On Approach to Increase Integration Rate of Elements of a Switched-capacitor Step-down DC-DC Converter

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ISSN 2581-3463
RESEARCH ARTICLE
On Approach to Increase Integration Rate of Elements of a Switched-capacitor
Step-down DC-DC Converter
E. L. Pankratov1,2
1
Department of Mathematics, Nizhny Novgorod State University, 23 Gagarin Avenue, Nizhny Novgorod,
603950, Russia, 2
Department of Mathematics, Nizhny Novgorod State Technical University, 24 Minin Street,
Nizhny Novgorod, 603950, Russia
Received: 15-05-2020; Revised: 01-06-2020; Accepted: 20-07-2020
ABSTRACT
In this paper, we introduce an approach to increase integration rate of elements of a switched-
capacitor step-down DC–DC converter. 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: Switched-capacitor step-down DC–DC converter, optimization of manufacturing,
analytical approach for modeling
INTRODUCTION
An actual and intensively solving problem of solid-state electronics is increasing of integration rate of
elements of integrated circuits (p-n-junctions, their systems et al.).[1-8]
Increasing of the integration rate
leads to necessity to decrease their dimensions. To decrease, the 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 gives a possibility to increase integration rate of elements of integrated circuits
through inhomogeneity of technological parameters due to generating inhomogenous distribution of
temperature. 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
sectionsbydiffusionorionimplantation.Thedopinggivesapossibilitytomanufacturefield-effecttransistors
framework a cascaded inverter circuit so as it is shown in Figure 1. The manufacturing gives a possibility to
increase density of elements of the switched-capacitor step-down DC-DC converter.[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 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]
Address for correspondence:
E. L. Pankratov
E-mail: elp2004@mail.ru

Pankratov: On approach to increase integration rate of elements
AJMS/Jul-Sep-2020/Vol 4/Issue 3 11










C x y z t
t x
D
C x y z t
x y
D
C x y z t
y
C C
, , , , , , , , ,
( )
=
( )





 +
( )






 +
( )










z
D
C x y z t
z
C
, , ,
(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 ,
( )
, , ,
0
z
x L
C x y z t
z =
∂
=
∂
, C (x,y,z,0)=f (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. Value of dopant diffusion coefficient could
be changed with changing materials 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 Refs.[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)
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 concentration of
radiation vacancies. Parameter V* describes the equilibrium distribution of concentration of vacancies.
The considered concentrational dependence of dopant diffusion coefficient has been described in details
in.[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
( )
∂ ( )
∂





 +
Figure 1: The considered cascaded inverter[4]

+
∂
∂
( )
∂ ( )
∂





 − ( ) ( )
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, ρ (x,y,z,0)=fρ
(x,y,z).(5)
Here ρ = I,V. The function I (x,y,z,t) describes the spatiotemporal distribution of concentration of
radiation interstitials; Dρ
(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 diinterstitials; 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 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
diinterstitials ΦI
(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
+ ( )
( )





 + ( )




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
∂ ( )
∂
=
=
Φ x y z t
x x
, , ,
0
0,
∂ ( )
∂
=
=
Φ x y z t
x x Lx
, , ,
0,
∂ ( )
∂
=
=
Φ x y z t
y y
, , ,
0
0 ,
∂ ( )
∂
=
=
Φ x y z t
y y Ly
, , ,
0 ,
∂ ( )
∂
=
=
Φ x y z t
z z
, , ,
0
0,
∂ ( )
∂
=
=
Φ 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 introduction leads to
transformation of Equation (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 Equation (8) with conditions (9) framework recently introduced approach,[18]
that is, 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 Equation (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
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 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
z
I I I
I
I
0 0
1 1
Φ Φ Φ Φ
Φ
∂
∂
ε
∂
∂
∂
∂
, , ,
, , ,
+
+ ( )

 

( )






−
ε
∂
∂
Φ Φ
Φ
I I
I
g x y z T
x y z t
z
, , ,
, , ,
− ( ) ( )
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
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 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
was used for analysis of radiation defects. To use the approach, we consider following transformation
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 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.Solutionsoftheequationshavebeencalculatedbystandardapproaches(see,forexample,[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 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 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 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 shall optimize annealing of dopant and/
or radiation defects. To do the optimization, we used recently introduced criterion.[26-33]
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 d z d y d x
x y z
L
L
L z
y
x
= ( )− ( )

 

∫
∫
∫
1
0
0
0
, , , , ,
Θ  .(12)
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 value of dopant diffusion coefficient in the substrate
a
Figure 2: (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 value of dopant diffusion coefficient in the substrate. Curve 1 corresponds
to homogenous sample and annealing time Θ = 0.0048 (Lx
2
+Ly
2
+Lz
2
)/D0
. Curve 2 corresponds to homogenous 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
b

