Method and circuit for determining characteristic parameters of an oscillator

Abstract

The invention relates to a method and to a circuit arrangement for determining characteristic parameters of an oscillator (OSZ), said method comprising the following steps: comparing (S1) an output signal (x) of the oscillator (OSZ) with a threshold value (x thr) in order to determine a comparison result; generating (S2) a feedback signal (FB) to which one of two different feedback levels (ACC+; ACC-) is applied depending on the determined comparison result; delaying (S3) the generated feedback signal (FB) by a delay time (tD) in order to establish a continuous periodic oscillation of the feedback-coupled oscillator; and determining (S4) the characteristic parameters of the oscillator (OSZ) on the basis of sampled values within a self-oscillation period of a triggered self-oscillation of the oscillator (OSZ).

Description

[0001] Description

[0002] title

[0003] Method and circuit for determining characteristic parameters of an oscillator

[0004] The invention relates to a method and a circuit for determining characteristic parameters of an oscillator, in particular characteristic parameters of an overdamped MEMS oscillator by triggering a self-oscillation.

[0005] State of the art

[0006] Most electronic systems require a clock source. Quartz resonators are frequently used for this purpose. However, MEMS (Micro-Electro-Mechanical System) oscillators, which combine a resonator with an oscillator IC to form a complete integrated clock source, offer certain advantageous technical characteristics compared to quartz crystals.

[0007] EP1706942B1 describes a compensated microelectromechanical oscillator. This oscillator includes a microelectromechanical resonator for generating an output signal, the output signal containing a first frequency. The oscillator further includes a frequency-adjustment circuit coupled to the microelectromechanical resonator for generating an output signal with a second frequency using the output signal of the microelectromechanical resonator and a set of values. The frequency-adjustment circuit includes a first-frequency multiplier circuit, the second frequency being higher than the first frequency, and the set of values ​​being determined using the frequency of the output signal of the microelectromechanical resonator. Oscillators (OSCs) are electronic circuits that generate a periodic output signal of a well-defined shape and frequency.A harmonic oscillator can be completely described by two parameters: its natural frequency fO and the damping of the oscillating output signal. The term "harmonic oscillator" is also used for damped harmonic oscillators, even though these, strictly speaking, do not exhibit harmonic oscillation but rather a damped oscillation.

[0008] A transient step analysis is unproblematic for a weakly damped device (Fig.1 A), since the relevant features of the vibration and the damping envelope in the measured transient curve, as sketched in Fig.1 A 1, can be easily observed or measured.

[0009] Figures 1A and 1B show a comparison of underdamped (Fig. 1A) and overdamped (Fig. 1B) oscillator relaxations. Figures 1A and 1B show the normalized amplitude x of the signal over time t. Despite the identical information content, the indicative signatures of oscillation and damping envelope are not discernible or measurable in Figure 1B.

[0010] Most sensors are of the overdamped type (as shown in Fig. 1B) to ensure uniform sampling of the applied external stimulus without overshoot in the signal path and at the user display. Therefore, it is currently difficult to accurately characterize most devices with respect to their natural frequency fO and quality factor Q or attenuation value DL through precise and rapid measurements.

[0011] A precise determination of these parameters using a transient step analysis is only possible if the exact time of the deactivation of the actuation is known. This time is followed by the downswing transient, as shown, for example, in the two subplots of Figures 1A and 1B.

[0012] The fundamental causality of this effect is illustrated in Fig. 2A. Fig. 2A shows an abstract representation of the regressive identification of the transient time instance tOff, the frequency fO, and the damping value DL. Fig. 2B shows an exemplary abstract representation of the regressive identification of the fundamental intrinsic causality (corresponding to Fig. 2A), the transient time instance tOff (stage), the frequency fO (oscillation), and the damping value DL (fall).

[0013] However, a disadvantage of a conventional transient step analysis is that, according to the relevant differential equation, for overdamped systems the natural frequency fO can only be observed in the very short time interval ZP after the switch-off time tOff (dashed line), as shown in Fig. 2B.

[0014] The following equation of motion can be given for a damped oscillation of a harmonic oscillator: x + 2 ■ D L ■ o) Q • x + Ü)O • x = a ext (t) (Eq. 1), which in the case of negligible velocity and without external acceleration can be approximated by: x + 6L>Q • x = 0 for x « 0 and a ext = 0 with a directly verifiable solution function: x~ COS(ÜJ0• t) = COS(2TT • f0■ t), where x is the deflection of the oscillation.

[0015] The given solution approximation (free oscillation) only applies when m • x = —k • x, i.e., after the initial holding stimulus (t>tOff) is switched off, and then only for a short time until the term —d ■ x also becomes quantitatively effective with increasing velocity x.

[0016] Figure 3B clearly illustrates the algorithmic determination of the parameters. The time interval ZP, indicated in Figure 2B as an extrapolation of the dashed curve, corresponds to the zero crossing of an undamped oscillator and thus to a quarter of an oscillation period: ZP ≤ ...

[0017] 4 4 / b'

[0018] Algorithm can estimate the natural frequency fO.

