Generalized Methodology Application for System Design
ALEXANDER ZEMLIAK
Department of Physics and Mathematics
Autonomous University of Puebla
Av. San Claudio y 18 Sur, Puebla, 72570
MEXICO
Institute of Technical Physics
National Technical University of Ukraine
UKRAINE
Abstract: - The design process for analogue circuit design is formulated on the basis of the optimum control
theory. The artificially introduced special control vector is defined for the redistribution of computational costs
between network analysis and parametric optimization. This redistribution minimizes computer time. The
problem of the minimal-time network design can be formulated in this case as a classical problem of the
optimal control for some functional minimization. There is a principal difference between the new approach
and before elaborated methodology. This difference is based on a higher level of the problem generalization. In
this case the structural basis of design strategies is more complete and this circumstance gives possibility to
obtain a great value of computer time gain. Numerical results demonstrate the effectiveness and prospects of a
more generalized approach to circuit optimization. This approach generalizes the design process and generates
an infinite number of the different design strategies that will serve as the structural basis for the minimal time
algorithm construction. This paper is advocated to electronic systems built with transistors. The main equations
for the system design process were elaborated.
Key-Words: - Circuit optimization, control theory formulation, controllable dynamic system, optimization
strategies, generalized methodology.
1 Introduction
One of the sources of overall improvement in design
quality is the reduction of CPU time in the design of
large systems. This problem has a great significance
because it has a lot of applications, for example on
VLSI electronic circuit design. Any traditional
system design strategy includes two main parts: the
mathematical model of the physical system that can
be defined by the algebraic equations or differential-
integral equations and optimization procedure that
achieves the optimum point of the design objective
function. Within the framework of this concept, it is
possible to change the optimization strategy and use
different models and different analysis methods, but
at each stage of the optimization process of the
circuit there is a fixed number of equations of the
mathematical model and a fixed number of
independent parameters when optimizing the circuit.
Some powerful techniques have been used to
reduce the time required to analyze the circuit.
Because a matrix of the large-scale circuit is a very
sparse, the special sparse matrix techniques are used
successfully for this purpose [1-2]. Other approach
to reduce the amount of computational required for
both linear and nonlinear equations is based on the
decomposition techniques. The partitioning of a
circuit matrix into bordered-block diagonal form can
be done by branches tearing as in [3], or by nodes
tearing as in [4] and jointly with direct solution,
algorithms gives the solution of the problem. The
extension of the direct solution methods can be
obtained by hierarchical decomposition and macro
model representation [5]. Other approach for
achieving decomposition at the nonlinear level
consists on a special iteration techniques and has
been realized in [6] for the iterated timing analysis
and circuit simulation. Optimization technique that
is used for the circuit optimization and design, exert
a very strong influence on the total necessary
computer time too. The numerical methods are
developed both for the unconstrained and for the
constrained optimization [7] and will be improved
later on. The practical aspects of these methods were
developed for the electronic circuits design with the
Received: June 29, 2021. Revised: November 3, 2021. Accepted: November 17, 2021. Published: January 3, 2022.
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DOI: 10.37394/23205.2022.21.2
Alexander Zemliak
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different optimization criterions [8-9]. The
fundamental problems of the development, structure
elaboration, and adaptation of the automation design
systems have been examine in some papers [10-11].
The ideas of designing the system described
above can be called the traditional approach or the
traditional strategy, since the method of analysis is
based on the laws of Kirchhoff. Other idea for the
problem of optimizing the circuit were developed at
a heuristic level several decades ago [12]. This idea
was based on the Kirchhoff laws ignoring for all the
circuit or for the circuit part. The special cost
function is minimized instead of the circuit equation
solving. This idea was developed in practical aspect
for the microwave circuit optimization [13] and for
the synthesis of high-performance analog circuits
[14] in extremely case, when the total system model
was eliminated. The authors of the last papers affirm
that the design time was reduced significantly. This
last idea can be named as the modified traditional
design strategy.
