INTERNATIONAL JOURNAL OF INTELLIGENT CONTROL AND SYSTEMS
VOL. 20, NO. 1 MARCH 2015, 26-34
Constrained Nonlinear Optimization of Unity Gain Operational
Amplifier Filters Using PSO, GA and Nelder-Mead
Ogri J. USHIE, Maysam ABBOD, Evans C. ASHIGWUIKE and Sagir LAWAN
Abstract-This work attempts to reduce component count in Low,
High, and All Pass active Filters. It also uses a lower order filter to
achieve same results as higher order ones in terms of the frequency
response. The optimizers used are Nelder-Mead, GA and PSO. The
filters are transformed into small signal analysis while nodal analysis
is used to translate a circuit to matrix form. Three different examples
are presented to illustrate the effectiveness of the approach. Results
have revealed that with a computer program, a lower order
operational amplifier filter can be used to achieve same results as a
higher order one. Also, PSO can achieve the best results as regard
frequency response for the three examples, followed by GA while
Nelder-Mead has the worst result.
Index Terms—Operational Amplifier, Optimization, Low, High, and
All-Pass Filter.
1.
INTRODUCTION
Circuit miniaturization is very important in the area of
electronics engineering as it reduces the size of appliance,
power consumption and thereby increases system
reliability. The essence of this work is to minimize stages
of operational amplifiers used as filter circuits. It is a wellknown fact that stages of operational amplifiers and the
related component arrangement in a filter determine its
orders. In this paper, lower order filters are used to achieve
the roles of higher order filters through the proposed
approach.
An operational amplifier circuit (op amp) is defined as
an electronics device that performs mathematical
operations such as addition, subtraction, integration and
differentiation [29]. Op amp is applied in all branches of
electronics, both digital and analog circuits. A filter is an
electronic circuit that passes electrical signals at certain
frequency ranges while preventing the passage of others
[4]. It finds usage in fields such as telecommunication.
Different approaches have been used to design filters.
For example, Cosine modulated filter banks are designed
by using an iterative Lagrange multiplier method described
in [43]. Design examples are shown to illustrate the
effectiveness of the new approach. In addition, the design
of multiplier-less non-uniform filter bank trans-multiplexes
is accomplished by the use of artificial bee colony
algorithm (ABCA) in [23]. Also, the design of digital
infinite impulse response (IIR) based on ABCA is
described in [13]. Different evolutionary approaches
applied to electronics filter design are compared in [40].
Operational trans-conductance amplifiers (OTA) and their
The authors are with Electronic and Computer Engineering, School of
Engineering and Design, Brunel University, London, UK
(Ogri.Ushie@brunel.ac.uk).
fundamental characteristics are discussed in [10]. Analog
circuit techniques and their usage in OTA and filter design
are presented in [5].
2.
OPTIMIZATION ALGORITHMS
2.1 Nelder-Mead
The Nelder-Mead algorithm is a direct search method
for the minimization of an objective function of n
variables. The method is shown to be computationally
compact and effective [26]. The Nelder-Mead simplex
method for multidimensional minimization is proved to
converge to a minimizer for convex functions in two
dimensions [17]. The detailed Nelder-Mead algorithm
using geometric operators (reflection, expansion,
contraction and shrinking) is given in [25]. Addition of a
penalty function to the Nelder-Mead algorithm makes it to
be extended to solve a constrained minimization problem
[9]. Despite the popularity of the Nelder-Mead algorithm,
it is defective because it is never the best method and
indeed it has no general convergence results [38].
2.2 Genetic Algorithm
Genetic Algorithm (GA) is a population–based
stochastic technique that makes use of the principle of
survival of the fittest to produce a better solution [27].
During iteration, individuals are selected for reproduction
according to their performance in the problem domain. A
group of individuals are generated in the processes, which
are better suited to their environment. The individuals are
then encoded accordingly as strings. After the decoding,
then the fitness is evaluated which serve as criteria for
selection of pairs of individuals for the next reproduction.
GA operators are; selection, mutation, and crossover.
The GA can be summarized as follows:
a)
b)
c)
d)
Formulate an objective function
Encode a solution into strings
Generate an initial population
Evaluate the fitness of every member of the
population with regard to the objective function
e) Specify GA parameters (crossover, mutation,
generation and population size)
f) Perform crossover with probability PC
g) Perform mutation with probability Pm
h) Select elite for the next generation
i) Update generation
j) End according to criteria
INTERNATIONAL JOURNAL OF INTELLIGENT CONTROL AND SYSTEMS
GA owns some advantages over traditional
optimization algorithms i.e., its ability to handle complex
problems and parallelism. Its disadvantages include;
setting its right parameters (mutation, crossover and
selection criteria) and formulation of population size and
proper fitness function [39].
