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This paper describes a novel design methodology using non-linear models for complex closed loop electro-mechanical sigma-delta modulators (EMΣΔM) that is based on genetic algorithms and statistical variation analysis. The proposed methodology is capable of quickly and efficiently designing high performance, high order, closed loop, near-optimal systems that are robust to sensor fabrication tolerances and electronic component variation. The use of full non-linear system models allows significant higher order non-ideal effects to be taken into account, improving accuracy and confidence in the results. To demonstrate the effectiveness of the approach, two design examples are presented including a 5th order low-pass EMΣΔM for a MEMS accelerometer, and a 6th order band-pass EMΣΔM for the sense mode of a MEMS gyroscope. Each example was designed using the system in less than one day, with very little manual intervention. The strength of the approach is verified by SNR performances of 109.2 dB and 92.4 dB for the low-pass and band-pass system respectively, coupled with excellent immunities to fabrication tolerances and parameter mismatch.

Embedding a micromachined sensing element in a closed loop, force feedback system is a technique commonly used to realise high performance MEMS (micro-electro-mechanical systems) sensors due to the many advantages attainable in terms of better linearity, increased dynamic range and bandwidth, and reduced parameter sensitivity to fabrication tolerances. In particular, MEMS inertial sensors employing a capacitive sensing element incorporated in sigma-delta modulator (ΣΔM) control systems with electrostatic feedback have gained popularity in the past due to their direct digital output signal, and avoidance of potential electro-static instability (due to the ‘pull-in’ effect). Earlier work used the micro-machined sensing element as the sole loop filter, and, since the sensing element is typically a second order mass-damper-spring low-pass filter, this resulted in a second order electro-mechanical ΣΔM (EMΣΔM) [

Linearized analytical models for ΣΔMs are described in for example [

In this work we present a novel design methodology for EMΣΔM based on genetic algorithms (GA) and Monte Carlo simulations, both using accurate non-linear models. Genetic algorithms are based on the mechanics of natural selection and genetics combining the fittest individuals in the population in order to search for the best solution [

In this work the genetic algorithm used is the gamultiobj() function in Matlab which is variant on the Non-dominated Sorting Genetic Algorithm-II (NGSA-II) [

As part of the overall design methodology, the GA facilitates multi-objective optimisation for the design of low-pass or band-pass EMΣΔM with a wide range of orders and with any architecture. Since the result of the optimisation is a large number of equally optimal solutions the design procedure subsequently carries out a robustness analysis based on statistical simulations to ensure stability of the design in the presence of fabrication tolerances, which can be substantial for micromachined sensing elements. Although numerous methods exist for output variation estimation, e.g., [

This paper is organised as follows: Section 2 describes the developed GA process in general; Section 3 gives an example for the design of a 5th order EMΣΔM MEMS accelerometer; Section 4 gives a second example of a band-pass EMΣΔM for a MEMS gyroscope; in Section 5 the design approach is discussed and in Section 6 conclusions are drawn.

An EMΣΔM consists of the following building blocks: (i) the micromachined sensing element; (ii) the pick-off circuit that capacitively measures the displacement of the proof mass in response to an inertial force and converts it to a voltage; (iii) a phase compensator (which may not be required if the sensing element is overdamped); (iv) an electronic loop filter comprising several integrators and minor feedback or feedforward loops; (v) a clocked one bit quantizer; (vi) a feedback block that converts the feedback voltage into an electrostatic force acting on the proof mass and counterbalancing the inertial force.

