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3.1 Introduction

Sigma-delta (also denoted as \(\Sigma\Delta\)) modulators are a class of data converters that have been in use for several decades [1]. Their main advantages are simplicity and robustness to component mismatch. The combination of the principles of oversampling, noise shaping and decimation allows this type of circuit to reach higher resolutions than in the case of other data converters, without the use of high-precision components. They can also be embedded in the control loop of sensors or actuators. In this way, two different goals are achieved: first, the closed-loop control of a given variable, and second, an implicit analogue-to-digital conversion of a given magnitude.

Sigma-delta modulators convert a time-sampled analogue input signal to a stream of bits (see the generic scheme of a \(\Sigma\Delta\) modulator in Fig. 3.1a). The quantised output stream must subsequently be filtered in order to achieve a good representation of the input signal. Therefore, it is necessary that the quantisation noise lie predominantly outside the signal band for the successful operation of such a modulator [2]. Sigma-delta modulators implement noise shaping by minimising an error between the input signal x and the feedback loop signal y. The difference between the two signals is passed to the loop filter. If the difference falls in the signal band, it passes to the output without attenuation. On the contrary, a difference that is out of the signal band attenuated by the filter. The signal from the filter is passed to the quantiser, which generates the next output value y. This output values is used in the next comparison step. The result of this strategy is a close match of input signal and quantised output in the pass-band of the filter and shaping of the quantisation noise outside the signal band.

Fig. 3.1
figure 1

a Block diagram of a generic sigma-delta modulator consisting of a filter and a quantizer (a comaprator). b Block diagram of a simplest first-order sigma-delta modulator. c Block diagram of a MEMS sigma-delta architecture

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There are a large number of different circuit topologies of sigma-delta modulators that include various loop filters or multiple feedback loop [2]. Continuous-time sigma-delta modulators that have analogue (continuous-time) elements in the feedback loop are also possible. However, the simplest topology is a first-order sigma-delta modulator shown in Fig. 3.1b, where the role of the loop filter is taken by a discrete integrator. Remarkably, the evolution of this feedback system is described by a simple discrete-time equation

$$u_{n+1}=u_{n}+x-\text{sgn} (u_n)$$
(3.1)

that is classified as a piecewise-smooth discontinuous map. Piecewise-smooth maps model many physical systems, including switching circuits and systems with oscillatory behavior [3–6]. In 3.1, x is the magnitude to convert, u n is the value that the integrator takes at the time \(t={\it nT}_s\) and \(\text{sgn}(x)\) is the signum function. It is assumed that the input x is constant or that its bandwidth is well below the sampling frequency. It can be shown that the average value of the output bitstream b n is x. To obtain the digital conversion of x, this bitstream is fed into a low-pass filter and decimated. We note here that in the analysis of \(\Sigma\Delta\) modulators, the input x is often considered as constant. Since standard converters operate at a rate many times greater than the highest frequency of input signal, the input to the converter can be indeed considered as ‘quasi'-constant. A remarkable property of sigma-delta modulators is that the output bits subsequently averaged retrieve a good approximation to the input. With ideal integrators and constant input, the output of the system when averaged over a ‘long enough’ time period equals exactly the input.

The field of the applications of \(\Sigma\Delta\) modulators has significantly expanded during recent decades. Closed loop architectures employing the elements of sigma-delta modulations have been introduced to the area of microelectromechanical systems. In such an architecture, a microelectromechanical system serving as a physical sensing element replaces the loop filter from Fig. 3.1a. Alternatively, we can say that a \(\Sigma\Delta\)-like structure serves as a feedback loop for the sensor. These topologies have been proposed, designed and implemented as an essential part of inertial sensors [7–11], gyroscopes [12], resonant sensors [13–15], air flow sensors [16–19] and capacitive MEMS [20, 21]. In all these applications, physical sensing elements are embedded in the control loop of sensors or actuators to keep constant, for example, the temperature of a component, or the position of an actuator. Among many advantages of this architecture, closed loop sensors which incorporate analogue-to-digital conversion within the loop produce a digital signal in the output.

Figure 3.1c represents the generalised idea of the systems that we will study in this chapter. It describes the basic principle of operation of a MEMS system that employs the feedback loop similar to those of sigma-delta modulators. Please note here that a MEMS is mechanical system, and as a result of applied forces, the input F in and/or the feedback \(F_{\it fb}\), it changes its position x and velocity v. A conversion to the electrical domain is required for x and v and it is carried out by a sensing mechanism/circuit [22]. The sensing of x and v and representing them as an electrical signal before feeding them to the comparator is shown by the corresponding block (\(x\to V\)). The reverse conversion of the resulting actuation voltage to the mechanical domain in the form of a force is carried out by the actuation mechanism or transducer and is shown as the corresponding block after the comparator (\(V \to x\)). In this chapter, we will consider a range of MEMS devices serving different aims and representing different MEMS applications. Their 1D topologies are shown in Fig. 3.2, and the mechanical components can deflect as shown in the figure if an actuation voltage and/or other mechanical force is applied. The detailed description of each topology will be given in later sections.

For these feedback topologies, it has been shown that periodic sequences (cycles) appear at the output of the comparator, and this is an intrinsic property of the system observed in all types of the system—conventional sigma-delta modulators and sigma-delta MEMS [13, 23–28]. In digital accelerometers, these cycles may be utilised for self-calibration purposes in the system without input, since the MEMS parameters may be extracted from their characteristics [26]. In resonant gravimetric sensors, the frequency of oscillations of the mechanical structure can be extracted from this cycle at the output [29].

Fig. 3.2
figure 2

Simple 1D models of MEMS mechanical structures from this study. a \(\Sigma\Delta\) accelerometer has a movable mass suspended between two electrodes. Voltage is applied to either top or bottom electrode depending on the position of the movable mass. b Pulsed digital oscillator uses a MEMS resonators that can be modelled as a mass-spring-damper system. The actuation force is applied to the resonator depending its position and causes its displacement up or down. c Electrostatic pulsed digital oscillator is similar to the conventional pulsed digital oscillator. The actuation force is created by applying a voltage to a fixed bottom electrode while the movable mass is made conducting. d Capacitive MEMS varactor/switch is a suspended electrode that can move and therefore change the capacitance of the device if a voltage applied to it or to the fixed electrode. Every devices is characterised by its mass m, spring constant k and damping coefficient b

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Our aim is to derive models in the form of a map (discrete-time system) in order to study this architecture as a dynamical system. We highlight here that due to the use of a 1-bit quantizer (a comparator) in the feedback loop, the resulting maps have the form of piecewise discontinuous maps. This approach explains the behaviour of the system from the standpoint of nonlinear dynamics and allows us not only to qualitatively explain the appearance of periodic cycles of the map but also to study the behaviour of the system over a wide range of control parameters. In particular, we introduce the plane spanned by parameters of the system where we define regions of admissibility of cycles. We outline similarities which arise between conventional \(\Sigma\Delta\) [24] modulation and MEMS \(\Sigma\Delta\) topologies.

We refer a reader who is interested in microtechnology and in practical aspects of MEMS to the books and articles cited in this chapter. This chapter discusses the statements of the problem that lead to piecewise-smooth discontinuous discrete-time systems and the most interesting results.

3.2 \(\Sigma\Delta\) Inertial MEMS Sensors

With the advance of microelectromechanical systems technologies, force-balanced electromechanical modulation incorporating micromechanical transducers has been employed in a number of inertial and force sensing applications, including accelerometers [6–11], gyroscopes [12] and pressure sensors [13]. Closed-loop feedback for such applications makes the sensor characteristics insensitive to mechanical properties of micromachined structures, which are often nonlinear and subject to significant variations, thereby improving the sensor scale factor accuracy.

A typical MEMS sigma-delta based accelerometer is shown in Fig. 3.1b, and many MEMS sensor systems are based on variations of this architecture. The MEMS responds to an external force (acceleration) F in by acquiring a displacement. The displacement is measured and, depending whether it is below or above the rest position, an appropriate force is applied to compensate this displacement. Therefore, the micromechanical structure oscillates around its rest position. A remarkable property of this system is that the displacement of the structure and the input can be calculated simply by processing the sequence of bits b n at the output of the comparator. In this section, we describe a MEMS accelerometer topology with the first order sigma-delta feedback loop (Fig. 3.1b), since it is an essential part of various inertial sensors, for example, digital accelerometers [7–9, 23, 25, 30, 31].

