A genetic algorithm for the optimal design of a multistage amplifier

International Journal of Electrical and Computer Engineering (IJECE)
Vol. 10, No. 1, February 2020, pp. 129~138
ISSN: 2088-8708, DOI: 10.11591/ijece.v10i1.pp129-138  129
Journal homepage: http://ijece.iaescore.com/index.php/IJECE
A genetic algorithm for the optimal design
of a multistage amplifier
El Beqal Asmae1
, Bachir Benhala2
, Izeddine Zorkani3
1,3
Faculty of Sciences Dhar el Mhraz, University of Sidi Mohamed Ben Abdllah, Morocco
2
Faculty of Sciences, University of Moulay Ismail, Morocco
Article Info ABSTRACT
Article history:
Received Sep 19, 2018
Revised Aug 8, 2019
Accepted Aug 29, 2019
The optimal sizing of analog circuits is one of the most complicated
processes, because of the number of variables taken into, to the number of
required objectives to be optimized and to the constraint functions
restrictions. The aim is to automate this activity in order to accelerate
the circuits design and sizing. In this paper, we deal with the optimization of
the three stage bipolar transistor amplifier performances namely the voltage
gain (AV), the input impedance (ZIN), the output impedance (ZOUT),
the power consumption (P) and the low and the high cutoff frequency
(FL,FH), through the Genetic Algorithm (GA). The presented optimization
problem is of multi-dimensional parameters, and the trade-off of all
parameters. In fact, the passive components (Resistors and Capacitors) are
selected from manufactured constant values (E12, E24, E48, E96, E192) for
the purpose of reduce the cost of design; also, the intrinsic parameters of
transistors (hybrid parameters and the junction capacitances) are considered
variables in order not to be limited in design. SPICE simulation is used to
validate the obtained result/performances.
Keywords:
Genetic algorithm
Metaheuristic
Optimization
Three-stage bipolar transistor
amplifier
Copyright © 2020 Institute of Advanced Engineering and Science.
All rights reserved.
Corresponding Author:
El Beqal Asmae,
Faculty of Sciences Dhar El Mhraz,
University of Sidi Mohamed Ben Abdllah,
Fez 30050, Morocco.
Email: asmae.elbekkal@gmail.com
1. INTRODUCTION
Despite the strong trend towards integrated circuits, discrete components are still used in analog
design especially for circuits that are not produced in large quantities. Discrete components such as
Resistors (R) and Capacitors (C) are produced according to the industrial series such as E12, E24, E48, E96
or E192. To reduce costs and make the design faster, discrete components are selected according to values
constants of the previous series. An exhaustive search of all possible combinations of values for selection of
an optimized design is not always feasible.
On other hand, almost all the design of analog circuits has been oriented towards MOS transistor-
based circuits mainly due to their low power consumption. Studies that address the sizing of circuits based on
bipolar transistors remain very scarce although they have better speed (switching times) and wider
bandwidths [1]. In addition, these studies deal with design considering the intrinsic parameters of
bipolar transistors as fixed, such as the works [2, 3] where an usual analog circuits are sized in which
the current gain (β) and the base- emitter and base-collector junction capacitances (Cπ) and (Cµ) are
considered as fixed which limits the design and subsequently reduces the performance of these circuits.
In order to overcome the aforementioned difficulties and limitations, an intelligent and efficient
optimization technique requires short computation time with high accuracy, must be used. Methods based on
the use of Meta-heuristics appeared then to resolve complex optimization problems, they always offer

 ISSN: 2088-8708
Int J Elec & Comp Eng, Vol. 10, No. 1, February 2020 : 129 - 138
130
approximate solutions for optimization problems at a very reasonable times [4]. They are used in many
engineering problems such as Scheduling Problem [5], Vehicle Routing Problem [6], Language Recognition
System [7] etc.
Some Meta-heuristics are used by the analog designers to solve the design problems of integrated
circuits and also of discrete component systems, such as Simulated Annealing (SA) [8], Genetic Algorithms
(GA) [9], Tabu Search (TS) [10], Particle Swarm Optimization (PSO) [11], Ant Colony Optimization
(ACO) [12-14] and Artificial Bee Colony (ABC) [15-17].
In this work, we propose the use of the Genetic Algorithm (GA), known by its effectiveness of
optimization, for the optimal sizing of three stages bipolar transistor amplifier. SPICE simulations are given
to show the validity of obtained results. The rest of the paper is organized as follows: The second part gives
an overview on the principle of the genetic algorithm. The third part deals with the application of
the proposed algorithm to the optimal design of a three stages bipolar transistor amplifier. The fourth part
shows the results of the optimal sizing. Finally, the fifth section, followed by a conclusion, presents how to
set SPICE parameters and shows the simulation results.
2. GENETIC ALGORITHM
The GA find their origins in the biological processes of survival and adaptation. Its principle
consists of sampling a population of potential solutions. A population of individuals is, initially, randomly
generated. The GA performs then operations of selection, crossover and mutation on the individuals,
corresponding respectively to the principal of survival of the fittest, recombination of genetic material and
random mutation observed in nature [18]. The optimization process is carried out through the generation of
successive populations until a stop criterion is met. The flowchart in Figure 1 provides an overview of a GA
procedure [18].
Figure 1. Flowchart of a GA
There are therefore 6 elements necessary for the running of the GA [18]:
1. We begin the process of fitting the problem to a GA by defining a chromosome as an array of variable
values to be optimized.
2. The user must fix a priori the sizing parameters of the algorithm, in particular the size of the population
and the number of generations (which is very often used as a condition for stopping the algorithm).

