On the Sizing of CMOS Operational Ampliﬁers by Applying Many-Objective Optimization Algorithms

: In CMOS integrated circuit (IC) design, operational ampliﬁers are one of the most useful active devices to enhance applications in analog signal processing, signal conditioning and so on. However, due to the CMOS technology downscaling, along the very large number of design variables and their trade-offs, it results difﬁcult to reach target speciﬁcations without the application of optimization methods. For this reason, this work shows the advantages of performing many-objective optimization and this algorithm is compared to the well-known mono- and multi-objective metaheuristics, which have demonstrated their usefulness in sizing CMOS ICs. Three CMOS operational transconductance ampliﬁers are the case study in this work; they were sized by applying mono-, multi- and many-objective algorithms. The well-known non-dominated sorting genetic algorithm version 3 (NSGA-III) and the many-objective metaheuristic-based on the R2 indicator (MOMBI-II) were applied to size CMOS ampliﬁers and their sized solutions were compared to mono- and multiobjective algorithms. The CMOS ampliﬁers were optimized considering ﬁve targets, associated to a ﬁgure of merit (FoM), differential gain, power consumption, common-mode rejection ratio and total silicon area. The designs were performed using UMC 180 nm CMOS technology. To show the advantage of applying many-objective optimization algorithms to size CMOS ampliﬁers, the ampliﬁer with the best performance was used to design a fractional-order integrator based on OTA-C ﬁlters. A variation analysis considering the process, the voltage and temperature (PVT) and a Monte Carlo analysis were performed to verify design robustness. Finally, the OTA-based fractional-order integrator was used to design a fractional-order chaotic oscillator, showing good agreement between numerical and SPICE simulations.


Introduction
In terms of the optimal sizing of CMOS analog integrated circuits (ICs), it remains a challenge to accomplish target specifications and, during recent years, metaheuristics have shown their usefulness in this task. For instance, some recent mono-objective metaheuristics applied in the optimization of analog ICs can be found in [1][2][3][4][5] and some multi-objective metaheuristics can be found in [6][7][8][9]. Nevertheless, the design of ICs using CMOS technology, especially in the analog domain, remains a complex process due to the many design variables, constraints and trade-offs among them. For that reason, this work shows the appropriateness of applying many-objective optimization algorithms to improve both fitness functions and trade-offs and the sized solutions are compared with those obtained with mono-objective and multi-objective optimization methods. to achieve; therefore, the analog IC design problem is quite suitable for applying metaheuristics. Recent OTA design issues include requirements such as low power [22], low voltage [23], improved linearity, high CMRR, high gain, high GBW and robustness to variations to enhance applications in active filters [24], sensor conditioning [25], bio-medical signal amplification [26] and radio frequency (RF) [27].
Among the currently available OTA topologies, this work is focused on sizing the CMOS ones shown in Figure 1. The topology given in Figure 1a shows a simple differential amplifier. Figure 1b shows a single-stage OTA and Figure 1c shows the recycled foldedcascode (RFC) OTA proposed in [28], which is a modification from [29], to double the slew rate (SR). All the OTAs have a differential input stage with source degeneration, which allows us to adjust the transconductance (g m ) and improve linearity [30], which is required to design an FOCO by using biquadratic OTA-filters. The variable resistors shown in Figure 1 were designed using two P-MOS transistors connected by their gates, as shown in Figure 2.
(a) Differential pair.  CMOS OTAs are characterized using an FoM, while other performances and constraints are included according to the application of mono-, multi-and many-objective metaheuristics. The FoM that is considered in this work is given in (1), where the subindex s means small signal. For the many-objective optimization FoM s , DCGain and CMRR were maximized and power consumption and total silicon area were minimized. During the sizing optimization process, the design variables were the channel width (W) and length (L) of all transistors and the value of the bias current (I b ), which were updated within a SPICE netlist-file. These design variables and the control voltage (V c ) were encoded into a vector x ∈ R n , where n represents the problem's dimension. The vector was defined by x = [a 1 , a 2 , ..., a n ] and each design variable was bounded to the values a i ∈ [2 µm, 1000 µm] for W i , a i ∈ [2 µm, 10 µm] for L i , a i ∈ [10 µA, 100 µA] for I b i and a i ∈ [−0.9 v, 0.9 v] for V c i . Finally, each constraint was handled as g(x) ≥ 0.
Other inherent constraints that depend on the circuit's topology were considered, such as, in a differential pair, the W/L sizes of both transistors were equal, as well as some MOS transistors had to be equal when they were part of a current mirror with unity gain.