Application of genetic learning for combinational logic
design which has a case-based memory of past problem
solving attempt that learnt to improve the quality of the
result for similar design problems is explained [22]. The
algorithm is applied to parity checker and the presented
result has improvement. Furthermore, the extension of GA
and its use to improve circuit’s parameters is presented
[37]. While an automated algorithm used for the
optimization of analog circuits is presented [6]. An
automated combinational circuit design using GA is
presented [14], and related filter design circuit using GA is
explained [15, 20, 21, 24]. The papers laid emphases on the
use of evolutionary methodologies in design of electronic
circuits. GA use in digital circuit design is also in [22]
[31].
In addition, a modified GA algorithm kernel for
efficiency improvement on the analog IC design cycle is
presented in [2]. Furthermore, competitive co-evolutionary
differential evolution (CODE), a new algorithm with
practical user-defined specifications is proposed to design
analog ICs in [19]. A directly performance-constrained
template-based automatic layout retargeting and
optimization for analog ICs is presented in [42]. In addition,
a new CMOS wideband low noise amplifier with gain
control is proposed in [36]. Besides, a new approach to an
optimal analog test point’s selection is presented in [11].
Furthermore, simulation-based approach in which the
simulator and the search algorithm are being optimized for
analog circuit synthesis is in [30].
2.3 Particle Swarm Optimization
Particle Swarm Optimization (PSO) is also a
population-based
stochastic
optimization
method
developed Kennedy and Eberhart 1995 [32] inspired by
social behavior of flock of bird or school of fish.
Compared to GA, PSO has no genetic operators such as
reproduction, crossover and mutation but dynamically
adjusts its velocity. Also, PSO does not implement
survival of the fittest and it has fewer parameters. In PSO,
particles are flown in search of the required solution in the
problem space and each potential solution is updated in the
process. The particles as a whole are called a swarm.
Detailed PSO algorithm and suitable parameter selection
guide and improved PSO algorithm with leadership are
shown in [32] [44]. PSO algorithm is used to power
electronics circuits (PECS) design as presented in [41].
Many studies on the use of PSO in analog circuit
designs are as follow. It is used to improve analog circuit
performance [7]. Optimal design of analog circuits using a
PSO technique is in [8]. The emphasis is on the suitability
27
of PSO to solve both multi-objective and single-objective
discrete optimization problem. In addition, the applications
of PSO in microwave amplifier are in [34] [33].
Furthermore, PSO for design of analog circuits is presented
in [16]. Analog signal processing has many applications in
op amp based amplifiers, mixers, filters and comparators.
The quality of the results shown compared to the one
simulated with SPICE, PSO gives accurate result and
promising method for device modeling in analog circuits.
Photonic band gap and its usage for low-pass filter
design are proposed in [28]. The first-time use of Minimax filter design applying electromagnetic simulations is
presented in [1]. Noise optimization in operational transconductance amplifier filters is investigated and compared
in terms of the filter parameters and topologies in [3].
Minimization of electronics circuit that helps to reduce
power consumption, and increase system reliability is
illustrated in [35].
Swarm intelligence application in the first attempt to
formulate an optimal power flow problem that consider
uncontrollable and controllable distributed generator in
power networks is presented in [12]. The use of adaptive
PSO based on clustering that help to solve the problem of
PSO algorithm being trapped into local optima while
solving complex multimodal function optimization
problems is given in [18].
Despite all these studies, there is no research that
attempts to optimize an operational amplifier filter circuit
in terms of component count reduction which this work
intends to address.
PSO Algorithm
a) Formulate an objective function
b) Initialize a population of particles with random
‘position’ and ‘velocity’ in n-dimensions of the
problem space i=0
c) Evaluate the fitness of each particle to obtain
pbest
d) Compare each particle’s fitness with its previous
best fitness obtained. If the new value is better
than pbest, then set the pbest as the new value and
pbest location as the new location in ndimensional space
e) Compare pbest of particle with each other and
update the gbest location with the highest fitness
f) Change the position and velocity of the particle
according to equation (1) and (2) respectively
g) Repeat step (c) to (f) until convergence is reached
based on designed criteria.
xt +1 = xt + vt
(1)
vt +1 = vt w + c1α1( pt − xt ) + c2α 2 (gt − xt )
(2)
where xt =position, vt = velocity, w = initial weight, c1 and
c2 acceleration constant, α1 and α2 are random variable, pt =
pbest and gt = gbest.
