- Email: [email protected]
Control System Design Study for a Micromachined Accelerometer
Copyright © IFAC New Trends in Design of Control Systems, Smolenice, Slovak Republic, 1997
CONTROL SYSTEM DESIGN STUDY FOR A MICROMACHINED ACCELEROMETER Michael Kraft, Christopher P. Lewis, Thomas G. Hesketh
Coventry University, School of Engineering Priory St., Coventry, CV1 5FB, UK Tel.: +44 1203 838842, Fax: +44 1203838949 email: [email protected]. uk
Abstract: In this paper an analogue, closed loop accelerometer based upon a bulk-micromachined, capacitive sensing element is described. The sensing element is embedded in a closed loop control system using electrostatic forces as a reset mechanism. A mathematical model is derived which allows to employ conventional linear control theory. The validity of the model is discussed. Particular attention is paid to the controller design; here a PID controller is used. The simulated results are compared to measurements on the actual hardware prototype. Keywords: Accelerometers, Microsystems, Control Theory, Closed Loop Sensor, Modelling
of the seismic mass. One method of reducing these nonlinearities and improving the performance is to use the sensing element in a closed loop control system in which a reset force is applied to the seismic mass to keep it at the central position between the electrodes. This force has to counterbalance the accelerating force; the magnitude of which provides an accurate measure for the acceleration. In micromachined sensing elements the dimensions are as such that electrostatic
1. INTRODUCTION The market for micromachined silicon accelerometers is continuously increasing, the majority of which are relatively simple open loop devices with a variety of mechanical structures. In this paper a 'servo' or closed loop transducer is described. A bulk-micromachined sensing element is depicted in fig. I. It typically consists of two fixed outer electrodes and a movable centre plate or seismic mass which acts as a common centre electrode of two capacitors electrically connected in series. Any acceleration causes the seismic mass to move from its central position, this results in a differential imbalance in capacitance which is proportional to the acceleration for small deflections of the seismic mass. Such an open loop accelerometer has limited performance in terms of bandwidth, linearity and dynamic range . Additionally, the device suffers from inherent nonlinear effects due to the damping, (Zhang et. al., 1992, Marco et. al., 1993), and the electrostatic forces which are introduced by the signal pick-off; these effects increase in magnitude with the deflection
Fig. 1: A typical bulk-micromachined, capacitive sensing element.
139
forces can be used to generate the restoring force. In the design approach chosen in this work the seismic mass or centre plate had no floating charge being effectively grounded; with this arrangement it is difficult to generate a linear feedback signal because of the dependency of the electrostatic force on the square of the voltage and the inverse square of the gap between the electrodes. The most common solution to this problem is to superimpose two electrostatic forces on the seismic mass which are in balance (van Kampen et. al., 1994, Zimmermann et. al., 1995, Boxenhorn and Greiff, 1990, Warren, 1991). However, this leads only to a linear feedback relationship if very small displacements of the seismic mass are assumed. A further complication is that the micromachined, capacitive sensing element had only three connections (top and bottom electrodes and seismic mass), care had to be taken that the feedback signals did not interact with the excitation signals required for signal pick-off. Generally two arrangements are possible: separation in either the time or the frequency domains, (Burstein and Kaiser, 1996). Usually, for an analogue, closed loop accelerometer the latter approach is employed; this relies on the use of a high frequency carrier applied to the outer plates of a capacitive half bridge, the centre plate of which is used for the signal pick-off, (Kroemer et. al. , 1995, Kuehnel and Sherman, 1994). The magnitude of the subsequent signal is proportional to the imbalance in capacitance, which is, in turn, proportional to the deflection of the seismic mass from its rest position for small motion. This signal is then subjected to a phase-sensitive demodulator and a lowpass filter. The output signal of the filter provides a measure of the imbalance in capacitance.
arrangement produces an electrostatic force on the seismic mass as in eq.(2). VI
(~sinwt - V +V
)2 2 B F A(-------2 2 0 (d-x) 1
- E
VI
(-~sinwt + V + V ) 2 B
2
F
)
... (2)
(d +X )2
where 2d is the nominal gap between the outer electrodes, x is the deflection of the seismic mass from the central position, A is the area of the plates and EO is the absolute permittivity of free space. Making the assumption that the seismic mass deflections are small, it can be shown that the feedback force is proportional to the applied voltage, VF, as in eq.(3), (van Kampen et. al. , 1994).