Mono-Objective Algorithms
Mono-objective optimization metaheuristics have the goal of finding the best minimum or maximum value, which corresponds to the solution of a single objective function. The sizing of an analog IC by a mono-objective algorithm provides information on the nature of the optimization problem, but, generally, the algorithm does not provide a set of workarounds that interchange different objectives with each other. However, it is well known that a single-objective metaheuristic provides a final solution or a set of solutions that are really good [31]. In this paper, we applied mono-objective metaheuristics, such as PSO, MOL and DE, to size the OTAs given in Figure 1.

PSO Algorithm
PSO is a metaheuristic that performs an iterative optimization based on the social behavior of birds or fishes and it is based on the mathematical models given in (2) and (3) [32]. PSO's behavior is based on a random initialization of a set of particles whose search space was defined and bounded in this work. This process is intended to give favorable initial position and velocity to the particles. The random initialization and the mathematical models given in (2) and (3) allow to change the particles' position at every iteration and depending on the value of some random parameters, the velocity is adjusted. In this manner, both the position p i and velocity v i are updated until a stop criterion is met. In the updating equations pbest and gbest provide information on the particle's best position and best global position that is measured among all particles. The parameter rand() returns a random and uniform real value between 0 and 1. The constant parameter c 1 denotes the confidence of a particle and c 2 denotes its confidence in the swarm. In this paper, we set the constant parameter c 1 and c 2 equal to 2, as recommended in [33].

MOL Algorithm
Basically, the MOL algorithm is a simplified version of PSO. It is derived from the original behavior of PSO, but, in this case, MOL neglects the best position of a particle [34]. This is conducted by eliminating pbest in (2), which updates to (4), meaning that there is more communication among the particles [35]. Therefore, the MOL algorithm tends to have a purely social behavior, since each particle follows the best among all without looking for another solution on its own. In addition, having a simplified model makes it easier to select the parameters in MOL, which is not easy in PSO. One of these parameters is the inertia weight (w), which is related to facilitating the particles to find better solutions by exploring different directions throughout the values in the search space. In this paper, the parameters in MOL were set to w = 0.95 and c 2 = 2.

DE Algorithm
DE belongs to the class of evolutionary algorithms (EA) that are based on the concepts of the theory of evolution of species proposed by Darwin. DE has two main steps [36], initialization and evolution. In the initialization phase, the population is randomly generated within a determined search space. In the evolution phase, the current population is handled by performing genetic operations, such as mutation, crossover and selection processes, that are executed in the optimization loop until a determined number of generations or a stop criterium is met. Some of the most important advantages of DE are: its simplicity of programming, versus other EAs; its much better performance, considering correctness, speed of convergence and robustness; and the fact that the script can be coded in relatively few lines and that the complexity of the search space is low, considering similar EAs [37].
After the initialization of the population, each individual is mutated and then a new solution v ij is generated from three randomly selected parents, as given in (5). Thereafter, the crossover operation is executed to create a new solution considered as a trial solution, as described in (6). This crossover operation is executed taking a target x ij and a mutant vector v ij , with a probability value P c between 0 and 1. The weighting factors P m and P c are the mutant and crossover probability, respectively. Finally, a replacement is carried out by elitist selection, in which the target and trial vectors are compared considering the fitness values. The individual with better fitness is taken and it survives to perform the same optimization process that follows in the next generation, as described in (7). In DE the mutation, crossover and selection process are executed at each generation until a stop criterion is met. v u j = v j if rand() < P c or j = j rand x j otherwise (6)

Multi-Objective Algorithms
In a given optimization problem, if one can observe that the targets are in conflict with each other, then the solution can be obtained by applying multi-objective metaheuristics. That is, if one objective improves, the other deteriorates and vice versa. The most representative multi-objective metaheuristics are NSGA-II and MOEA/D.

