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.

1. www.ajms.com 54
ISSN 2581-3463
RESEARCH ARTICLE
On Approach to Increase Integration Rate of Elements of an Operational
Amplifier Circuit
E. L. Pankratov1,2
1
Nizhny Novgorod State University, 23 Gagarin Avenue, Nizhny Novgorod, 603950, Russia, 2
Nizhny Novgorod
State Technical University, 24 Minin Street, Nizhny Novgorod, 603950, Russia
Received: 12-08-2018; Revised: 20-09-2018; Accepted: 05-12-2018
ABSTRACT
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.
Key words: Integrator operational amplifier, increasing integration rate of field-effect heterotransistors,
optimization of manufacturing
INTRODUCTION
An actual and intensively solving problems of solid-
state electronics are increasing of integration rate of
elements of integrated circuits (p-n junctions).[1-8]
Increasing of the integration rate leads to necessity
to decrease their dimensions. To decrease, the
dimensions are using several approaches. They are
widelyusinglaserandmicrowavetypesofannealing
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 leads to generating inhomogeneous
distribution of temperature. In this situation several
parameters (such as diffusion coefficient and other)
became inhomogeneous. Due to the inhomogenety
Address for correspondence:
E. L. Pankratov
E-mail: elp2004@mail.ru
one can obtain a possibility to increase integration
rateofelementsofintegratedcircuitsduetochanging
of diffusion speed with coordinate. In this situation,
one can obtain decreasing dimensions of elements
of integrated circuits[18]
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
an integrator circuit so as it is shown in Figure 1.
The manufacturing gives a possibility to increase

2. Pankratov: On increasing of dencity of elements of an amplifier circuit
AJMS/Oct-Dec-2018/Vol 2/Issue 4 55
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 the 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.
Method of solution
In this section, we determine spatiotemporal
distributions of concentrations of infused
and implanted dopants. To determine these
distributions, we calculate appropriate solutions
of the second Fick’s law.[1,3,18-23]


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 0
, ( , , , ) ( , , )
C f
x y z x y z (2)
The function C(x,y,z,t) describes the spatiotemporal
distribution of concentration of dopant; T is the
temperature of annealing; DС
is the dopant diffusion
coefficient.Valueofdopantdiffusioncoefficientcould
bechangedwithchangingmaterialsofheterostructure,
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.[20-22]
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)
Figure 1: The considered amplifier circuit[4]

3. Pankratov: On increasing of dencity of elements of an amplifier circuit
AJMS/Oct-Dec-2018/Vol 2/Issue 4 56
Here, the function DL
(x,y,z,T) describes the
spatial (in heterostructure) and temperature
(due to Arrhenius 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 concentrational
dependence on dopant diffusion coefficient has
been described in details[20]
. It should be noted that
using diffusion type of doping did not generation
radiation defects. In this situation ζ1
= ζ2
= 0,
we determine spatiotemporal distributions of
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
, , ,
( )
− ( ) ( )
k x y z T I x y z t
I I
, , , , , , ,
2
(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
∂ ( )
∂
=
=
 x y z t
x x
, , ,
0
0 ,
∂ ( )
∂
=
=
 x y z t
x x Lx
, , ,
0 ,
∂ ( )
∂
=
=
 x y z t
y y
, , ,
0
0,
∂ ( )
∂
=
=
 x y z t
y y Ly
, , ,
0 ,
∂ ( )
∂
=
=
 x y z t
z z
, , ,
0
0 ,
∂ ( )
∂ =
 x y z t
z z Lz
, , ,
= =
0 0
, ( , , , ) ( , , )
r x y z x y z
fr (5)
Here, ρ =I,V. The function I (x,y,z,t) describes the
spatiotemporal distribution of the concentration of
radiation interstitials; Dρ
(x,y,z,T) is the diffusion
coefficientsofpointradiationdefects;termsV2
(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 ΦV
(x,y,z,t)
and di-interstitials ΦI
(x,y,z,t) by solving the
following system of equations.[21,22]
∂
∂
∂
∂
∂
∂
∂
∂
ΦI
I
I
I
x y z t
t
x
D x y z T
x y z t
x
y
D x
, , ,
, , ,
, , ,
( )
= ( )
( )






