A genetic algorithm for the optimal design of a multistage amplifier

ABSTRACT


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 transistorbased 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 baseemitter 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 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.
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.

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].  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.

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.
h 11 , ρ, β 1 are the hybrid parameters for the first transistor, h' 11 , ρ' , β 2 for the second transistor and h" 11 , ρ" , β 3 for the third transistor. The input impedance: With: The output impedance: With: 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: Where R iS is the resistance at the terminals of the i th capacitor with all the other capacitors are shorted, in our case we have: With: The small signal equivalent circuit at high frequencies is as bellow in Figure 3: 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).
The decision variables are the resistors, the capacitors, the hybrid parameters of the transistors and the supply voltage V CC , 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).

RESULT AND DISCUSSION
The collector current at the Q-point I C 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. The serial components values are calculated as follows: Where [p, q, r, s] are real numbers that are the design variables for each i th 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 F H .

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: The ideal "maximum forward beta" without correction for Early effect is given by: d. Compute the "forward Early voltage" using: Where I C , is the bias current at which the h-parameters were measured. V CB 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, V CB , at which the measurement was made. g. Determining CJE: For C π , SPICE determines the base-emitter junction capacitance C je and the diffusion capacitance C b and add these: 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 C π by the junction capacitance alone.

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 V BE vs. I C for the used transistor [26]. Figure 4 shows the Base − emitter voltage.  A DC analysis reveals that V CB 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.  Figure 5. Frequency response curve of the voltage gain for the three-stage amplifier

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.