NSGA-II Algorithm
NSGA-II was proposed by Kalyanmoy Deb in 2002 [38], as a version that improved the original NSGA proposed by Goldberg [39], in which an initial population is partitioned into fronts according to a non-dominance criterion. NSGA-II performs an iterative optimization process, starting from an initial population or a set of individuals; after an evaluation, the individuals with better fitness are selected as parents and evolve according to the concept of natural selection applying genetic operators, such as crossover, mutation and selection, to generate new offsprings. This process is called generation and is repeated until a stop criterion is met.
Its main characteristics are fast non-dominated sorting, computational complexity in the order of O(MN 2 ), fast ranking function, crowding distance assignment and elitism. It determines a rank value that considers the number of individuals that dominates each solution. NSGA-II performs crowding distance to keep the solution's diversity in the Pareto front to ensure that each feasible solution is kept at a good crowding distance. The elitism is an operator that executes the fast non-dominated sort approach and crowding distance and selects the individual with the highest ranking. [40]. It transforms a multi-objective optimization problem into a set of single optimization problems. Each subproblem is associated to an individual solution and all of them are simultaneously optimized and quantified using utility functions, weight vectors and a reference point. In general, each subproblem is evaluated using only the information of its neighbors as a mechanism of elitism. This characteristic allows us to reduce the computational complexity without affecting the performance and even outperforms NSGA-II.

Many-Objective Algorithms
The optimization of problems with more than three objectives is an open research area and shows a great opportunity to improve its performance. The definition of manyobjective optimization was introduced by Purshouse and Fleming [41] after observing that multi-objective algorithms based on Pareto optimality presented serious problems as the number of objectives increased.

NSGA-III Algorithm
NSGA-III is a many-objective metaheuristic proposed in 2013 [42], where, unlike NSGA-II, the solution's diversity among all individuals in the population is enhanced by supplying and adaptively updating a number of well-spread reference points. According to [43], NSGA-III has a similar framework to NSGA-II, the only difference being the inclusion of representative changes in the selection process. The operations are executed as follows: First, the parent population is randomly initialized; afterwards, the binary tournament selection, crossover and mutation genetic operations take place to generate an offspring population. In the next step, both populations are combined to duplicate the number of individuals that are sorted taking into account their domination level. Half of the best individuals in the combined population are selected to update the population that becomes parent one, ready to execute the same genetic operations in the next generation. The selection process starts with a set of reference points, which are generated by spanning a hyperplane with unit intercepts associated to the objective axis by applying the method for normal boundary intersection, as detailed in [44].

MOMBI-II Algorithm
MOMBI-II was proposed by Raquel Hernández in [45] and performs a hierarchical ranking of the individuals of a population. MOMBI-II is a metaheuristic capable of handling two or more objectives, since it uses indicators as a selection mechanism to keep the solution's diversity and also serves, during the optimization process, as a guide for the search or as a stopping criterion. The R2 indicator is a very reliable approach that is useful for problems formulated with multi-and many-objective metaheuristics. It is also suitable to compare approximations of different multi-objective optimizers by means of utility functions. One of the main advantages of the R2 indicator is the low computational cost and the compatibility with the Pareto front. Therefore, MOMBI-II has the advantage to optimize problems formulated with objectives higher than three and without having problems with Pareto dominance.

Sizing OTAs by NSGA-III, MOMBI-II and MOEA/D
The pseudo-codes provided in [46] were modified to maximize FoM s , differential mode gain and CMRR and minimize the power consumption and the total MOS area. In Algorithms 1-3, the adaptations of the NSGA-III, MOMBI-II and MOEA/D algorithms are shown, respectively. All the algorithms linked SPICE to evaluate the circuit characteristics related to the objective functions, taken from the * .lis output file generated by SPICE, and, to reduce execution time, the command .MEASUREMENT was used within the input file * .sp.
The * .sp SPICE input file was encoded by means of subcircuits to perform a direct current (.DC), alternating current (.AC) and transient (.TRAN) analyses. In addition, to avoid modifying this file at each generation, a * .lib file was linked with the original netlist. This * .lib file had the design variables W, L, I b and V c . Therefore, the variables obtained from the NSGA-III, MOMBI and MOEAD algorithms were replaced within the * .lib file without compromising any analysis in the * .sp file.
Algorithm 1 NSGA-III. The optimization process is generally the same for each algorithm; for all of them, it is necessary to specify the number of individuals (P), generations (Gen), crossover constant (P c ), mutation constant (P m ), number of design variables, restrictions and limits for each design variable. However, each algorithm has its own method of sorting, ranking and selecting the best individuals.