For our design methodology the user must first choose an architecture and the order, which can either be taken from the literature on EMΣΔM, an architecture for a purely electronic A/D ΣΔM, or a novel architecture developed by the user. The next step is to develop a Simulink model. The model can be as simple or as complex as deemed necessary by the user. Second order effects may be included. A few examples include: (i) the pick-off circuit can be modelled simply as a gain constant, or the nonlinear relationship between displacement and differential change in capacitance may be included; (ii) The micromachined sensing element may be simply modelled as a second order lumped parameter system with mass, damping and mechanical spring constant as the only parameters, or higher order modes e.g., from the dynamics of the sense fingers can be included [

The proposed methodology is represented by the flow-chart in

The GA is then initialized with a user specified number, N_{R} of, within the constraints, randomly chosen parameter sets; which is termed a population. Each parameter set is termed an individual. This initial population represents generation 1. The system then runs N_{R} simulations (one for each individual) and records the performance objectives for each individual, for example SNR and proof mass displacement as previously discussed. Once the first generation has been simulated, the result is stored as a table where each row consists of the parameter set for one individual and its performance. As explained in the previous section, the GA sorts the results and then performs a number of functions including picking the very best individuals (elite preservation), generating a certain number of new random individuals (mutation) and cross fertilising good individuals to create new offspring. This last step actually involves taking different parameters from different good individuals and combining them to create a new individual (child). These three steps create generation 2, which again consists of N_{R} individuals. The whole process continues until either a specified maximum number of generations has been reached or the user monitoring the evolution determines that sufficient convergence has been achieved. Although it would be possible to automate the convergence detection, for example by calculating bit string affinity, we have found in practice that the insight gained from making this an interactive decision is very valuable.

Simulation length is an important consideration during the GA process and introduces a trade-off between accuracy and total optimisation time. Often systems can appear initially stable, only to lose stability a short time later and therefore it is possible to unwittingly promote unstable systems forward in the evolutionary process if the simulation time is too short. However, long simulation times can result in excessively time-consuming optimisation periods, given the large number of simulations involved. This issue has been addressed in this work by typically running a small number of simulations initially to establish a ‘quick’ simulation period that represents a reasonable trade-off between the chance of missed instability and computation time. When the final solution is chosen at the end of the whole process, a more extensive simulation is performed with a ‘long’ simulation period to verify stability beyond doubt. Values for the ‘quick’ simulation periods will depend strongly on the type of architecture being designed but typically lie in the region of a few seconds, and the ‘long’ simulation period is typically 8 times longer than the ‘quick’ period. Both of these parameters are defined alongside all the goals, parameters and constraints in the system file.

The next step in the methodology is to consider robustness, which is an important measure of how parameter variation will affect performance or stability and a key contribution of this work. It cannot be assumed that the individuals in the final population of the GA step are the most robust, since they have only been optimized for SNR and RMS displacement, not tolerance to variation. For example, an individual in one of the earlier populations may have only slightly lower performance than one in the final population but may be far more robust. For this reason, the robustness stage of the process must consider the full history of individuals, which we refer to as a census. Theoretically, Monte Carlo simulations could be performed on every individual in the census, however, with hundreds of simulations per individual required for the robustness analysis this would be too time consuming.

Therefore, before Monte Carlo robustness analysis is performed, the census needs to be filtered to discard all individuals that do not meet the objectives (

To demonstrate the design procedure we present a 5th order low-pass ΣΔM for a MEMS accelerometer with a sensing element fabricated in SOI (Silicon on Insulator). The main specifications of the sensor are listed in

In this example the ten parameters shown in

For the ΣΔM the oversampling ratio (OSR) needs to be specified. The OSR is related to the sensor bandwidth, BW and the sampling frequency fs by OSR = fs/(2 × BW). Here, we choose OSR = 64 resulting in a sampling frequency of 128 kHz. Furthermore, the criteria (

The GA is then run using 200 individuals, which are, within the specified range, randomly chosen parameter sets; for each individual a simulation is carried out and the SNR is calculated. This calculation is performed by a function ‘calcSNR’ available through the Delta Sigma Toolbox for Matlab [

Examining the SNR column (2nd from the right), it can be seen that some SNR values are negative in the first block; this indicates an unstable system, as the SNR is a good indicator for system stability [

The next step in the design process is robustness analysis which starts with a thinning and filtering algorithm. All individuals of all generations are stored in a matrix with 200 × 15 rows (number of individuals times number of generations). For the robustness analysis the same goal function values (a minimum SNR of 100 dB and a maximum RMS deflection of 40 nm) are chosen; if required these values can be modified at this point. The filtering algorithm simply discards the individuals that have a SNR < 100 dB or an RMS proof mass deflection >40 nm. The thinning algorithm now finds the most distinct individuals in the remaining design space, as explained in Section 2.