+
Φ
Φ
Φ
,
, , ,
, , ,
y z T
x y z t
y
I
( )
( )






∂
∂
Φ
+ ( )
( )






+ ( )
∂
∂
∂
∂
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
( )
− ( ) ( )
, , , , , ,

4. Pankratov: On increasing of dencity of elements of an amplifier circuit
AJMS/Oct-Dec-2018/Vol 2/Issue 4 57
Boundary and initial conditions for these equations
are
∂ ( )
∂
=
=
Φρ x y z t
x x
, , ,
0
0 ,
∂ ( )
∂
=
=
Φρ x y z t
x x Lx
, , ,
0,
∂ ( )
∂
=
=
Φρ x y z t
y y
, , ,
0
0 ,
∂ ( )
∂
=
=
Φρ x y z t
y y Ly
, , ,
0,
∂ ( )
∂
=
=
Φρ x y z t
z z
, , ,
0
0 ,
∂ ( )
∂
=
=
Φρ x y z t
z z Lz
, , ,
0,
ΦI
(x,y,z,0) = fΦI
(x,y,z), ΦV
(x,y,z,0)=fΦV
(x,y,z)(7)
Here, DΦρ
(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 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:
Dρ
(x,y,z,T)=D0ρ
[1+ερ
gρ
(x,y,z,T)], where D0ρ
is the
average values of diffusion coefficients, 0≤ερ
1,
|gρ
(x,y,z,T)|≤1, ρ=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+εI,V
gI,V
(x,y,z,T)], kI,I
(x,y,z,T)
=k0I,I
[1+εI,I
gI, I
(x,y,z,T)] and kV, V
(x,y,z,T)=k0V,V
[1+εV,V
gV,V
(x,y,z,T)], where k0ρ1,ρ2
is the their
average values, 0≤ εI,V
1, 0≤εI,I
1, 0≤εV,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
, , , , , , *
( ) = ( ) ,
 = L k D D
I V I V
2
0 0 0
, , Ω  
= L k D D
I V
2
0 0 0
, ,
 = D D t L
I V
0 0
2
, χ = x/Lx
, η = y/Ly
, φ = z/Lz
. The
introductionleadstotransformationofEquations (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
, , ,
, , ,
× + ( )

 
 ( )
− + ( )
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

5. Pankratov: On increasing of dencity of elements of an amplifier circuit
AJMS/Oct-Dec-2018/Vol 2/Issue 4 58
 
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 conditions (9) gives us possibility to obtain
equations for initial order approximations of
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 theAppendix.
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 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: DΦρ
(x,y,z,T)=D0Φρ
[1+
εΦρ
gΦρ
(x,y,z,T)], where D0Φρ
is the average values
of diffusion coefficients. In this situation, the
Equation (6) could be written as
∂
∂
∂
∂
ε
∂
∂
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










+ ( ) ( )+
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
I
I I
I
I
0
0
1
Φ
Φ Φ
Φ
Φ
∂
∂
ε
∂
∂
, , ,
, , ,
∂
∂
∂
ε
∂
∂
z
g x y z T
x y z t
z
I I
I
1+ ( )

 

( )










Φ Φ
Φ
, , ,
, , ,
− ( ) ( )
k x y z T I x y z t
I , , , , , ,
∂
∂
∂
∂
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
V
V V
V
V
0
0
1
Φ
Φ Φ
Φ
Φ
∂
∂
ε
∂
∂
, , ,
, , ,
∂
∂
∂
ε
∂
∂
z
g x y z T
x y z t
z
V V
V
1+ ( )

 

( )










Φ Φ
Φ
, , ,
, , ,
− ( ) ( )
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
ρ ρ ρ
ε
x y z t x y z t
i
i
i
, , , , , ,
( ) = ( )
=
∞
∑ Φ
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 Φρ0
(x,y,z,t), corrections for
them Φρi
(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.
Now, we calculate distribution of concentration of
dopant in space and time using the approach, which
wasusedforanalysisofradiationdefects.Tousethe
approach,weconsiderthefollowingtransformation
of approximation of dopant diffusion coefficient:
DL
(x,y,z,T) = D0L
[1 + εL
gL
(x,y,z,T)], where D0L
is
the average value of dopant diffusion coefficient,
0 ≤ εL
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 Equation (1) and
conditions (2) leads to obtain equations for the
functionsCij
(x,y,z,t)(i≥1,j≥1),boundaryandinitial
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,whichhavebeenusedinappropriateseries.
Usually, the second-order approximations are enough
good approximations to make qualitative analysis and