Figure 3: (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 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 value of dopant diffusion coefficient in nearest sections
a
b
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 number of curve corresponds
to increasing of annealing time
a

Figure 4: (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
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
a
Figure 5: (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
b

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 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.[19-23]
CONCLUSIONS
In this paper, we introduce an approach to increase integration rate of element of a switched-capacitor
step-down DC–DC converter. The approach gives us possibility to decrease area of the elements with
smaller increasing of the element’s thickness.
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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
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
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
I V I V
g T I V I
, , , , , , , , , , ,
   χ
, , , , , ,
( ) ( )

 


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
, , ,
, , , , , , 0
000 100
, , , , , ,
( ) ( )

 
 ×

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
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
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
, , , , , ,
∂ ( )
∂
=
∂ ( )
∂
+
∂
  
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
, , , , , , ;
∂ ( )
∂
=
∂ ( )
∂
+
∂
  
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
, , , , , ,
∂ ( )
∂
=
∂ ( )
∂
+
∂
  
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
, , ,
, , ,
( ) ( ) ( ) ( ) ( )
, , 000 , , 000 001
1 , , , , , , 1 , , , , , , , , ,
V V V V I V I V
g T V g t I V
ε χ η ϕ χ η ϕ ϑ ε χ η ϕ 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 d wd vd u
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 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


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
, , ,
( ) = − ( ) ( ) ( ) ( ) −
( ) ( ) 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 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
(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 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 ρ
001 2
, , ,
( ) = − ( ) ( ) ( ) ( ) −
( ) ( ) ( )
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 ρ
002 2
, , ,
( ) = − ( ) ( ) ( ) ( ) −
( ) ( ) ( )
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
, , ,
( ) = − ( ) ( ) ( ) ( ) −
( ) ( ) 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
× ( ) ( ) ( )− ( ) ( ) ( ) ( ) −
( ) ( ) ( )
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
, , ,
( ) = − ( ) ( ) ( ) ( ) −
( ) ( ) 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
, , ,
( ) = − ( ) ( ) ( ) ( ) −
( ) ( ) 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
( ) ( ) ( ) ( ) ×
=
∞
∑
× −
( ) ( ) ( ) ( ) ( )
∂ ( )
∂
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
nI nI n n n
I V I V
ϑ τ
ε
1
10
, ,
, , ,

0
0 000
0
1
0
1
0
1
0
u v w V u v w d wd vd u d
, , , , , ,
τ τ τ
ϑ
( ) ( )
∫
∫
∫
∫ 

V
D
D
nc c c e e s u
V
I
n n n nV nV n
101
0
0
2
, , ,
( ) = − ( ) ( ) ( ) ( ) −
( ) ( ) c
c v g u v w T
n V
n
( ) ( ) ×
∫
∫
∫
∫
∑
=
∞
, , ,
0
1
0
1
0
1
0
1
ϑ
× ( )
∂ ( )
∂
− ( ) ( ) ( )
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
( ) ( ) ( ) ( ) ×
=
∞
∑
× −
( ) ( ) ( ) ( ) ( )
∂ ( )
∂
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
nV nV n n n
I V I V
ϑ τ
ε
1
10
, ,
, , ,

0
0 000
0
1
0
1
0
1
0
u v w V u v w d wd vd u d
, , , , , ,
τ τ τ
ϑ
( ) ( )
∫
∫
∫
∫  ;

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,







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
∂ ( )
∂
=
=
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; Φρ
0(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,
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
, , ,
( )
( )
( ) ( )
( )
( )
00 10 00 10
0
, , , , , , , , , , , ,
, , , , , ,
L
C x y z t C x y z t C x y z t C x y z t
D
x P x y z T x y P x y z T y
γ γ
γ γ
    
∂ ∂
∂ ∂

+ + +
   

∂ ∂ ∂ ∂
   
    

+
∂
∂
( )
( )
∂ ( )
∂











+
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

 
 
= − + +
 
 
 
 
 
, ( ) ( ) ( ) ( )
0 0 0
, ,
y
x z
L
L L
nC n n C n
F c u c v f u v w c w d wd vd u
= ∫ ∫ ∫ ;
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
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
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
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


, , ,
, , ,
01
d
d vd u d
L
L
L
t
n
z
y
x

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
− ( ) ( ) ( ) ( ) −
( ) ( ) ( ) ( )
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
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
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

On Approach to Increase Integration Rate of Elements of a Switched-capacitor Step-down DC-DC Converter

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On Approach to Increase Integration Rate of Elements of a Switched-capacitor Step-down DC-DC Converter