3.2.1 Statement of the Problem

The block diagram in Fig. 3.1b shows a MEMS embedded into a feedback loop. The MEMS represents a movable conducting plate/mass suspended between two electrodes. A simple 1D model of the MEMS mechanical structure is shown in Fig. 3.2a. When a voltage is applied to one of the electrodes, there is an attracting electrostatic force between the movable plate and this electrode. If a voltage is applied to the ‘bottom’ electrode the plate will move down, while if a voltage is applied to the ‘top’ electrode the plate will move up. The feedback loop processes the current position of the MEMS at every sampling time \({\it nT}_s\), and if this position is below zero (the decision bit from the comparator is \(b_n = -1\)), the voltage is applied to the top electrode (we symbolically denote this event as the application of \(+V\)) to compensate this position displacement, and vice versa if this position is above zero (the decision bit from the comparator is \(b_n = +1\)), the voltage is applied to the bottom electrode (-V). Thus, the system oscillates around its rest position. A typical waveform of the feedback voltage/force is shown in Fig. 3.3a where during the time interval from \({\it nT}_s\) to \((n+1)T_s\) a constant voltage depending on the position of movable mass is applied to it.

Fig. 3.3
figure 3

In all examples from this chapter, the actuation force/voltage depends on the state of the MEMS mechanical structure in a similar way. However, the actuation force waveform will be specific for each example. a Voltage waveform for the actuation of a \(\Sigma\Delta\) accelerometer. The ‘polarity’ of the applied voltage is defined by the sampled position of the mechanical structure at a sampling time \((n-D)T_s\), and a constant voltage is turn on for the entire interval \({\it nT}_s<t<(n+1)T_s\). The similar actuation waveform is used for capacitive MEMS to control dielectric charging. b Force waveform of the pulsed digital oscillator (PDO) consists of a train of very short (delta) pulses whose polarity is defined by the sampled position of the MEMS resonator. c Voltage waveform of the electrostatic pulsed digital oscillator. The oscillator is constantly biased. When a ‘negative’ pulse must be applied to the resonator, the bias voltage is turned off for time interval dt and then turned on for the rest of the sampling interval time (‘anti-pulse’). When a ‘positive’ pulse must be applied to the resonator, the bias voltage does not change

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The displacement of the mechanical structure (in this case, a micromechanical resonator) from Fig. 3.1b as a function of time \(\xi(t)\) is described by the well-known mass-spring-damper equation

$$m\ddot{\xi}(t) + b \dot{\xi}(t) + k \xi(t) = F_{in}(t) + F_{\it fb}(t)$$
(3.2)

where m is the mass of the plate, b is the damping factor and k is the effective spring constant. The net force F(t) consists of the feedback \(F_{\it fb}(t)\) and the input \(F_{in}(t)\). Here, we assume that the feedback actuation takes place through electrostatic force, which is the case for many systems. For Eq. (3.2), \(F_{\it fb}(t)\) is written in the following form:

$$\begin{aligned}F_{\it fb}(t)= \left\{\begin{array}{@{}l} - \varepsilon_{0} {\it AV}_{0}^{2}/\!\!\left(2[g +\xi (t)]^2\right) \text{for} \;{\it nT}_{s} <t < (n+1) T_{s} \;\text{if} \;\xi((n-D)T_{s})>0\,\\ + \varepsilon_0 {\it AV}_0^2/\!\!\left(2[g -\xi (t)]^2\right) \text{for} \;\;{\it nT}_s <t < (n+1) T_s \;\text{if} \;\xi((n-D)T_s)<0\end{array}\right.\end{aligned}$$
(3.3)

where ϵ0 is the permittivity in vacuum, A is the electrode area, V 0 is the control voltage and D is a delay in the feedback loop (\(D\geq 1\)). As shown in [32, 33], the number of delays is an effective control parameter that affects the dynamics of this system, and therefore we give here a generalised model accounting for an arbitrary number of delays D.

The nature of the electrostatic force is nonlinear, though it is usual to linearise it by dropping off the small displacement \(\xi(t)\) compared to the equilibrium gap g. As an alternative, expanding the nonlinear part of the electrostatic force, one can use the following representation for linearised \(F_{\it fb,lin}(t)\):

$$F_{fb,lin}(t) \approx -\text{sgn}(\xi((n-D)T_s)F_0 \, \Pi(t)+ 2F_0\xi(t)/g$$
(3.4)

which substituted in (3.2), produces a shift in the natural frequency \(\omega = \sqrt{k/m}\). Thus, the new frequency now is \(\omega_0 = 2\pi f_0 = \omega \sqrt{1-2 F_0/g\omega^2}\) where \(F_0 = \varepsilon_0 A V_0^2/2g^2\). The linearised feedback force, therefore, can be simply written as \(F_{fb, lin}(t) = -\text{sgn}(\xi(t_{n-D}))F_0 \Pi(t)\). The symbol \(\Pi (t)\) denotes a square pulse of unit magnitude and length T s .

Let us introduce the following variables: time \(\tau = \omega_0 t\), displacement \(x = m \omega_0^2 \xi /F_0\), dissipation \(\beta = b/(2m\omega_0)\) and normalised input \(a = F_{in}/F_0\). We will consider in this chapter that a is a constant. Indeed, since the natural and sampling frequencies are high, we may assume a time-varying input to be constant over relatively large sampling events.

Thus, we obtain the following normalised equation instead of (3.2):

$$x^{\prime\prime} + 2\beta{x}^\prime + x = a - \text{sgn}(x_{n-D})\Pi(t)\,$$
(3.5)

where now \({x}^\prime\) represents the derivative with respect to dimensionless time τ. In the right-hand side of equation, the term x n-D , the position at the time instance \((n-D)T_s\), is present, reflecting the presence of multiple delays in the feedback loop.

The solution of a mass spring damper equation with the left part as in (3.5) and, in the most general case, with some F(t) in the right part consists of two terms, namely, the decaying free oscillations \(x_1(t)\) and the forced ones \(x_2(t)\). While the expression for \(x_1 (\tau)\) is well-known, we note that for arbitrary F(t) the forced oscillations may be found as follows:

$$x_2(\tau) = \int\limits^{\tau}_{0} \frac{F(t)e^{-\beta(\tau-t)}}{\sqrt{1-\beta^2}} \sin(\sqrt{1-\beta^2}(\tau - t)) dt\,$$
(3.6)

where the integral can be easily solved in the interval \([0, \tau]\) for certain functions F(t), for instance, if it is a constant F 0. Thus, solving (3.6) for \(F(t)=F_0\) and introducing the new variable \(y = -(\beta x/\sqrt{1-\beta^2}) - v/\sqrt{1-\beta^2}\) (where \(v=\dot{x}\) is the velocity), one obtains

$$\begin{array}{@{}l@{}} x_{1}(\tau) = e^{-\beta\tau}\left( x_0 \cos (\sqrt{1-\beta^2}\tau)- y_0\sin (\sqrt{1-\beta^2}\tau)\right) \\y_{1}(\tau) = e^{-\beta\tau}\left( y_0 \cos (\sqrt{1-\beta^2}\tau) + x_0\sin (\sqrt{1-\beta^2}\tau)\right)\\x_{2}(\tau) = F_0 \left(1 - e^{-\beta \tau} \cos (\sqrt{1-\beta^2}\tau)- \frac{\beta e^{-\beta \tau}}{\sqrt{1-\beta^2}}\sin(\sqrt{1-\beta^2}\tau)\!\!\right)= F_0\,\zeta (\beta,\tau)\\y_{2}(\tau) = F_0 \left(\!\!\frac{-\beta}{\sqrt{1-\beta^2}} + \frac{\beta e^{-\beta \tau}}{\sqrt{1-\beta^2}} \cos (\sqrt{1-\beta^2}\tau) - e^{-\beta \tau}\sin(\sqrt{1-\beta^2}\tau)\!\!\right) = F_0\,\eta(\beta,\tau)\end{array}$$
(3.7)

where ζ and η are defined as the expressions in the brackets in the latter two equations. Knowing the solutions, it is easy now to obtain an iterative map from (3.7) by simply assuming that \(x_{n} = x(n\tau_s)\) and \(y_n = y(n\tau_s)\) where \(\tau_s = \omega_0 T_s\) and \(\tau_n = n \tau_s\).

Introducing the feedback force in the form

$$F_n = a - \text{sgn}(x_{n-D})$$
(3.8)

(F n will be constant during one sampling event), we write the sampled system as the following equations [34]:

$$\begin{aligned}\left(\begin{array}{@{}c@{}c@{}} x_{n+1}\\ y_{n+1}\end{array} \right)= & \alpha \mathbf{R} (2\pi\, f \sqrt{1-\beta^2})\left(\begin{array}{@{}c@{}c@{}} x_n \\ y_n\end{array} \right)+ F_n \left(\begin{array}{@{}c@{}c@{}} \zeta (\beta, f) \\ \eta (\beta, f)\end{array} \right)\,\end{aligned}$$
(3.9)

where we introduce the parameters \(\alpha = {\rm exp}(-\beta\tau_s) = {\rm exp}(-2\pi\beta f)\) (or, returning to the original variables, \(\alpha = {\rm exp}[-b T_s/(2m)]\)) and the normalised frequency \(f = \tau_s /(2\pi)\) (\(f =f_0/f_s\)). We also used the notation \(\mathbf{R}(\alpha)\) in (3.9) to denote the rotation matrix

$$\mathbf R (\alpha) = \left(\begin{array}{@{}ll@{}} \cos \alpha & -\sin \alpha \\\sin \alpha & \cos \alpha\end{array}\right)$$
(3.10)

The terms ζ and η in (3.9) caused by the presence of F(t) in the right part of equation (3.5) are functions of the parameters α, f and τ s and not of x n and y n .