At the same time, all these ideas can be
generalized to reduce the total computer design time
for system design. This generalization can be done
on the basis of the control theory approach and
includes the special control function to control the
design process. This approach consists of the
reformulation of the total design problem and
generalization of it to obtain a set of different design
strategies inside the same optimization procedure
[15]. The number of the different design strategies,
which appear in the generalized theory of the first
level, is equal to M
2
for the constant value of all the
control functions, where M is the number of
dependent parameters. These strategies serve as the
structural basis for more strategies construction with
the variable control functions. The main problem of
this new formulation is the unknown optimal
dependency of the control function vector that
satisfies to the time-optimal design algorithm. One
way to solve this problem is to use the Lyapunov
function of the design process [16].
However, the developed theory [15] is not the
most general. In the limits of this approach only
initially dependent system parameters can be
transformed to the independent but the inverse
transformation is not supposed. The next more
general approach for the system design supposes
that initially independent and dependent system
parameters are completely equal in rights, i.e. any
system parameter can be defined as independent or
dependent one. In this case we have more vast set of
the design strategies that compose the structural
basis and more possibility to the optimal design
strategy construct.
In this case the new objective function would be
introduced to take into account the corresponding
information about the system. The number of the
different design strategies, which appear in this new
generalized theory, is equal to ∑
=
+
M
i
i
MK
C
0
for the
constant value of all the control functions, where M
is the number of dependent parameters and K is the
number of independent parameters. These strategies
serve as the structural basis for other strategies
construction with the variable control functions. The
almost infinite number of the different design
strategies appears for this methodology in contrast
to the results [15] where only one particular case
was studied. The characteristic curves of the
transistor must be taken into account in order to
obtain both a good and a real design.
2 Problem Formulation
In accordance with the last design methodology [15]
the design process is defined by the optimization
procedure, which can be determined in continuous
form as:
( )
dx
dt
f X U
i
i
=,, Ni ,...,2,1
=
(1)
and by the analysis of the electronic system
model in next form:
(
)
( )
1 0− =u g X
j j , (2)
where N=K+M, K is the number of independent
system parameters, M is the number of dependent
system parameters, X is the vector of all variables
(
)
N
xxxX ,...,, 21
=
; U is the vector of control
variables
(
)
M
uuuU ,...,, 21
=
, where
u
j
∈
Ω
;
{
}
Ω = 0 1; .
The functions of the right part of the system (1)
can be determined for the gradient method for
instance as:
( )
( ) ( )
f X U b xC X u g X
i
i
j j
j
M
,= − +
=
∑
δ
δ ε
12
1
(3)
for
i
K
=
1
2
,
,
.
.
.
,
,
j
M
=
1
2
,
,
.
.
.
,
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( )
( ) ( )
( )
( )
{ }
f X U b u xC X u g X
u
dt
x X
i i K
i
j j
j
M
i K
ii
,
'
=− ⋅ +
+−− +
−
=
−
∑
δ
δ ε
η
1
1
2
1
(3')
for
i
K
K
N
=
+
+
1
2
,
,
.
.
.
,
,
where C(X) is the traditional objective function of
the design process. By this formulation the initially
dependent parameters for NKKi ,...,2,1
+
+
=
can be
transformed to the independent ones when 1
=
j
u
and it is dependent when 0
=
j
u. On the other hand
the initially independent parameters are independent
ones always. The optimal behavior of the control
functions for the minimal-time problem can be
found by means of some approximate methods of
the control theory [17]-[19].
We develop the new approach that permits to
generalize more the design methodology [15]. We
suppose now that all system parameters can be
independent or dependent ones. In this case we need
to change the equations (2) and (3). The equation (2)
is transformed to the next one:
(
)
(
)
01 =− Xgu ji (4)
Ni ,...,2,1
=
and j
∈
J
where J is the index set of all those functions for
which. ui = 0, J = {j1, j2, . . .,jz}, js
∈
Π
with s = 1,
2, . . ., z, where
Π
is the set of indexes from 1 to M,
Π
= {1, 2, . . ., M}, z is the number of equations that
will be left in the system (4), z ∈{0, 1. . ., M}. The
right hand side of the system (1) is defined now as:
( )
( ) ( )
{ }
Xx
dt
u
UXF
xd
d
ubUXf
i
i
i
i
ii
η
+−
−
+
⋅−=
'
1
),(,
(5)
Ni ,...,2,1
=
where F(X,U) is the generalized objective function
and it is defined as:
( ) ( ) ( )
∑
Π∈
+=
Jj
jj XguXCUXF
\
2
1
,
ε
(6)
This definition of the design process is more
general than in [15]. It generalizes the methodology
for the system design and produces more
representative structural basis of different design
strategies.