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INTERNATIONAL JOURNAL OF INTELLIGENT CONTROL AND SYSTEMS
3.
METHODOLOGY
The same approach is used in all the three examples
presented next. Original high, low and all pass filter
circuits are simulated in PSPICE and used as a standard to
guide in the program and comparison. The operational
amplifiers are analyzed by using small signal analysis. The
circuits are transformed into a matrix form by using nodal
analysis. Matlab programs are coded to realize PSO and
GA algorithms discussed above. Each program called an
objective function subprogram. In the case of NelderMead, the objective function is called directly from its
optimization toolbox. Also, components value ranges are
specified by setting their upper and lower limit. In order to
rank the result and normalize the function, Multi-objective
Pareto Ranking is applied. The best result among ranked
particle for a specified iteration is taken as a solution. In
order to locate the quality factor at the corresponding
frequency specified by the original circuit, the slope of the
curves is taken and the programs are directed by locating
Q-factor to 0.707 point with the corresponding frequency
defined by original circuit. The algorithm flowchart is
shown in Figure 1.
Start
Filter circuit for
optimization
SPICE Simulation
Minimization of
Component
Small Signal Module
Generation of
MATLAB code
Optimization Using GA,
PSO and Nelder-Mead
Simulation
no
Simulation
satisfies the
Objective
function?
yes
Simulation of the optimized
filter circuit in PSPICE
Stop
Figure 1. The proposed algorithm.
3.1 Specifications of the objective function
The Pareto ranked multi-objective optimization function is
based on:
f = cf1 + cf 2
(3)
where f is the objective function, cf1 is the difference
between the targeted and achieved lower frequency band at
-3dB, and cf2 is the difference between the targeted and
achieved upper frequency band at -3dB. Their set values
used in the minimization are as follow:
For the high-pass filter,
cf1 = targeted (cf1) – 1 kHz
For the low-pass filter,
cf2 = targeted (cf2) – 33 kHz
For an all-pass filter,
cf1 = targeted (cf1) – 31 kHz
cf2 = targeted (cf2) – 302 kHz
The constants for this simulation are: Voltage source
v1=1 vac and op amp (but in terms of component count, the
total number is being reduced during minimization).
Table 1. A High-pass filter component value ranges.
Component
Minimum
Maximum
name
value
value
R1 (Ω)
35E3
40E3
C1 (F)
3E-9
20E-9
Table 2. A Low-pass filter component value ranges.
Component
Minimum
Maximum
name
value
value
R1( Ω)
0.5E3
1E3
R2 (Ω)
0.01E3
0.2E3
R3 (Ω)
3E3
6.5E3
C1 (F)
0.1E-9
5E-9
C2 ( F)
500E-12
800E-12
C3 (F)
0.01E-9
0.2E-9
Table 3. An All-pass filter component value ranges.
Component
Minimum
Maximum
name
value
value
R1(Ω)
0.5E3
2E3
R2 (Ω)
0.5E3
2E3
R3 (Ω)
0.5E3
2E3
R11 (Ω)
0.5E3
2E3
R12 (Ω)
0.5E3
2E3
R13 (Ω)
0.5E3
2E3
R22 (Ω)
0.5E3
2E3
R23 (Ω)
0.5E3
2E3
R32 (Ω)
0.5E3
2E3
R33 (Ω)
0.5E3
2E3
C1 (F)
1E-9
2E-9
C2 ( F)
1E-9
2E-9
C3 (F)
1E-9
2E-9
INTERNATIONAL JOURNAL OF INTELLIGENT CONTROL AND SYSTEMS
29
Using the low-pass filter for illustration, the formulas
used in the simulation are:
𝐴=[
$%
&'
; 0; 0; 0; 0]
⎛
⎜ b 11
⎜
⎜
⎜ b 21
B = ⎜ 0
⎜
⎜ 0
⎜
⎜ 0
⎜
⎝
b 12
0
b 22 b 23
=13, component Maximum Values = 2 2 2 2 1 1 2 2 2
2 2 2 2, component Minimum Values = 0.5 0.5 0.5
0.5 0.5 0.5 0.5 0.5 0.5 0.5 1 1 1 and acceleration
constants (c1 = c2 =1.49618)
Each particle generates the random component values
within the given range according expression:
(4)
0
0
b 32 b 33 b 34
0
b 43 b 44
0
b 53 b 54
0 ⎞⎟
⎟
0 ⎟
⎟
⎟
b 35 ⎟
⎟
b 45 ⎟
⎟
b 55 ⎟⎠
X P = MinValue + (MaxValue − MinValue) × rand ( N × NC )
(7)
(5)
c.