The system (3.9) is a piecewise-smooth discontinuous mapping

$$\mathbb{R}^2\times\mathbf{ B}^D \longrightarrow \mathbb{R}^2\times\mathbf{ B}^D$$
(3.11)

where \(\mathbb{R}\) is the set of real numbers and \(\mathbf B\) is a two-element set such that \(\mathbf B = \{-1, 1\}\). Map (3.9) belongs to the class of contracting mappings considered in [35]. It is important to note that it has been formally shown there that the output of contracting mapping is always a stable cycle, and therefore map (3.9) displays only stable cycles.

We can define the binary output sequence \(b_n\in \mathbf B\) as

$$b_n = \text{sgn} (x_n)$$
(3.12)

and, according to the notes made in the introduction, this sequence represents the digital output of the system.

Strictly speaking, (3.9) is written for the underdamped case when \(\beta<1\). For the overdamped case, \(\beta>1\), the expression in the root \(\sqrt{1-\beta^2}\) becomes negative and f itself complex. The map (3.9) can still be written in this form if, considering that the expression \(2\pi f\) is now complex, we recall that \(\cos (ix) = \cosh (x)\) and \(\sin (i x) = i\sinh (x)\) in the rotation matrix. Since the root \(\sqrt{1-\beta^2}\) was also used in the definition of y, the second variable must be transformed \(y^* \rightarrow -i y\). (After these changes, all variables and parameters in (3.9) will be still real).

The form of the map (3.9) resembles the map proposed for the pulsed digital oscillator (PDO) topology [14]. Due to the difference in driving of the PDO and the studied system, the iterative map for the PDO may be seen as a particular case of the map (3.9) with \(\zeta = 0\) and \(\eta = \text{const}\) studied in detail in references [14, 28, 32]. The PDO dynamics is very different from that of first-order sigma-delta, although it shares some common features such as, for example, after a nonlinear bitstream conversion (edge detection), noise shaping [29].

3.2.2 Periodic Solutions

Independently from the study [35], it is well-known that the conventional \(\Sigma\Delta\) architecture displays periodic sequences (cycles) in the output [24]. A microresonator embedded into this type of structure displays periodic behaviour as well [26]. In this section, we study periodic sequences \(\{(x_n,y_n)\}\) that are produced by the map (3.9).

For a given N-periodic sequence of signs b n

such that \(x_{n}=x_{n+N}\), the following sequence determines the N-cycle of map (3.9)

$$\begin{aligned}\left(\begin{array}{@{}c@{}c@{}} x_{n}\\ y_{n}\end{array}\right) = \left( \mathbf{I} - \alpha ^N \mathbf{R}( N 2\pi f\sqrt{1-\beta^2})\right)^{-1} \sum\limits_{j=1}^{N}\alpha^{N-j} \mathbf R((N-j)2\pi f\sqrt{1-\beta^2}) \times \nonumber\\ \times (a - \text{sgn}(x_{n-D+j-1}))\left(\begin{array}{@{}c@{}c@{}} \zeta \\ \eta\end{array}\right)\end{aligned}$$
(3.13)

A cycle given by (3.13) will be asymptotically stable. Indeed, let us consider the evolution of a small disturbance of some point \(x_n, y_n\) that belongs to the N-cycle. Let also this disturbance lies in the δ-neighbourhood of this point such that \(\delta = \text{min} |x_n|>0\) for all N. In this case, evolution after k iterations of \(\tilde{x}_n = x_n + \delta x\) and \(\tilde{y_n}=y_n + \delta y\) is defined as

$$\left(\begin{array}{@{}c@{}c@{}} \tilde{x}_{n+k}\\\tilde{y}_{n+k}\end{array}\right) = \left(\begin{array}{@{}c@{}c@{}} x_{n+k}\\y_{n+k}\end{array}\right) + \alpha^{k}\mathbf{R}(2\pi k f\sqrt{1-\beta^2}) \left(\begin{array}{@{}c@{}c@{}} \delta x\\\delta y\end{array}\right)$$
(3.14)

As is seen from this formula, the disturbed trajectory approaches the initial one since \(\alpha<1\).

To validate the existence and find the area of admissibility of a particular cycle over a range of control parameters of (3.9), we can apply the following strategy:

  • According to [35], the output of this map is always a cycle.

  • Assume the sequence b n  (3.12) for the sign of the position. For instance, studies [24, 27, 29] describe the approaches to determine sequences b n .

  • Fix the parameters of the map (namely β, f and a) and calculate \(\{(x_n, y_n)\}\) using Eq. (3.13).

  • Check if the condition (3.12) is fulfilled. It is worth noting that although the sequence (3.12) is used to generate the cycle (3.13), the resulting cycle may have a signature sequence that differs from the desired sequence. In this case we say that at this parameters, the cycle is not admissible.

Varying β and f over wider regions with implementation of the above strategy allows one to obtain the parameter plane (\(\beta,f\)) with regions of cycles admissibility—so called tongues.

3.2.3 Results

3.2.3.1 System with No Input

First, we consider the system with no input, i.e. a = 0. The reasons of this study are: (a) for certain types of inertial sensors, resonant accelerometers, a proof mass changes the strain of an attached resonator, hence changing its resonant frequency. The scheme may also be used as a part of a self-sustained oscillations system [11]. The parameter of interest in this case is the change of the frequency, and in terms of the model (3.9) we have only two parameters that entirely control the dynamics of the system—β and f; (b) the topology with no input displays periodic sequences (cycles) at the comparator output which may be used for self-testing purposes.

For a linear underdamped resonator, the output waveform in this case will be a sinusoid, and, therefore, one can obtain the sequence b n for the expression (3.13) as the sign of a sampled sinusoid

$$b_n = \text{sgn}(\cos(2\pi k M/N + \varphi_0))$$
(3.15)

where φ0 is an arbitrary phase and \(1\leq k\leq N\). The ratio \(M/N\) defines the rotation number of a cycle, i.e. the number of loops around the origin a trajectory makes in one iteration. The expression (3.15) is valid not only for the case of high-Q resonators but also for cycles in the overdamped case (with no input for the both cases). The most general discussion on admissible sequences is given in [27]. The sequence b n in the case of a linear over damped resonator can be found using the algorithm described in [24].

Fig. 3.4
figure 4

\(\Sigma\Delta\) MEMS accelerometer described by map (3.9), underdamped case. a Regions of admissibility (with overlapping) for N-cycles with \(1\leq N\leq 20\) in the parameter plane calculated from (3.13). The insert of the figure shows the form of the tongues that correspond to odd (by the example of the 3-cycle) and even (by the example of 4-cycle) cycles. b Parameter plane with tongues that correspond to different cycles obtained from numerical simulations of the map (3.9) with the zero initial conditions

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Fig. 3.5
figure 5

\(\Sigma\Delta\) MEMS accelerometer described by map (3.9), overdamped case. a Regions of admissibility (with overlapping) for the 2, 4, 8, 12, 16 and 24-cycles. b Parameter plane with tongues that correspond to different cycles obtained from numerical simulations of the map (3.9) with the zero initial conditions

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The parameter plane \((\beta,f)\) for tongues with rotation numbers \(M/N\), \(0<N<15\) is shown in Fig. 3.4 (for the underdamped case) and Fig. 3.5 (for the overdamped case). The grey areas show possible values of parameters at which a specific cycle can be observed in the system. As is seen from the figure, the tongues overlap (shown by darker gray shades): at the same β and f there coexist several cycles and which one of them will eventually be displayed by the system depends only on initial conditions. For example, the planes of parameters calculated for the zero initial conditions are shown in Fig. 3.4b and 3.5b. For the areas of tongues overlapping, one can plot basins of attraction, i.e. the areas on the plane spanned by the initial conditions x 0 and y 0 (Fig. 3.6c). In general, the picture that one observes for the underdamped case is very similar to the PDO dynamics [28, 32] in the sense that tongues and overlapping areas are very typical for the system.