Int J Elec & Comp Eng ISSN: 2088-8708 
A genetic algorithm for the optimal design of a multistage amplifier (El Beqal Asmae)
131
3. Then the Generation of the initial population (set of possible solutions) can be random or from known
approximate solution(s).
4. Each chromosome has a cost found by evaluating the cost function f at the variables. The higher this cost,
the greater is the chance of an individual (solution) being selected for reproduction.
5. Now is the time to decide which chromosomes in the initial population are fit enough to survive and
possibly reproduce offspring in the next generation, the costs and associated chromosomes are ranked
from lowest cost to highest cost .The rest die off.
6. Then recombination/reproduction is achieved through two genetic operators, namely crossover and
mutation.
 Crossover that combines (mates) two chromosomes (parents) to produce a new chromosome
(offspring). The idea behind crossover is that the new chromosome may be better than both of
the parents if it takes the best characteristics from each of the parents.
 Mutation is usually considered as an auxiliary operator to extend the search space and causes release
from a local optimum when used cautiously with the selection and crossover systems.
Operations of selection, crossover, and mutation are repeated until a favorable number of
individuals for the new generation is created, and the objective function is calculated again for all of
the individuals in the new generation. The best individual in the new generation according to its fitness is
kept to continue to the next generation. Thus, the fitness of the entire population will be decreased with
the reproduction of the generation.
In the literature, the number of application studies of the GA technique is uncountable and the fields
of application are very diverse. These include for example: Power Supply System [19], Electric
Vehicles [20], Traffic Light Signal Parameters Optimization [21], Dynamic Optimization Problems [22],
Resolution university course schedules [23], Power factor improvement in the industry [24], etc.
In the following, we present an application of the GA to the optimal design of a three-stage amplifier.
3. APPLICATION: THREE-STAGE BIPOLAR TRANSISTOR AMPLIFIER CIRCUIT
We propose in this section, the optimal sizing of three stage bipolar transistor amplifier.
The schematic of this amplifier is given in Figure 2.
Figure 2. The three-stage amplifier
According to the study of the equivalent circuit of this amplifier in small signals in the mid band
where all the capacitances are neglected, we have obtained the following equations for AV, ZIN and ZOUT:
The voltage gain:
   
     1βRhR1βRhh
1ββRR1βR
A
11th112th3
"
11
'
11
122th1th3
V


 (1)

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132
With:
43B2 RRR  (2)
65B3 RRR  (3)
1EI1 RρR  (4)
2C
'
I2 RρR  (5)
L3E RR"ρR  (6)
'
112Be2 hRz  (7)
2e1Ith1 zRR  (8)
3B2Ith2 RRR  (9)
h11, ρ, β1 are the hybrid parameters for the first transistor, h’11, ρ’ , β2 for the second transistor and
h”11, ρ” , β3 for the third transistor.
The input impedance:
 








 1βhRRhRZ 111
'
1I2B111BIN
(10)
With:
21B1 RRR  (11)
The output impedance:
 
1β
hRR
RZ
3
''
112B1I
3IOUT


 (12)
With:
"
3EI3 ρRR  (13)
An estimate for the lower cutoff frequency for an amplifier with multiple coupling and bypass
capacitors is given by the sum of the reciprocals of the "short-circuit" time constants:


5
1
L
1
2
1
i iiS CR
F

(14)
Where RiS is the resistance at the terminals of the ith
capacitor with all the other capacitors are shorted, in our
case we have:
  1βRh1RBR 11th111S  (15)










1β
h
RzR
1
11
1I2e2S
(16)

133
    
  '
112B1I
2
'
'
3e2C'
112B1I2E3S
hRR
βρ
1
ρzR
hRRRR




(17)
  


  1βRhRRR 3
"
113B2IS4
(18)











