Abstract
The performance of the conventional SigmaDelta modulator can be improved by cascading electrical integrator behind the mechanical sensing element. In this paper, a novel SigmaDelta modulator with a fractionalorder $P{I}^{\lambda}{D}^{\mu}$ controller is proposed and thus extending the traditional SigmaDelta modulator to a new fractionalorder in the SigmaDelta ADC design field. The fractionalorder $P{I}^{\lambda}{D}^{\mu}$ controller allows fine tuning accuracy in seeking the coefficients for the proposed SigmaDelta modulator and provides the robust stability in the placement of poles and zeros for compensating the phase delays. The parameters of the proposed fractionalorder SigmaDelta modulator is designed by using swarm intelligent algorithm, which offers opportunity to simplify the process of tuning parameters and further improve the noise performance. Simulation from the proposed fractionalorder SigmaDelta modulator, results of SNR > 134 dB and the noise floor under –170 dB are obtained in frequency range of [5150 Hz].
1. Introduction
Embedding a micromachined sensing element in a closed loop system is a technique, which is commonly used to achieve high performance MEMS (microelectromechanical 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. MEMS sensors employing a capacitive sensing element incorporated in SigmaDelta modulator control systems with electrostatic feedback have gained popularity in the past. The structure of SigmaDelta modulator composes of the mechanical sensitivity unit and the poststage detection circuit in the MEMS sensor system, which is able to directly convert acceleration to a digital signal and avoid the potential electrostatic instability. The feedback structure of SigmaDelta modulator effectively suppresses the quantization noise in low frequency region and emphasizes the high frequency quantization noise which can be simply removed by low pass filter. In recent years, in order to improve the performance, capacity MEMS sensor usually takes advantages of the single loop, high order SigmaDelta modulator [1]. The designing of the high order SigmaDelta modulator for highperformance MEMS sensor has been investigated comprehensively in [25]. At present, to the author’s knowledge, SigmaDelta modulator all focused on employing additional electronic integrator in series with the mechanical sensing element to form the high order SigmaDelta modulator [2, 3]. The stability, however, is worsen as the system order increasing, and the parameters design is much more complicated. As discussed, designing a stable and robust sensor, with high performance, is a challenging problem and is much desired to be researched.
Fractionalorder calculus belongs to the branch of mathematics, which is concerned with differentiations and integrations of nonintegerorder [6]. According to ref. [7], The remarkable advantage of fractionalorder integrator over its counterpart integerorder one is that the stability and robustness of the fractionalorder integrator is much stronger. As we all know that SigmaDelta modulators that applied so far were all considered as integerorder modulators. Whereas, the proposed SigmaDelta modulator with the fractionalorder digital loop integrator combines some characteristics of systems between the order $N$ and ($N+1$). Therefore, we will have more possibilities for an adjustment of the poles or zeros of the noiseshaping integrator according to special requirements through changing the system order as a real (not only integer) value. In this paper, a 3rd order digital loop integrator is employed to perform noiseshaping of quantization noise from the 1bit comparator to improve $SNR$, and furthermore a $P{I}^{\lambda}{D}^{\mu}$ controller is inserted immediately afterwards to provide the weak or strong fractionaltype poles, or zeros to improve the robust stability for the proposed fractionalorder SigmaDelta modulator. The parameters of the proposed 3rd order digital loop integrator and the $P{I}^{\lambda}{D}^{\mu}$ controller are optimized by using PSO algorithm, and the optimization details will be shown in the following parts.
2. Mathematical model of the proposed fractionalorder SigmaDelta modulator
The system block diagram of the proposed fractionalorder SigmaDelta modulator is shown in Fig. 1.
Fig. 1Block diagram of the proposed SigmaDelta modulator system
The loop is consisted of an electromechanical sensing element, a charge amplifier, a digital loop filter, a fractionalorder $P{I}^{\lambda}{D}^{\mu}$ controller and a 1bit quantizer. In Fig. 1, ${k}_{0}$ is the gain of the charge amplifier, ${H}_{1}\left(z\right)$ is the digital loop integrator, ${H}_{2}\left(z\right)$ is the fractionalorder controller, ${k}_{1}$ is the 1bit quantizer gain, ${Q}_{1}$ is the quantization noise of 1bit quantizer, and ${k}_{2}$ is the equivalent linear model of 1bit DAC feedback. The sensing element can be modeled as a massspringdamper system, a 2nd order dynamics with transfer function $M\left(s\right)$ that can be expressed as Eq. (1):
where $m$ is the proof mass, $b$ is the damping coefficient and $k$ is the spring constant.
