∂
y z
C x y z t
P x y z T
C x y z t
z
00 01
γ
γ
, , ,
, , ,
, , ,
;
∂ ( )
∂
=
∂ ( )
∂
+
∂ ( )
∂
+
C x y z t
t
D
C x y z t
x
D
C x y z t
y
L L
11
0
2
11
2 0
2
11
2
, , , , , , , , ,
D
D
C x y z t
z
L
0
2
11
2
∂ ( )
∂
, , ,
+
∂
∂
( )
( )
( )
∂ ( )
∂
−
x
C x y z t
C x y z t
P x y z T
C x y z t
x
10
00
1
00
, , ,
, , ,
, , ,
, , ,
γ
γ
+
∂
∂
( )
( )
( )
−
y
C x y z t
C x y z t
P x y z T
10
00
1
, , ,
, , ,
, , ,
γ
γ
×
∂ ( )
∂
+
∂
∂
( )
( )
−
C x y z t
y z
C x y z t
C x y z t
P x y z
00
10
00
1
, , ,
, , ,
, , ,
, , ,
γ
γ
T
T
C x y z t
z
D L
( )
∂ ( )
∂
00
0
, , ,
+
∂
∂
( )
( )
∂ ( )
∂
+
∂
∂
D
x
C x y z t
P x y z T
C x y z t
x y
C
L
0
00 10 0
γ
γ
, , ,
, , ,
, , , 0
0 10
γ
γ
x y z t
P x y z T
C x y z t
y
, , ,
, , ,
, , ,
( )
( )
∂ ( )
∂
+
∂
∂
( )
( )
∂ ( )
∂
+
z
C x y z t
P x y z T
C x y z t
z
D L
00 10
0
γ
γ
, , ,
, , ,
, , , ∂
∂
∂
( )
∂ ( )
∂
x
g x y z T
C x y z t
x
L , , ,
, , ,
01
+
∂
∂
( )
∂ ( )
∂
+
∂
∂
( )
∂
y
g x y z T
C x y z t
y z
g x y z T
C x y
L L
, , ,
, , ,
, , ,
,
01 01 ,
, ,
z t
z
( )
∂
;
∂
∂
C x y z t
x
ij
x
, , ,
( )
=
=0
0,
∂
∂
C x y z t
x
ij
x Lx
, , ,
( )
=
=
0,
∂
∂
C x y z t
y
ij
y
, , ,
( )
=
=0
0,
∂
∂
C x y z t
y
ij
y Ly
, , ,
( )
=
=
0,
∂
∂
C x y z t
z
ij
z
, , ,
( )
=
=0
0,
∂
∂
C x y z t
z
ij
z Lz
, , ,
( )
=
=
0, i≥0, j≥0;
C00
(x,y,z,0)=fC
(x,y,z), Cij
(x,y,z,0)=0, i³1, j³1.
Functions Cij
(x,y,z,t) (i³0, j³0) could be approximated by the following series during solutions of the
above equations.
C x y z t
F
L L L L L L
F c x c y c z e t
C
x y z x y z
nC n n n nC
n
00
0
1
2
, , ,
( ) = + ( ) ( ) ( ) ( )
=
∞
∑
∑ .