1β
hR
RRR
3
"
112th
3IL5S
(19)
With:
  1βRhRz 33L
"
113Be3  (20)
The small signal equivalent circuit at high frequencies is as bellow in Figure 3:
Figure 3. Equivalent circuit of a transistor at high frequencies
At high frequencies, impedances of coupling and bypass capacitors are small enough to be
considered short circuits. Open-circuit time constants associated with impedances of device capacitances are
considered instead.
The higher cutoff frequency:



6
1i
iio
H
CR
1
ω
(21)
π2
ω
F H
H  (22)
Where Rio is resistance at terminals of ith capacitor Ci with all other capacitors open-circuited, for our circuit
we have:
 
 1βRhr
Rrh
R
11th111x
1th1x11
1πC


 (23)
  
 1βRhr
1βRhr
R
11th111x
11th111x
1μC


 (24)
   '
112x2B1I2πC hrRRR  (25)
  








'
11
22πC
4e2I2πC2μC
h
βR
1zRRR (26)

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134
 
 1βRRhr
RRrh
R
32th
"
113x
2th3x
"
11
3πC


 (27)
 
 
2th3x
3
"
11
3
"
11
3μC
Rr
1βRh
1
1βRh
R





(28)
With:
 1βRhz 3
"
11e4  (29)
To have a maximum excursion of the output signal, we should check the following constraint for all
the transistors and the power consumption equation is expressed in (31).
2
V
V CC
CE  (30)
 3E2EC1E
2
CC
CC RRRRII
2
V
3P  (31)
The decision variables are the resistors, the capacitors, the hybrid parameters of the transistors and
the supply voltage VCC, they present the chromosome of our GA, and the discrete components must have
a value of the standard series (E12, E24, E48, E96, and E192).
4. RESULT AND DISCUSSION
The collector current at the Q-point IC is fixed at 0.5mA. The studied algorithm parameters are given
in Table 1. The optimization technique works on MATLAB codes and the circuit is simulated in SPICE to
obtained frequency response.
Table 1. GA parameters
Population size Selection Probability Mutation Probability Generation
900 0.5 0.0001 1000
The serial components values are calculated as follows:
Ω10100pR iq
ii  (32)
F10100rC is
ii  (33)
Where [p, q, r, s] are real numbers that are the design variables for each ith
component.
The following two tables present the different optimal values given by the application of the genetic
algorithm. The Table 2 presents the optimal values of the hybrid parameters and the supply voltage.
The Table 3 presents the optimal values, linear and those following the different series, of resistors and
capacitors forming the studied amplifier. The Table 4 gives the corresponding performances to optimal
values presented in the Table 2 and Table 3. According to the results in Table 4, we notice that the
performances are almost the same for all series with a slight advantage for the series E192 which presents the
best gain Av and the best higher cutoff frequency FH.
Table 2. Optimal values for hybrid parameters
β1 β2 β3 ρ(Ω) ρ’ (Ω) ρ” (Ω) h11 (Ω) h'11 (Ω) h"11 (Ω) VCC(V)
Linear values 300 103 192.41 1038 1085.88 1623.55 1516.66 1856.58 1435.28 5

135
Table 3. Optimal values of R and C
Linear values E12 E24 E48 E96 E192
R1 (KΩ) 44.85 47 43 44.2 45.3 44.8
R2 (KΩ) 68.87 68 68 68.1 68.1 69
R3 (KΩ) 45.50 47 47 46.4 45.3 45.3
R4 (KΩ) 72.10 68 75 71.5 71.5 72.3
R5 (KΩ) 41.30 39 43 42.2 41.2 41.2
R6 (KΩ) 85.73 82 82 86.6 86.6 85.6
RC2 (KΩ) 1.01 1 1 1 1 1
RE1 (KΩ) 5.38 5.6 5.6 5.36 5.36 5.36
RE2 (KΩ) 5.91 5.6 6.2 5.9 5.9 5.9
RE3 (KΩ) 5.52 5.6 5.6 5.62 5.49 5.49
RL (KΩ) 142.07 150 150 140 143 142
C1 (µF) 23.00 22 22 22.6 23.2 22.9
C2 (µF) 64.12 68 62 64.9 63.4 64.2
C3 (µF) 80.24 82 82 78.7 80.6 80.6
C4 (µF) 12.55 12 13 12.7 12.4 12.6
C5 (µF) 69.38 68 68 68.1 69.8 69
rx1 (Ω) 15.02 15 15 14.7 15 15
rx2 (Ω) 11.92 12 12 12.1 11.8 12
rx3 (Ω) 11.54 12 12 11.5 11.5 11.5
Cµ1 (pF) 3.08 3.3 3 3.01 3.09 3.09
Cµ2 (pF) 2.85 2.7 2.7 2.87 2.87 2.84
Cµ3 (pF) 5.21 5.6 5.1 5.11 5.23 5.23
Cπ1 (pF) 9.95 8.2 9.1 9.53 9.76 9.88
Cπ2 (pF) 6.40 6.8 6.2 6.49 6.34 6.42
Cπ3 (pF) 16.35 15 16 16.2 16.2 16.4
Table 4. Performances associated to the optimal values
AV (dB) ZIN (KΩ) ZOUT (Ω) FL (Hz) FH (MHz) P (mW)
Linear values 44.85 47 43 44.2 45.3 44.8
E12 68.87 68 68 68.1 68.1 69.0
E24 45.50 47 47 46.4 45.3 45.3
E48 72.10 68 75 71.5 71.5 72.3
E96 41.30 39 43 42.2 41.2 41.2
E192 85.73 82 82 86.6 86.6 85.6
5. COMPUTING SPICE PARAMETERS AND SIMULATION
5.1. Computing SPICE parameters
The following step-by-step procedure leads to the required spice parameters, indicated by boldface
characters in the equations [25].
a. Compute the “transport saturation current” using:





 

T
BE
C
V
V
expIIS (34)
Where: 
q
KT
VT
b. The ideal “maximum forward beta” without correction for Early effect is given by:
βBF
(35)
c. Compute h11 from:
C
T
11
I
Vβ
h  (36)
d. Compute the “forward Early voltage” using:
CIρVAF (37)
Where IC, is the bias current at which the h-parameters were measured.

 ISSN: 2088-8708
136
e. Compute the value of the “zero-bias base resistance” using:
xrRB (38)
f. Determining CJC:
For Cµ, SPICE determines collector-base capacitance from:
MJC
VJC
CJC








CBV
1
Cμ
(39)
VCB is the Q-point collector-base voltage that SPICE will determine during the dc analysis.
We need to specify MJC, VJC, and CJC so that when SPICE runs a simulation, the resulting Cµ will match
the desired value.
Reasonable values for MJC and VJC are MJC = 0.5, VJC =0.7 V.
To find CJC the “base-collector zero-bias depletion capacitance”, the value of Cµ, will be given as
well as the voltage, VCB, at which the measurement was made.
g. Determining CJE:
For Cπ, SPICE determines the base-emitter junction capacitance Cje and the diffusion capacitance Cb
and add these:
bjeπ CCC  (40)
TFCJE
11
π
h
β
2C  (41)
Here TF is the forward transit time. We need to specify CJE and TF, so that when SPICE runs a simulation,
the resulting Cπ will match the desired value. To find CJE, we set TF = 0s, and modeling Cπ by the junction
capacitance alone.
CJE 2Cπ (42)
5.2. Simulation
For our simulation we use the 2N2222A NPN BJT, the data sheet of the transistor contain
the information needed to find IS, below is a plot of VBE vs. IC for the used transistor [26]. Figure 4 shows
the Base − emitter voltage.
Figure 4. Base − emitter voltage
From the plot above, for IC= 0.5mA we have VBE = 0.62 V at 25°C, and VT = 26mV at
the same temperature. From (34), IS= 22 × 10-15
A. The following Table 5 presents VAF calculated from (37)
corresponds to each transistor.

137
Table 5. Values of VAF
Transistor 1 Transistor 2 Transistor 3
VAF (V) 0.52 0.54 0.81
A DC analysis reveals that VCB for the circuit is 1.56 V, from (39) and (42), we find CJC and CJE
correspond to the three transistors, as shown in Table 6. After setting SPICE parameters, we simulate
the three-stage amplifier and we have the frequency response curve of the voltage gain for E12 as shown in
Figure 5, we notice that the mid-band gain is 19.12 dB, the upper cutoff frequency is 14.11 MHz and
the lower cutoff frequency is 33.56 Hz, that we give a mid-band equal to 14.10MHz.
Table 6. Values of CJC and CJE for E12
Transistor 1 Transistor 2 Transistor 3
CJC (pF) 6.33 5.18 10.75
CJE (pF) 4.10 3.40 7.50
Figure 5. Frequency response curve of the voltage gain for the three-stage amplifier
6. CONCLUSION
In this paper, we have presented an application of the Genetic Algorithm for the optimal design of
three-stage bipolar transistor amplifier. We selected the optimal values of discrete components from different
manufactured series and we gave the optimal values for the hybrid parameters of the transistors. The design
of the amplifier with the targeted performances is successfully realized by using the GA method, validity of
the proposed technique was proved via SPICE simulation.
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[26] 2N2222A Small Signal Switching Transistor Datasheet, [Online], Available: http://onsemi.com.

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