In the simplest case the feedback voltage, VF, is proportional to the output voltage of the low-pass filter and, in turn, the position of the seismic mass. The bias voltage is chosen according to the desired dynamic range of the accelerometer; to support a force produced by an acceleration 'a' the minimum bias voltage required is given by eq.(4).
It is obvious that the design of a closed loop analogue accelerometer is by no means a trivial task, each of the problems outlined above has to be carefully addressed. In this paper the particular problems associated with the development of an analogue servo accelerometer are considered. For brevity only the small signal conditions are discussed in detail here, however particular problems arising under large signal operation are identified.
VB =
~ mgd: EA o
... (4)
m is the mass of the centre plate. Eq. (4) simplifies to VB/ ..[a= 8.7 V/..[m/s for the accelerometer under development. It was then possible to plot the net electrostatic force
acting on the seismic mass as a function of the deflection for differertt bias voltages as in fig. 2. The solid line shows the accurate solution of eq. (2), the dashed line the linearized solution according to eq. (3). The pick-off gain used was 1.6 V/J.lm (this value was obtained from measurement of the actual hardware), a proportional gain factor of 6 was assumed.
2. MA THEMATIeAL MODELLING In order to provide suitable feedback force, Fel, a bias voltage, VB, an excitation voltage, VI , and the feedback signal, VF were applied to the electrodes as in eq. (1). VI - VB + V F -VI + VB + V F
It is obvious that the electrostatic force has a useful
linear range for deflections of approximately ±0.5 Ilm . If the seismic mass is deflected further, the
... (1)
Here, the resultant voltages applied to the top and bottom plates are VTP and VBP respectively. This
140
It was now possible to determine the open loop and closed loop transfer functions, the feedback gain being abbreviated by KF=2eOA VB/d0 2 :
t. to
Open Loop TF : F OL(S)
=K
K s 2+K s +K mH (s) D P I po F ms 3 +bs 2+ks
. . . (Sa)
where HF(S) is the filter transfer function .
-0.' ., '---~--~-~--~-~----'
·3
·2
·t
0 Dcnccli<>"II'"q
Closed loop TF :
Fig. 2: Net electrostatic force on the seismic mass for different bias voltages and proportional feedback. The solid line shows the exact solution according to eq. (2), the dashed line the approximated solution for small deflections according to eq. (3).
F CL(s)
= Kpom(KDs2 +Kps+KI) /(ma2s5+
(ma 1 +ba 2)s 4 +(m +ba 1 +ka 2)s 3 + (b +ka 1 +Kr"po f( K D)s 2+(k+Kr f("poK p)s+Kr'po f( K) I ... (Sb)
where ai , a2 are filter parameters. feedback relationship becomes nonlinear, ultimately the feedback changes polarity and the seismic mass is attracted to one of the outer electrodes. This is a fundamental problem and a serious drawback of an analogue, closed loop system. In a real system such a situation could arise if either of the two drive amplifiers supplying the voltage to the top and bottom electrodes saturate due to an acceleration magnitude larger than the dynamic range of the accelerometer; this could occur as a result of a shock condition, (Kraft, 1997).
Flcqu c,KY IIW/I)
If small deflections of the seismic mass are assumed the mathematical model for the sensing element can be assumed to be a second order system, (Lewis et. al., 1996), since the nonlinear damping effects due to viscous flow or ' squeeze film' damping can be neglected. As a first approximation the signal pick-off can be assumed to be a simple gain factor. A second order Butterworth low-pass filter was used with a corner frequency of 600 Hz. A PlO controller was introduced for compensation, the output signal of which constituted the output signal of the accelerometer. Fig.3 shows the mathematical model of the analogue, closed loop accelerometer where m, b and k are sensing element parameters (b: damping coefficient, k: spring constant); KP, KI and KD are the controller settings.