Here, e t n D t
L L L
nC C
x y z
( ) = − + +
exp π 2 2
0 2 2 2
1 1 1
, F c u c v f u v w c w d wd vd u
nC n n C n
L
L
L z
y
x
= ( ) ( ) ( ) ( )
∫
∫
∫ , ,
0
0
0
;
C x y z t
L L L
n F c x c y c z e t e s u
i
x y z
nC n n n nC nC n
0 2
2
, , ,
( ) = − ( ) ( ) ( ) ( ) −
( )
π
τ (
( ) ( ) ( )
∫
∫
∫
∫
∑
=
∞
c v g u v w T
n L
L
L
L
t
n
z
y
x
, , ,
0
0
0
0
1
19. Pankratov: On increasing of integration rate of elements of hybrid comparator circuit
AJMS/Jul-Sep-2019/Vol 3/Issue 3 19
× ( )
∂ ( )
∂
− ( ) ( )
−
c w
C u v w
u
d wd vd u d
L L L
n F c x c y c z
n
i
x y z
nC n n n
10
2
2
, , ,τ
τ
π
(
( ) ( ) −
( )
∫
∑
=
∞
e t e
nC nC
t
n
τ
0
1
× ( ) ( ) ( ) ( )
∂ ( )
∂
−
∫
c u s v c v g u v w T
C u v w
v
d wd vd u d
n n n L
i
L
L z
, , ,
, , ,
10
0
0
τ
τ
y
y
x
L
x y z
nC nC
n
L L L
n F e t
∫
∫ ∑
− ( )
=
∞
0
2
1
2π
× ( ) ( ) ( ) −
( ) ( ) ( ) ( ) ( )
∂ −
c x c y c z e c u c v s v g u v w T
C u v
n n n nC n n n L
i
τ , , ,
,
10 ,
, ,
w
w
d wd vd u d
L
L
L
t z
y
x
τ
τ
( )
∂
∫
∫
∫
∫ 0
0
0
0
, i³1;
C x y z t
L L L
n F c x c y c z e t e s u
x y z
nC n n n nC nC n
01 2
2
, , ,
( ) = − ( ) ( ) ( ) ( ) −
( )
π
τ (
( ) ( ) ( )
∫
∫
∫
∫
∑
=
∞
c v c w
n n
L
L
L
t
n
z
y
x
0
0
0
0
1
×
( )
( )
∂ ( )
∂
−
C u v w
P u v w T
C u v w
u
d wd vd u d
L L L
x y z
00 00
2
2
γ
γ
τ τ
τ
π
, , ,
, , ,
, , ,
n
n F c x c y c z e t
nC n n n nC
n
( ) ( ) ( ) ( )
=
∞
∑
1
× −
( ) ( ) ( ) ( )
( )
( )
∂
e c u s v c w
C u v w
P u v w T
C u v w
nC n n n
τ
τ τ
γ
γ
00 00
, , ,
, , ,
, , ,
(
( )
∂
− ( )
∫
∫
∫
∫ ∑
=
∞
v
d wd vd u d
L L L
n e t
L
L
L
t
x y z
nC
n
z
y
x
τ
π
0
0
0
0
2
1
2
× ( ) ( ) ( ) −
( ) ( ) ( ) ( )
( )
F c x c y c z e c u c v s w
C u v w
P u
nC n n n nC n n n
τ
τ
γ
γ
00 , , ,
,v
v w T
C u v w
w
d wd vd u d
L
L
L
t z
y
x
, ,
, , ,
( )
∂ ( )
∂
∫
∫
∫
∫
00
0
0
0
0
τ
τ ;
C x y z t
L L L
n F c x c y c z e t e s u
x y z
nC n n n nC nC n
02 2
2
, , ,
( ) = − ( ) ( ) ( ) ( ) −
( )
π
τ (
( ) ( ) ( )
∫
∫
∫
∫
∑
=
∞
c v c w
n n
L
L
L
t
n
z
y
x
0
0
0
0
1
× ( )
( )
( )
∂ ( )
∂
−
C u v w
C u v w
P u v w T
C u v w
u
d wd v