[0019] However, an unknown switch-off time makes it necessary to consider this as an unknown along with the natural frequency f0 and the damping value D. L to determine. Due to the high information content of the initial decay of the signal after time t. wA conventional mathematical algorithm tends to place the time tOff very close to the pronounced inflection point. Consequently, the barely visible and difficult-to-evaluate cosine-like transient period ZP is identified as being too short, and the algorithm assigns an excessively high natural frequency fO to the measured curve. Furthermore, a conventional algorithm attempts to fit the overall curve well, and this is only achieved by assigning an excessively high damping value DL to the further decline of the curve due to the simultaneously over-assigned natural frequency fO.

[0020] In many products and devices, the measurement of the switch-off time tOff is not implemented for various reasons. As a result, the calculation of the natural frequency fO and the attenuation value DL is subject to a high degree of measurement uncertainty.

[0021] Disclosure of the invention

[0022] According to a first aspect, the invention provides a method for determining characteristic parameters of an oscillator comprising the following steps:

[0023] Comparing an output signal of the oscillator with a threshold value to determine a comparison result;

[0024] Generating a feedback signal that is subjected to one of two different feedback levels depending on the determined comparison result;

[0025] Delaying the generated feedback signal with a delay time to establish a continuous periodic oscillation of the feedback oscillator; and

[0026] Determining the characteristic parameters of the oscillator based on sampled values ​​within a self-oscillation period of a triggered self-oscillation of the oscillator. The two feedback levels can be fixed accelerations or fixed voltage values ​​applied to the sensor capacitances to generate different acceleration forces, or two control parameters that produce two different, predefined force profiles on the sensor over time.

[0027] In one possible embodiment of the method, the specific characteristic parameters have a natural frequency or a resonance frequency (fO) and a quality factor (Q) of the oscillator.

[0028] The Q factor, or quality factor, is an important characteristic of oscillators. It indicates the ratio between the energy stored in the resonant circuit and the energy lost per oscillation period. The higher the quality factor (Q), the more stable the output frequency and phase of the oscillator's output signal. The inverse Q value of an oscillator represents a measure of the energy loss per period, or, in the case of an overdamped system, the damping DL.

[0029] The most fundamental parameter of an oscillator is the frequency f of its output signal. This is the repetition rate (cycle) of the signal output by the oscillator and is measured in Hertz (Hz).

[0030] A key characteristic of an oscillator is the constancy of its frequency.

[0031] Knowing these properties is equivalent to knowing the frequency dependence of the device's sensitivity and noise susceptibility, and can be used to determine corrections. Furthermore, determining the natural frequency fO allows for the prediction of the device's key trim parameters and can help avoid costly trimming procedures.

[0032] Knowing the quality factor Q or the damping value DL is an important parameter for assessing the integrity of the component, since a defective seal and the ingress of air, or even steam or water, can alter the damping value in the component and lead to degradation and corrosion. In one possible embodiment of the method, the threshold is set at half the oscillation range of the oscillator's output signal.

[0033] This reduces the noise disturbance.

[0034] In one possible embodiment of the method for determining characteristic parameters, the output signal of the oscillator is sampled during a sampling period extending over several cycles to generate sample values.

[0035] In one possible embodiment of the method for determining characteristic parameters of an oscillator, a period of oscillation of the feedback self-oscillating oscillator is determined on the basis of the sample values ​​generated during the sampling period.

[0036] Alternatively, the electrical circuit can store the switching times of the feedback circuit and thus make them available for later period determination of the sampled signal waveform.

[0037] The resulting period depends (in a non-simple functional relation) causally on the resonance frequency fO of the oscillator, the damping DL = 1 / (2*Q) and the threshold xThr and the delay tD, and is generally not identical to the resonance frequency fO.

[0038] In one possible embodiment of the method for determining characteristic parameters of an oscillator, the system is parameterized with fixed parameters xThr, tD, FBBias and FBGain (see Fig. 3B).

[0039] When the process is switched on, the oscillator moves from its resting state (xO=O). <xThr, v0=0) insbesondere bei höher gedämpften Systemen nach wenigen Oszillationen in eine periodische Oszillation um den Schwellwert xThr entsprechend der gesetzten festen Parameter (Einschwingen: Fig. 4A). Eine bestimmte Periodendauer stellt sich automatisch ein. Die Fig. 4.B zeigt einen Ausschnitt der Oszillationen (den sog. periodisch stationären Zustand PSS) nach Erreichen der stabilen Selbstoszillation. Bei einer möglichen Ausführungsform des Verfahrens zum Bestimmen charakteristischer Parameter eines Oszillators wird während der periodischen Selbstoszillation (PSS) jeweils immer wieder ein Schaltzeitpunkt erreicht, bei dem das Rückkopplungsniveau umgeschaltet wird, in Abhängigkeit eines Schwellenwert-Zeitpunktes, bei dem das Ausgangsignal des Oszillators den Schwellenwert erreicht, und der Verzögerungszeit

[0040] In one possible embodiment of the method for determining characteristic parameters of an oscillator, a deflection of the oscillator's output signal and a time derivative of the deflection of the output signal at the calculated switching time are determined as starting conditions for an iterative optimization by interpolating sampled values.

[0041] In one possible embodiment of the method for determining characteristic parameters of an oscillator, a transient response curve is calculated based on initial values ​​of characteristic parameters of the oscillator and the determined initial conditions.