Optimal Solutions Applying Mono-, Multi-and Many-Objective Metaheuristics
A sized solution provided by a metaheuristic is said to be feasible when it achieves all the target specifications. This section summarizes the sizing of the OTAs applying PSO, MOL and DE as mono-objective; NSGA-II and MOEA/D as multi-objective; and NSGA-III, MOEA/D and MOMBI-II as many-objective metaheuristics. The mono-objective sizing optimization considered the FoM described in Section 2 as the single objective function, with the other electrical characteristics as constraints. The multi-objective sizing considered the optimization of three objectives simultaneously, namely, FoM, DC Gain and power consumption. The many-objective optimization process included five objectives: FoM, DC Gain, power consumption, CMRR and the total MOS area. For each metaheuristic 10 runs were executed, with 100 generations and a population equal to 70 individuals/particles to size the OTAs shown in Figure 1. The target specifications are listed in Table 1. Table 1. Target specifications for the sizing of the OTAs shown in Figure 1.
Although the optimization process is similar for each metaheuristic, the behavior is quite different. In PSO, the particles search in their vicinity both individually and as a group in order to find better positions according to their updating equations defined in (2) and (3), while, the behavior of MOL, despite being a variation of PSO, is purely social, that is, all the particles search together for the global best position. DE is an evolution-based algorithm, finding better solutions through genetic processes such as mutation, crossover and selection. The main advantage of applying mono-objective optimization such as PSO, MOL and DE is that the cost of the objective function is taken to the limit. This means that they are capable of reaching very good values; however, their great disadvantage is that the other characteristics are under a threshold, defined as constraints, limiting their performance. In multi-and many-objective algorithms, the main goal is devoted to enhancing the trade-off among different objectives. The most popular multi-objective algorithm in recent decades is the NSGA-II, whose main characteristics are the diversity, the convergence and the robustness given to the solutions that can be sketched in the Pareto front. However, its performance is limited by the number of objectives. For this reason, this work highlights the advantages of the many-objective metaheuristics to size CMOS amplifiers, which have a huge number of target specifications. The main characteristics of the many-objective algorithms applied herein are: NSGA-III is a variation of the wellknown NSGA-II; MOEA/D is a decomposition algorithm with the advantage of handling more than three objectives at the same time; and MOMBI-II has the advantage of using performance indicators as a selection mechanism.
The best solutions of 10 runs for each metaheuristic are shown in Tables 2-4, for sizing the differential pair, one-stage and RFC OTA, respectively. The five objectives functions FoM, DC gain, CMRR, power consumption and total MOS area values are listed in the first five lines. The first three characteristics are maximized while the other two are minimized. The other performances, such as GBW, phase margin, PSRR±, SR± and the maximum and minimum output voltage, were handled as constraints. Below Vmin, we list the best sizes of the MOS transistors for each metaheuristic, the bias current and voltage control. In each Table, the best solution values are given in bold face. Table 2. Best solutions (in boldface) obtained by applying mono-, multi-and many-objective algorithms to size the differential pair shown in Figure 1a.

Mono-Objective
Multi  Table 3. Best solutions (in boldface) obtained by applying mono-, multi-and many-objective algorithms to size the one-stage OTA shown in Figure 1b.  Table 4. Best solutions (in boldface) obtained by applying mono-, multi-and many-objective algorithms to size the RFC OTA shown in Figure 1c. As it can be seen, after analyzing these solutions, the metaheuristics with the best performance were, in general, the many-objective ones, since at least one improved the performance of the mono-and multi-objective metaheuristics. They also improved the trade-offs among the five objectives. It can also be pointed out that the mono-objective metaheuristics had a better performance than the multi-objective ones and even their solutions were quite close to the solutions found by the many-objective metaheuristics.

Mono-Objective
The execution times of the three types of optimization algorithms, known as mono-, multi-and many-objective, are given in Tables 5-7 for the differential pair, one-stage OTA and RFC-OTA, respectively.

