AJMS

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.
REFERENCES
1. Lachin VI, Savelov NS. Electronics. Rostov-na-Donu: Phoenix; 2001.
2. Alexenko AG, Shagurin II. Microcircuitry. Moscow: Radio and Communication; 1990.
3. Avaev NA, Naumov YE, Frolkin VT. Basis of Microelectronics. Moscow: Radio and Communication; 1991.
4. Huang S, Diao S, Lin F. An energy-efficient high-speed CMOS hybrid comparator with reduced delay time in 40-nm
CMOS process. Analog Integr Circ Sig Process 2016;89:231-8.
5. Fathi D, Forouzandeh B, Masoumi N. New enhanced noise analysis in active mixers in nanoscale technologies.
Nano 2009;4:233-8.
6. Chachuli SA, Fasyar PN, Soin N, Kar NM, Yusop N. Pareto ANOVA analysis for CMOS 0.18 µm two stage Op-amp.
Mater Sci Sem Proc 2014;24:9-14.
7. Ageev AO, Belyaev AE, Boltovets NS, Ivanov VN, Konakova RV, Kudrik YY, et al. Au-TiBx-n6HSiC schottky barrier
diodes: specific features of charge transport in rectifying and nonrectifying contacts. Semiconductors 2009;43:897-903.
8. Li Z, Waldron J, Detchprohm T, Wetzel C, Karlicek RF, Chow TP Jr. Monolithic integration of light-emitting diodes and
power metal-oxide-semiconductor channel high-electron-mobility transistors for light-emitting power integrated circuits
in GaN on sapphire substrate. Appl Phys Lett 2013;102:192107-9.
9. Tsai JH, Chiu SY, Lour WS, Guo DF. High-performance InGaP/GaAs pnp δ-doped heterojunction bipolar transistor.
Semiconductors 2009;43:971-4.
10. Alexandrov OV, Zakhar’in AO, Sobolev NA, Shek EI, Makoviychuk MM, Parshin EO. Formation of donor centers upon
annealing of dysprosium-and holmiumimplanted silicon. Semiconductors 1998;32:1029-32.
11. Kumar MJ, Singh TV. Quantum confinement effects in strained silicon MOSFETs MOSFETs. Int J Nanosci 2008;7:81-4.
12. Sinsermsuksakul P, Hartman K, Kim SB, Heo J, Sun L, Park HH, et al. Band alignment of SnS/Zn(O,S) heterojunctions
in SnS thin film solar cells. Appl Phys Lett 2013;102:53901-5.
13. Reynolds JG, Reynolds CL, Mohanta A Jr., Muth JF, Rowe JE, Everitt HO, et al. Shallow acceptor complexes in
p-type ZnO. Appl Phys Lett 2013;102:152114-8.
14. Ong KK, Pey KL, Lee PS, Wee AT, Wang XC, Chong YF. Dopant distribution in the recrystallization transient at the
maximum melt depth induced by laser annealing. Appl Phys Lett 2006;89:172111-4.
15. Wang HT, Tan LS, Chor EF. Pulsed laser annealing of Be-implanted GaN. J Appl Phys 2005;98:94901-5.
16. Shishiyanu ST, Shishiyanu TS, Railyan SK. Shallow p-n junctions in Si prepared by pulse photon annealing.
Semiconductors 2002;36:611-7.
17. Bykov YV, Yeremeev AG, Zharova NA, Plotnikov IV, Rybakov KI, Drozdov MN, et al. Diffusion processes in
semiconductor structures during microwave annealing. Radiophys Quantum Electron 2003;43:836-43.
18. Pankratov EL, Bulaeva EA. Doping of materials during manufacture p-n junctions and bipolar transistors. Analytical
approaches to model technological approaches and ways of optimization of distribution of dopants. Rev Theor Sci
2013;1:58-82.
19. Erofeev YN. Pulse Devices, Higher School. Moscow: Higher School; 1989.
20. Kozlivsky VV. Modification of Semiconductors by Proton Beams. Sant-Peterburg: Nauka; 2003.
21. Gotra ZY. Technology of Microelectronic Devices. Moscow: Radio and Communication; 1991.
22. Vinetskiy VL, Kholodar GA. Radiative Physics of Semiconductors. Kiev: Naukova Dumka; 1979.
23. Fahey PM, Griffin PB, Plummer JD. Point defects and dopant diffusion in silicon. Rev Mod Phys 1989;61:289-388.

9. Pankratov: On increasing of integration rate of elements of hybrid comparator circuit
AJMS/Jul-Sep-2019/Vol 3/Issue 3 9
24. Tikhonov AN, Samarskii AA. The Mathematical Physics Equations. Moscow: Nauka; 1972.
25. Carslaw HS, Jaeger JC. Conduction of Heat in Solids. Oxford: Oxford University Press; 1964.
26. Pankratov EL. Dopant diffusion dynamics and optimal diffusion time as influenced by diffusion-coefficient nonuniformity.
Russ Microelectron 2007;36:33-9.
27. Pankratov EL. Redistribution of dopant during annealing of radia-tive defects in a multilayer structure by laser scans for
production an implanted-junction rectifiers. Int J Nanosci 2008;7:187-97.
28. Pankratov EL. On approach to optimize manufacturing of bipolar heterotransistors framework circuit of an operational
amplifier to increase their integration rate. Influence mismatch-induced Stress. J Comput Theor Nanosci 2017;14:4885-99.
29. Pankratov EL. On optimization of manufacturing of two-phase logic circuit based on heterostructures to increase density
of their elements. Influence of miss-match induced stress. Adv Sci Eng Med 2017;9:787-801.
30. Pankratov EL, Bulaeva EA. On increasing of density of transistors in a hybrid cascaded multilevel inverter. Multidiscip
Model Mater Struct 2017;13:664-77.
31. Pankratov EL, Bulaeva EA. An approach to manufacture of bipolar transistors in thin film structures. On the method of
optimization. Int J Micro Nano Scale Transp 2014;4:17-31.
32. Pankratov EL, Bulaeva EA. An analytical approach for analysis and optimization of formation of field-effect
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 Fi
(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
× ( ) ( ) ( )
( )
( )
∂
−
c v c w C u v w
C u v w
P u v w T
C u v w
n n 10
00
1
00
, , ,
, , ,
, , ,
, ,
τ
τ
γ
γ
,
,τ
τ
π
( )
∂
−
∫
∫ ∑
=
∞
u
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 s v c w
C u v w
nC n n n nC nC n n n
τ
γ
00
1
, , ,
,
, , ,
, , ,
τ τ
γ
( )
( )
∂ ( )
∂
∫
∫
∫
∫ P u v w T
C u v w
v
L
L
L
t z
y
x
00
0
0
0
0
× ( ) − ( ) ( ) ( ) ( )
C u v w d wd vd u d
L L L
n F c x c y c z e t e
x y z
nC n n n nC n
10 2
2
, , ,τ τ
π
C
C n
L
t
n
c u
x
−
( ) ( )
∫
∫
∑
=
∞
τ
0
0
1
× ( ) ( ) ( )
( )
( )
∂
−
c v s w C u v w
C u v w
P u v w T
C u v w
n n 10
00
1
00
, , ,
, , ,
, , ,
, ,
τ
τ
γ
γ
,
,τ
τ
( )
∂
∫
∫ w
d wd vd u d
L
L z
y
0
0