As far as the tongues defined by (3.13) are concerned, firstly we note that the 2-cycle (the area shown in blue) can formally exist everywhere in the plane (it is shown by the lightest grey shade in Figs. 3.4a and 3.5a). In practice, for certain parameters it would be almost impossible to obtain it since the initial conditions demanded for it may be unrealsitic. We also note that only even cycles exist for the overdamped resonator.

Secondly, we draw the attention to the different forms of tongues that correspond to odd and even N for the underdamped case (see the insert in Fig. 3.4a where the tongues with N = 3 and N = 4 are shown). Though the odd tongues have conventional form, the even tongues are rather unusual and ‘cut off’ large areas in the plane. To prove it, we obtained an explicit form for the 3- and 4-cycles directly from (3.13) (which we do not give here due to its complex form).

The points x n as functions of the parameter f at a fixed β for the 3- and 4-cycles are shown in Fig 3.6a and b, respectively. Since \(\text{sgn}(x_n)\) must be the same as the sequence b n which generated the cycle, we highlight the interval of the f axis over which these cycles are admissible and this area is precisely the cross section of the tongues from the plane 3.4a.

Fig. 3.6
figure 6

\(\Sigma\Delta\) MEMS accelerometer described by map (3.9). a Points x n of the 3-cycle of map (3.9) as a function of the normalised frequency f. b Points x n of the 4-cycle as a function of the normalised frequency f. The 3-cycle is admissible if \(x_1>0\), \(x_2>0\) and \(x_3 <0\). The interval of f where this condition is fulfilled is highlighted by grey. This interval is larger at larger β and tends to zero at smaller β. The 4-cycle is admissible if \(x_1, x_2>0\) while \(x_3, x_4 <0\). This is the case for any \(f<0.25\) and does not depend on β. This explains why the odd cycles have the regions of admissibility in the form of tongues originating from a specific point at the axis f (\(\beta = 0\)) while all even tongues cut off large areas in the \((\beta, f)\) plane. c Example of the basins of attraction of different cycles: at the same values of parameters various output is possible depending on initial conditions

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3.2.3.2 System with Input

In this section, we briefly discuss results for the topology of a \(\Sigma\Delta\) MEMS accelerometer which is used with an input (map (3.9) with \(a\neq 0\)). Here, we present results only for the overdamped case, though the topology may be used also with high-Q resonators [9]. We also restrict ourselves to the case when the sampling frequency is much higher than the natural frequency of the resonator and the input a is constant.

As noted before, the system output represents a cycle with a frequency that depends on the input and on the parameters of the device. Since this topology is based on the same ideas as a \(\Sigma\Delta\) modulator, the input is obtained as an average of the output cycle. First, we note that the average output x out that is the average on the bit sequence b n as a function of the input a is close to a linear dependance, but a magnified part of the plot reveals that it consists of a number of steps (Fig. 3.7). We recall that the same situation is observed in a conventional \(\Sigma \Delta\) modulator [24] and in the PDO [14, 28]. Such steps appear due to frequency locking and indicate that particular cycles exist in a finite interval of a control parameter.

The output of the system can be presented in the plane of parameters \((\beta,a)\), see Fig. 3.8a (similar to Fig. 3.4 and 3.5). To plot this plane, we choose those sequences b n for the formula 3.13 that correspond to the widest steps in Fig. 3.7. Now, the steps in \(x_{out}(a)\) can be considered as cross sections of areas in which cycles are admissible. As is seen from the plane (3.8), the width of steps depends on the dissipation β: the higher the dissipation, the smaller the width (and, consequently, the higher the resolution of the system). Figure 3.8(b) shows the average output coded by different colours: from 0 (blue) to 1 (red).

Fig. 3.7
figure 7

\(\Sigma\Delta\) MEMS accelerometer described by map (3.9). Average output of a \(\Sigma\Delta\) accelerometer as a function of the input a and the magnified part of the plot. The plot displays a set of plateaus and resembles devil’s staircase. The magnified plot also consists of a set of plateaus

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Fig. 3.8
figure 8

\(\Sigma\Delta\) MEMS accelerometer described by map (3.9). a Tongues that correspond to the widest steps of the plot (3.7) in the (\(a,\beta\)) plane. b The average output for different normalised inputs a and dissipations β presented by different colours: from 0 (blue) to 1 (red)

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3.3 MEMS PDO

In this section, we describe how a map that is very similar to the one derived above appears as a model of another MEMS-based application. This application is called as a PDO. It belongs to the class of resonant sensors [36] in which an oscillating mechanical structure (put in resonance) responds to an external stimulus such as an environment change in pressure, concentration of a specific compound, viscosity, etc. These sensors typically detect shifts in the frequency or amplitude of the oscillation of a MEMS device as a result of this external influence.

The PDO is a micromechanical structure that is embedded into a sigma-delta type feedback loop with appropriate control circuitry. As a result of the feedback force applied to the microresonator, it maintains self-sustained oscillations [13–15]. Such structures can be used in resonant mass sensors. It detects the change in the environment by changing its oscillation characteristics (mainly the frequency of oscillations). These sigma-delta feedback not only allows to sustain self-oscillations but also to monitor changes in the resonant frequency of the resonator simply by processing the binary sequence b n generated at their output. In references [33, 37] it was demonstrated how these circuits allow the actuation of multiple vibrational modes of the mechanical structure and, therefore, increasing the sensitivity of the measurements [38, 39].

3.3.1 Statement of the Problem

Similar to the previously discussed system, the PDO can be described through the diagram shown in Fig. 3.1b. In this case, we assume that there is no external input to the resonator (i.e. \(F_{in}=0\)) and the system is self-oscillating. The MEMS mechanical structure in the feedback loop is schematically shown in Fig. 3.2b. The position of the MEMS resonator is evaluated at each sampling time T s , and very short pulses of force are applied to the resonator. A typical actuation waveform is presented in Fig. 3.3b. In the case of a sigma-delta accelerometer whose actuation waveform is shown in Fig. 3.3a, the feedback force is continuously applied to the microstructure during the entire time interval T s . In the case of the PDO, only a short pulse of a very small duration is applied. This pulse can be modelled as a Dirac delta pulse.

Let us briefly discuss a dynamical model of such system. The derivation of the governing equation is similar to those described in the previous section (Sect. 3.2). Detailed study of the model has been carried out in [13, 28, 33]. The position x(t) of the MEMS resonator is described by the second-order differential equation

$$m\ddot{x}(t) + b \dot{x}(t) + k x(t) = F_{\it fb}(t)$$
(3.16)

where m is the mass of the movable plate, b is the damping factor, k is the spring factor, \(F_{\it fb}(t)\) is the force that acts on the resonator. According to the actuation principle, the position of the MEMS resonator is evaluated discretely every sampling time instant \({\it nT}_s\) and depending on its value, the corresponding feedback force is applied:

$$F_{\it fb}(t)= \left\{\begin{array}{@{}l@{}} - F_0 \, \delta (t-{\it nT}_s) \text{if} x((n-D-1)T_s)>0\,\\+ F_0 \, \delta (t-{\it nT}_s) \text{if} x((n-D-1)T_s))<0\,\end{array} \right.$$
(3.17)

where D is the number of delays in the feedback loop (\(D\geq 0\)). The feedback force \(F_{\it fb}\) represents a train of delta-pulses with a constant amplitude F 0 depending on the sign of the resonator position. For simplicity, we will use the notation \(t_n = n T_s\) and \(x_n = x({\it nT}_s )\), we also use Dirac delta function \(\delta(t)\) to denote a short pulse.

General solutions of inhomogeneous differential equations consist of a superpositions of decaying eigen oscillations of the free resonator and the forces oscillations. The form of the decaying oscillations is known from Sect. 3.2, and we give it here in matrix form using the rotation matrix \(\mathbf R\)

$$\begin{aligned} \left(\begin{array}{@{}l@{}} x (t) \\ y (t)\end{array}\right) = {\rm exp} (- \omega_0 \beta \, t) \mathbf{R}(\omega_0 \sqrt{1-\beta^2}\,t) \left(\begin{array}{@{}l@{}} x (t_0) \\ y (t_0)\end{array}\right)\end{aligned}$$
(3.18)

where \(\beta = b/(2m\omega_0)\) and \(\omega_0 = \sqrt{k/m}\). In this equation, we also introduced a new variable y that is a linear combination of the displacement x and the velocity v: \(y(t) = -\beta x/\sqrt{1-\beta^2} - v/(\omega_0\sqrt{1-\beta^2})\).