6. Pankratov: On increasing of dencity of elements of an amplifier circuit
AJMS/Oct-Dec-2018/Vol 2/Issue 4 59
to obtain quantitative results. All analytical results
have been checked by numerical simulation.
DISCUSSION
In this section, we analyzed spatiotemporal
distributions of concentrations of dopants. Figure 2
shows typical spatial distributions of concentrations
of dopants in neighborhood of interfaces of
heterostructures. We calculate these distributions
of 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 concentration of dopant at one time.
Changingrelationbetweenvaluesofdopantdiffusion
coefficients leads to opposite result [Figure 3].
It should be noted that framework the considered
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 values of dopant
diffusion coefficient in layers of heterostructure increases with increasing of 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 concentration of dopant, implanted in heterostructure from Figure 1, on coordinate in direction, which is perpendicular
to interface between epitaxial layer substrate. Difference between values of dopant diffusion coefficient in layers of
heterostructure increases with increasing of 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
Figure 3: (a) Distributions of 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
values of dopant diffusion coefficients increases with increasing of number of curves. Value of dopant diffusion coefficient
in this section is smaller than the value of dopant diffusion coefficient in the 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
a b

7. Pankratov: On increasing of dencity of elements of an amplifier circuit
AJMS/Oct-Dec-2018/Vol 2/Issue 4 60
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 ψ(x,y,z)
[Figure 4]. Farther, the required values of
optimal annealing time have been calculated by
minimization the following mean squared error:
U
L L L
C x y z
x y z
dzdydx
x y z
L
L
L z
y
x
=
( )
− ( )








∫
∫
∫
1
0
0
0
, , ,
, ,
Θ

(12)
We show optimal values of annealing time as
functions of parameters on 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 layers of heterostructure. If the
dopant has no enough time to achieve the interface,
it is practicable to anneal the dopant additionally.
Figure
5b shows the described dependences of
optimal values of additional annealing time for the
sameparametersasforFigure 5b.Necessitytoanneal
radiation defects leads to smaller values of optimal
Figure 4: (a) Distributions of 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 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 number of curve corresponds to increasing of annealing time
a b
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 ξ = γ = 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
ε for a/L=1/2 and ξ = γ = 0. Curve 3 describes the dependence of the annealing time on value of parameter ξ for a/L=1/2
and ε = γ = 0. Curve 4 describes the dependence of the annealing time on value of parameter γ for a/L=1/2 and ε = ξ =
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 ξ = γ = 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 ε
for a/L=1/2 and ξ = γ = 0. Curve 3 describes the dependence of the annealing time on value of parameter ξ for a/L=1/2 and
ε = γ = 0. Curve 4 describes the dependence of the annealing time on value of parameter γ for a/L = 1/2 and ε = ξ = 0
a b

8. Pankratov: On increasing of dencity of elements of an amplifier circuit
AJMS/Oct-Dec-2018/Vol 2/Issue 4 61
annealing of implanted dopant in comparison with
optimal annealing time of infused dopant.
CONCLUSIONS
In this paper, we introduce an approach to increase
integration rate of element of an operational
amplifier circuit. The approach gives us possibility
to decrease area of the elements with smaller
increasing of the element’s thickness.
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9. Pankratov: On increasing of dencity of elements of an amplifier circuit
AJMS/Oct-Dec-2018/Vol 2/Issue 4 62
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,
∂ ( )
∂
=
∂ ( )
∂
+
∂
  
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 ;

10. Pankratov: On increasing of dencity of elements of an amplifier circuit
AJMS/Oct-Dec-2018/Vol 2/Issue 4 63
∂ ( )
∂
=
∂ ( )
∂
+
∂
  
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
× + ( )

 

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 , , ,

11. Pankratov: On increasing of dencity of elements of an amplifier circuit
AJMS/Oct-Dec-2018/Vol 2/Issue 4 64
+
∂
∂
( )
∂ ( )
∂