The number of individuals to be considered for the robustness analysis is specified as a user defined parameter, and is set to 40 here. Another user defined parameter sets the number of Monte Carlo simulations that will be performed for each individual; in this example 100. For each design parameter the user specifies a standard deviation providing a measure of parameter variation. The robustness analysis typically varies more design parameters than those explored by GA; for example the parameters of the sensing elements (its mass, damping coefficient and spring constant) were considered as fixed for the GA, but for the robustness analysis were varied by 2%, 25% and 5%, respectively, whereas the electronic gain constants optimized by the GA were varied only by 2%. This reflects the considerable fabrication tolerances that a micromachined sensing element typically exhibits. A function in the program generates 100 Gaussian distributed parameter sets based on a particular individual’s parameters (as the means) and the user supplied standard deviations. For each individual therefore, 100 simulations are run and the SNR and RMS displacement performance recorded. A yield value is calculated representing the percentage of the simulations for each individual that exceed the specified goal values. The user can then review the yield and performance of the investigated individuals and choose one as the final design. Here, the final parameter set is shown in

A further example of the proposed design methodology is now presented for a continuous time, 6th order band-pass EMΣΔM for a vibratory rate MEMS gyroscope fabricated in SOI technology, as described in [

The Simulink model is shown in

The genetic algorithm is then executed, which again creates and simulates an initial population of individuals, determining their fitness by combining their SNR and sense mode proof mass displacement. A combination of cross fertilisation, mutation, and elite preservation is used to create a new population and the evolution continues. After the specified number of generations, the algorithm halts, storing the entire multi generation census for the next step. Filtering and thinning is performed on the census results to ignore individuals which do not pass the specified goal values of 70 dB SNR and 20 nm RMS displacement, or are too close to one another. The individuals remaining after the thinning algorithm are feasible and distinct solutions, and are then used as the input to the Monte Carlo based robustness analysis of the next stage. A total of 200 Monte Carlo simulations are performed on each of these solutions using realistic standard deviations for all electrical and mechanical parameters, and from this a simulated yield is calculated and documented against the solution point. The designer then has the opportunity to choose a solution from this final list, trading off performance against yield for their particular application. In this case the parameters for the chosen design are shown in

The two examples presented illustrate the usefulness of the methodology for the design of arbitrarily complex EMΣΔM. To the best of the authors’ knowledge this is the first time such a design methodology has been presented based on non-linear models. The design methodology for the majority of EMΣΔM presented in the literature is not described, which is an indication that a manual process relying on trial and error was used. This requires a considerable experience in ΣΔM and MEMS sensor design and hence has a high initial knowledge threshold. Even after sufficient knowledge has been gained it can often take weeks to develop a satisfactory system design, with no real certainty that the system is robust or even optimal. The proposed approach greatly expedites the design process and gives much greater confidence that the results are both optimal and robust. In some literature sources a design methodology for EMΣΔM is described based on root locus techniques [

The design methodology described here circumvents both drawbacks: it is based on a full non-linear system model and it yields a design solution that is very close to the optimum, as it takes into account both SNR and proof mass displacement as performance measures. Additional performance parameters, such as dynamic range and maximum input signal could be additionally included as optimization metrics as required. Another advantage is the designer’s total freedom in the initial choice of the control system architecture; whereas the EMΣΔM described in the literature to date all are adapted architectures of ΣΔM analogue to digital converters. Therefore, our methodology facilitates the exploration of novel architectures for EMΣΔM; one example is to have a two-channel ΣΔM for the sense mode of a gyroscope, one channel for the signal, the other for the quadrature error. Furthermore, the GA design parameter set could be extended such that different architectures, or loop orders could be available as part of the GA evolution, allowing extremely diverse design space exploration.