where b11 = 1/ R1 + 1/ R6 + j × ω × C1
b12 = −1/ R6
b21 = −( Av / Ro + 1/ R6 + j × ω × C1 )
b22 = 1/ R2 + Av / Ro + 1/ R6 + 1/ Ro
b23 = −1/ R2
b32 = −1/ R2
b33 = 1/ R2 + 1/ R3 + j × ω × C3
b34 = −1/ R3
b35 = − j × ω × C3
b43 = −1/ R3
b44 = 1/ R6 + 1/ R3 + j × ω × C2
b45 = −1/ R6
b53 = − j × ω × C3
b54 = −( Av / Ro + 1/ R6 )
b55 = 1/ R0 + Av / Ro + 1/ R6 + j × ω × C3
𝐶 = 𝐵 .' ×𝐴
d.
e.
f.
(6)
where C contains unknown voltages in all the nodes to be
determined. 𝐶0 is the voltage across the load resistor being
analyzed to get its frequency response within a certain
range of frequency in this case 1 Hz to 10 MHz. The
frequency response curves (Figures 6 - 8) are the plots of
gain in magnitude against frequency. The same approach is
applied to both high and all pass filters.
3.2 Use of PSO to minimize filter circuit
The following steps were implemented while using PSO
to minimize filter circuit:
a. Formulation an objective function is discussed in
section 3.1.
b. Initialization of particles using All-pass filter
component value ranges as example. The same
parameters were used for other examples except
component specification.
Number of iteration (Nit) = 30, number of particle (N)
= 20, number of components to be minimized (NC)
where Xp is the randomly generated particles of size
N by NC, MinValue is the component Minimum
Values, MaxValue is the component Maximum
Values while N and NC as defined in (b) above.
Evaluation the fitness value of each particle.
The personal best (pbest) of each particle is evaluated
with respect to the objective function that specifies
the frequency response of the circuit to be optimized.
Compare each particle’s fitness with its previous best
fitness obtained. If the new value is better than pbest,
then set the pbest as the new value and pbest location
as the new location in n-dimensional space
Compare pbest of particle with each other and update
the gbest location with the highest fitness
The position and velocity of the particle are being
changed according to equation (1) and (2)
respectively.
Repeat step (c) to (f) until convergence is reached
based on designed criteria. The termination criteria
used was number of iteration. The summaries of
parameters used are in Table 4.
Table 4. Summary/definition of PSO’s symbols used
Value
Symbols
Meaning
Remark
used
c1 & c2
acceleration
1.4962
the value that gives
the best result
constant
random
0 -1
randomly generated
α1 & α2
variable
between 0
&1
initial
0.7298
the value that gives
𝟂
weight
the best result
Nit
number of
iterations
30
N
number of
particles
20
number of
components
to be
minimized
13
NC
number of iterations
that give the best
result
number of particles
that give the best
result
number of
components to be
minimized
3.3 Use of GA and Nelder-Mead to minimize filter
circuit
In the case of GA, the objective function is being called
directly from the optimization toolbox in the fitness
30
INTERNATIONAL JOURNAL OF INTELLIGENT CONTROL AND SYSTEMS
function option. Also, components value ranges are
specified by setting their upper and lower limit. While GA
parameters include; crossover = 0.8, population size = 50,
mutation = 0.05 and the number of generations = 100. The
number of components to be optimized is specified in the
number of variable option. The code is automatically
generated using generate code option in the file menu to
enable results to be ranked. The summaries of parameters
used are in Table 5.
Table 5. Summary/definition of GA’s symbols used
Value
Symbols Meaning
Remarks
used
PC
crossover
0.8
the crossover that
gives best result
mutation
0.05
The mutation that
Pm
gives the best result
number of
100
number of iterations
Nit
iteration
that give the best
result
number of
50
number of individuals
N
individuals
that give the best
result
number of
13
number of
NC
components to
components to be
be minimized
minimized
Also in Nelder-Mead (fminco-constrained nonlinear
minimization), the objective function is being called
directly from the optimization toolbox in the objective
function option. The algorithm option was set at interior
point. Derivative option was set at approximated by solver.