3 Numerical Results
3.1 Example 1
In Fig. 1 there is a circuit that has 3 independent
variables as admittance 321 ,, yyy (K=3) and 3
dependent variables as nodal voltages 321 ,, VVV
(M=3) at the nodes 1, 2, and 3 respectively.
Fig. 1 Three-node circuit
Kirchhoff´s law applied to this circuit includes
three equations that can be written as follows:
(
)
0
1111
=
⋅
−
−
=
yVEIg B
0
222 =⋅−= yVIg E (7)
(
)
0
3323 =⋅−−= y
VEIg C
The X vector includes seven components defined
by the following formulas: 1
2
1yx =, 2
2
2yx =,
3
2
3yx =, x4 = V1, x5 = V2 and x6 = V3, IE, IC and IB
can be obtained in four regions by Ebers-Moll static
model, implemented in SPICE2 [20]. The system
model is determined by the following equations:
(
)
2
1411 )( x
xEIXg B⋅−−=
(
)
5
2
22 )( xxI
Xg E⋅−= (8)
(
)
2
3623 )( x
xEIXg C⋅−−=
The optimization procedure includes six
equations in this case:
( ) ( )
( ) ( ) ( ){ }
Xdttx
dt
u
UXF
x
ubUXf
i
i
i
ii
η
δ
δ
+−−
−
+
⋅−=
1
,,
(9)
6,...,2,1
=
i.
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3.1.1 Strategy (111000). This is the traditional
design strategy. Only three first equations of the
system (10) compose the optimization procedure
with objective function F(X)=C(X) and with three
equations (8) that permit to calculate all the
coordinates of the vector X. Equations (8) are solved
by the Newton-Raphson method. Having
characterized the transistor, selecting one operation
point (e.g. VBC = -2.2 V., VCE = 2.9 V. and VBE =
0.7 V.), the characteristic for this amplifier is to has
the Collector voltage similar to a constant value then
the function objective is defined as
(
)
2
16
)( mmxx
C−= , but in order to study all the
trajectories arriving to the same final point, we add
the terms
(
)
2
254 mmx
x−− and
(
)
2
364 mmx
x−− ,
mm2 and mm3 are the voltages of union of the
transistor, therefore the function ordinary objective
C(X) is defined by the following formula:
(
)
(
)
(
)
2
364
2
254
2
16 mm
xxmmxxmmxxC −−+−−+−=)( .
3.1.2 Strategy (111111). This is the modified
traditional design strategy. The six equations of the
system (9) compose the optimization procedure with
the objective function F(X) but the equations (8)
disappear from the system’s model. The objective
function F(X) is defined by the following form:
( ) ( )
(x)
j
gXCXF
j
∑
=
+=
3
1
2 (10)
3.1.3 Intermediate strategies. Others strategies are
intermediate ones. Some of these are the strategies
that appear in the previously developed
methodology and the others are the strategies that
appear inside the new generalized approach. Only
some of the total number of the different design
strategies are shown in Table 1, because of the
number of strategies for this example are equal to
∑
=
3
0
6
i
i
C= 32 strategies. Table 1 corresponds to the
“old” strategies that have been analyzed in previous
papers. Table 2 corresponds to the new strategies
that appear in limits of the proposed approach.
Table 1. Strategies of the “old” structural basis.
Strategy Iterations Time (ms)
1 111000 9311 7977.00
2 111001 7514 4989.11
3 111010 75635 43053.10
4 111011 324 60.10
5 111100 25079 10970.1
6
111101
243
40.11
7 111110 10232 2398.5
8
111111
2418
196.21
Table 2. Some strategies of the “new” structural
basis.
Strategy Iterations Time (ms)
1
101111
30
5.00
2 110111 778 139.10
3
101110
55992
25094.21
4 011100 12850 10992.33
5 011110 30015 10998.24
6
011101
47
19.73
7 110011 174 60.01
8
110101
606
220.21
3.2 Example 2
In Fig. 2 there is a circuit that has 5 independent
variables as admittance 54321 ,,,, yyyyy (K=5) and
5 dependent variables as nodal voltages
54321 ,,,, VVVVV (M=5). The total number of
variables are N = M + K = 10.