[0042] In one possible embodiment of the method for determining characteristic parameters of an oscillator, a difference between a measured curve, which includes sampled values, and the calculated self-oscillation is minimized in an iterative optimization process by varying oscillator parameters in order to determine the associated optimized characteristic parameters of the oscillator.

[0043] In one possible embodiment of the method for determining characteristic parameters of an oscillator, the oscillator has a MEMS oscillator.

[0044] In one possible embodiment of the method for determining characteristic parameters of an oscillator, the feedback reference times are also stored, and synchronization for averaging individual periods then takes place based on the stored reference times.

[0045] Alternatively, synchronization can also be achieved through subsequent statistical data processing (post-processing) of a measurement curve. Synchronization can be accomplished through statistical post-processing of the raw measurement curve (e.g., by determining the inflection points = points of steepest transients = points of least noise).

[0046] According to a further aspect, the invention provides a circuit for determining characteristic parameters of an oscillator, comprising a comparator designed to compare a displacement output signal of the oscillator with a threshold value for determining a comparison result; a feedback signal generation unit designed to generate a feedback signal which, depending on the comparison result determined by the comparator, is subjected to one of two different feedback levels; and a feedback signal delay unit designed to delay the generated feedback signal by a delay time in order to establish a continuous periodic oscillation of the feedback oscillator.and with an evaluation unit designed to determine the characteristic parameters of the oscillator based on sampled values ​​within a self-oscillation period of a triggered self-oscillation of the oscillator.

[0047] The additional circuit engineering effort caused by the circuit according to the invention is extremely low and essentially consists in most applications of providing the comparator and the feedback signal delay unit, which in turn can be implemented by a simple flip-flop.

[0048] In one possible embodiment of the circuit for determining characteristic parameters of an oscillator, the feedback signal delay unit features a D flip-flop.

[0049] In one possible embodiment of the circuit for determining characteristic parameters of an oscillator, the characteristic parameters have a natural frequency (fO) and a quality factor (Q) of the oscillator.

[0050] Furthermore, according to another aspect, the invention provides a device with at least one oscillator, which is trimmed based on the characteristic parameters of the oscillator determined by the method according to the invention.

[0051] Preferred embodiments of the method and circuit according to the invention will be explained in more detail below with reference to the figures.

[0052] They show:

[0053] Fig. 1 A,1 B Transient signal waveforms of a weakly damped and a strongly damped oscillator to illustrate a technical challenge underlying the inventive method and circuit;

[0054] Fig. 2A, 2B Transient signal characteristics of a strongly damped harmonic oscillator shown in detail to explain a technical challenge underlying the inventive method and circuit, wherein Fig. 2B illustrates the signal characteristics of Fig. 2A using an oscillator response to a shutdown stage of a strongly damped oscillator;

[0055] Fig,3A,3B shows an embodiment of a circuit arrangement and a method according to the invention;

[0056] Figs. 4A, 4B, 4C are signal diagrams to illustrate the operation of a circuit arrangement and a method according to the invention; Fig. 5 is a signal diagram to illustrate the operation of a circuit arrangement and a method according to the invention;

[0057] Fig. 6 shows a section of the signal diagram according to Fig. 5;

[0058] Fig. 7A-7C Evaluation results to illustrate the accuracy achievable with the aid of the circuit arrangement and the method according to the invention.

[0059] Fig. 3B shows a block diagram illustrating a possible embodiment of a circuit arrangement proposed according to the invention for determining characteristic parameters of an oscillator OSZ, in particular a MEMS oscillator, with thermal noise as a realistic distortion, embedded in a positive feedback system.

[0060] Fig. 3A shows a flowchart illustrating a possible embodiment of the method according to the invention. The method serves to determine characteristic parameters of an oscillator (OSZ). In the illustrated embodiment, the method comprises several main steps.

[0061] In a first step S1, an output signal (x) of the oscillator OSZ is compared with a threshold value (x thr) to determine a comparison result.

[0062] In a further step S2, a feedback signal (FB) is generated, which is subjected to one of two different feedback levels (ACC+; ACC-) depending on the determined comparison result.

[0063] The two feedback levels (ACC+; ACC-) do not necessarily have to be fixed accelerations or forces. They could, for example, be two fixed control parameters, such as voltages, which would then generate two different ACC+ / ' (t) curves with different amplitudes. In this case, too, a periodic self-oscillation will occur that can be evaluated with respect to the oscillator parameters fO and DL. In a further step S3, the generated feedback signal FB is delayed by a delay time (tD) to establish a continuous periodic oscillation of the feedback oscillator OSZ.

[0064] In a further step S4, the characteristic parameters of the oscillator OSZ are determined based on sampled values ​​within a self-oscillation period of a triggered self-oscillation of the oscillator OSZ.

[0065] In one possible embodiment of the method, the characteristic parameters determined in step S4 have a natural frequency (fO) and a quality factor (Q) of the oscillator OSZ.

[0066] In one possible embodiment of the method for determining characteristic parameters, the output signal x of the oscillator OSZ is sampled during a sampling period extending over several cycles to generate sample values.

[0067] In one possible embodiment of the method for determining characteristic parameters of the oscillator OSZ, in step S4 a period (TRep) of an oscillation of the feedback self-oscillating oscillator OSZ is determined on the basis of the sample values ​​generated during the sampling period.