 +
∂
∂
(
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
, , , , , , ;
∂ ( )
∂
=
∂ ( )
∂
+
∂
  
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 ≥1, j ≥1, k ≥1).
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 dwdvdu
n n
ρ ρ
ρ
π π π
= ( ) ( ) ( ) ( )
∫
∫
∫
1
0
1
0
1
0
1
*
cos cos cos , , , cn
(χ) = cos (π n χ),
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 dwdvdud
D
D
nc c c e e
n I
I
V
n nI nI
( ) ( ) − ( ) ( ) ( ) ( ) −
, , , τ π χ η ϕ ϑ τ
2 0
0
(
( ) ( ) ( ) ×
∫
∫
∫
∑
=
∞
c u s v
n n
n 0
1
0
1
0
1
ϑ

12. Pankratov: On increasing of dencity of elements of an amplifier circuit
AJMS/Oct-Dec-2018/Vol 2/Issue 4 65
× ( ) ( )
∂ ( )
∂
−
−
∫c w g u v w T
I u v w
v
dw dv du d
D
D
n I
i I
, , ,
, , ,

100
0
1
0
0
2
τ
τ π
V
V
n nI nI
n
nc 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
dwdvdu d
n n n I
i
, , ,
, , ,

100
0
1
0


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
× ( )
∂ ( )
∂
− ( ) ( ) ( ) ( )
−
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
c w g u v w T
V u
v
dw dv du d
D
D
n
n V
i V
I
, , ,
,

c
c c c e
n nV
n
χ 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
(χ) = sin (π n χ);

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
d τ ;

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
   )
) ( )

 


V u v w d wd vd u d
010 , , ,  ;

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
d τ ;

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
dw dv du d
D
D
nc c
I
i I
V
n
, , ,
, , ,

100 0
0
2
τ
τ π χ n
n n nI
n
c e
η ϕ ϑ
( ) ( ) ( )
=
∞
∑
1

13. Pankratov: On increasing of dencity of elements of an amplifier circuit
AJMS/Oct-Dec-2018/Vol 2/Issue 4 66
× −
( ) ( ) ( ) ( ) ( )
∂ ( )
∂
−
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
× ( ) ( ) ( )− ( ) ( ) ( ) ( ) −
( ) ( ) ( )
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
   )
) ( )

 


V u v w d wd vd u d
100 , , , 

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
, , , , , , , , , , ,
   w
w d wd vd u d
, 
( )

 
 ;

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
nI nI n n n I V I V
ϑ τ ε
1
0
1
0
1
, , , , ,
0
0
1
0
100 000
∫
∫
( ) ( )
ϑ
τ τ τ
 
I u v w V u v w d wd vd u d
, , , , , ,

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

14. Pankratov: On increasing of dencity of elements of an amplifier circuit
AJMS/Oct-Dec-2018/Vol 2/Issue 4 67
× ( )
∂ ( )
∂
− ( ) ( ) ( )
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
( )
( )
∫
∫
∫
∫ 0
1
0
1
0
1
0
000
 ;

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 u d
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 u d
001 000
, , , , , ,
τ τ
τ
Equations for functions Φρi
(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 , , ,
+ ( )
( )











−




z
g x y z T
x y z t
z
I
I i
Φ
Φ
, , ,
, , ,
1
, i≥1,

15. Pankratov: On increasing of dencity of elements of an amplifier circuit
AJMS/Oct-Dec-2018/Vol 2/Issue 4 68







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 takes the form
∂ ( )
∂
=
=
Φi
x
x y z t
x
, , ,
0
0 ,
∂ ( )
∂
=
=
Φi
x L
x y z t
x
x
, , ,
0 ,
∂ ( )
∂
=
=
Φi
y
x y z t
y
, , ,
0
0 ,
∂ ( )
∂
=
=
Φi
y L
x y z t
y
y
, , ,
0 ,
∂ ( )
∂
=
=
Φi
z
x y z t
z
, , ,
0
0 ,
∂ ( )
∂
=
=
Φi
z L
x y z t
z
z
, , ,
0 , i≥0; Φρ0
(x,y,z,0)=fΦρ
(x,y,z),
Φρi
(x,y,z,0)=0, i≥1.
Solutions of the above equations could be written as
Φ Φ Φ
  