OTA-Based Fractional-Order Chaotic Oscillator
The usefulness of sizing OTAs by many-objective metaheuristics is demonstrated herein by selecting an optimized OTA to design a fractional-order integrator, whose design robustness is verified by performing process, voltage and temperature (PVT) variations and Monte Carlo analysis. The robust OTA-based fractional-order integrator was employed in the design of a fractional-order chaotic oscillator (FOCO) taken from [47] and given in (8). This FOCO had chaotic oscillation when a = 2.05, b = 1.12 and c = 0.42, with initial conditions x 0 = 0.1 and y 0 = z 0 = 0. Figure 3 shows the portraits of (8), in which the state variables are plotted against each other to generate the attractors. It can be appreciated that the amplitudes of the state variables are below 1, so that it is suitable for CMOS design using a UMC technology at 180 nm. Considering the system of equations from (8), one can see that the electronic implementation can be performed using amplifiers, adders/subtractors and one multiplier to evaluate X(s) * X(s). In addition, a fractional-order integrator is also necessary, which, according to [48], can be approximated by H(s) = 1/s 0.9 , which is modeled by (9). Figure 4 shows the block diagram description to implement (8), where the blocks labeled as H(s) describe the fractional-order integrator H(s) = 1 s 0.9 , given in (9). The blocks labeled a, b and c represent the constants of the system in (8). These blocks could be replaced with OTAs, which converted an input voltage into an output current; these currents, in turn, were added to the currents of the other blocks and converted back to voltage at the adder block output. Most of the blocks needed to design the FOCO could be implemented using an OTA with adjustable g m [49]. In this section, we use the one-stage OTA depicted in Figure 1b. The convolution X(s) * X(s) observed in the state variable Z(s) could be evaluated by the CMOS four-quadrant multiplier designed in [49] and is shown in Figure 5.

Fractional-Order Integrator Design Using an OTA-C Biquadratic Filter and an OTA-C Low-Pass Filter
To carry out the design of the fractional-order integrator, the transfer function in (9) was decomposed into two functions, to have H(s) = H 1 (s)H 2 (s), one of second-order and one of first-order, as given in (10) and (11), respectively. These functions were designed using an OTA-C biquadratic and first-order active filters, as shown in Figure 6, whose topologies were taken from [21]. The circuit transfer functions are given in (12) for the biquadratic and in (13) From H 1 (s) and (12) and by setting V 1 = V 2 = 0 and V 3 = V 4 = V 5 = V i , the transfer function of the OTA-based circuit shown in Figure 6a was rewritten as given in (14).
The design of the biquadratic filter was performed as follows: If gb 0 = 500 µA/V, the capacitor values are obtained as C 2 = gb 1 /216.6920 = 2.3 µF and C 1 = gb 0 gm 2 / 278.2968C 2 = 650 µF. In this manner, gm 2 = 361.5540C 2 = 830 µA/V and gm 1 = 774.1476C 1 C 2 /gm 2 = 1.4 m A/V. For the low-pass filter shown in Figure 6b, if gm 1 = 500 µA/V, the capacitor value is C 1 = gm 1 /2.2675 = 220 µF; therefore, R 1 = 1/0.01292C = 350 KΩ. In order to achieve a high resistance value, the resistor shown in Figure 7 was implemented [50]. The main advantage of this resistor relies on reaching high resistance values with a good linearity range. In practice, the value of R in Figure 8 was adjusted to 400 KΩ to adjust the frequency response of the fractional-order integrator.
(b) Low-pass filter.  We propose the OTA-based design of the fractional-order integrator shown in Figure 9. It includes a biquadratic OTA-C filter in cascade connection with the low-pass OTA filter. As mentioned in [49], it was necessary to consider that the input port had to include the filter embedding the largest zero and the largest pole. The frequency response of the fractional-order integrator is sketched in Figure 9, which shows the comparison between the ideal response simulated in MatLab of the transfer function given in (9) and its CMOS design using 180 nm UMC technology using the three OTA topologies shown in Figure 1. The sizes of the OTA were taken for the one-stage OTA applying MOEA/D for five objectives. To measure the symmetry of the output signal of the OTA, we used the methodology proposed in [51], in which the output signal must have an absolute minimum value higher than 90% of the absolute maximum signal. In Figure 9, the good agreement between the MatLab and the CMOS implementations of the fractional-order integrator can be appreciated. . Frequency response of the fractional-order integrator, considering ideal transfer function approximation in MatLab, macro-modeling and HSPICE simulation using the CMOS differential pair (DP), one-stage and RFC OTAs.