Fig. 4: Open loop Bode-plot for the analogue accelerometer for small deflections and for different settings for the proportional and integral gain. From inspection of the closed loop transfer function it can be seen that the smaller the parameters of the sensing element (spring constant k and damping factor b) the more significant the influence of the PID parameters become, hence for closed loop operation a micromachined sensing element with small spring and damping constants is desirable. I
An accelerometer having a dynamic range of 2 g, (which necessitated a bias voltage, VB = 12.3 V according to eq. (4) ) has a Bode-plot as in fig. 4; this shows the simulated results with proportional gain constant KP values of2, 6 and 10. The integral gain constant was set to Kp/ I 0 for each run and zero differential gain was assumed. Inspection of fig. 4 reveals that the -3 dB cut-off frequency is about 500 radls (80 Hz). Additionally, it can be seen that the phase margin is sufficient large for all three simulation runs, consequently the closed loop system is predicted to be stable.
forte
Fig. 3: Small signal model for an analogue, closed loop accelerometer.
141
•
~I =
f
·'10
• -
.,..
..
jlq
I.'
I.'
Fig. 5: Simulated, closed loop frequency response for the analogue accelerometer for small deflections and for different settings of the proportional and integral gain.
Fig. 6: Measured frequency response of the analogue, closed loop accelerometer. The dashed line shows the simulated response for KP = 6 and KI = 0.6.
For verification the corresponding simulated, closed loop frequency responses are shown in fig. 5. It can be seen that the bandwidth of the accelerometer was increased considerably. As expected the higher the proportional gain setting, the higher the bandwidth and the flatter the frequency response. A proportional gain of 2 resulted in unsatisfactory performance, this was too low to counteract the integral gain required for accurate steady state performance. For a proportional gain of 6 the bandwidth was increased to 465 Hz and for gain of 10 the bandwidth was 357 Hz, this was due to a resonant peak. Obviously, derivative gain was not required here.
square of the frequency. Consequently, it is difficult to generate sufficient acceleration at low frequencies and at higher frequencies care has to be taken that the transducer does not saturate. The frequency response analyser used (Voltech TF 2000) related the signal from the accelerometer under test to the reference transducer to get a measured frequency response directly referring to acceleration .. An obvious discrepancy between the simulated and measured results is that in the simulation a slight decrease of gain is predicted at low frequencies due to the integral action of the controller where as the measured frequency response shows an increase of gain. Since the acceleration magnitudes are very small at low frequencies, the increase in gain at low frequencies is believed to originate from the way the measurement was taken. A mechanical resonance of the mounting of the sensing element and the reference accelerometer on the vibration table can also cause mid-range measurement errors.
3. MEASUREMENT RESULTS The micromachined accelerometer was tested dynamically by mounting it, together with a reference, Sundstrand Q-flex accelerometer on a vibration table. The reference accelerometer used had a flat frequency response up to 300 Hz. The measured, closed loop response is shown in fig .6 together with the simulated response for KP = 6 and KI = 0.6, the entire frequency range from 0 to 300 Hz lies within the 3 dB window, consequently the cut-off frequency is increased from 80 Hz in the open loop condition to above 300 Hz. Generally, the bandwidth of the device was satisfactory for the requirements specified in this research project.
Further experiments with different PID gain settings were carried out, these did not result in any improvement in performance. Another discrepancy between the hardware implementation and the mathematical model was that if the gain settings of the PI controller were increased considerably, the system started to exhibit ~n unstable oscillation; such behaviour is believed to originate from effects which are not considered in the mathematical model, e.g. cross-axis sensitivity, non-ideal parallel electrodes etc.
The measured frequency response agreed quite well with the theoretically predicted performance. With the test method used it is difficult to quote exact figures for the degree of agreement since the measured frequency response is only valid up to the bandwidth of the reference accelerometer. A further drawback this test method is the use of a vibration table without closed loop control of acceleration; the acceleration magnitude does not stay constant over the frequency range tested but increases with the
4. CONCLUSIONS In this paper a mathematical model for an accelerometer based upon a micromachined, capacitive sensing element is developed. In order to apply conventional control theory the assumption that the seismic mass stays close at the central position between
142
the electrodes has to be made. There was good agreement between measured and theoretical results. A closed loop control system, as expected, showed improved performance, the bandwidth was increased from 80 Hz to above 300 Hz. Under certain conditions the polarity of the feedback signal can change, this drawback can only be overcome by applying a different control strategy, e.g. use of oversampling conversion or sigma-delta modulation. Such a device not only results in an improved stability but also yields a direct digital output signal, (Lewis and Kraft, 1996, Smith et. ai., 1994, de Coulon et. al., 1993, Henrion et. ai., 1990, Lu et. aI., 1995). However, the design of the control system cannot be approached any longer with conventional control theory as these systems have an inherently nonlinear, discrete data structure.