[0068] In one possible embodiment of the method for determining characteristic parameters of an oscillator, a switching time (tstart) at which the feedback level (ACC+; ACC-) is switched is calculated as a function of a threshold time (tthr) at which the output signal (x) of the oscillator OSZ reaches the threshold (xthr), the delay time (tD) and as a function of the determined period (TRep) of the oscillation of the feedback oscillator OSZ.

[0069] In one possible embodiment of the method for determining characteristic parameters of the oscillator OSZ, in step S4, a deflection of the output signal x of the oscillator OSZ and a time derivative (dx / dt) of the deflection of the output signal x at the calculated switching time (tstart) are determined as starting conditions for an iterative optimization by interpolating sample values.

[0070] In one possible embodiment of the method for determining characteristic parameters of the oscillator OSZ, a transient curve is calculated in step S4 based on initial values ​​of characteristic parameters of the oscillator OSZ and the determined initial conditions.

[0071] In one possible embodiment of the method for determining characteristic parameters of the oscillator OSZ, in an iterative optimization process in step S4, a difference between a measured curve, which includes sampled values, and the calculated transient curve is minimized by varying oscillator parameters in order to determine the associated optimized characteristic parameters of the oscillator OSZ.

[0072] In one possible embodiment of the method for determining characteristic parameters of an oscillator, the oscillator OSZ is a MEMS oscillator.

[0073] The invention further provides a circuit for determining characteristic parameters of an oscillator OSZ.

[0074] Fig. 3B shows a block diagram illustrating a possible embodiment of the circuit according to the invention.

[0075] The circuit features a comparator KOMP designed to compare a displacement output signal (x) of the oscillator OSZ with a threshold value (xthr) to determine a comparison result.

[0076] The circuit according to the invention further comprises a feedback signal generation unit FBEE, which is designed to generate a feedback signal (FB) which, depending on the comparison result determined by the comparator KOMP, is subjected to one of two different feedback levels (ACC+; ACC-);

[0077] The circuit according to the invention further comprises a feedback signal delay unit FBVE, which is designed to delay the generated feedback signal (FB) in time by a delay time tD in order to establish a continuous periodic oscillation of the feedback oscillator OSZ. In one possible embodiment of the circuit, the feedback signal delay unit FBVE is implemented using a D flip-flop as a simple implementation.

[0078] The circuit according to the invention further comprises an evaluation unit AE, which is designed to determine the characteristic parameters of the oscillator OSZ on the basis of sampled values ​​within a self-oscillation period of a triggered self-oscillation of the oscillator OSZ.

[0079] In one possible embodiment, the evaluation unit AE can be supplied with the signal waveform of the comparator KOMP (e.g., sig_FB) in addition to the sampled values, in order to, for example, detect the switching times independently of the xDefl signal. The AE block stores the data (here xDefl and sign_FB). In the operational system, this AE block represents the hardware component "memory" for the self-oscillation and the reference signals.

[0080] The block selfExc_acc_gE shown in Fig. 3B corresponds to the direct acceleration injection into the system. The selfExc_acc_gE block is a redundant unit that represents a variant of the AE block scaled to gravitational acceleration. The signals arriving there are not readily observable in reality. This storage device is intended solely for observing the acceleration injection during the development process of the method and is not typically present in practice, nor is it relevant to the method according to the invention. The output signal xDefl, however, is directly measurable.

[0081] The scope shown schematically in Fig. 3B serves solely for evaluation and visualization purposes in model and process development. The scope and the selfExc_acc_gE memory block are optional and can be omitted in other embodiments.

[0082] In one possible embodiment of the circuit according to the invention for determining characteristic parameters of an oscillator OSZ, the characteristic parameters of the oscillator OSZ determined by the circuit include a natural frequency (fO) and a quality factor (Q) of the oscillator OSZ. The invention further provides a device with at least one oscillator OSZ, which is trimmed based on the characteristic parameters of the oscillator determined by the method according to the invention.

[0083] The deflection ("deflection") The output signal x of the oscillator OSZ shown in Fig. 3B is used as the output signal x of the oscillator OSZ by the comparator KOMP with an adjustable or predefined threshold value x. Thr compared. The comparison results in the output signal x of the oscillator OSZ being subjected to one of two different feedback acceleration levels:

[0084] Finj = FB Bias ± FB Bias.

[0085] The feedback is delayed by a time tO and therefore transitions into a phase-span mode. Consequently, the system establishes a continuous periodic oscillation.

[0086] This effect is also known as the Barkhausen effect. Barkhausen's stability criterion defines the conditions under which an electrical circuit, consisting of an amplifier and suitable feedback, can oscillate independently. Utilizing the Barkhausen effect for the inventive method shown in Fig. 3A and the inventive circuit shown in Fig. 3B results in the generation of nearly precise periodic oscillations. The period of the oscillations is highly reproducible and is only minimally affected by thermal noise within the system.

[0087] The accuracy of the periodicity can be optimized, in particular, by placing the trigger event at a high-speed point in time, i.e., with a high time derivative of the displacement signal x, to make it least susceptible to noise injection. This can be controlled by adjusting the FBras gain, which determines the threshold xThr.

[0088] The details of the signal evaluation are described in detail below. The evaluation of the signals by the inventive method shown in Fig. 3A, using the circuit arrangement shown in Fig. 3B, allows the parameters of an oscillator (OSZ), in particular a MEMS oscillator, to be determined with a high accuracy of 0.1% or better.