In Fig. 1, ${H}_{1}\left(z\right)$ is the 3rd order digital loop integrator which is expressed as:
where ${a}_{3}$, ${a}_{2}$, ${a}_{1}$, ${a}_{0}$, ${b}_{3}$, ${b}_{2}$, ${b}_{1}$, ${b}_{0}$ are the coefficients of ${H}_{1}\left(z\right)$. ${H}_{2}\left(z\right)$ is the $P{I}^{\lambda}{D}^{\mu}$ controller, the transfer function of the $P{I}^{\lambda}{D}^{\mu}$ controller is expressed as:
where $\lambda $, $\mu $ are the integral order, differential order of the $P{I}^{\lambda}{D}^{\mu}$ controller, respectively.
To achieve high SNR and robust stability of the overall system, ${a}_{3}$, ${a}_{2}$, ${a}_{1}$, ${a}_{0}$, ${b}_{3}$, ${b}_{2}$, ${b}_{1}$, ${b}_{0}$ in Eq. (2) and ${K}_{p}$, ${K}_{i}$, ${K}_{d}$ in Eq. (3) are sensitively optimized by using PSO algorithm, respectively.
3. Stability analysis for fractionalorder SigmaDelta modulator system
In this paper, the proposed fractionalorder SigmaDelta modulator is described as a FOLTI (fractionalorder linear time invariant) system which generally expressed as the following form:
where ${a}_{k}\mathrm{}(k=0,\cdot \cdot \cdot n)$, ${b}_{k}\mathrm{}(k=0\cdot \cdot \cdot m)$are constants, $\alpha $ is the fractional commensurate order and $\alpha <1$. Eq. (4) can be rewritten as:
As shown in ref [8], when matrix $A$ is deterministic without uncertainty, the stability condition for Eq. (5) is clearly expressed by $\underset{i}{\mathrm{m}\mathrm{i}\mathrm{n}}\left\mathrm{a}\mathrm{r}\mathrm{g}\left({\lambda}_{i}\right(A)\right>\alpha \pi /2$, $i=\mathrm{1,2},\dots ,N$. In this paper, $\lambda $, $\mu $ are both set as 0.5 to inspect the effects of the halforder integrator. When $\lambda =\mu =\text{0.5}$, the fractional commensurate order of the proposed fractionalorder SigmaDelta modulator is $\alpha =\text{0.5}$. Therefore, the stability domain of the proposed fractionalorder SigmaDelta modulator is $\left\mathrm{a}\mathrm{r}\mathrm{g}\left({\lambda}_{i}\right(A)\right>\pi /4$, $i=\mathrm{1,2},\dots ,N$ as shown in Fig. 2.
Fig. 2Stable domain of the proposed fractional–order SigmaDelta modulator
It can be seen from Fig. 2, the stability domain of the proposed fractionalorder SigmaDelta modulator is wider than that of the traditional integerorder one which stability is only in the lefthalf $s$plane. Therefore, the gain margin range of the proposed fractionalorder SigmaDelta modulator is broader.
4. PSO Algorithm for fractionalorder SigmaDelta modulator
In this paper, the Simulink model of the proposed fractionalorder SigmaDelta modulator is developed shown in Fig. 3.
Fig. 3Simulink model of fractionalorder SigmaDelta modulator
In this paper, we consider ${X}_{i}=({a}_{2},{a}_{1},{a}_{0},{b}_{2},{b}_{1},{b}_{0},{k}_{p},{k}_{i},{k}_{d})$ as the position vector of PSO algorithm for the proposed fractionalorder SigmaDelta modulator. The population size of the PSO algorithm is set as 40, and the individual numbers are 1000 corresponding to the dimension 9, and the iterations is 30, $\omega =\text{0.9,}\text{}{c}_{1}={c}_{2}=\text{2}$. A typical objective for MEMS sensor is high $SNR$, which can be maximized and is calculated based on the power spectral density of the output bitstream. The oversampling ratio ($OSR$) needs to be specified in the proposed fractionalorder SigmaDelta modulator system. Here, we choose sample frequency is 128 kHz, and $OSR=$64.