Mathematical Model for a Micromachined Accelerometer. Trans. Inst. of Meas. and Control., Vol. 18, No.2, pp. 92-98, 1996. Lewis, c.P. and Kraft, M. Simulation of a Micromachined Digital Accelerometer in SIMULINK and PSPICE. UKACC 1nl. Con! on Control" Voll , pp. 205 - 209, Exeter UK, 1996. Lu, c., Lemkin, M. and Boser, B. A monolithic surface micromachined accelerometer with digital output. IEEE J. Solid-State Circuits, Vol. 30, No. 12, pp. 1367-1373, 1995. Marco, S., Samitier, J., Ruiz, 0., Herms, A. and Morante, J. Analysis of electrostatic damped piezoresistive silicon accelerometer. Sensors and Actuators, A 37-38, pp. 317-322, 1993.
REFERENCES Smith, T., Nys, 0., Chevroulet, M., de Coulon, Y. and Degrauwe, M. E1ectro-mechanical sigmadelta converter for acceleration measurements. IEEE International Solid-State Circuits Conference, pp. 160-161 , 1994.
Boxenhorn, B. and Greiff, P. Monolithic silicon accelerometer. Sensors and Actuators, A21-A23, pp. 273-277, 1990. Burstein, A. and Kaiser, W. J. Mixed analog-digital highly sensitive sensor interface circuit for lowcost microsensors. Sensors and Actuators, A52, pp. 193-197,1996. de
van Kampen, R. P., Vellekoop, M., Sarro, P. and Wolffenbuttel, R. F. Application of electrostatic feedback to critical damping of an integrated silicon accelerometer. Sensors and Actuators, A 43, pp. 100-106, 1994.
Coulon, Y., Smith, T., Hermann, J., Chevroulet, M. and Rudolf, F. Design and test of a precision servoaccelerometer with digital output. 7th Int. Corif. Solid-State Sensors and Actuators (Transducer '93), Yokohama, pp. 832835, 1993.
Warren, K. Electrostatically force-balanced silicon accelerometer. J. of the Inst. ofNav., V. 38, No. 1, pp. 91-99, 1991. Zhang, L., Cho, D., Shiraishi, H. and Trimmer, W. Squeeze-film damping in micromechanical systems. ASME, Micromechanical Systems, DSCVol. 40, pp. 149-160,1992.
Henrion, W., Disanza, L., lp, M., Terry S. and Jerman, H. Wide dynamic range direct digital accelerometer. IEEE Solid State Sensor and Actuator Workshop, pp. 153-157, Hilton Head Island, 1990.
Zimmermann, L., Ebersohl, J., Le Hung, F., Berry, J. P., Baillieu, F., Rey, P., Diem, B., Renard, S., Caillat, P. Airbag application: a microsystem including a silicon capacitive accelerometer, CMOS switched capacitor electronics and true self-test capability. Sensors and Actuators, A 46-47, pp. 190-195, 1995.
Kraft, M. Closed loop digital accelerometer employing oversampling conversion. Coventry University, Ph.D. dissertation, 1997. Kroemer, 0., Fromhein, 0., Gemmeke, H., Kiihner, T., Mohr, J. and Strohrmann, M. High-precision readout circuits for LIGA acceleration sensors. Sensors and Actuators, A46-A47, pp. 196-200, 1995 . Kuehnel, W. and Sherman, S. A surface micromachined silicon accelerometer with onchip detection circuitry. Sensors and Actuators, A4S, pp. 7-16,1994. Lewis, C. P., Kraft, M. and Hesketh, T. G.