Fig. 2 Five-node circuit
Kirchhoff´s law applied to this circuit includes
three equations that can be written as follows:
(
)
0
1111
1
=
−
⋅
−
=
B
IyVE
g
0
1222 =−⋅= E
IyVg
(
)
0
2133
23
=
−
−
⋅
−
=
BC IIyV
Eg (11)
0
2444
=
−
⋅
=
E
IyVg
(
)
0
2552
5
=
−
⋅
−
=
C
IyVE
g
The X vector includes ten components defined by
the following formulas: 1
2
1yx =, 2
2
2yx =,
3
2
3yx =, 4
2
4yx =, 5
2
5y
x=, x6 = V1, x7 = V2, x8 =
V3, x9 = V4, x10 = V5. The model of the system
includes five equations:
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(
)
0)( 1
2
1611 =
−⋅−= B
IxxEXg
0)( 1
2
272 =−⋅= E
IxxXg
(
)
0)( 21
2
38
23 =−−⋅−= BC IIxxEXg (12)
0)( 2
2
494 =−⋅= E
IxxXg
(
)
0)( 2
2
510
25 =−⋅−= C
IxxEXg
IE, IC and IB can be obtained in four regions by
Ebers-Moll static model, implemented in SPICE2.
The optimization procedure includes ten equations
in this case:
( ) ( )
( ) ( ) ( ){ }
Xdttx
dt
u
UXF
x
ub
UXf
i
i
i
ii
η
δ
δ
+−−
−
+
⋅−=
1
,,
(13)
10,...,2,1
=
i
3.2.1 Strategy (1111100000). This is the traditional
design strategy. Only five first equations of the
system (13) compose the optimization procedure
with objective function F(X)=C(X) and with five
equations (12) that permit to calculate all of the
coordinates of the vector X. Equations (12) are
solved by the Newton-Raphson method. Having
characterized the transistor, selecting one operation
point (e.g. VBC1 = -2.2 V., VCE1 = 2.9, VBE1 = 0.7 V.,
VBC2 = -2.2 V., VCE2 = 2.9 V. and VBE2 = 0.7 V.),
the characteristic for this amplifier is to has the
Collector voltage similar to a constant value then the
function objective it is defined as
(
)
2
110
)( mm
xXC −= , but in order to study all the
trajectories arriving to the same final point, we add
the terms
(
)
2
276 mmx
x−− ,
(
)
2
386 mmx
x−− ,
(
)
2
498 mmx
x−− and
(
)
2
5108 mm
xx −− then the
traditional objective function C(X) is defined by the
following form:
(
)
(
)
( ) ( )
( )
2
5108
2
498
2
386
2
276
2
110
)(
mmxx
mmxxmmxx
mmxxmmxxC
−−+
−−+−−+
−−+−=
(14)
where mm1 = 7.8, mm2 = VBE1 = 0.7, mm3 =VBC1= -
2.2, mm4 = VBE2= 0.7 and mm5 = VBC2 = -2.2
3.2.2 Strategy (1111111111). This is the modified
traditional design strategy. The ten equations of the
system (16) compose the optimization procedure
with the objective function F(X) but the equations
(12) disappear from the system’s model. The
objective function F(X) is defined by the following
form:
( ) ( )
(x)
j
gXCXF
j
∑
=
+=
52
1
(15)
3.2.3 Intermediate strategies. Others strategies are
intermediate ones. Some of these are the strategies
that appear in the previously developed
methodology and the others are the strategies that
appear inside the new generalized approach. Only
some of the total number of the different design
strategies are shown in Table 3, and Table 4 because
of the number of strategies for this example are
equal to ∑
=
5
0
10
i
i
C=512 strategies. Table 3
corresponds to the “old” strategies that have been
analyzed in previous papers. Table 4 corresponds to
the “new” strategies that appear in limits of the
proposed approach.
Table 3. Some “old” strategies.
Strategy Iterations Time (s)
1 1111100000
83402 333.6
2 1111100011
6695 8.990
3
1111100111
3395
4.007
4 1111101111
253 1.290
5
1111110001
70887
125.994
6 1111110011
93677 92.018
7 1111110111
588 2.700
8 1111111001
148299 158.038
9 1111111011
24678 15.945
10
1111111100
56464
57.015
11 1111111101
496 2.400
12
1111111110
5583
2.007
13 1111111111
614 1.699
Table 4. Some “new” strategies.