The only result of the application of a delta-pulse is a change in the velocity of the MEMS resonator:

$$\begin{aligned}\begin{array}{@{}l@{}} \displaystyle y(t_n+)-y(t_n -)=\frac{ F_{0}}{\omega_0^2 \sqrt{1-\beta^2}} \text{sgn} (x_{n-D-1}).\end{array}\end{aligned}$$
(3.19)

Collecting these two solutions, we can write discrete-time equations to describe the evolution of the PDO. Assuming that \(x_n = x({\it nT}_s)\) and \(y_n = y ({\it nT}_s +)\), we write the equation in discrete-time matrix form

$$\begin{aligned}\left(\begin{array}{@{}l@{}} x_{n+1} \\ y_{n+1}\end{array}\right) = \alpha \mathbf{R}(2\pi f) \left(\begin{array}{@{}l@{}} x_{n} \\ y_{n}\end{array}\right) + b_{n-D} \left(\begin{array}{@{}l@{}} 0 \\ \zeta\end{array}\right)\end{aligned}$$
(3.20)

and

$$b_n = \text{sgn} (x_{n})$$
(3.21)

In Eq. (3.20), we have introduced the following parameters: contraction coefficient \(\alpha = {\rm exp} (-2\pi f /\sqrt{1-\beta^2})\), normalised sampling frequency \(f=T_s \omega_0 \sqrt{1-\beta^2}/2\pi\) and normalised amplitude of the pulses \(\zeta=F_0/(\omega_0 m \sqrt{1-\beta^2})\). Similarly to (3.9), the system (3.20) is a piecewise-smooth discontinuous mapping of the form

$$\mathbb{R}^2\times\mathbf{ B}^{D+1} \longrightarrow \mathbb{R}^2\times\mathbf{ B}^{D+1}$$
(3.22)

where \(\mathbb R\) is the set of real numbers and \(\mathbf B\) is a two-element set such that \(\mathbf B = \{-1, 1\}\).

3.3.2 Periodic Solutions

In [28], it was shown that periodic solutions of the mapping (3.20) are stable cycles. More generally, the system (3.20) represents a piecewise contractive map and, according to [35], always displays periodic stable sequences or cycles.

The procedure to investigate limit cycles is described in detail in the previous Sect. 3.2. By assuming a specific sequence \(b_n = \text{sgn} (x_{n})\), we can obtain the explicit form of the N-cycles as

$$\left(\begin{array}{@{}c@{}c@{}} x_{n}\\y_{n}\end{array}\right) = \left( \mathbf{I} - \alpha ^N \mathbf{R}( 2\pi N f)\right)^{-1} \sum\limits_{j=1}^{N}\alpha^{N-j} \mathbf R((N-j)2\pi f) \text{sgn}(x_{n-D+j-1}))\left(\begin{array}{@{}c@{}c@{}} 0 \\\zeta\end{array}\right)$$
(3.23)

The stability analysis can be done in the same way as it was carried out in Sect. 3.2 and every cycle produced by (3.23) is stable.

To validate the existence of a particular cycle, we can apply the following strategy:

  • Assume the sequence b n  (3.21) for the sign of the position.

  • Fix the parameters of map (3.20) (namely β, f and a) and calculate \(\{(x_n, y_n)\}\) using Eq. (3.13).

  • {Check if the condition (3.21) is fulfilled.} It is worth noting that although the sequence (3.21) is used to generate the cycle (3.13), the resulting cycle may have a signature sequence that differs from the desired sequence. In this case we say that at this parameters, the cycle is not admissible.

Varying β and f over wider regions with implementation of the above strategy allows one to obtain the parameter plane (\(\beta,f\)) with regions of cycles admissibility—so called tongues.

3.3.3 Results

To analyse the output sequences of map (3.20), we note that the MEMS waveform in this case will be a sinusoid, and, therefore, one can obtain the sequence of signs b n as the sign of a sampled sinusoid

$$b_n = \text{sgn}(\cos(2\pi k M/N + \varphi_0))$$
(3.24)

where ϕ0 is an arbitrary phase and \(1\leq k\leq N\). The ratio \(M/N\) defines the rotation number of a cycle, i.e. the number of loops around the origin a trajectory makes in one iteration.

The ‘parameter’ of interest in the case of the PDO is its oscillation frequency and it can be calculated using two approaches. Since the PDO is a digital oscillator, the oscillation frequency can be directly calculated from the output bitstream [14, 29]. We will use the route suggested in [29] to generate the auxiliary sequence and obtain the digital frequency of oscillations f D . The q n sequence is defined as follows:

$$q_{n} = \left\{\begin{array}{@{}ll@{}} 1, & \text{if $b_{n} \neq b_{n+1}$,} \\0, & \text{if $b_{n} = b_{n+1}$.}\end{array}\right.$$
(3.25)

The digital frequency of oscillations can now be written as

$$f_{D} = \frac 1 2 \cdot \frac{\sum\limits_{n=1}^{N}q_{n}}{N}$$
(3.26)

On the other hand, one can calculate the rotation number of the map. If α n is the angle made by the point \((x_n, y_n)\) and the origin of the \(x,y\) plane, the rotation number is defined as [40]

$$\rho = \lim\limits_{n\to\infty} \frac{\alpha_n} {2\pi\, n}$$
(3.27)

Both, the digital frequency and the rotation number, are equivalent, and one can ensure that they give the same result. Therefore, the resulting oscillation frequency, in the oversampling regime (\(f_{\it osc}/f_s<1/2\)), is

$$f_{\it osc}=f_s f_D = f_s \rho$$
(3.28)

where \(f_s = 1/T_s\) is the sampling frequency.

The rotation number (related to the resulting oscillation frequency) as a function of the normalised frequency f (related to the sampling and natural frequency) is presented in Fig. 3.9a. In the ideal case, this should be a straight line displaying that any small change in the sampling or natural frequency will result in a change of the oscillation frequency detected by the electronics. However, this plot resembles a devil’s staircase and consists of discrete steps. The oscillations are frequency-locked in this case with one frequency being a rational number times the other frequency. Practically this means that the system (MEMS sensor) has a limited resolution: if the frequency is locked, the system is not responding to a small change in the environment and this change cannot be detected from the bitstream sequence b n . This phenomenon is only important when using resonators with low-quality factors, i.e. with high losses. From the figure, one can see a magnified part of the plot shown in Fig. 3.9b that also resembles a devil’s straicase. The size of steps is defined by the dissipation parameter β: the smaller the β, the more straight and smooth the plot is.

Fig. 3.9
figure 9

Pulsed digital oscillator described by map (3.20). Rotation number ρ (or digital frequency f D , both related to the resulting oscillation frequency \(f_{\it osc}\)) as a function of the normalised frequency f (related to the sampling frequency and/or the natural frequency. \(\beta=0.05\)

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Fig. 3.10
figure 10

Pulsed digital oscillator described by map (3.20). a Regions of admissibility (with overlapping) for N-cycles with \(1\leq N\leq 15\) in the parameter plane calculated from (3.23). b Parameter plane with tongues that correspond to different cycles obtained from numerical simulations of the map (3.20) with zero initial conditions

Full size image

The parameter plane \((\beta,f)\) for tongues with rotation numbers \(M/N\), \(0<N<16\) is shown in Fig. 3.10. Plot 3.10a shows the domains of existence of specific cycles. These domains can overlap (shown by a darker shade of gray): at the same β and f there coexist several cycles and which one of them will eventually be displayed by the system depends only on initial conditions. The planes of parameters calculated for zero initial conditions are shown in Fig. 3.10b.

3.4 Modification of the PDO: Electrostatic MEMS Oscillator

PDOs described in Sect. 3.3 are implemented using thermoelectric actuation that allows one to emulate ‘positive’ and ‘negative’ pulses. Modern MEMS devices are often implemented using electrostatic actuation. As it was described in Sect. 3.2, the main difficulty of electrostatic actuation is that the generated force is always attractive and it is not possible to emulate a negative pulse. Therefore, a modification of the actuation technique from Sect. 3.3 is required and we describe it in this section. The corresponding oscillator is called the electrostatic PDO or e-PDO [41].

The e-PDO has the same application as the conventional PDO. Namely, it is used as a resonant sensor that detect small shifts of the resonant frequency of a mechanical structure as a result of changes in the variable to be measured. We give here the map for this system as it has quite a complex form compared to the two studied cases, but displays very similar features to the conventional PDO map.