143
CONTROL SYSTEM DESIGN STUDY FOR A MICROMACHINED ACCELEROMETER Michael Kraft, Christopher P. Lewis, Thomas G. Hesketh
Coventry University, School of Engineering Priory St., Coventry, CV1 5FB, UK Tel.: +44 1203 838842, Fax: +44 1203838949 email: [email protected]. uk
Abstract: In this paper an analogue, closed loop accelerometer based upon a bulk-micromachined, capacitive sensing element is described. The sensing element is embedded in a closed loop control system using electrostatic forces as a reset mechanism. A mathematical model is derived which allows to employ conventional linear control theory. The validity of the model is discussed. Particular attention is paid to the controller design; here a PID controller is used. The simulated results are compared to measurements on the actual hardware prototype. Keywords: Accelerometers, Microsystems, Control Theory, Closed Loop Sensor, Modelling
of the seismic mass. One method of reducing these nonlinearities and improving the performance is to use the sensing element in a closed loop control system in which a reset force is applied to the seismic mass to keep it at the central position between the electrodes. This force has to counterbalance the accelerating force; the magnitude of which provides an accurate measure for the acceleration. In micromachined sensing elements the dimensions are as such that electrostatic
1. INTRODUCTION The market for micromachined silicon accelerometers is continuously increasing, the majority of which are relatively simple open loop devices with a variety of mechanical structures. In this paper a 'servo' or closed loop transducer is described. A bulk-micromachined sensing element is depicted in fig. I. It typically consists of two fixed outer electrodes and a movable centre plate or seismic mass which acts as a common centre electrode of two capacitors electrically connected in series. Any acceleration causes the seismic mass to move from its central position, this results in a differential imbalance in capacitance which is proportional to the acceleration for small deflections of the seismic mass. Such an open loop accelerometer has limited performance in terms of bandwidth, linearity and dynamic range . Additionally, the device suffers from inherent nonlinear effects due to the damping, (Zhang et. al., 1992, Marco et. al., 1993), and the electrostatic forces which are introduced by the signal pick-off; these effects increase in magnitude with the deflection
Fig. 1: A typical bulk-micromachined, capacitive sensing element.
139
forces can be used to generate the restoring force. In the design approach chosen in this work the seismic mass or centre plate had no floating charge being effectively grounded; with this arrangement it is difficult to generate a linear feedback signal because of the dependency of the electrostatic force on the square of the voltage and the inverse square of the gap between the electrodes. The most common solution to this problem is to superimpose two electrostatic forces on the seismic mass which are in balance (van Kampen et. al., 1994, Zimmermann et. al., 1995, Boxenhorn and Greiff, 1990, Warren, 1991). However, this leads only to a linear feedback relationship if very small displacements of the seismic mass are assumed. A further complication is that the micromachined, capacitive sensing element had only three connections (top and bottom electrodes and seismic mass), care had to be taken that the feedback signals did not interact with the excitation signals required for signal pick-off. Generally two arrangements are possible: separation in either the time or the frequency domains, (Burstein and Kaiser, 1996). Usually, for an analogue, closed loop accelerometer the latter approach is employed; this relies on the use of a high frequency carrier applied to the outer plates of a capacitive half bridge, the centre plate of which is used for the signal pick-off, (Kroemer et. al. , 1995, Kuehnel and Sherman, 1994). The magnitude of the subsequent signal is proportional to the imbalance in capacitance, which is, in turn, proportional to the deflection of the seismic mass from its rest position for small motion. This signal is then subjected to a phase-sensitive demodulator and a lowpass filter. The output signal of the filter provides a measure of the imbalance in capacitance.
arrangement produces an electrostatic force on the seismic mass as in eq.(2). VI
(~sinwt - V +V
)2 2 B F A(-------2 2 0 (d-x) 1
- E
VI
(-~sinwt + V + V ) 2 B
2
F
)
... (2)
(d +X )2
where 2d is the nominal gap between the outer electrodes, x is the deflection of the seismic mass from the central position, A is the area of the plates and EO is the absolute permittivity of free space. Making the assumption that the seismic mass deflections are small, it can be shown that the feedback force is proportional to the applied voltage, VF, as in eq.(3), (van Kampen et. al. , 1994).
In the simplest case the feedback voltage, VF, is proportional to the output voltage of the low-pass filter and, in turn, the position of the seismic mass. The bias voltage is chosen according to the desired dynamic range of the accelerometer; to support a force produced by an acceleration 'a' the minimum bias voltage required is given by eq.(4).
It is obvious that the design of a closed loop analogue accelerometer is by no means a trivial task, each of the problems outlined above has to be carefully addressed. In this paper the particular problems associated with the development of an analogue servo accelerometer are considered. For brevity only the small signal conditions are discussed in detail here, however particular problems arising under large signal operation are identified.