The upper limits of the component are fed at start point.
Components value ranges are specified by setting their
upper and lower limit. The function tolerance was set at
1E-4 while X-tolerance was set at 1E-8. The time required
for the three different circuits are specified in their
respective Tables
4.
RESULTS AND DISCUSSION
The original and optimized circuits are simulated in
PSICE to obtain frequency response. The 0.707 of the
maximum gain is taken to locate cut-off frequency that
serves as a reference. Example 1 illustrates a high-pass
filter; Example 2 describes a low-pass one and Example 3
an all-pass one.
4.1 Example 1: High-pass filter circuit
The original high pass filter circuit [4] is in Figure 2.
While the optimized Nelder-Mead constrained nonlinear
minimization circuit is in Figure 3. The original and
optimized circuit’s frequency response curve with different
line style is shown in Figure 6. The component values for
optimized GA, PSO circuit and summary of simulated
results for the high pass filter are in Table 6.
For low and high unity gain filters,
AV = 1 = −2Q 2
(8)
which implies that Q = 0.707. The analysis simply
illustrates that, the quality factor is located at 0.707.
C1
R2
U1
+
C2
100n
1Vac
0Vdc
V1
OUT
R1
-­
C3
100n
U2
100n
1.65k
+
Vout
OUT
OPAMP
R3
3.16k
2.1k
-­
OPAMP
0
Figure 2. Original High-pass filter circuit [4].
C1
U1
+
3.55n
1Vac
0Vdc
Vout
OUT
V1
R1
39.55k
-­
OPAMP
0
Figure 3. High-pass filter circuit using Nelder-Mead.
0V
0V
0A
Figure 4. High-pass filter frequency response curve.
Table 6. Optimized results of the high-pass filter circuit.
Circuit
Initial
NelderGA
PSO
element
Circuit
Mead
CC1 (nF)
100
3.55
3.16
3.60
CC2 (nF)
100
CC3 (nF)
100
R1 (kΩ)
2.1
39.55
38.16
37.00
R2 (kΩ)
1.65
R3 (kΩ)
3.16
Op amp U1
1
1
1
1
Op amp U2
Ground
V1 (ac volt)
No. of
Components
Component
reduction
percentage
Elapsed time
1
1
1
10
1
1
5
1
1
5
1
1
5
-
50%
50%
50%
-
-
Objective
function error
-
4.82E3
3.36
second
s
3.2E3
0.66
second
s
3.2E3
INTERNATIONAL JOURNAL OF INTELLIGENT CONTROL AND SYSTEMS
31
4.2 Example 2: Low-pass filter circuit
The original low pass filter circuit [4] is in Figure 5.
While the optimized Nelder-Mead constrained nonlinear
minimization circuit is in Figure 6. The original and
optimized circuit’s frequency response curve with different
line style is shown in Figure 7. The component values for
optimized GA, PSO circuit and summary of simulated
results for the high pass filter are in Table 7.
R1
C3
U1
+
3.16k
OUT
1Vac
0Vdc
V1
-­
C1
R2
R3
1.87k
4.42k
1.5n
C5
+
OUT
OPAMP
C2
1n
4.7n
U2
-­
820p
OPAMP
R4
R5
1.47k
4.53k
U3
+
Vout
OUT
C4
-­
330p
OPAMP
0
Figure 5. Original low-pass filter circuit [4].
R1
C3
U1
+
0.775k
OUT
1Vac
0Vdc
V1
-­
C1
R2
R3
0.12k
5.51k
OPAMP
4.01n
0.12n
U2
+
OUT
C2
799.01p
-­
Vout
OPAMP
0
Figure 6. Low-pass filter circuit using Nelder-Mead circuit.
Table 7. Optimized results of the low-pass filter circuit.
