2. Theory and Method
2.5 Neural Network
2.5.1 Error Back-Propagation Network
EBP is multilayer feed-forward neural network (FFNN) with the back-propagation learning algorithm which is including input layer, hidden layer and output layer. The way of this operating is transmitting the input signal forward to the hidden layer through the calculating of activation function and then estimates from the hidden layer to the output layer. Fig13 is shown as a general FFNN figure. Every big circle is considered a neuron consisting of summer and TF (f1 or f2). The input target is shown as
Then according to the gradient-descent method, the least value of E is estimated by k
/ ( / )( / )( / ) f '(2 )O
, = ,
η is a constant known as the learning factor from 0 to 1. In Fig13 it is observed that the influence of weight v will extend to all output. Hence we need all the value of ji errors
{
e e1, 2,⋅⋅⋅,ek,⋅⋅⋅,eK}
. To obtain the optimum value of v we should calculate jiSimulated annealing (SA) is a generic probabilistic meta-algorithm for the global optimization problem, namely locating a good approximation to the global optimum of a given function in a large search space. SA has demonstrated to be a good technique for solving hard combinatorial optimization problems. In SA method, each
point of the search space is analogous to a state of some physical system, and the objective function E(s) to be minimized is analogous to the internal energy of the system in that state. The goal is to bring the system from an initial state to a new state with the minimum possible energy.
2.6.1 Acceptance probabilities
There are two conditions about accepting rule of SA. One is that the value of the objective function is decreased. When the value of the objective function is increased the other accepts moves with probability
( )E
p e =
− Δ T (56)where Δ denotes variation of the objective function, T is a control parameter called E the temperature . Next, a random number generated uniformly on the interval (0,1) is sampled, and if the sample is less than p the move is accepted. It follows that the system may move to the new state even when it is worse than the current one. It is this feature that prevents the method from staying in a local minimum—a state that is worse than the global minimum, but better than any of its neighbors. Initially the high temperature T causes the high probability of accepting a move that increases the objective function. When the search progresses the temperature is gradually decreased.
Finally, the probability of accepting a move that increases the objective function becomes vanishingly small. In general, the temperature is lowered in accordance with an annealing schedule.
2.6.2 Cooling course
The most generally employed annealing schedule is called exponential cooling which begins at some initial temperature T and decreases the temperature in steps 0 according to
k 1 k
T+ =αT (57)
where 0< <α 1 a cooling constant. Typically, a fixed number of moves must be accepted at each temperature before proceeding to the new state. A course of SA action is stopped either when the temperature reaches some final value T or the f system is not transformed to a new state after some times. A good value for α suggested experientially is 0.95 and that T should be chosen so that the initial 0 acceptance probability is higher than 0.8. The initial solution is generated typically at random.
3. Design optimization of diaphragm pattern
Model of microspeakers
Traditionally, lumped parameters methods with EMA analogy are commonly exploited to model loudspeakers. Despite the simplicity, the conventional methods are applicable to model the dynamics in low frequency regime especially in the neighborhood of fundamental resonance. However, this may not be sufficient for microspeaker analysis. The lumped parameter model can not predict well the high frequency responses of microspeakers that may play an important role in overall performance such as output level and roll-off frequency. In this work, the diaphragm-coil assembly of microspeaker will be modeled using FEA. The FEA model will be combined with the EMA analogous circuit to establish a fully coupled model for the microspeaker.
EMA analogous circuit of microspeaker
A sample of moving-coil microspeaker with a 16.4 mm diameter and 4.3 mm thickness is shown in Fig. 14. The front and rear view of the microspeaker are shown in Figs. 14 (a) and (b), respectively. The EMA analogous circuit of this microspeaker can be established in Fig. 3. The coupling of the electrical domain and the mechanical domain is modeled by a gyrator, whereas the coupling of the mechanical domain and the acoustical domain is modeled by a transformer [9]. The T-S parameters can be identified via electrical impedance measurement [9] and [10], as summarized in Table 2. The dynamic response of the microspeaker can be simulated on the platform of this model.
Loop equations can be written for the preceding FEA-lumped parameter circuit of the microspeaker as follows [29]:
0
The symbol “ ” denotes parallel connection of circuit. The loop equations can be solved for the current and velocity of the diaphragm for each frequency. From the current and velocity, the electrical impedance and the on-axis SPL responses of the microspeaker can be simulated.