[0089] An advantageous feature of the method according to the invention is that the accuracy can be achieved relatively easily without increased equipment and control effort by extending the measurement time or repeating it, and thereby simply and reproducibly increasing the number of oscillations. The storage requirements can optionally be minimized by simultaneously averaging the cycles.

[0090] In addition to the high achievable accuracy, a major advantage of the method according to the invention is that excellent conditions for the spectral analysis of the oscillator OSZ (highly reproducible vibrations) are created with minimal additional electronic effort.

[0091] All that is needed is a comparator KOMP and a time delay unit FBZE (e.g. a simple D flip-flop) and no complicated and expensive pattern generation, as is typically used for the spectral analysis of devices.

[0092] Diagram Fig. 4A shows the onset of the oscillation process when the feedback circuit is activated. The oscillator is initially below the threshold xthr (x=0, v=0) and is therefore set into displacement after the preset delay tD, passing the threshold xthr. Subsequently, a periodic oscillation is established by the feedback mechanism.

[0093] The diagrams in Fig. 4B and 4C show the scope output of the system shown in Fig. 3 for a long period (Fig. 4B) and for a zoom-out (Fig. 4C).

[0094] In one possible embodiment, the delay tO can be set to approximately 1 / 3 of the oscillation period TO = 1 / f0 of the undisturbed oscillator OSZ: tO = 1 / 3 (1 / fO). The apparent result is highly accurate periodicity, but some noise is present in the amplitude fluctuations. The oscillation shown in Figs. 4A and 4B is not sinusoidal.

[0095] Fig. 5 shows a single vibration and the main features for identifying its properties.

[0096] Preferably, the threshold xthr of the comparator KOMP is set sufficiently high to minimize noise interference at the threshold transition time tthr.

[0097] A system delay tD is assigned. Due to the feedback system shown in Fig. 3B, the system switches its power with a delay of tD after the threshold value x is undershot. thr where this is at time t thr _ occurs. This point in time forms the threshold transition time tthr-, as shown in Fig. 5.

[0098] After falling below or reaching the threshold value xthr at the threshold transition time tthr, the signal is below the threshold value xthr, so that the feedback system forces a positive acceleration ACC+= FB Bias+ FB Gain after the delay tD.

[0099] The duration of each force cycle ACC+ / - and the total duration TRep depend on the oscillator properties to be characterized (fO, DL=1 / (2*Q)) and the preset conditions (xthr, tD) and the type of force mechanism acting.

[0100] In the special case of constant forces ACC+ / -, the duration of each force cycle ACC+ / - is half the period of the observed oscillation TRep.

[0101] The period TRep is unknown. However, the period TRep of the oscillation can be determined from a sufficiently long sampling period (100 ms contains ~n-10). 2cycles) with correspondingly high accuracy (STRep ~1 / n*100* (1 / f0)).

[0102] Another way to determine the period is by evaluating the switching signals sig_FB, which are optionally stored for evaluation, shown in Fig. 3B. Using the determined values ​​of the period TRep of the oscillation and the threshold transition time t th r(+ / -) (via sig_FB, Fig. 3B) and the selected delay time tD.

[0103] For each period (i) identified in this way, the characteristic times (tStarti = tsw-, and tthr + / -) and the signal waveform can be determined as shown in Fig. 5. tStartj = t t hr,i-i+ tD

[0104] The wide track at the signal points in Fig. 5 is caused by the superposition of all ~n- 10 2The cycles of a complete measurement are realized and illustrate the signal disturbance caused by noise. The points and circles in Fig. 5 result from averaging the superimposed individual curves. This can be advantageously evaluated by averaging the errors of the individual measurements:

[0105] The next step of the method according to the invention consists of determining the characteristic parameters of the oscillator OSZ based on the sampled values ​​of a self-oscillation period.

[0106] The basic motion sequence follows the system differential equation of the system - in the simplest case, for example, the Newtonian equation of motion of the harmonic oscillator with parameters fö and quality factor Q= 1 / (2 • DL) respectively externally acting acceleration as in Eq. 1.

[0107] For the following explanations, it is initially assumed for simplification that the values ​​of the natural frequency f0= — 271 )0 and the quality factor Q = — 2DL are known.

[0108] The situation in Fig. 5 with known natural frequency f0 and with known quality factor Q for a given time interval [tstart, tstart + T ßep ] corresponds to an initial value calculation of the dynamics, which depends only on the displacement x and its time derivative at the start time [x, x] at t = t start depends (or on other state variables at the start time in the case of higher system complexity). If the mathematical system description is known (as a differential equation (DE) as in GI.1 or as a more complex system of DEs), then the motion sequence can be calculated numerically (with appropriate software). In special situations, the system of DEs can be reduced to an LTI system and known analytical basic solutions can be applied.

[0109] First, let us consider such a special case, which can also be advantageously implemented with MEMS oscillator components:

[0110] To do this, the external acceleration (Fig. 5) is adjusted to ACC~ = 0 by means of the force input FBias — FGain = 0. Only in the second segment of the period [t start , istart + ^Rep] then the deflection force FBias + FGain acts and in the subsegment [t st art> tsw ] is nur The restoring spring force is effective, and the known basic solutions of free damped harmonic oscillation apply there: at D L < 1 Eq. 2

[0111] (or corresponding exponential solutions for the supercritical damping case D) L > and the so-called asymptotic limiting case D L = 1.)