The robustness analysis performed following the GA is a key contribution of the work, giving confidence in a design and ensuring manufacturability. Without this it is possible to design a system which may easily become unstable due to inevitable fabrication tolerances. As with any multi objective optimisation, there is no single optimal solution but instead a range of equally optimal solutions, which is why it is important for the designer to choose the final design solution based on a performance

Many design flows have been performed by the authors for a wide range of EMΣΔM architectures and they are confident to claim that the GA explores the design space well and finds an excellent design solution even for complex and non-linear design spaces with multiple objectives. A side benefit of the approach is the insight gained from its use by those who have little experience in the area. With typical design times of a single day for complex architectures, the approach offers an extremely efficient alternative to manual design procedures which often take weeks.

The presented methodology allows the system level design of arbitrarily complex EMΣΔM with ease and in a short period of time. The design process relies on a GA that varies a set of system parameters and records the performance for each set. After a filtering and thinning step a robustness analysis is carried out to ensure system stability in the presence of fabrication tolerances, which can be considerable especially for micromachined sensors. The usefulness of the approach has been illustrated through two design examples including a 5th order low-pass EMΣΔM for a MEMS accelerometer, and a 6th order band-pass EMΣΔM for the sense mode of a MEMS gyroscope. In both cases the described methodology delivers near optimum system level design parameters. Compared with previously described design of EMΣΔM our methodology provides the users with greater confidence that the final design solution is near optimum and robust, ensuring stability in the presence of fabrication tolerances.

The authors would like to thank ICUK for partly funding this project.

Block diagram of an electro-mechanical sigma-delta modulator.

Generic process flow for the GA-based design algorithm.

Simulink model of a 5th order EMΣΔM for a MEMS accelerometer.

Scatter plot for the EMΣΔM gain constants (k1–k2 and kf1–kf2) from the entire GA set of individuals including unfeasible designs.

Scatter plot for the EMΣΔM gain constants (k1–k2 and kf1–kf2) of the 40 individuals remaining after thinning and filtering.

Power spectral density of the individual chosen as final solution.

Simulink model of a six order continuous time, band-pass EMΣΔM for the sense mode of a MEMS gyroscope.

Power spectral density of the final gyroscope solution.

MEMS Accelerometer Parameters.

Mass [kg] | 1.7e−6 |

Damping coefficient [N/ms] | 3.5e−4 |

Spring constant [N/m] | 5.5 |

Nominal capacitance [pF] | 5.5 |

Nominal electrode gap [um] | 6 |

Bandwidth [kHz] | 1 |

Max. acceleration [G] | +/−2.5 G |

Design Parameters for the Genetic Algorithm.

Boost gain kbst [V/V] | 20–400 |

Minor feedback loop gain kf1 [V/V] | 0.1–2 |

Minor feedback loop gain kf2 [V/V] | 0.1–2 |

Minor feedback loop gain kf3 [V/V] | 0.1–2 |

Integrator gain k1 [V/V] | 0.1–2 |

Integrator gain k2 [V/V] | 0.1–2 |

Integrator gain k3 [V/V] | 0.1–2 |

Feedback voltage [V] | 10–30 |

Compensator zero frequency [kHz] | 0.5–50 |

Compensator pole frequency [kHz] | 10–1,000 |

Example Individuals in the Evolutionary Process.

| |||||||||||||
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50/1 | 234.49 | 0.96328 | 0.97271 | 1.13 | 1.97 | 1.21 | 0.70887 | 14.42 | 9,160 | 200,632 | −19.21 | 326.20 | |

