3.1.2. Objective Function

The problem consists in minimizing supply current consumption for all three selected audio input signals. Despite the multi-objective approach, we study as a monoobjective problem using the aggregation approach [16]. It is one of the most often used methods for generation of Pareto optimal solutions. Our optimization algorithms allow us to minimize the objective function expressed as:

(4)

where, Ii represents the supply current consumption for each input signal, i Î [1, 3] the number of the objective and wi the weighting coefficient. In our case, wi = 1 because no preference between each objective is made.

3.1.3. Constraints

Constraints are conditions that must be satisfied in order to find a feasible design. Inequality constraints are used: each of the three ODGs has to be above -0.5 with the three input signals.

3.2. Optimization Algorithm

Once the problem has been formulated, we must choose the best optimization algorithm that allows us to minimize the objective function under the constraints. The Genetic Algorithm (GA) is one of the most popular and robust algorithms. It is based on natural genetic and natural selection mechanisms and some fundamental ideas are borrowed from genetics in order to artificially construct an optimization procedure. The GA acts over a population of potential solutions, applying intensification (crossover) and diversification (mutation) operators to explore the problem space. The fittest individuals are selected and give birth to a new population in the hope of improving the solution quality. More details on the mechanism of GAs can be found in [17]. GA is useful for a global search solution. However it is very slow and poor in a localized search. The direct search algorithm, Pattern Search (PS), on the contrary, is often able to ﬁnd local optima for constrained optimization problems, but it cannot guarantee that the solution is the global optimum of the problem. It ensures computational robustness when it starts from a feasible initial solution [18]. By combining GA with PS, an algorithm referred to as the GA-PS hybrid algorithm is formulated in this paper. In other words, the GA looks at the whole solution space to obtain a quasi-optimal solution. Then, the PS is used to increase the quality and speed of convergence to the optimal solution.

3.3. Cascade Simulation-Based Optimization

To optimize the Class-G amplifiers for different types of input signal simultaneously, we use a multiple simulation-based optimization to find the optimal solution with respect to the three audio signals. Figure 6 presents the concept of our approach.

For each iteration of the optimization loop, a simulation of our model, presented in Section 2, was performed for each audio input signal to find the performance such as current consumption I_{BAT} and the quality factor ODG.

4. Results

4.1. Model Validation

An existing circuit [6] was modeled with our proposed method (Section 2). Several measurements on [6] have been done to find the input parameters of the simplified electrical equations given in Equation (3) and to find the switching algorithm (α, β, etc).

In Figure 7, the measurement test bench is presented and allows us to compare the current found by the measurement of [6] and our proposed model. The setup configuration is a 47 Ω load, 3.6 V power supply and signal n°2. The error is less than 5% over all the output power range. Other input signals gave the same results. The comparison was also made with other existing Class-G [6] and it showed that the error is less than 10% with the same conditions. These results confirm that the Class-G amplifier model gives a reliable current consumption and can be used by the optimizer.

4.2. Optimizer Algorithm Comparison

For our application, we compared three algorithm optimizations to show the effectiveness of the proposed hybrid GA-PS algorithm. GA-PS is compared to GA and another hybrid algorithm GA-SQP, which is based on a GA coupled with the local search algorithm: Sequential Quadratic Programming (SQP). The optimization algorithms used in this study are part of the MATLAB optimization toolbox. In Table 2, we compared the output power of 1mW with signal 1, 2 and 3. The objective function is the mean of the consumption for these three input signals. The best result is obtained using a GA-PS optimization, in terms of minimizing the objective function while keeping an acceptable time of optimization. We therefore used this solution in order to reduce the consumption of our amplifier.

Indeed, over 100 000 simulations would be necessary with an exhaustive search. Moreover, extrapolating the results of a GA-PS optimization with a model using a transistor and macro-model would require 134 years to

Figure 6. Cascade simulation-based optimization.

Figure 7. Comparison between [6] and our model.