01
00
1
00
, , ,
, , ,
, , ,
, , ,
τ
τ τ
γ
γ
d
d u d
L L L
F c x c y
x y z
nC n n
n
τ
π
− ( ) ( )
=
∞
∑
2
2
1
× ( ) ( ) −
( ) ( ) ( ) ( )
( )
−
nc z e t e c u s v C u v w
C u v w
P
n nC nC n n
τ τ
τ
γ
01
00
1
, , ,
, , ,
γ
γ
τ
u v w T
C u v w
v
L
L
L
t z
y
x
, , ,
, , ,
( )
∂ ( )
∂
∫
∫
∫
∫
00
0
0
0
0
× ( ) − ( ) ( ) ( ) ( ) −
( )
c w d wd vd u d
L L L
n F c x c y c z e t e c
n
x y z
nC n n n nC nC n
τ
π
τ
2
2
u
u c v
n
L
L
t
n
y
x
( ) ( )
∫
∫
∫
∑
=
∞
0
0
0
1
× ( ) ( )
( )
( )
∂ ( )
∂
−
s w C u v w
C u v w
P u v w T
C u v w
n 01
00
1
00
, , ,
, , ,
, , ,
, , ,
τ
τ τ
γ
γ
w
w
d wd vd u d
L L L
n c x
L
x y z
n
n
z
τ
π
0
2
1
2
∫ ∑
− ( )
=
∞
× ( ) ( ) ( ) −
( ) ( ) ( ) ( ) ( )
∂
F c y c z e t e s u c v c w C u v w
C
nC n n nC nC n n n
τ τ
01
00
, , ,
u
u v w
u
L
L
L
t z
y
x
, , ,τ
( )
∂
∫
∫
∫
∫ 0
0
0
0
×
( )
( )
− ( ) (
−
C u v w
P u v w T
d wd vd u d
L L L
n F c x c y
x y z
nC n n
00
1
2
2
γ
γ
τ
τ
π
, , ,
, , ,
)
) ( ) ( ) −
( ) ( )
∫
∫
∑
=
∞
c z e t e c u
n nC nC n
L
t
n
x
τ
0
0
1
20. Pankratov: On increasing of integration rate of elements of hybrid comparator circuit
AJMS/Jul-Sep-2019/Vol 3/Issue 3 20
× ( ) ( ) ( )
( )
( )
∂
−
s v c w C u v w
C u v w
P u v w T
C u v w
n n 01
00
1
00
, , ,
, , ,
, , ,
, ,
τ
τ
γ
γ
,
,τ
τ
π
( )
∂
−
∫
∫ ∑
=
∞
v
d wd vd u d
L L L
n
L
L
x y z n
z
y
0
0
2
1
2
× ( ) ( ) ( ) ( ) −
( ) ( ) ( ) ( ) (
F c x c y c z e t e c u c v s w C u v w
nC n n n nC nC n n n
τ τ
01 , , , )
)
( )
( )
−
∫
∫
∫
∫
C u v w
P u v w T
L
L
L
t z
y
x
00
1
0
0
0
0
γ
γ
τ
, , ,
, , ,
×
∂ ( )
∂
− ( ) ( ) ( ) ( )
C u v w
w
d wd vd u d
L L L
F c x c y c z e t
x y z
nC n n n nC
00
2
2
, , ,τ
τ
π
e
e s u
nC n
L
t
n
x
−
( ) ( )
∫
∫
∑
=
∞
τ
0
0
1
× ( ) ( )
( )
( )
∂ ( )
∂
n c v c w
C u v w
P u v w T
C u v w
u
d wd vd u d
n n
00 01
γ
γ
τ τ
, , ,
, , ,
, , ,
τ
τ
π
0
0
2
1
2
L
L
x y z
n nC
n
z
y
L L L
c x e t
∫
∫ ∑
− ( ) ( )
=
∞
× ( ) −
( ) ( ) ( ) ( )
( )
( )
∂
F c y e c u s v c w
C u v w
P u v w T
C
nC n nC n n n
τ
τ
γ
γ
00 0
, , ,
, , ,
1
1
0
0
0
0
u v w
v
d wd vd u d
L
L
L
t z
y