3.1 Simulation and Measurement of Frequency Responses
Simulations and experiments are undertaken in this paper to validate the aforementioned integrated micro-speaker model. The frequency response from 20 Hz to 20 kHz of the micro-speaker is measured using a 2 Vrms sweep sine input.
Figure 15 (a) shows the experimental arrangement for measuring voice-coil impedance (with symbols defined in the figure):
s
Figure 15 (b) shows the experimental arrangement for measuring the on-axis SPL response by using a microphone positioned at 5 cm away from the micro-speaker.
Next, simulation of the diaphragm response was carried out using the integrated FEA-lumped parameter model mentioned above. Figures 16 (a) and (b) compare the voice-coil impedance and the on-axis SPL obtained from the simulation and the experiment, respectively. It can be observed that response predicted by conventional lumped parameter model is in good agreement with the measurement in low frequencies. In high frequencies, the conventional approach fails to capture the response due to the flexural modes of the diaphragm. However, the response simulated by the integrated FEA-lumped parameter model matches the measured response quite well.
3.2 Diaphragm Optimization using the Taguchi method
As mentioned previously, the diaphragm pattern has major impact on the micro-speaker response. To pinpoint the optimal pattern design, the Taguchi method and sensitivity analysis are exploited in this study. The Taguchi method [18] is very useful for experimental design, particularly for problems with finite number of discrete levels of design factors and thus reduction of the number of experiments is highly desired. In the following, the dimensions of diaphragm-voice coil assembly of micro-speakers will be optimized by using the Taguchi method. Table 4 shows the L9(3 )4 orthogonal array to be used in the Taguchi procedure. Here, nine observations and four factors are involved. The factors, each discretized into three levels, include the height of inner arc (H), the height of outer arc (h), the bandwidth of outer arc (d) ,and the thickness of diaphragm (t), as summarized in Table 4.
The following procedure aims to find the optimal parameters for the micro-speaker diaphragm design according to the cost function:
0 1
where f is the lower cutoff frequency of micro-speaker, 0 f is the upper cutoff 1 frequency of micro-speaker, SPL denotes the mean SPL in the piston bandwidth (defined in Fig. 17) and STD denotes the standard deviation of SPL in the piston bandwidth that serves as a flatness measure
( )
2where n is the number of frequency components of SPL in the band and SPL is the i ith SPL in the band. The symbol Δ signifies the difference of the performance parameters between the original design and the Taguchi design, e.g.,
0 0,Taguchi 0,original
f f f
Δ = − , ,w jj =1 ~ 4, is the weight for the performance parameter i.
In order to accommodate more design objectives, we consider using four kinds of weighting schemes, and summarized in Table 5. In scheme 1, the weights for the performance parameters are equal. Larger weights are used to emphasize the lower cutoff frequency and the mean of SPL in evaluating schemes 2 and 3, respectively.
In the weighting scheme 4, however, more emphasis is placed on the lower cutoff frequency, the upper cutoff frequency and the mean of SPL than the standard deviation of SPL.
Figures 18 (a)-(d) show the simulated voice-coil impedance and the SPL response of Run 1 to 9 in the L9(3 )4 orthogonal array. The values of calculated cost function for all weighting schemes are summarized in Table 6. The cost function of Run 7 has attained the highest value among all schemes. In Run 7, the lower cutoff frequency is reduced to 567.3 Hz, the upper resonance frequency is increased to 20 kHz, the SPL of the resonance frequency is increased to 84.8 dB, and the standard deviation is 1.88 dB. The optimal design result indicates that the height of inner arc (H) and the width of the outer arc (d) should be as large as possible, and
the height of outer arc (h) should be as small as possible, which will maximize the cost functions.
3.3 Sensitivity Analysis of Corrugation
Sensitivity analysis of diaphragm corrugation is undertaken to examine the effect of corrugation number on the micro-speaker performance. The analysis is based on the optimal diaphragm dimensions obtained in Run 7 of the preceding Taguchi procedure. The simulated voice-coil impedance and SPL response of Run 7 for different corrugation numbers are shown in Figs. 19 (a)-(b). The values of performance indices and the resulting cost function for different corrugation numbers are summarized in Table 7. It can be observed that corrugation tends to reduce the fundamental resonance frequency, but the relation is not linear. Further, corrugation tends to increase the mean and the standard deviation of the SPL response. In another word, increasing the number of corrugations will decrease the flatness of SPL during the effective frequency range. Nevertheless, the corrugation does not seem to affect the upper cutoff frequency significantly.