3.4.1 Statement of the Problem

A schematic structure of the e-PDO can be described using the block diagram from Fig. 3.1b with no external input force (\(F_{in} =0\)). A simplified 1D mechanical structure of the e-PDO is shown in Fig. 3.2c. The displacement of the microresonator ξ is described by the following equation:

$$m\ddot{\xi}(t)+b\dot{\xi}(t)+k\xi(t)=F_{\it fb}(\xi,t)$$
(3.29)

where m is its mass, b is the damping factor and k is the effective spring constant. The force acting on the resonator will consist only of the feedback force \(F_{\it fb}\). The e-PDO is actuated electrostatically and therefore \(F_{\it fb}\) is an electrostatic force. The electrostatic force is always attractive, and it is not possible to create a repulsive pulse required for sigma-delta feeedback. To overcome the similar issue, inertial sensor/accelerometers described in Sect. 3.2 employ two electrodes that located at the two opposite sides of a movable mass. However, this is not always possible because of size or design limitations. In the case of the e-PDO, we use the following actuation scheme:

$$V_{\it fb}(t)= \left\{\begin{array}{@{}ll@{}} V_0 (1- \Pi(t)) & \text{for} \quad{\it nT}_s <t < (n+1) T_s \quad\text{if}\quad \xi((n-D)T_s)>\xi_{av}\,\\V_0 & \text{for} \quad{\it nT}_s <t < (n+1) T_s \quad\text{if}\quad \xi((n-D)T_s)<\xi_{av}\,\end{array}\right.$$
(3.30)

where \(\Pi(t)\) denotes a square-shaped pulse

$$\Pi(t)= \left\{\begin{array}{@{}ll@{}} 0 &\text{for}\quad t>d t,\\1 & \text{for}\quad 0<t<d t\end{array}\right.$$
(3.31)

In the above formula, \(\xi_{av}\) is the current average position of the resonator, V 0 is a constant voltage applied, D is the number of delays (\(D\geq 1\)) and dt is a short interval of time (\(dt< T_s\)). We can summarise this actuation scheme as follows. If the position of the resonator at a given sampling time is below the time averaged position (\(\xi<\xi_{av}\)), a constant bias voltage is held \(V_{\it fb} = V_0\) (no pulse applied). If the position of the resonator at a given sampling time is above the time averaged position (\(\xi>\xi_{av}\)), an ‘anti'-pulse is applied: the constant biasing voltage V 0 is turned off for a very short time d t and then turn on again. A typical actuation waveform of the e-PDO is presented in Fig. 3.3c.

The system is described by the following equation:

$$\begin{aligned} \ddot{\xi}(t) + ({b}/{m}) \dot{\xi}(t) &+ \omega_{0}^{2} \xi(t) = \frac{\varepsilon_0 A V_{0}^{2}}{2 m g^2 (1-\xi(t)/g)^2} \left(1-\sum\limits_n b_n \Pi (t - {\it nT}_s)\right)\end{aligned}$$
(3.32)

where the natural frequency \(\omega_0 = \sqrt{k/m}\), ϵ0 is the vacuum permittivity, A is the area of the actuating electrode and g is the rest gap between the movable mass and the electrode. The sequence of the position sign b n defines the application of an anti-pulse:

$$b_n = \frac 1 2 \left[1 + \text{sgn}(\xi((n-D)T_s) - \xi_{av})\right]$$
(3.33)

Since most of the time a constant voltage is applied and it turned off only for short time instances dt, the resonator will oscillate around an equilibrium position ξ0. We introduce a small dimensionless deflection x from the electrostatic equilibrium position such that \(\xi=g(x_0 + x)\) and dimensionless time \(\tau = \omega_0 t\). A linearised equation describing the deflection x will have form

$$x'' + 2 \beta x' + x = \frac{2\psi_0 \,x}{(1-x_0)^3}\left( 1 - \sum\limits_n b_n \Pi (\tau - n\tau_s) \right) - \frac{\psi_0}{(1-x_0)^2} \sum\limits_n b_n \Pi (\tau - n\tau_s)$$
(3.34)

where \(\beta = b/(2m\omega_0)\) is the normalised dissipation coefficient and \(\psi_0 = \varepsilon_0 A V_0^2/\) \((2 m \omega_0^2 g^3 )\) is the normalised force amplitude and the prime sign denotes the derivative with respect to dimensionless time τ. The electrostatic equilibrium x 0 is the solution of the equation

$$x_0 - \frac{\psi_0}{(1-x_0)^2}=0$$
(3.35)

and it is worth mentioning that \(x_0 \approx x_{av}\) and with good accuracy we can assume that \(x_{av} = x_0\). To obtain a map, one has to solve differential Eq. (3.34). Let us consider two cases.

No anti-pulse is applied. This case corresponds to \(b_n = 0\), and the system is constantly biased. Equation (3.34) can be written into a simpler form

$$\begin{aligned}0\leq \tau <\tau_s: \qquad &x''+ 2\beta x' + x (1 - 2\psi_0/(1-x_0)^3) = 0\end{aligned}$$
(3.36)

where \(\tau_s = \omega_0 T_s\). Denoting \(\omega_1 = \sqrt{1-\beta^2}\) and \(\omega_2 = \sqrt{1 - \beta^2 - 2\psi_0/(1- y_0)^3}\), we can write the solution of (3.36) in the form:

$$\left(\begin{array}{@{}l@{}} x (\tau_s) \\y (\tau_s)\end{array}\right)= e^{-\beta \tau_s} \mathbf{R}(\omega_2 \tau_s) \left(\begin{array}{@{}l@{}} x (\tau_0)\\y (\tau_0)\end{array}\right)$$
(3.37)

where \(\mathbf R\) is the rotation matrix. In the above equation, we used the variable \(y=-\beta x/\omega_2 - x'/w_2\).

Anti-pulse is applied. This corresponds to the case \(b_n=1\). Equation (3.34) can be split into two equations

$$\begin{aligned} 0\leq \tau <d\tau: \qquad\qquad\qquad x''+ 2\beta x' + x = - \frac{\psi_0}{(1-x_0)^2} \nonumber\\ d\tau\leq \tau <\tau_s: \qquad x''+ 2\beta x' + x (1 - 2\psi_0/(1-x_0)^3) = 0\end{aligned}$$
(3.38)

The solution of (3.38) is

$$\begin{aligned}\left(\begin{array}{@{}l@{}} x (\tau_s) \\ y (\tau_s)\end{array}\right) = e^{-\beta \tau_s} \mathbf{R}(\omega_2 (\tau_s - d\tau)) \mathbf{A} \mathbf{R}(\omega_1 d\tau) \left(\begin{array}{@{}l@{}} x (\tau_0) \\ (\omega_2/\omega_1) y (\tau_0)\end{array}\right) + \nonumber\\ + e^{-\beta \tau_2} \mathbf{R}(\omega_2 (\tau_s - d\tau))F_0 \left(\begin{array}{@{}l@{}} \zeta\\ (\omega_1/\omega_2)\eta\end{array}\right)\end{aligned}$$
(3.39)

where \(F_0 = -\psi_0/((1-y_0)^2)\) and \(\mathbf A = \left(\begin{array}{@{}lll@{}} 1 & 0 \\ 0 & (\omega_1/\omega_2)^{b}\end{array} \right)\). The coefficients ζ and η are the same as described in Sect. 3.2 and using the notation introduced in this section, they can be presented in the following form:

$$\begin{aligned} &\zeta = 1 - e^{-\beta\tau_1}\cos(\omega_1d\tau)- \frac{\beta\,e^{-\beta d\tau}}{\sqrt{1- \beta^2}}\sin(\omega_1d\tau), \nonumber\\ &\eta = -\frac{\beta}{\sqrt{1-\beta^2}} + \frac{\beta\,e^{-\beta d\tau}}{\sqrt{1- \beta^2}}\cos(\omega_1d\tau)- e^{-\beta\tau_1}\sin(\omega_1d\tau)\,\end{aligned}$$
(3.40)

An iterative system of equations can be obtained from (3.37) and (3.39) by assuming that \(x_n = x({\it nT}_s)\) and \(y_n=y ({\it nT}_s)\). In its compact form, the resulting map can be written as

$$\begin{aligned}\left(\begin{array}{@{}l@{}} x_{n+1} \\ y_{n+1}\end{array}\right) = \alpha \mathbf{R}( \omega_2 (\tau_s - d\tau)) \mathbf A \mathbf{R} (\omega_2 (\omega_1/\omega_2)^{b_n} d\tau) \left(\begin{array}{@{}l@{}} x_n\\ (\omega_2/\omega_1)^{b_n} y_n\end{array}\right) + \nonumber\\ + \alpha_\tau \mathbf{R}(\omega_2 (\tau_s - d\tau))F_0 b_n \left(\begin{array}{@{}l@{}} \zeta\\ (\omega_1/\omega_2)^{b_n}\eta\end{array}\right)\end{aligned}$$
(3.41)

where the sequence of signs b n is defined as follows

$$b_n = \frac 1 2 \left[1 + \text{sgn}(x((n-D)T_s))\right]$$
(3.42)

In these equations, \(\alpha = {\rm exp}(-\beta\tau_s)\) and \(\alpha_\tau = {\rm exp} (-\beta (\tau_s - d\tau))\). This is a piecewise-smooth discontinuous map

$$\mathbb{R}^2\times\mathbf{ B}^D \longrightarrow \mathbb{R}^2\times\mathbf{ B}^D$$
(3.43)

where \(\mathbb R\) is the set of real numbers and \(\mathbf B\) is a two-element set such that \(\mathbf B = \{0, 1\}\). Note that the this map transforms into to the conventional PDO map if \(\omega_1 = \omega_2\) and \(d\tau \ll \tau_s\) [41].