x
, , ,τ
τ
( )
∂
∫
∫
∫
∫
× ( )− ( ) ( ) ( ) ( ) −
( ) ( ) (
n c z
L L L
n F c x c y c z e t e c u c v
n
x y z
nC n n n nC nC n n
2
2
π
τ )
) ( )
∫
∫
∫
∫
∑
=
∞
s w
n
L
L
L
t
n
z
y
x
0
0
0
0
1
×
( )
( )
∂ ( )
∂
C u v w
P u v w T
C u v w
w
d wd vd u d
00 01
γ
γ
τ τ
τ
, , ,
, , ,
, , ,
;
C x y z t
L L L
n F c x c y c z e t e s u
x y z
nC n n n nC nC n
11 2
2
, , ,
( ) = − ( ) ( ) ( ) ( ) −
( )
π
τ (
( ) ( ) ( )
∫
∫
∫
∫
∑
=
∞
c v c w
n n
L
L
L
t
n
z
y
x
0
0
0
0
1
× ( )
∂ ( )
∂
− ( ) (
g u v w T
C u v w
u
d wd vd u d
L L L
n F c x c y
L
x y z
nC n n
, , ,
, , ,
01
2
2
τ
τ
π
)
) ( ) ( )
=
∞
∑ c z e t
n nC
n 1
× −
( ) ( ) ( ) ( ) ( )
∂ ( )
∂
e c u s v c w g u v w T
C u v w
v
d wd vd u d
nC n n n L
τ
τ
τ
, , ,
, , ,
01
0
L
L
L
L
t
x y z
z
y
x
L L L
∫
∫
∫
∫ −
0
0
0
2
2π
× ( ) −
( ) ( ) ( ) ( ) ( )
∂ ( )
∂
n e t e c u c v s w g u v w T
C u v w
w
d w
nC nC n n n L
τ
τ
, , ,
, , ,
01
d
d vd u d
L
L
L
t
n
z
y
x
τ
0
0
0
0
1
∫
∫
∫
∫
∑
=
∞
× ( ) ( ) ( )− ( ) ( ) ( ) ( ) −
F c x c y c z
L L L
F c x c y c z e t e
nC n n n
x y z
nC n n n nC nC
2
2
π
τ
(
( ) ( ) ( )
∫
∫
∫
∑
=
∞
s u c v
n n
L
L
t
n
y
x
0
0
0
1
× ( )
( )
( )
∂ ( )
∂
∫
n c w
C u v w
P u v w T
C u v w
u
d wd vd u d
n
Lz
00 10
0
γ
γ
τ τ
τ
, , ,
, , ,
, , ,
−
− ( ) ( )
=
∞
∑
2
2
1
π
L L L
n F c x c y
x y z
nC n n
n
× ( ) ( ) −
( ) ( ) ( ) ( )
( )
( )
c z e t e c u s v c w
C u v w
P u v w T
n nC nC n n n
τ
τ
γ
γ
00 , , ,
, , ,
∂
∂ ( )
∂
∫
∫
∫
∫
C u v w
v
d wd vd u d
L
L
L
t z
y
x
10
0
0
0
0
, , ,τ
τ
21. Pankratov: On increasing of integration rate of elements of hybrid comparator circuit
AJMS/Jul-Sep-2019/Vol 3/Issue 3 21
− ( ) ( ) ( ) ( ) −
( ) ( ) ( ) ( )
2
2
π
τ
L L L
n F c x c y c z e t e c u c v s w
C
x y z
nC n n n nC nC n n n
0
00
0
0
0
0
1
γ
γ
τ
u v w
P u v w T
L
L
L
t
n
z
y
x
, , ,
, , ,
( )
( )
∫
∫
∫
∫
∑
=
∞
×
∂ ( )
∂
− ( ) ( ) ( ) (
C u v w
w
d wd vd u d
L L L
n F c x c y c z e t
x y z
nC n n n nC
10
2
2
, , ,τ
τ
π
)
) −
( ) ( )
∫
∫
∑
=
∞
e s u
nC n
L
t
n
x
τ
0
0
1
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