[0112] The coefficients of the solution are given by the following equations with the known descriptive quantities f0 and D. L = — 2Q of the oscillator differential equation linked:

[0113] "d2 = "o 2 - ("o • Ö ) 2

[0114] Y = a)0■ D L

[0115] By adjusting the sizes fo = ~ and D L Can the solution equation (GL 2) be applied to the measurement data in the segment of free oscillation [t]? st art> tsw ] in Fig- 5 by curve fitting (iterative regression) and thus obtain the values ​​of f0 and D belonging to the test subject from the experimental course. L = — .

[0116] 2Q

[0117] The method remains valid even if the above idealization cannot or is not desired. If the feedback forces are controlled via electrical voltages L7(t) across the sensor capacitances and the feedback is set in the start segment [t], the following applies: st art> tsw ] not a At zero (FBias — FGain 0) relatively complicated forces also act there and the describing equation changes into an inhomogeneous ODE with x-dependent “right side”:

[0118] In this case, no simple basic solutions are known; however, solution curves can be calculated numerically without any fundamental difference from the previous procedure. As before, the free parameters of the oscillator are varied, and the curve to be fitted is calculated numerically in each iteration, thus obtaining the values ​​of f0 and D corresponding to the device under test from the experimental data. L = — .

[0119] 2Q

[0120] The duration of the self-oscillation and the starting point of the force switching can be determined with high accuracy.

[0121] To better understand this, it can be helpful to visualize that the switching point of the force does not coincide with the peak maximum in Figures 5 and 6. Figure 6 illustrates the evaluation of the starting conditions for self-oscillation (at tStart). The switching occurs slightly earlier, and the maximum is reached later. This is because the momentum of the previous period leads to a further upward surge until the downward force of the period under consideration bends the curve downwards. This must be compensated for or taken into account by the initial condition, which is derived as described below.

[0122] Therefore, there are several possibilities for choosing the initial conditions:

[0123] A) It is possible to artificially interpolate the measurement curve to obtain the displacement x precisely at the force switching point tStart. Furthermore, additional measurement points or samples are available before the start of the self-oscillation. These earlier measurement points can be used and fitted with a smooth curve, e.g., a parabola, to determine the time derivative precisely at the force switching point tStart.

[0124] In this way, a very accurate and robust estimate of the velocity v= x' osc is obtained for the starting condition of the dynamics calculation after the force switching point tstart.

[0125] This velocity or time derivative of the deflection x is symbolically represented in Fig. 6 together with the previous measurement points as a tangential arrow.

[0126] In this way, highly precise ODE start conditions [x,x] are achieved at t = t start .

[0127] B) Alternatively, the starting point can also be defined as the maximum of the oscillation curve (t tart , M ax) and interpolate the displacement at this point from the measurement data. This variant has the convenient property that the initial velocity can then be set to zero. The following equally valid and precise starting condition is then obtained: [x, 0] at t tstart,Max'

[0128] The frequency f0 of the system is, as in connection with GI.1, again in the first measurement samples after the transition point x(t). start Max ) is encoded dominantly because x is approximately 0 there.

[0129] These initial conditions allow the dynamics of the system to be calculated after the force is switched to the new low and constant value at tStart.

[0130] The actual situation is that the values ​​of the natural frequency fO and the quality factor Q = 1 / (2 • DL) are unknown and are calculated from the sampled self-oscillation data. Since the starting conditions and the period of the oscillation Trep, or the time intervals of the force segments, can be derived with high accuracy from the measurement data, the values ​​can be calculated by iterative optimization in one possible embodiment of the method according to the invention.

[0131] For this purpose, plausible starting assumptions or starting values ​​fO-start, Qstart for the natural frequency fö and the quality factor Q= 1 / (2 • DL) are used and then the transient curve is calculated according to these starting values ​​fO-start and Qstart.

[0132] In Matlab, the corresponding commands are as follows: sol = ode45 (xvdot, tSpan, xvO, opts); xSol = deval (sol, tFDown, 1); with: and with: function xwtot = dxv_dt(t a xv^ fO Frjj DL Tryj acc) xvdot = [v; acc]; end

[0133] This represents a first-order system differential equation of the spring-mass-damper system, which conforms to the conventions of the numerical solver functions of typical numerical packages.

[0134] ACC is set to the constant 'ACC-' level, as shown in Fig. 5.

[0135] The external influences introduced into the dxv_st(...,acc) ODE function via the input acc can, in one possible embodiment, be fixed feedback levels ACC+ / -. However, they can also be fixed voltage values ​​applied to the sensor capacitances to generate different acceleration forces, or even two control parameters that result in two different, fixed force profiles on the sensor that are not constant over time in the two segments.

[0136] All these generalized options regarding force input can be incorporated into the dxv_st(...,acc) ODE function via function references and evaluated there, for example, depending on the sensor state elements x and v. All these options lead to evaluable self-oscillations and allow the determination of the sensor characteristic parameters (e.g., fO and DL). tFDown is the time axis of the experimental sampling points in the sampling range tSpan with the sampling period ts. = J / fs.