The best ${X}_{i}$ is obtained by running the PSO algorithm, and the proposed fractionalorder SigmaDelta modulator achieved about 105 dB to 135 dB of $SNR$ by yielding the different values for ${X}_{i}$ during the optimization process. The corresponding $SNR$ of the proposed fractionalorder SigmaDelta modulator is 134.011 dB which is 10X improved comparing to only 5th order SigmaDelta modulator. The PSD plot of the fractionalorder SigmaDelta modulator is built with selected coefficients and shown in Fig. 4. Comparing to the 5th order SigmaDelta modulator, the proposed fractionalorder SigmaDelta modulator not only performs better $SNR$, but also the wider noise floor (Bandwidth: 110 Hz in the proposed fractionalorder SigmaDelta modulator, 102 Hz in the 5th order SigmaDelta modulator) and the sharper slope (Amplitude gain: 71.2 dB/decade in the proposed fractionalorder SigmaDelta modulator, 60.4 dB/decade in the 5th order SigmaDelta modulator). Fig. 5 is the plot of the measured $SNR$ versus the input signal power, the two fitted curves are extended by the test data to get the estimated DR of the proposed fractionalorder and the 5th order SigmaDelta modulator.
Fig. 4PSD of the fractionalorder and the integerorder SigmaDelta modulator
Fig. 5Measured SNR versus input signal power
5. Robust stability analysis
The root locus of the proposed fractionalorder and the 5th order SigmaDelta modulator are built and shown in Fig. 6.
In Fig. 6(b), the Root locus of the 5th order SigmaDelta modulator pass through the right half $s$plane, whereas, the poles and zeros of the fractionalorder SigmaDelta modulator, as shown in Fig. 6(a), are allocated along the negative axis of the $s$plane and the farthest pole achieves –1.2×10^{7} which induce the root locus leave the right half $s$plane. This leads to better stability of the proposed fractionalorder SigmaDelta modulator to the continuous domain frequency response.
Fig. 6Root locus comparison
a) The proposed fractionalorder SigmaDelta modulator
b) 5th order SigmaDelta modulator
Similar results are also observed at system response for parametric yield errors, for instance, taking different spring constants ${k}_{s1}=$ 50 N/m, ${k}_{s2}=$ 100 N/m, ${k}_{s3}=$ 150 N/m in MEMS transfer function to verify the robustness of the proposed fractionalorder SigmaDelta modulator, Fig. 7 shows the PSD plot of the proposed fractionalorder SigmaDelta modulator with different spring constant ${k}_{s1}$, ${k}_{s2}$, ${k}_{s3}$.
Fig. 7PSD of the fractionalorder SigmaDelta modulator with different spring constant k
Fig. 8PSD of the proposed fractionalorder SigmaDelta modulator with different Damping coefficient b
Also, taking ${b}_{1}=$ 1.8×10^{3}^{}Ns/m, ${b}_{2}=$ 2.4×10^{3}^{}Ns/m, ${b}_{3}=$ 3.2×10^{3}^{}Ns/m in MEMS transfer function, Fig. 8 shows the PSD plot of the proposed SigmaDelta modulator with different damping coefficient $b$.
Simulation results show that $SNR$ and the Noise floor of the three sensing elements with different $k$ and three sensing elements with different $b$ only present an acceptable slight fluctuation, which indicate that the proposed fractionalorder SigmaDelta modulator also has strong robustness against to the sensitivity of MEMS devices. The response for the sensitivity of MEMS devices is very important to close the loop up in the real systematical construction.
6. Conclusions
A fractionalorder SigmaDelta modulator structure is proposed in this paper. A fractionalorder $P{I}^{\lambda}{D}^{\mu}$ controller is cascaded behind the integerorder modulator to adjust the stability and SNR in fine way in SigmaDelta modulator system. The orders of $P{I}^{\lambda}{D}^{\mu}$controller is both considered one value $\lambda =\mu =$ 0.5 as commensurate order. The proposed fractionalorder SigmaDelta modulator achieved $SNR=$134.011 dB in simulation and noise floor under –170 dB in frequency range of [5150 Hz]. The simulated root locus shows the improved stability comparing to the pure integerorder system with good potential to progress further. This study can promote the development of high performance of MEMS sensor, also provide a scientific and technical supports for the application of fractionalorder theory to practical system.
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