The values of cost function in relation to the corrugation number for different weighting schemes are also summarized in Table 7. The values of cost function are derived from the result of Run 7 of the Taguchi method. The results reveal that the optimal corrugation number is 30, in which the cost function is within 0.0493 - 0.0616 for the four weighting schemes.
4. Bass-enhancement and Optimization Using FEA-Lumped
In this section, the simulation of the vented-box system designed to enhance the bass response of the micro-speaker is carried out using the integrated FEA-lumped parameter model mentioned above. This will be explored in a more general context of vibration absorber theory. In this paper, vibration absorber theory will not be discussed. The detail is clearly discussed in [28]. Next, the Sequential Quadratic Programming (SQP) suggested in References [24]-[27] is utilized to design the optimized vented-box system.
4.1 Modeling the Vented-box Acoustical System
The general diagram of a vented-box system is shown in Fig. 20 The system primarily consists of an enclosure of volume VAB and a port with a cross-sectional area SP with radius aP and length LP. The mechanism of low-frequency enhancement lies in the Helmholtz resonator comprised of the acoustic mass in the vent and the acoustic compliance in the enclosure. More precisely, the vent can be modeled as an acoustic mass and an acoustic resistance. The acoustic resistance and acoustic mass of the port, and acoustic compliance of the enclosure are given by [9]
0
The mechanical impedance obtained using FEA mentioned above is changed into a lumped-parameter model. Therefore, the overall EMA analogous circuit of vented-box is shown in Fig. 21.
4.2 Optimal Design of the Vented-box
The design variables are selected to be the port radius (a ), the duct length (P L ) P
and the volume of cavity (VABC). The Helmholtz frequency of the vented box system is selected to be 400Hz. To initiate the SQP constrained optimization procedure, the lower resonance frequency of the coupled speaker-enclosure system is also selected to be 400 Hz. The design variables are selected to be the port radius, the duct length and the volume of cavity. To make circuit like as a parallel second-order oscillator circuit, the acoustic system is simplified to Fig. 22. And the cost function is chosen as the maximum sound pressure level at the frequency 400Hz.
This can be written in terms of the following optimization formalism:
6 6
reflect to mechanical system respectively. Vp is the volume of the duct. The results obtained using constrained optimization are also summarized in Table 8. Fig 23 shows the on-axis SPL obtained using constrained optimization. Result reveals that the SPL response after optimization at 400Hz is higher than original design.
4.3 Simulation and Measurement of Frequency Responses
A mockup was made for validating the vented-box design obtained previously using constrained optimization. The frequency response from 20 Hz to 20 kHz of the micro-speaker is measured using a 2 Vrms sweep sine input. Fig 24 (a) and (b) the voice-coil impedance and the on-axis SPL with the vent open are compared, respectively. The solid line is the result of experiment. The dot is the result of
simulation. The result of SPL response reveals that simulation is larger obviously than experiment at 400Hz about 5dB. The higher frequency range of on-axis SPL can be modeled nearly as a result of the aforementioned hybrid FEA-lumped parameter model.
5. Intelligent modeling and Optimization of the Diaphragm Geometry
In this section intelligent modeling with neural network is used to predict the performance of micro-speaker SPL response. A number of performance measures concerning the lower cutoff frequency f , mean SPL in the piston range 0 SPL, and the flatness of SPL response STD are weighted and summed to set up the cost function. Next, the SA method is utilized to design optimization of the diaphragm geometry.
5.1 Predicted System Using Neural Network
A set of the diaphragm geometry as inputs and the corresponding micro-speaker performance as outputs summarized in Table 9 and Table 10 respectively are normalized by formula where Index denotes the input or output data. A training set of input-output pair normalized in the range of 0-1 is summarized in Table 11. For the problems, we study here, we chose a four-layer feedforward network shown in Fig. 25 with four neurons in the input layer, the two hidden layers of NH neurons and MH neurons, and an output layer of four neurons corresponding to the number of performance variables.