[0137] After integration, a solution curve xSol (f0, Try, DL, Try) is obtained which generally does not match the measured sampling curve xmeas (ts) over the time interval tSpan= [tStart, tStart+ TRep / 2], so that one does not immediately start with the correct PT2 or oscillator parameters { / 0,Try, DL, Try}.

[0138] The process is therefore subjected to iterative optimization, which minimizes the difference between the measured and the calculated curve by varying the parameters {f0, Try, DL, Try}:

[0139] The result is the characteristic values ​​of the oscillator { / 0, DL}. op tim under noise and measurement conditions.

[0140] The method of self-oscillation by positive delayed feedback used in the inventive method leads to a large number of oscillations within a relatively short sampling period, as shown, for example, in Fig. 4A, resulting in -290 periods in 100ms.

[0141] Since the period and switching times are known, the entire sampling process can be decomposed into its individual periods and superimposed on the same time axis. This is shown in the preceding figures and is illustrated in Figures 4 and 5 as a superposition of all individual periods.

[0142] There are various synchronization options for superimposing and averaging the individual periods. As outlined previously, the times of threshold I-value exceedance and fall below the threshold can be captured with minimal effort, and the switching times of the feedback force can be determined via the set delay (using t). start i = t^ hr i-1 + t D ) and save it along with the measurement curve.

[0143] If, for technical reasons, no stored reference times are available, synchronization can also be achieved retrospectively using the measurement curve, for example by determining the inflection point of the downslope curves and synchronizing the individual curves for superposition at that point. The inflection point is characterized by the steepest derivative and thus represents the time with the least disturbance from noise.

[0144] In the method according to the invention, synchronization therefore preferably takes place at the time of the stored switching times, but can also be carried out statistically based on the data. This noise variation is unavoidable and can also be observed as amplitude variation in the time series graph of the sampling in Fig. 4A. The circles in Fig. 5 represent the calculated curves with the optimal identification of the optimally derived oscillator parameters {f0, DL}.

[0145] Fig. 7A corresponds to Fig. 5 and shows the evaluation period of the self-oscillation including realistic noise distortion effects for an accelerometer with {f0 = 3580 Hz, DL = 1.2}. The accuracy when repeating a measurement with different noise levels is in the range of 0.1%.

[0146] Conventional methods based on the evaluation of individual step transients exhibit repeatability accuracies of only 5–10% in the range of supercritically damped oscillators. Transient step experiments with additional timing acquisition can improve accuracy down to the 1% range—however, they require an extended electronic circuit to access the timing information in the MHz digital domain and a synchronization effort for sampling and force in the digital domain.

[0147] Statistical stabilization of step experiments through averaging necessitates repetition of the measurement processes at sufficiently long intervals, ensuring a steady-state resting condition during each shutdown pulse (mathematical consistency). Within the same experimental time, the self-oscillation method presented here allows for a significantly higher number of averaging operations. Mathematical consistency is ensured in the case of self-oscillation by the precise mathematical determination of the initial conditions of the respective evaluation period [x, x] at t = t. start , as described above.

[0148] However, in the method according to the invention, any high-frequency synchronization effort can be avoided and only a minimum of additional circuit technology is required (namely a comparator KOMP and a delay unit FBVE), which does not have to meet any timing conditions.

[0149] The evaluation shows that accuracies in the range of 0.1% can be achieved, which is even sufficient to derive the sensitivity tuning of the devices, which previously required time-consuming and costly direct physical stimuli and correspondingly bulky and expensive devices.

[0150] The method according to the invention proves successful for various {f0, DL} ranges, which also include underdamped regimes with equally good accuracy.

[0151] Fig. 7B shows an example distribution of the determined parameters fO and Q from the individual periods of self-oscillation as bar charts in a histogram representation. The individual periods represent an observed total variation of 15 Hz for fO. The visual width of the bell curve corresponds approximately to 6 sigma of the Gaussian distribution. Therefore, a sigma value of 15 Hz / 6 ~ 3 Hz is inferred, which corresponds to a measurement accuracy of 0.1%. Thus, evaluating all superimposed curves of a 100 ms measurement cycle yields the mean value of fO ~ 3522 Hz, visible in the histogram, with a sigma uncertainty of 3 Hz ~ 0.1% of the true component value.

[0152] The correlation diagram in Fig. 7C still shows the misleading correlation between the two oscillator parameters (fO, DL). However, this correlation is only present within a very narrow range. As demonstrated in previous studies, the effect on the unavoidable link between time (e.g., t) is start ) and frequency parameters f0,DL ) is, however, significantly minimized with the aid of the method according to the invention. This is due to the fact that the error variation of the reference times (t) start , t~^) is minimized by the inherent periodicity and self-oscillation and the large number of evaluated periods.

[0153] The results show a systematic shift of the {f0, DL} values ​​of approximately 1% towards lower values. This can be attributed to the limited sampling rate of the device, which in turn causes a slight low-pass effect. This effect can be corrected computationally, as its origin is well understood.

[0154] In addition to its application in MEMS accelerometers, the inventive method and circuit arrangement can also be used for various analysis tasks in MEMS gyroscopes. Here, too, a significant advantage lies in the fact that the method can generate the desired periodic behavior simply by introducing basic comparator and delay structures.