Circuit
Initial
NelderGA
PSO
Circuit
Mead
elements
CC1 (nF)
1
4.01
0.24
0.1
CC2 (pF)
820
799.01
786.66
793.55
CC3 (nF)
1.5
0.12
0.02
0.2
CC4 (pF)
330
CC5 (nF)
4.7
R1 (kΩ)
3.16
0.775
0.54
0.54
R2 (kΩ)
1.87
0.12
0.09
0.02
R3 (kΩ)
4.42
5.51
3.19
3
R4 (kΩ)
1.47
R5 (kΩ)
4.53
Ground
1
1
1
1
Op amp U1
1
1
1
1
Op amp U2
1
1
1
1
Op amp U3
V1 (ac volt)
No. of
Components
Component
reduction
percentage
Elapsed time
1
1
15
1
10
1
10
1
10
-
33.33%
33.33%
33.33%
-
-
Objective
function error
-
843.03
4.98
seconds
792.03
0.99
seconds
792.03
R1
R3
C3
R5
OPAMP
-­
OUT
1Vac
0Vdc
1n
1k
1k
V1
R4
C2
2k
1n
-­
+
U1
R2
1k
R7
1k
OPAMP
1k
R6
OUT
+
U2
-­
1k
OPAMP
OUT
R8
C1 1n
+
U3
1k
0
C5
1n
R9
C4
2k
1n
R10
R12
1k
OPAMP
-­
1k
R11
OUT
1k
+
U4
Figure 7. Low-pass filter frequency response curve.
-­
OPAMP
OUT
R13
+
U5
1k
C7
4.3 Example 3: All pass filter
The original all pass filter circuit [4] is in Figure 8.
While the optimized Nelder-Mead constrained nonlinear
minimization circuit is in Figure 9. The original and
optimized circuit’s frequency response curve with different
line style is shown in Figure 10. The component values for
optimized GA, PSO circuit and summary of simulated
results for the all-pass filter are in Table 8.
1n
R15
R14
C6
2k
1n
R17
1k
-­
OPAMP
OUT
+
U6
1k
R16
1k
-­
OPAMP
OUT
R18
1k
Vout
+
U7
Figure 8. Original 7th order all-pass filter circuit [4].
INTERNATIONAL JOURNAL OF INTELLIGENT CONTROL AND SYSTEMS
32
Table 8. Optimized Results of the all-pass filter circuit.
Circuit
Initial
NelderGA
PSO
element
Circuit
Mead
C1 (nF)
1
1.55
1.76
1.82
C2 (nF)
1
1.55
1.52
1
C3 (nF)
1
1.55
1.52
1
C4 (nF)
1
1.55
1.58
1.09
C5 (nF)
1
1.55
1.58
1.09
C6 (nF)
1
C7 (nF)
1
R1(kΩ)
1
1.33
0.65
1.01
R2 (kΩ)
1
1.33
0.64
0.54
R3 (kΩ)
1
1.33
0.65
1.01
R4 (kΩ)
2
0.78
0.84
1
R5 (kΩ)
1
1.33
1.76
1.59
R6 (kΩ)
1
1.33
0.58
0.5
R7 (kΩ)
1
1.33
0.59
0.5
R8 (kΩ)
1
1.33
0.59
0.5
R9 (kΩ)
2
0.78
0.99
1
R10 (kΩ)
1
1.33
0.74
1.32
R11 (kΩ)
1
1.33
1.76
2
R12 (kΩ)
1
1.33
1.97
1.43
R13 (kΩ)
1
1.33
1.97
1.43
R14 (kΩ)
2
R15 (kΩ)
1
R16 (kΩ)
1
R17 (kΩ)
1
R18 (kΩ)
1
Ground
1
1
1
1
Op amp U1
1
1
1
1
Op amp U2
1
1
1
1
Op amp U3
Op amp U4
Op amp U5
Op amp U6
Op amp U7
V1 (ac volt)
No. of
Components
Component
reduction
percentage
Elapsed time
1
1
1
1
1
1
34
1
1
1
1
25
1
1
1
1
25
1
1
1
1
25
-
26.47%
26.47%
26.47%
-
-
Objective
function error
-
3.377E5
18.55
seconds
2.188E5
4.05
seconds
2.188E5
Table 9 illustrates the frequencies of all the quality
factor points. The presented results have suggested that,
this approach can further reduce components in low, high
and all pass filter especially in area where a phase angle
change has no effect on appliance.
5.