According to EBP theory above-mentioned we can obtain NN system given by
3
and second hidden layer respectively, bI , b and J bK are bias units, θIJ, W JK and V are weight, ˆKm x is input about diaphragm geometry H, ,h ,d and t. We refer m this network as 4-NH-MH-3 NN. The error values resulting from the difference between target output obtained by measuring and actual output obtained by NN system are shown in Table 12. The NN system predict correctly so that all differences are very small. We produce others new set of input-output pair normalized in the range of 0-1 summarized in Table 13. Then we can obtain actual output with NN system. All errors obtained from the difference between target output and actual output are very small summarized in Table 14. It follows that this NN system has very high accuracy.
5.2 Performance Optimization Using the Simulated Annealing (SA)
In the present study, we choose simulated annealing for this purpose because of its simplicity and ability to produce global optimal solutions for complex problems which outweigh its relatively large computational requirements. The SA algorithm above-mentioned is detailed. Differing procedure above-mentioned, we study here, is solve the maximization problem. According to the NN system that can predicts the performance of microspeaker SPL response the objective function E is chosen as follow:
E= −0.5f0+0.4SPL−0.1STD (71)
The parameters of SA algorithm T , 0 T and f α are 1, 0.1 and 0.95 respectively.
Fig. 26 reveals the converge profile of SA algorithm with 4-10-6-3 NN system. The maximum value of objective function E is 0.4625. Then we can obtain the optimal performance of microspeaker using the hybrid method of neural network and simulated annealing (NNSA). The geometry of diaphragm summarized in Table15.
To prove the precision of optimal result obtained by the hybrid method NNSA we
utilize the geometry of diaphragm to simulate the SPL response of microspeaker by ANSYS again. Results obtained via ANSYS and NNSA are shown in Table15. It shows that the difference of performance has very high drop height as a result of too less training set of input-output pairs.
6. Conclusions
FEA-lumped parameter model has been presented for electroacoustical simulation of microspeakers. The mechanical impedance obtained from the FEA model of the diaphragm-voice coil assembly is incorporated into the lumped parameter model of the microspeaker system. The integrated EMA model provides better prediction for the voice-coil impedance and the SPL response at high frequencies than the conventional lumped parameter approach that neglects the higher-order flexural modes of the diaphragm.
On the basis of the proposed model, the dimensions of diaphragm are optimized with the aid of the Taguchi method. Using the results of the Taguchi method as a starting point, the optimal number of diaphragm corrugation is determined using sensitivity analysis. According to the optimized design of Table 6, the fundamental resonance frequency has been decreased and the flatness of the SPL response has been improved over the non-optimized design. In terms of the SPL frequency responses of Run 7 for different number of corrugation in Fig 18(b) reveals that SPL will increase with adding number of corrugation. FEA-lumped parameter model is also employed to the simulation of vented-box system that can predict the high frequency behavior of SPL response. Constrained optimization techniques were also employed to find the design that maximizes the sound pressure output of the vented-box system under practical constraints.
Another optimization method of dimensions of diaphragm geometry is NNSA.
Using the intelligent modeling of NN system which can predict the performance of SPL response, the optimal dimensions of diaphragm geometry is estimated via SA method. According to the optimized dimensions of diaphragm geometry, the performance of SPL response is calculated again to prove the accuracy of prediction.
The high difference shown in Table 15 reveals that dimension of diaphragm geometry which is not involving in the training set of input-output pair causes the high variation of performance of SPL response. This phenomenon bringing about high difference is due to the insufficient of the training set of input-output pairs which aren’t easy to produce via ANSYS.
7. A
PPENDIXThe measurement of loudspeaker efficiency
Loudspeaker efficiency is defined as the sound power output divided by the electrical power input. Most loudspeakers are actually very inefficient transducers;
about 1% of the electrical energy sent by an amplifier to a typical home loudspeaker is converted to the acoustic energy we can hear. The remainder is converted to heat, mostly in the voice coil and magnet assembly. The main reason for this is the difficulty of achieving proper impedance matching between the acoustic impedance of the drive unit and that of the air into which it is radiating. The efficiency of loudspeaker drivers varies with frequency as well. For instance, the output of a woofer driver decreases as the input frequency decreases.
I. Calculate Loudspeaker Efficiency based on TS parameters
The reference on-axis sensitivity of a driver is defined as the SPL at 1m away from the loudspeaker for a voice-coil voltage 1Vrms or R Vrms. The latter is the E
The reference on-axis sensitivity of a driver is defined as the SPL at 1m away from the loudspeaker for a voice-coil voltage 1Vrms or R Vrms. The latter is the E