Strategy
Iterations
Time (s)
1 0000011111 55 0.159
2 0000111110 7912 23.985
3
0000111111
209
0.429
4 0001111100 57245 229.963
5
0001111111
420
0.560
6 0011111011 25884 52.022
7
0011111101
232
0.309
8 0011111110 138426 230.014
9 0011111111 381 0.319
10
0101010111
201
0.400
11
0101110100 47186 190.979
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12
0101110111 242 0.329
13
0101111111
371
0.319
14
0110110111 338 0.440
15
0110111111
414
0.340
16
0111010111 156 0.209
17
0111011111 480 0.409
18
0111110110 8511 11.998
19
0111110111 68 0.080
20
0111111011
22381
26.012
21
0111111100 31525 55.060
22
0000011111
55
0.159
23
0000111110 7912 23.985
24
0000111111 209 0.429
25
0001111100 57245 229.963
26
0001111111 420 0.560
27
0011111011
25884
52.022
28
0011111101 232 0.309
29
0011111110
138426
230.014
30
0011111111 381 0.319
31
0101010111 201 0.400
32
0101110100 47186 190.979
33
0101110111 242 0.329
34
0101111111
371
0.319
35
0111111110 9264 8.961
36
0111111111
205
0.0906
37
1000001111 98 0.290
38
1000011111 150 0.309
39
1001101100
40121
165.00
40
1001101111 286 0.379
41
1001111101
170
0.239
42
1001111111 547 0.479
3.3 Example 3
In Fig. 3 there is a circuit that has 7 independent
variables as admittance 7654321 ,,,,,, yyyyyyy
(K=7) and 7 dependent variables as nodal voltages
7654321 ,,,,,, VVVVVVV (M=7).
Fig. 3 Seven-node circuit
Kirchhoff law applying for this circuit the seven
equations can be writing in following form:
(
)
0
1111
1
=
−
⋅
−
=
B
IyVE
g
0
1222 =−⋅= E
IyVg
(
)
0
2133
23
=
−
−
⋅
−
=
BC IIyV
Eg
0
2444
=
−
⋅
=
E
IyVg (16)
(
)
0
3255
25
=
−
−
⋅
−
=
BC IIyV
Eg
0
3666
=
−
⋅
=
E
IyVg
(
)
0
3772
7
=
−
⋅
−
=
C
IyVE
g
The X vector includes fourteen components
defined by the following formulas: 1
2
1yx =, 2
2
2yx =,
3
2
3yx =, 4
2
4yx =, 5
2
5yx =, 6
2
6yx =, 7
2
7yx =,
x8 = V1, x9 = V2, x10 = V3, x11 = V4, x12 = V5, x13 =
V6, x14 = V7, E1 = 5V y E2 = 10V, The model of the
system is:
(
)
0)( 1
2
1811 =
−⋅−= B
IxxEXg
0)( 1
2
292 =−⋅= E
IxxXg
(
)
0)( 21
2
31023 =−−⋅−= BC IIxxEXg
0)( 2
2
4114 =−⋅= E
IxxXg (17)
(
)
0)( 32
2
51225 =−−⋅−= BC IIxxEXg
0)( 3
2
6136 =−⋅= E
IxxXg
(
)
0)( 3
2
714
27 =−⋅−= C
IxxEXg
The optimization procedure includes fourteen
equations in this case:
( ) ( )
( ) ( ) ( ){ }
Xdttx
dt
u
UXF
x
ub
UXf
i
i
i
ii
η
δ
δ
+−−
−
+
⋅−=
1
,,
(18)
14,...,2,1
=
i.
3.3.1 Strategy (11111110000000). This is the
traditional design strategy. Only seven first
equations of the system (18) compose the
optimization procedure with objective function
F(X)=C(X) and with five equations (17) that permit
to calculate all of the coordinates of the vector X.