51/1 | 94.01 | 0.85257 | 1.91 | 1.33 | 1.1 | 0.68428 | 1.17 | 25.96 | 8,819 | 163,056 | −19.11 | 920.04 | |

52/1 | 155.32 | 1.37 | 1.8 | 0.43793 | 1.22 | 1.23 | 1.47 | 22.28 | 27,150 | 139,900 | −22.79 | 846.42 | |

53/1 | 96.58 | 1.84 | 1.52 | 0.71009 | 0.57052 | 1.19 | 0.52997 | 25.5 | 46,26 | 391,552 | −25.01 | 852.42 | |

54/1 | 109.55 | 1.76 | 1.88 | 0.33214 | 0.72423 | 1.86 | 1.21 | 16.43 | 10,567 | 497,281 | −22.71 | 1,030.00 | |

55/1 | 195.72 | 0.4096 | 0.69417 | 0.17444 | 0.90656 | 1.49 | 1.63 | 13.92 | 24,260 | 230,138 | −20.5 | 856.28 | |

56/1 | 109.55 | 1.76 | 1.88 | 0.33214 | 0.72423 | 1.86 | 1.21 | 16.43 | 10,567 | 497,281 | −22.71 | 1,030.00 | |

57/1 | 51.13 | 0.54297 | 1.93 | 0.86304 | 1.28 | 1.08 | 1.64 | 15.49 | 1,664 | 98,432 | −30.21 | 234.18 | |

58/1 | 114.42 | 1.94 | 1.15 | 1.79 | 0.74525 | 1.77 | 1.72 | 25.95 | 11,089 | 503,748 | −22.29 | 451.83 | |

59/1 | 173.05 | 1.61 | 0.42045 | 1.75 | 1.37 | 1.81 | 0.99297 | 12.17 | 9,593 | 860,350 | −26.48 | 634.67 | |

| |||||||||||||

50/8 | 189.17 | 0.87123 | 1.61 | 0.28068 | 0.7491 | 0.65026 | 1.31 | 20.68 | 2,301 | 130,096 | −29.06 | 321.07 | |

51/8 | 183.63 | 1.07 | 1.36 | 1.32 | 0.96871 | 0.70766 | 0.96289 | 23.03 | 4,139 | 51,573 | −20.11 | 66.59 | |

52/8 | 205.03 | 1.15 | 1.2 | 1.75 | 0.36596 | 0.78318 | 0.87712 | 23.05 | 5,466 | 96,872 | 104.63 | 0.0342 | |

53/8 | 205.03 | 1.15 | 1.2 | 1.75 | 0.36596 | 0.78318 | 0.87712 | 23.05 | 5,466 | 96,872 | 104.63 | 0.0342 | |

54/8 | 205.03 | 1.15 | 1.2 | 1.75 | 0.36596 | 0.78318 | 0.87712 | 23.05 | 5,466 | 96,872 | 104.63 | 0.0342 | |

55/8 | 185.93 | 0.56781 | 0.72001 | 1.92 | 0.64756 | 0.8026 | 1.16 | 13.05 | 13,159 | 412,664 | −16 | 82.72 | |

56/8 | 22.41 | 0.59207 | 0.20479 | 1.59 | 0.25323 | 1.62 | 1.47 | 10.09 | 5,441 | 242,054 | −23.79 | 11.16 | |

57/8 | 27.41 | 0.7697 | 0.47903 | 1.58 | 0.32926 | 0.4721 | 1.44 | 13.98 | 5,234 | 251,351 | 60.91 | 0.8161 | |

58/8 | 92.89 | 0.8883 | 1.49 | 1.7 | 0.58791 | 0.61525 | 0.67127 | 18.02 | 4,174 | 309,443 | −1.46 | 193.27 | |