Table 2. Algorithm comparison.

find an optimal solution, since a single simulation with an audio signal during ten seconds takes three months.

4.3. Optimization Results

At the moment, the minimum supply voltage of a ClassG amplifier is 1.2 V [6]. However, because of the real power needed in general, we made an optimization allowing the constraint to be above 700 mV (which is the actual limit for the power amplifier) in order to see the leverage of optimization by tuning while keeping the other constraints as before. The test signal used is signal 4. It is used to perform the optimization in order to prove the robustness of the proposed optimization. The results shown in Figure 8 presents the gain obtained compared to [6] from 20 µW until 20 mW and highlight the need to lower the supply voltage of the power amplifier, since this reduction reduces the consumption for low power without degrading the consumption at high power. Moreover, the algorithm used in this optimization always respects the condition for the audio quality. Figure 7 also shows that the gains in consumption start to reduce after 2 mW for > 1.2 V and 5 mW for > 0.7 V. This result is explained by the facts that highest is the output power, lower are the switches of the amplifiers until the blocking of the upper supply.

We can note that for a Class-G amplifier with two power supplies, the reduction of under 700 mV is not justified. Indeed, the optimization performed did not show a reduction in consumption.

This conclusion leads us to perform the optimization with constraints above 700 mV, as shown in

Figure 8. Current consumption vs. V_{DD_LOW} for signal n°4.

Table 3. Here, we present the results for one music used for the optimization (signal n°2) and two signals not used for the optimization (signal n°4 and 5), to prove the robustness of the proposed optimization.

It can be noted that the results presented in Table 3 do not reduce the audio quality compared to the initial configuration of [6], which means that the ODG is always above –0.5. The results found by the optimizer are summarized in Table 4 and compared to two industrial circuits [6,7]. These parameters are not directly found in the datasheet but are obtained using reverse engineering on their test board. This table shows that the parameters of current industrial circuits are oversized. The threshold voltage can be placed closer to the supply voltage, the decay time has to be reduced and the lowest power supply should be minimized even if the highest power supply is at 1.9 V. However, like the industrial circuits [6,7], the attack time is not reliable to gain in consumption without deteriorate the audio quality. But this parameter has to be tried for Class-G with more than two power supplies in order to see if the results could be better. Indeed, all the parameters present in Table 4 can’t be applied for Class-G with more than two power supplies.

Figure 9 shows the gain in the current consumption from 20 µW to 5mW between the initial configuration from [6] and the optimization. 18% at 100 µW and at least 25% at 1 mW are saved with an optimal configuration of the switching power supply algorithm. Even the two signals not used for the optimization give a gain in con-

Table 3. Comparison for different class-G amplifiers.

Table 4. Value of the optimized parameters.

Figure 9. Gain in current consumption because of optimization.

sumption at low and nominal power. Like previously, for high power, they are few switch of the amplifier and the gain start to decrease after 2 mW. It is well to remember that the same level detector than [6] is used (based on logic gates and comparator which are few consuming).

5. Conclusion

In this paper, an original equation-based model has been introduced and associated with a hybrid optimization algorithm in order to reduce the current consumption of audio Class-G amplifiers by choosing the best parameters of the power supply switching algorithm. The proposed model has been validated. It also saves simulation time to predict the power consumption and keeps audio quality with various input signals. The optimizer coupled to this model allows us to find the best power supply switching strategy for a Class-G amplifier with two supplies by giving the optimal value of the parameters (, α, β, decay time and attack time). At least 25% of power consumption can be saved by optimizing the switching algorithm compared to an existing Class-G circuit with the same electrical implementation. In addition, the model is robust for operating conditions since the optimization was done for multiple input signals without loss of audio quality thanks to the PEAQ method. In future work, this approach will be used with Class-G amplifiers with more than two-power supplies, in order to optimize on all the range of power.

6. Acknowledgements

The authors gratefully acknowledge the standard linear division of ST Microelectronics for their valuable technical help.

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