CONCLUSIONS
Operational amplifier filter circuits are optimized. In
the high-pass filter circuit, third order filter is minimized
by using one stage operational amplifier to obtain an
equivalent result of third order that has a component count
R1
R3
C3
R5
OPAMP
-­
OUT
1Vac
0Vdc
V1
R4
C2
0.78k
1.55n
-­
+
U1
R2
1.33k
1.55n
1.33k
1.33k
R7
1.33k
OPAMP
1.33k
R6
OUT
+
U2
C1 1.55n
1.33k
-­
OPAMP
OUT
R8
+
U3
1.33k
0
C5
1.55n
R9
C4
0.78k
1.55n
R10
R12
1.33k
OPAMP
-­
1.33k
R11
OUT
+
U4
1.33k
-­
OPAMP
OUT
R13
1.33k
Vout
+
U5
Figure 9. All-pass filter of optimized Nelder-Mead 7th order circuit.
Figure 10. All-pass filter Frequency response curve.
Table 9. The location of Q-factor in the low, high and all pass
filter for the original circuit and the optimized circuit
Circuit
High-pass Low-pass All-pass filter circuit
type
filter
filter
Low-pass
High-pass
Original
1 kHz
33 kHz
10.30 kHz
100.5 kHz
NilderMead
GA
1.1 kHz
29 kHz
10.32 kHz
100.2 kHz
1.3 kHz
34 kHz
10.31 kHz
100.3 kHz
PSO
1.2 kHz
34 kHz
10.30 kHz
100.4kHz
reduction of five. In the low-pass filter circuit, a fifth order
filter circuit minimized by using a three stage operational
amplifier filter to obtain an equivalent result of a fifth order
one that has a component count reduction of five. In the
all-pass filter, a seventh order filter is minimized by using
a five stage operational amplifier that has a component
count reduction of nine. It implies that with a computer
program, a lower order operational amplifier filter can be
programmed in such a way to achieve a higher order
operational amplifier filter by locating the quality factor of
0.707 with its corresponding frequency as specified by the
original circuit. Also, PSO offers the best results as regard
INTERNATIONAL JOURNAL OF INTELLIGENT CONTROL AND SYSTEMS
frequency response for the three examples, followed by
GA while Nelder-Mead has the worst result.
ACKNOWLEDGEMENT
The first author would like to acknowledge the financial
support from Tertiary Trust Fund through University of
Calabar, Calabar, Nigeria. The authors also would like to
thank the comments provided by the anonymous reviewers
and editor, which help the authors improve this paper
significantly.
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Ogri J. Ushie received a B. Eng. in 2002 and
MEng in 2008 in Electrical Electronics
Engineering from Federal University of
Technology-Yola and Abubakar Tafawa
Belewa University-Bauchi both in Nigeria
respectively. He is currently a PhD research
student (Electrical Engineering and Electronics
Research) at Brunel University London. He is a
lecturer at the University of Calabar, Calabar-Nigeria. He teaches
courses in Electronics and Computer Technology in the same
University. He is also a graduate student member of IEEE. He is
a Cooperate member Nigerian Society of Engineers and a
registered Engineer with Council for the Regulation of
Engineering in Nigeria. His research interest is in area of
‘Intelligence Optimisation of Analog Electronic Circuit using
Machine Learning.’
Dr Maysam F. Abbod received the BSc degree
in electrical engineering from the Baghdad
University of Technology, Baghdad, Iraq, in
1987, and the PhD degree in Control
Engineering from the University of Sheffield,
Sheffield, UK in 1992. He is currently a reader
in Intelligent Systems at the College of
Engineering, Design, and Physical Sciences,
Brunel University London, Uxbridge, London UK. His main
research interests are intelligent systems for modelling, control
and optimization. Dr Abbod is a member of IET and a UK
Chartered Engineer.
Dr Evans C. Ashigwuike received a B. Eng.
(Hons) and M. Eng. from the department of
Electrical Electronics Engineering, Nnamdi
Azikiwe University, Nigeria in 1999 and 2005,
respectively. He worked as a lecturer in the
department
of
Electrical/Electronic
Engineering at the University of Abuja, Nigeria
between 2005 and 2011. He just completed his
PhD at Brunel University London. Dr Evans’s
research interests are in area of Computer Modelling,
Electromagnetic sensors for NDT, Ultrasonic signal processing
and Long Range Ultrasonic Testing.
Sagir Lawan was born in Gumel, Nigeria, in
1962. He received B. S. degree in
electronics/electrical engineering and the M.S.
degree in telecommunication and electronic
engineering
from
Obafemi
Awolowo
University Ile-Ife in 1998 and 2010
respectively. He is currently a candidate for
PhD degree in electronic and computer
engineering, Brunel University London. His
research interests include image processing, temporal error
concealment, and multi-view video coding.