Equations (17) are solved by the Newton-Raphson
method. Having characterized the transistor,
selecting one operation point (e.g. VBC1 = -1.7 V.,
VBE1 = 0.6 V., VBC2 = -1.0 V., VBE2 = 0.6 V., VBC3 =
-1.2 V. and VBE3 = 0.7 V.), .), the characteristic for
this amplifier is to has the Collector voltage similar
to a constant value then the function objective it is
defined as
(
)
2
114
)( mmxXC −= but in order to study
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all the trajectories arriving to the same final point,
we add the terms
(
)
2
298 mmx
x−− ,
(
)
2
3108 mm
xx −− ,
(
)
2
41110 mmxx −− ,
(
)
2
51210 mmxx −− ,
(
)
2
61312 mmxx −− y
(
)
2
71412 mmxx −− then the traditional objective
function C(X) is defined by the following form:
(
)
(
)
( ) ( )
( ) ( )
( )
2
71412
2
61312
2
51210
2
41110
2
3108
2
298
2
114
)(
mmxx
mmxxmmxx
mmxxmmxx
mmxxmmxXC
−−+
−−+−−+
−−+−−+
−−+−=
(19)
3.3.2 Strategy (11111111111111). This is the
modified traditional design strategy. The fourteen
equations of the system (18) compose the
optimization procedure with the objective function
F(X) but the equations (17) disappear from the
system’s model. The objective function F(X) is
defined by the following form:
( ) ( )
(x)
j
gXCXF
j
∑
=
+=
72
1
(20)
3.3.3 Intermediate strategies. Others strategies are
intermediate ones. Some of these are the strategies
that appear in the previously developed
methodology and the others are the strategies that
appear inside the new generalized approach. Only
some of the total number of the different design
strategies are shown in Table 3, because of the
number of strategies for this example are equal to
∑
=
7
0
14
i
i
C=16384 strategies. Table 5 corresponds to
the old strategies that have been analyzed in
previous papers. Table 6 corresponds to the new
strategies that appear in limits of the proposed
approach.
Table 5 Some “old” strategies.
Strategy Iterations
Time (s)
1 11111110000000
38775 351456.6
2 11111110000001
100843 742993.0
3 11111110000100
45407 440014.0
4 11111110010000
2643 29002.0
5 11111110100000
82811 1163987.0
6 11111110111111
1127 1020.0
7 11111111000000
10454 89019.0
8 11111111011111
540 955.0
9 11111111101111
53880 61040.0
10
11111111110111
1008 1007.0
11
11111111111011
5647 6012.0
12
11111111111101
226 1885.0
13
11111111111110
7441
7999.0
14
11111111111111
3979 4005.0
Table 6 Some “new” strategies.
Strategy
Iterations
Time (s)
1 00000001111111
72 549.0
2
00000011111111
235
1030.0
3 00000111111111
506 1030.0
4
00001111111111
891
2980.0
5 00011111111111
660 1050.0
6 00111111111111
1262 2002.0
7
01111111111111
504
953.0
8 10111111111111
351 380.0
9
11011111111111
316
350.0
10
11101111111111
662 709.3
11
11110111111111
801 986.0
12
11111011111111
532 956.0
13
11111100000001
11993 129003.0
14
11111101111111
308
30.10
Table 7 summarizes the integral information about
the computer gain for two levels of generalized
optimization for all examples.
Table 7 Summary of Gain
Example
Gain, Old
Strategy
Gain, New
Strategy
1 198.8 1595.4
2 258.60 4170
3
368.01
11676
In Fig. 4 we show the behavior of gains of the first
and second level of generalization for active
circuits.
Fig. 4 Gain in time for active circuits
0
2000
4000
6000
8000
10000
12000
14000
0 2 4 6 8
Nodes
G a in
0
50
100
150
200
250
300
350
400
Last Approach
New Approach
WSEAS TRANSACTIONS on COMPUTERS
DOI: 10.37394/23205.2022.21.2
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4 Conclusion
We developed a new and more complete approach
to the electronic system design with transistors. We
have checked that this approach generates more
broadened structural basis of different design
strategies. The total number of the different
strategies, which compose the structural basis by
this approach, is equal to ∑
=
+
M
i
i
MK
C
0
and the
previous methodology produced 2M strategies only.
Some new strategies have better convergence and
lesser computer time than the strategies that
appeared in before developed methodology. We can
observe that the new theory has a greater growth in
the gain when the number of nodes increases. We
can observe that the gains are greater when it is
active circuit.
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DOI: 10.37394/23205.2022.21.2
Alexander Zemliak
E-ISSN: 2224-2872
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