3.4.2 Results

Here we briefly discuss some of the results. Since the standard application for an e-PDO is as part of a resonant sensor and it is important to extract the change in the oscillation frequency in this system, the important parameter is the digital frequency of oscillations f D or the rotation number ρ (recall that the actual frequency of oscillation is \(f_{\it osc}=f_D f_s = \rho f_s\). The digital frequency can be calculated using the approach discussed in Sect. 3.3 and formula (3.26). The digital frequency of oscillation as a function of the normalised frequency \(f=T_s \omega_0 \sqrt{1-\beta^2}/2\pi\) is shown in Fig. 3.11a. This figure shows the same fractalised characteristic already seen in sigma-delta accelerometers and PDOs. The smaller the dissipation parameter β, the more straight is the plot. We also note that the system displays multistability, and a large number of stable cycles are admissible in the output at the same parameters. To demonstrate this, we present the basins of attractions corresponding to different cycles (Fig. 3.11b). Depending on the initial conditions \((x_0, y_0)\), the system converges to different periodic solutions.

Fig. 3.11
figure 11

Electrostatic pulsed digital oscillator described by map (3.41). a Digital frequency of oscillation (related to the actual oscillation frequency of the e-PDO) as a function of a normalised frequency f (rested to the sampling frequency and the natural frequency). The plot consists of a number of plateaus that are more visible when β is larger. b Example of the basins of attraction of different cycles in the plane of initial conditions \((x_0, y_0)\)

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3.5 Control of Dielectric Charge for Capacitive MEMS

So far we have considered examples that are related to the actuation and control of movable mechanical components of MEMS. In this section, we discuss a very different example that is related to MEMS reliability and does not directly involve the control of MEMS mechanics.

The electrostatic mechanism of actuation already mentioned in this study is very common in MEMS. However, in some types of MEMS that utilise dielectric materials, it leads to the accumulation of charge in these dielectrics. It is known to be a major reliability problem for these devices and especially a problem for radio frequency (RF) MEMS [42]. As is reported in recent reviews [43, 44], the accumulation of charge by dielectrics is very common and all typical dielectric materials are prone to it. In recent years, an alternative approach that consists of bipolar [45] and smart actuation techniques [20, 46] is suggested to address the problem of dielectric charging.

In references [20], a smart actuation method based on a feedback control was proposed for the actuation of capacitive MEMS with dielectrics. The aim of the actuation and control method is to ensure that the charge accumulated in the dielectric is not increasing and stays fixed at a desired level by applying a bipolar actuation voltage. We have investigated this method for a simple 1D model suitable for MEMS positioners and varactors which operate below pull-in of the MEMS structure. (Pull-in is a phenomenon when the movable suspended electrode collapses onto the fixed electrode, since the electrostatic force can no longer be compensated by the restoring mechanical force). In addition, we experimentally demonstrated that the method can be applied to switches which operate beyond pull-in. The proposed closed-loop feedback technique is based on the measurement of the MEMS capacitance at fixed time instances \({\it nT}_s\). The total capacitance of the device is a function of the position x(t) which is directly linked to the accumulated total charge in the dielectric Q d . Thus, the value of the total capacitance for a given voltage can be treated as an ‘indicator’ of the value of the total accumulated dielectric charge.

3.5.1 Statement of the Problem

A simple 1D model of a variable capacitor is shown in Fig. 3.2d. The deflection of the top electrode y is described by a mass-spring-damper ordinary differential equation

$$m\ddot{x}(t)+b\dot{x}(t)+kx(t)=F_{\text{el}}(V,x)$$
(3.44)

where m is the mass of the movable electrode, b is the damping factor and k is the spring coefficient. The value of the damping coefficient is large as the top electrode is moving in air, so the first two terms can be neglected. Thus, the system will reach the steady state relatively fast. Then, the position x(t), for a given value of voltage, is obtained analytically by solving the equation which expresses the balance of forces

$$ky=F_{el}(V,x)$$
(3.45)

where

$$\begin{aligned} F_{\text{el}} = \frac{\epsilon_0 A}{2} \frac{(V-V_{\text{shift}})^2}{(g-x+\frac{d}{\epsilon_d})^2}.\end{aligned}$$
(3.46)

Here, V is the applied voltage, ϵ0 is the vacuum permittivity, ϵ d is the relative permittivity of the dielectric, g is the gap distance between the upper electrode and the dielectric, d is the thickness of the dielectric layer, A is the area of the device and \(V_{\text{shift}} = Q_d/C_d\) is the voltage shift due to the accumulated charge Q d into the dielectric. The capacitance C d associated with the dielectric layer is given by \(C_d= \epsilon_0\epsilon_d A/d\). From (3.45), it follows that there is a critical value of \(F_{\text{el}}\) that exceeds the restoring spring force ky. In this case, the equilibrium position of the device cannot be maintained any longer. This results in the instantaneous collision of the top electrode onto the dielectric layer and the bottom electrode. Such an event (the collapse of the movable electrode) is called the pull-in event and the corresponding voltage is known as the pull-in voltage V PI .

To make the above equations self-consistent, we must supply the equations that define the evolution of the dielectric charge with the applied voltage and time. The electrostatic force in (3.46) depends on the voltage shift

$$V_{\text{shift}}(t)=Q_d(t)/C_d,\;\;V_{\text{eff}}(t)=V-V_{\text{shift}}(t)$$
(3.47)

that is a function of the charge accumulated in the dielectric. The device ‘sees’ that effective voltage which deviates from the applied voltage V. The evolution of the dielectric charge Q d is related to the actuation voltage and time. In principle, positive and negative charge can be accumulated into the dielectric. There are various proposed models which describe that evolution taking into account charge injection, dipole orientation, charge trapping along with others or combination of those [42]. The exact mechanism depends also on the device characteristics. Here, we will use a semi-empirical multi-exponential charging model where positive and negative charging components are included [20]. The time evolution of each component is

$$\begin{aligned} &Q^p(t)=\left\{\begin{array}{@{}ll@{}} Q_{\it max}^p \sum_{i}\zeta_i^p e^{-t/t_{Di}^p}& V>0\\ Q_{\it max}^p(1- \sum_{i}\zeta_i^p e^{-t/t_{Ci}^p}) & V<0\end{array}\right.&\nonumber\\ &Q^n(t)= \left\{\begin{array}{@{}ll@{}} Q_{\it max}^n(1- \sum_{i}\zeta_i^n e^{-t/t_{Ci}^n})& V>0\\ Q_{\it max}^n \sum_{i}\zeta_i^n e^{-t/t_{Di}^n} & V<0\end{array}\right.\end{aligned}$$
(3.48)

where \(Q_{\it max}^p\) and \(Q_{\it max}^n\) are the maximum values of positive and negative charges, respectively, t Ci and t Di the charging and discharging time constants and ζ i coefficients which express the weight of each exponential to the total charge (thus \(\sum_{i}\zeta_i=1\) for each component). The coefficients ζ i and the characteristic times τ i are can be determined from experiment data and are different, in general, between the positive and negative components. Now, the evolution of the total charge will be

$$Q_d(t)=Q^p(t)+Q^n(t)$$
(3.49)

The control method from [20] monitors the parasitic charge in the dielectric by comparing the capacitance C n of the device in each sampling time n T s with a threshold value \(C_{\text{th}}\), where

$$C_n = \frac{C_{g,0}}{1-\frac{x_n} g + \frac{C_{g,0}}{C_d}}$$
(3.50)

its discretised expression with \(C_{g,0}=A\varepsilon_{0}/g\) the capacitance of the gap and x n the discretised position of the up electrode in sampling time n T s . As long as the value of C n is less than \(C_{\textit{th}}\) a positive actuation voltage \(V^+\) is applied. When the value of C n exceeds the threshold value the polarity in the next time instant will be inversed and negative voltage V - will be applied. Summarising, the control method follows the scheme:

$$V_n= \left\{\begin{array}{@{}ll@{}} V^+ & \text{if}\quad C_n<C_{\it th}\,\\V^- &\text{if}\quad C_n> C_{\it th}\end{array}\right.$$
(3.51)

with V n the applied voltage at the nth sampling time. The capacitance threshold value \(C_{\it th}\) biuniquely corresponds to a charge target value \(Q_{\it th}\) that the control method aims to fix into the dielectric. The voltage actuation waveform will be somewhat similar to the waveform shown in Fig. 3.3a.