PVT and Monte Carlo Analysis of the Fractional-Order Integrator
The robustness of the fractional-order integrator was verified by performing a PVT analysis in the five corners: typical-typical (TT), fast-fast (FF), slow-slow (SS), slowN-fastP (SNFP) and fastN-slowP (FNSP). These corner analyses were simulated for the three OTAs and their results are shown in Figure 10 for the differential pair, Figure 11 for the one-stage OTA and Figure 12 for the RFC OTA. For each corner, the voltage supply was varied by ±10%. Likewise, the temperature was varied with values of 20 (T−), 60 (T) and 120 (T+).          Figure 11. PVT simulations of the CMOS fractional-order integrator given in Figure 8 using the one-stage OTA given in Figure 1.      Figure 8 using the RFC OTA given in Figure 1.
In addition to PVT analyses, a Monte Carlo analysis was performed to observe the variability due to matching conditions of the MOS transistors. This analysis was carried out assuming a deviation of 10% in the W's and L's of all MOS transistors. These two analyses were more than enough to appreciate the robustness of the CMOS design of the fractional-order integrator; therefore, it was used to design an FOCO. Figure 13 shows the histogram for each OTA topology for 1000 runs.

Fractional-Order Chaotic Oscillator Design Using OTAs
Once the fractional-order integrator was designed, the FOCO design was carried out; its OTA-based implementation is shown in Figure 14. It is noted the requirement of three fractional-order integrators (H(s)), five OTAs and one multiplier were required for its implementation. From (8), one can determine the value of the constants a, b and c. If the OTA has g m = 500 µA/V, the constant a is equal to 2.05(500 µA/V) = 1025 µA/V, b = 1.12(500 µA/V) = 560 µA/V and c = 0.42(500 µA/V) = 210 µA/V. The three constants multiplying the convolution X(s) * (X(s) in (8) leads to a transconductance value of 3(500 µA/V) = 1500 µA/V.
As mentioned above, the sizes of the OTAs were taken from the optimal solutions provided by the many-objective optimization with MOEA/D for five objectives. Finally, the portraits of the FOCO given in Figure 14 are shown in Figure 15, where the CMOS design was obtained using UMC technology at 180 nm. The bias was established at ±0.9. As can be seen, the MatLab simulation of the portraits of the FOCO shown in Figure 3 are in very good agreement with the portraits generated by the CMOS FOCO shown in Figure 15; therefore, we conclude confirming the suitability of performing many-objective optimization in the sizing of CMOS OTAs.

Conclusions
Sizing CMOS OTAs is a challenging task, mainly due to the large number of design variables and trade-offs among the target specifications. Fortunately, one can take advantage of applying metaheuristics to optimize such kinds of analog integrated circuits. In this manner, this paper shows the application of mono-, multi-and many-objective optimization algorithms to size three different CMOS OTA topologies. The many-objective optimization process was performed considering five different electrical characteristics taken as objective functions, namely, FoM, DC gain, power consumption, CMRR and total MOS area. The optimization results provided by the mono-, multi-and many-objective optimization algorithms to the three OTAs show that the many-objective metaheuristic not only found better sizing solutions in the main objective functions but also improved the trade-offs among the electrical characteristics, which lead to a more robust design. The robustness of the optimized OTAs was verified through PVT variation analyses and Monte Carlo simulations. That way, an optimized OTA was used to design a fractionalorder integrator using OTA-C active filters. Finally, the CMOS OTA-based fractional-order integrator was used in the design of a CMOS fractional-order chaotic oscillator (FOCO). As a result of the optimization process, the MatLab simulations of the FOCO were in very good agreement with the CMOS design using UMC technology at 180 nm. This application of an optimized OTA in the design of an FOCO leads us to conclude that, although all metaheuristics are a good option to size CMOS OTAs, the application of manyobjective optimization algorithms can provide much better results compared to mono-and multi-objective metaheuristics.