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
0
0
0
0
L
L
L
t z
y
x
∫
∫
∫
∫
− ( ) ( )

k u v w T I u v w d wd vd u d
I , , , , , ,  ,
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
 
= ( ) ( ) ( ) ( )
∫
∫
∫ , ,
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 (π 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
× −
( ) ( ) ( ) ( ) ( )
( )
−
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,

16. Pankratov: On increasing of dencity of elements of an amplifier circuit
AJMS/Oct-Dec-2018/Vol 2/Issue 4 69
where sn
(x) = sin (π 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


, , ,
, , ,
, , ,
y
y
C x y z t
P x y z T
C x y z t
y
00 00


, , ,
, , ,
, , ,
( )
( )
∂ ( )
∂





 +
+
∂
∂
( )
( )
∂ ( )
∂






D
z
C x y z t
P x y z T
C x y z t
z
L
0
00 00


, , ,
, , ,
, , ,
;
∂ ( )
∂
=
∂ ( )
∂
+
∂ ( )
∂
+
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
, , ,
, , ,
, , ,
, , ,


∂
∂





 +
∂
∂
( )
( )
( )
×






−
x y
C x y z t
C x y z t
P x y z T
01
00
1
, , ,
, , ,
, , ,





×
∂ ( )
∂


 +
∂
∂
( )
( )
−
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
, , ,
, , ,
, , ,
, , ,


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
, , ,
, , ,
, , ,
, , ,


T
T
C x y z t
z
D
x
C x y z t
P x y z
L
( )
∂ ( )
∂











+
∂
∂
( )
00
0
00
, , , , , ,
, , ,


T
T
( )
×








×
∂ ( )
∂


 +
∂
∂
( )
( )
∂
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
, , , , , ,
, , ,
, , ,


(
( )
∂





 +
∂
∂
( )
( )
∂ ( )
∂




y z
C x y z t
P x y z T
C x y z t
z
00 01


, , ,
, , ,
, , ,








;
∂ ( )
∂
=
∂ ( )
∂
+
∂ ( )
∂
+
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
, , ,
, , ,
, , ,
, , ,







 +
∂
∂
( )
( )
( )
×








−
y
C x y z t
C x y z t
P x y z T
10
00
1
, , ,
, , ,
, , ,


×
∂ ( )
∂


 +
∂
∂
( )
( )
−
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
, , ,
, , ,
, , ,
, , ,


T
T
C x y z t
z
D L
( )
∂ ( )
∂











+
00
0
, , ,

17. Pankratov: On increasing of dencity of elements of an amplifier circuit
AJMS/Oct-Dec-2018/Vol 2/Issue 4 70
+
∂
∂
( )
( )
∂ ( )
∂





 +
∂
∂
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


, , ,
, , ,
, , , 0
0 10


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


, , ,
, , ,
, , , ∂
∂
∂
( )
∂ ( )
∂





 +




 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, i ≥1, j ≥1.
Functions Cij
(x,y,z,t) (i ≥ 0, j ≥ 0) 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, ( ) 2 2
0 2 2 2
1 1 1
 
 
= − + +
 
 
 
 
 
nC C
x y z
e t exp n D t
L L L
 , F c u c v f u v w c w d wd v d 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
× ( )
∂ ( )
∂
− ( ) ( )
−
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
 , , ,
,
10 ,
, ,
w
w
d wd vd u d
L
L
L
t z
y
x


( )
∂
∫
∫
∫
∫ 0
0
0
0
, i ≥1;
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

18. Pankratov: On increasing of dencity of elements of an amplifier circuit
AJMS/Oct-Dec-2018/Vol 2/Issue 4 71
× ( ) ( ) ( ) −
( ) ( ) ( ) ( )
( )
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
 
01
00
, , ,
u
u v w
u
L
L
L
t z
y
x
, , ,
( )
∂
×
∫
∫
∫
∫ 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
× ( ) ( ) ( )
( )
( )
∂
−
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
, , ,
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∫
∫
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c v c w
n n
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L
L
t
n
z
y
x
0
0
0
0
1

19. Pankratov: On increasing of dencity of elements of an amplifier circuit
AJMS/Oct-Dec-2018/Vol 2/Issue 4 72
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