By defining the quantities \(\alpha_{C} = e^{-T_s/\tau_{C}}\) and \(\alpha_{D} = e^{-T_s/\tau_{D}}\) the discretised equations for charge evolution Eq. 3.48 for the case i = 1 will have form

$$\begin{aligned} Q_{n+1} &= \alpha_{D} \left( \frac{\alpha_{C}}{\alpha_{D}} \right)^{b_n} Q_n + Q_{\it max} (1-\alpha_{C}) b_n = \Phi (Q_n, b_n)\end{aligned}$$
(3.52)

where we introduced the decision bit sequence b n as

$$b_n = \frac{1}{2}(1 + \text{sgn} (C_{\it th} - C_{n})).\,$$
(3.53)

The complete model that describes the evolution of charge can be expressed as

$$Q_{n+1} = \sum\limits_i \Phi_i ( Q_n^{(i)}, b_n) = \Theta (Q_n, b_n)$$
(3.54)

The general form of the map which expresses the evolution of the charge dynamics under the operation of this control algorithm can be written for the state vector \((Q, b)\)

$$\begin{aligned}\left(\begin{array}{@{}l@{}} Q_{n+1} \\ b_{n+1}\end{array}\right) = \left(\begin{array}{@{}l@{}} \qquad\qquad\Theta(Q_n,b_n)\, \\ \frac{1}{2} \left[ 1+\text{sgn}\left(C_{\it th} - \frac{C_{g,0}}{1 + \gamma - x (Q_n,b_n) /g}\right) \right] \,\end{array}\right)\end{aligned}$$
(3.55)

and this is a piecewise-smooth mapping of the form

$$\mathbb{R}\times \mathbf B \to \mathbb{R}\times \mathbf B$$
(3.56)

where \(\mathbb{R}\) is the set of real numbers and \(\mathbf B = \{0,1\}\).

3.5.2 Periodic Solutions

According to [35], the output of the map (3.55) are stable periodic sequences, or cycles. By considering for simplicity i = 1 in (3.48) and introducing the parameters \(\nu = \alpha_{C}/\alpha_{D}\), \(\mu = Q_{\it max}(1-\alpha_{C})\) and the sum

$$S^q_p = \sum\limits_{j=p}^{q}b_{n+j}$$
(3.57)

for \(p\leq q\), the kth iteration of the map is

$$Q_{n+k} = \alpha_{D}^k\nu^{S^{k-1}_0}Q_n - \mu\sum\limits_{j=1}^{k-1}\alpha_{D}^{k-j}\nu^{S^{k-1}_{j}}b_{n+j-1} - \mu b_{n+k-1}\,$$
(3.58)

Since the sequence \(\{b_n\}\) is periodic [35], \(Q_n = Q_{n+N}\). Using N instead of k and \(S^{N} = S_{0}^{N-1}\) as the sum of all b n in the sequence, we write that

$$Q_{n} = \frac{-\mu}{1-\alpha_{D}^N\nu^{S^N}}\left(b_{n+N-1} + \sum\limits_{j=1}^{N-1}\alpha_{D}^{N-j}\nu^{S^{N-1}_{j}}b_{n+j-1} \right)\,$$
(3.59)

and this equation defines the cycles of charge displayed by map (3.55) if the sequence b n for \(1\geq n\leq N\) is given. The stability analysis of a cycle can be done using the approach discussed in Sect. 3.2 and one can ensure that this cycle is asymptotically stable.

3.5.3 Results

An important property of this control method is that subsequent discharging events are not possible, i.e. the bitstream sequence cannot contain two subsequent zeros. At the moment the polarity of the applied voltage changes (when the total capacitance exceed the \(C_{\it th}\) value) the total capacitance drops below the \(C_{\it th}\) value. Thus, the next event will be a charging event (of the opposite polarity). This means that the minimum amount of charge the control algorithm is able to fix in the device is restricted to one charging and one discharging events defining the two-cycle

$$\begin{aligned}\hat{Q}_{1}& = -\frac{\alpha_{D}\mu}{(1-\beta)}, \qquad ( b_{1} = 1)\,\nonumber\\ \hat{Q}_{2}& = -\frac{\mu}{(1-\beta)}, \qquad ( b_{2} = 0)\,\end{aligned}$$
(3.60)

where we denoted \(\beta=\alpha_{C}\alpha_{D}\).

Fig. 3.12
figure 12

Control of dielectric charge for capacitive MEMS described by map (3.55). Charge Q d fixed by the control method in the dielectric as a function of a desired charge amount \(Q_{\it th}\). a corresponds to a short sampling time (\(T_s = 0.01\) s) while b corresponds to a larger sampling time (\(T_s = 5\) s). The large plateau seen in both figures corresponds to the 2-cycle. The parameters of the device used in this simulation are taken from [20]

Full size image

Therefore, the minimum amount of charge in the dielectric is defined by this two-cycle. If the target charge \(Q_{\it th}\) set for the algorithm is less than the charge defined by the two-cycle, the control algorithm will not be able to fix the desired dielectric charge. Instead, the charge corresponding to the two-cycle will be fixed. Thus, depending on the system parameters, the control method will yield either the sequence of charge Q n given by (3.59) with \(N>2\) whose average value is equal to the desired amount of charge \(Q_{\it th}\) or a two-cycle (3.60) whose average value will be, in the most general case, larger than the desired amount. Therefore, we say that the control algorithm operates successfully if the output bit sequence b n corresponds to a cycle with \(N>2\).

For the illustration of the control method, the parameters given in [20] are used in this study. The parameter of interest in such a system is the time-averaged charge in the dielectric controlled by the algorithm. In order to show the performance of the control method, in Fig. 3.12 we plot the average dielectric charge as a function of the target charge \(Q_d(Q_{\it th})\). In the ideal case, this must be a straight line: a small change in \(Q_{\it th}\) must result in the small change of Q d that algorithm fixes in the dielectric. This plot is shown in Fig. 3.12. Figure 3.12a corresponds to a very short sampling time T s while Fig. 3.12b corresponds to a very long sampling time. For a relatively large T s , the plot is not a straight line, but resembles rather a devils staircase plot. The devils staircase has already appeared in other examples from this study. Practically, the presence of devils staircase in this system means that the algorithm will have a finite resolution/accuracy of fixing the dielectric charge when T s is large. There is a large plateau seen in both figures for smaller \(Q_{\it th}\). This large plateau corresponds to a two-cycle. In this case, some charge is fixed in the dielectric, but it does not correspond to a desired level.

Fig. 3.13
figure 13

Control of dielectric charge for capacitive MEMS described by map (3.55). Simulated capacitance and dielectric charge transients applying the charge control method

Full size image

To obtain a broader understanding of the system dynamics, we can consider a plane of parameters spanned by the target capacitance \(C_{\it th}\) (related to the desired charge \(Q_{\it th}\)) and the ratio of the discharging and charging times \(\tau_D/\tau_C\) that strongly affect the behaviour of the system. We can define the area where the algorithm is able to fix \(Q_{\it th}\) (successful operation of the algorithm) and where the algorithm is not able to fix \(Q_{\it th}\) (the output of the algorithm is a two-cycle). The two boundary is given by the following equation:

$$-\frac{\alpha_{D}\mu}{(1-\beta)} = Q_{\it th}$$
(3.61)

Figure 3.13 shows such a plane with the above boundary plotted. Practically, this means that for realistic devices one should select the algorithm parameters above the dashed line shown in the figure.

3.6 Conclusions

We have considered a number of examples describing a MEMS, actuated or controlled, by a feedback inspired by sigma-delta modulation. In the introduction, we described the principles of sigma-delta conversion and showed how they can be applied to actuated or controlled MEMS. We discussed three examples: an inertial sensor, a self-sustained oscillator and a charge control actuation method. In all cases, we have shown how a piecewise-smooth discontinuous map arises as a model of these systems. Although belonging to different class of devices, all the examples display a number of similar features, for instance, a devil’s staircase as the representation of frequency or charge locking. For all these models, the output is a stable sequence (cycle) and frequency/charge locking corresponds to a specific cycle of the map that persists or dominates in a large area of control parameters. A very similar phenomenon was found in conventional leaky sigma-delta modulators and it reflects the finite resolution in representing the input signal. In a self-sustained oscillator, the persistence of limit cycles give rise to frequency locking, i.e. the inability of the oscillator to change its frequency for certain values of control parameters. Finally, in the problem of charge control in capacitive MEMS, we observe charge locking. Charge locking is very similar to frequency locking and it displays the finite resolution of charge control in MEMS. The persistence of limit cycles and frequency and charge locking are parasitic phenomena. Since we present the study of the system dynamics over wide range of parameters, we can estimate the values of parameters in order to avoid these undesirable effects. The models we present in this chapter that describe realistic systems and behaviour were experimentally validated in a number of cited studies. We refer a reader who is interested in further investigation of MEMS, and in particular of sigma-delta inspired MEMS to the literature cited in this chapter.