59/8 | 148.94 | 0.9982 | 1.53 | 0.55979 | 1.66 | 0.57027 | 1.41 | 20.18 | 4,478 | 50,853 | −16.32 | 63.21 | |

| |||||||||||||

50/15 | 205.03 | 1.15 | 1.2 | 1.75 | 0.36596 | 0.78318 | 0.87712 | 23.05 | 4,442 | 78,727 | 109.24 | 0.03086 | |

51/15 | 205.03 | 1.15 | 1.2 | 1.75 | 0.36596 | 0.78318 | 0.87712 | 23.05 | 4,423 | 78,390 | 109.21 | 0.03096 | |

52/15 | 205.03 | 1.15 | 1.2 | 1.75 | 0.36596 | 0.9726 | 0.87712 | 23.05 | 4,508 | 79,906 | −16.5 | 57.76 | |

53/15 | 203.02 | 1.13 | 1.2 | 1.75 | 0.37593 | 0.77241 | 0.87096 | 23.05 | 4,404 | 78,053 | 109.15 | 0.03076 | |

54/15 | 203.02 | 1.13 | 1.2 | 1.75 | 0.37593 | 0.77241 | 0.87096 | 23.05 | 4,551 | 80,664 | 108.15 | 0.03076 | |

55/15 | 203.02 | 1.13 | 1.2 | 1.75 | 0.37593 | 0.77241 | 0.87096 | 23.05 | 4,551 | 80,664 | 108.15 | 0.03076 | |

56/15 | 203.78 | 1.14 | 1.2 | 1.75 | 0.3703 | 0.78191 | 0.87604 | 23.05 | 4,162 | 73,759 | 107.81 | 0.03092 | |

57/15 | 203.78 | 1.14 | 1.2 | 1.75 | 0.3703 | 0.78191 | 0.87604 | 23.05 | 4,162 | 73,759 | 107.81 | 0.03092 | |

58/15 | 205.03 | 1.15 | 1.2 | 1.75 | 0.36596 | 0.78318 | 0.87477 | 23.05 | 5,252 | 93,083 | 107.18 | 0.03228 | |

59/15 | 205.03 | 1.15 | 1.2 | 1.75 | 0.36596 | 0.78318 | 0.87712 | 23.05 | 4,420 | 78,348 | 109.19 | 0.03096 |

Final Design Parameters.

Boost gain kbst [V/V] | 204.92 |

Minor feedback loop gain kf1 [V/V] | 1.14 |

Minor feedback loop gain kf2 [V/V] | 1.2 |

Minor feedback loop gain kf3 [V/V] | 1.75 |

Integrator gain k1 [V/V] | 0.37 |

Integrator gain k2 [V/V] | 0.78 |

Integrator gain k3 [V/V] | 0.87 |

Feedback voltage [V] | 23.05 |

Compensator zero frequency [kHz] | 4.413 |

Compensator pole frequency [kHz] | 78.22 |

Gyroscope and ΣΔM Parameters.

Mass of proof mass [kg] | 2e−6 | 2e−6 |

Mechanical spring constant [N/m] | 1,268 | 1,328 |

Resonant frequency [Hz] | 4,027 | 4,073 |

Quality factor | 216 | 85 |

Pick-off gain [V/m] | - | 1e6 |

Sampling frequency [Hz] | - | 32,768 |

Oversampling ratio | - | 256 |

Frequency of input angular rate [Hz] | - | 32 |

Max. input angular rate [°/s] | - | 200 |

Design Parameters for the Genetic Algorithm.

Boost gain, kbst [V/V] | 834.08 |

Minor feedback loop gain kf1 [V/V] | 2.38 |

Minor feedback loop gain kf2 [V/V] | 0.819 |

Minor feedback loop gain kf3 [V/V] | 3.45 |

Minor feedback loop gain kf4 [V/V] | 1.37 |

Feedback voltage [V] | 11.61 |

Compensator zero frequency [Hz] | 769 |

Compensator pole frequency [Hz] | 29,970 |