[0155] This is even more relevant for gyroscopes, since in a gyro environment a large part of the signal conditioning and processing is already implemented in the circuits, and further increases in complexity are generally avoided. Gyroscope oscillators are implemented as high-Q / low-damping oscillators. As a result, the self-driven oscillation is a nearly ideal sinusoidal oscillation, and data evaluation is in some aspects even simpler than in the case of an overdamped accelerometer. In the case of a gyroscope oscillator, the method according to the invention can even be carried out without an additional delay element.

[0156] The method according to the invention has been described as a characterization method for supercritical accelerometers. However, the method according to the invention can also be used as a basis for characterizing high-quality resonators. The method according to the invention can also be used with more complex and expensive MEMS gyroscopes.

[0157] Due to the simplicity of the required circuitry and the fact that today's sensor modules generally have sufficiently powerful microcontrollers, the method according to the invention is also particularly suitable for integrated installation in a module and for autonomous operation throughout the device's lifetime. Therefore, the method according to the invention can be used to implement trimming corrections for lifetime effects or more precise consistency and integrity checks of a module during operation.

Claims

TI Patent claims:

1. Procedure for determining characteristic parameters of an oscillator (OSC) with the following steps: Comparing (S1) an output signal (x) of the oscillator (OSC) with a threshold value (x thr) to determine a comparison result; Generating (S2) a feedback signal (FB) which, depending on the determined comparison result, is subjected to one of two different feedback levels (ACC+; ACC-); Delaying (S3) the generated feedback signal (FB) with a delay time (tD) to establish a continuous periodic oscillation of the feedback oscillator (OSZ); and Determine (S4) the characteristic parameters of the oscillator (OSC) using samples within a self-oscillation period of a triggered self-oscillation of the oscillator (OSC).

2. Method for determining characteristic parameters of an oscillator according to claim 1, wherein the characteristic parameters comprise a natural frequency (fO) and a quality factor (Q) of the oscillator.

3. Method for determining characteristic parameters of an oscillator according to claim 1 or 2, wherein the feedback reference times are stored and synchronization for averaging individual periods is performed based on the stored reference times, or wherein synchronization for averaging individual periods is performed by subsequent statistical data processing of the measurement curve.

4. Method for determining characteristic parameters of an oscillator according to one of claims 1 to 3, wherein the output signal (x) of the oscillator (OSZ) is sampled during a sampling period extending over several cycles to generate sample values.

5. Method for determining characteristic parameters of an oscillator according to claim 4, wherein a period (TRep) of an oscillation of the feedback self-oscillating oscillator (OSZ) is determined on the basis of the sample values ​​generated during the sampling period.

6. Method for determining characteristic parameters of an oscillator according to one of claims 1 to 5, wherein a switching time (tstart) at which the feedback level (ACC+; ACC-) is switched is calculated as a function of a threshold time (tthr) at which the output signal (x) of the oscillator (OSZ) reaches the threshold (xthr), the delay time (tD) and as a function of the determined period (TRep) of the oscillation of the feedback oscillator (OSZ).

7. Method for determining characteristic parameters of an oscillator according to claim 6, wherein by interpolation of sampled values ​​a deflection of the output signal (x) of the oscillator (OSZ) and a time derivative of the deflection of the output signal (x) at the calculated switching time (tstart) are determined as starting conditions for an iterative optimization.

8. Method for determining characteristic parameters of an oscillator according to claim 7, wherein a transient response curve is calculated based on initial values ​​of characteristic parameters of the oscillator (OSC) and the determined initial conditions.

9. Method for determining characteristic parameters of an oscillator according to claim 8, wherein in an iterative optimization process a difference between a measured curve comprising sampled values ​​and the calculated transient curve is minimized by varying oscillator parameters in order to determine the associated optimized characteristic parameters of the oscillator (OSC).

10. Method for determining characteristic parameters of an oscillator according to Claim 1 or 9, wherein the oscillator (OSC) comprises a MEMS oscillator.

11. Circuit for determining characteristic parameters of an oscillator, comprising a comparator (KOMP) designed to compare a displacement output signal (x) of the oscillator (OSZ) with a threshold value (xthr) to determine a comparison result; a feedback signal generation unit (FBEE) designed to generate a feedback signal (FB) which, depending on the comparison result determined by the comparator (KOMP), is subjected to one of two different feedback levels (ACC+; ACC-); and a feedback signal delay unit (FBVE) designed to delay the generated feedback signal (FB) in time with a delay time tD to establish a continuous periodic oscillation of the feedback oscillator (OSZ).and with an evaluation unit (AE) designed to determine the characteristic parameters of the oscillator (OSC) based on sampled values ​​within a self-oscillation period of a triggered self-oscillation of the oscillator (OSC).

12. Circuit for determining characteristic parameters of an oscillator according to claim 11, wherein the feedback signal delay unit (FTR) comprises a D flip-flop or another suitable delay circuit.

13. Circuit for determining characteristic parameters of an oscillator according to claim 11, wherein the characteristic parameters include a natural frequency (fO) and a quality factor (Q) of the oscillator (OSZ).

14. Device with at least one oscillator (OSC) which is trimmed according to one of claims 1 to 10 based on the characteristic parameters of the oscillator determined by the method according to the invention.

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