VB =
~ mgd: EA o
... (4)
m is the mass of the centre plate. Eq. (4) simplifies to VB/ ..[a= 8.7 V/..[m/s for the accelerometer under development. It was then possible to plot the net electrostatic force
acting on the seismic mass as a function of the deflection for differertt bias voltages as in fig. 2. The solid line shows the accurate solution of eq. (2), the dashed line the linearized solution according to eq. (3). The pick-off gain used was 1.6 V/J.lm (this value was obtained from measurement of the actual hardware), a proportional gain factor of 6 was assumed.
2. MA THEMATIeAL MODELLING In order to provide suitable feedback force, Fel, a bias voltage, VB, an excitation voltage, VI , and the feedback signal, VF were applied to the electrodes as in eq. (1). VI - VB + V F -VI + VB + V F
It is obvious that the electrostatic force has a useful
linear range for deflections of approximately ±0.5 Ilm . If the seismic mass is deflected further, the
... (1)
Here, the resultant voltages applied to the top and bottom plates are VTP and VBP respectively. This
140
It was now possible to determine the open loop and closed loop transfer functions, the feedback gain being abbreviated by KF=2eOA VB/d0 2 :
t. to
Open Loop TF : F OL(S)
=K
K s 2+K s +K mH (s) D P I po F ms 3 +bs 2+ks
. . . (Sa)
where HF(S) is the filter transfer function .
-0.' ., '---~--~-~--~-~----'
·3
·2
·t
0 Dcnccli<>"II'"q
Closed loop TF :
Fig. 2: Net electrostatic force on the seismic mass for different bias voltages and proportional feedback. The solid line shows the exact solution according to eq. (2), the dashed line the approximated solution for small deflections according to eq. (3).
F CL(s)
= Kpom(KDs2 +Kps+KI) /(ma2s5+
(ma 1 +ba 2)s 4 +(m +ba 1 +ka 2)s 3 + (b +ka 1 +Kr"po f( K D)s 2+(k+Kr f("poK p)s+Kr'po f( K) I ... (Sb)
where ai , a2 are filter parameters. feedback relationship becomes nonlinear, ultimately the feedback changes polarity and the seismic mass is attracted to one of the outer electrodes. This is a fundamental problem and a serious drawback of an analogue, closed loop system. In a real system such a situation could arise if either of the two drive amplifiers supplying the voltage to the top and bottom electrodes saturate due to an acceleration magnitude larger than the dynamic range of the accelerometer; this could occur as a result of a shock condition, (Kraft, 1997).
Flcqu c,KY IIW/I)
If small deflections of the seismic mass are assumed the mathematical model for the sensing element can be assumed to be a second order system, (Lewis et. al., 1996), since the nonlinear damping effects due to viscous flow or ' squeeze film' damping can be neglected. As a first approximation the signal pick-off can be assumed to be a simple gain factor. A second order Butterworth low-pass filter was used with a corner frequency of 600 Hz. A PlO controller was introduced for compensation, the output signal of which constituted the output signal of the accelerometer. Fig.3 shows the mathematical model of the analogue, closed loop accelerometer where m, b and k are sensing element parameters (b: damping coefficient, k: spring constant); KP, KI and KD are the controller settings.
Fig. 4: Open loop Bode-plot for the analogue accelerometer for small deflections and for different settings for the proportional and integral gain. From inspection of the closed loop transfer function it can be seen that the smaller the parameters of the sensing element (spring constant k and damping factor b) the more significant the influence of the PID parameters become, hence for closed loop operation a micromachined sensing element with small spring and damping constants is desirable. I
An accelerometer having a dynamic range of 2 g, (which necessitated a bias voltage, VB = 12.3 V according to eq. (4) ) has a Bode-plot as in fig. 4; this shows the simulated results with proportional gain constant KP values of2, 6 and 10. The integral gain constant was set to Kp/ I 0 for each run and zero differential gain was assumed. Inspection of fig. 4 reveals that the -3 dB cut-off frequency is about 500 radls (80 Hz). Additionally, it can be seen that the phase margin is sufficient large for all three simulation runs, consequently the closed loop system is predicted to be stable.
forte
Fig. 3: Small signal model for an analogue, closed loop accelerometer.
141
•
~I =
f
·'10
• -
.,..
..
jlq
I.'
I.'
Fig. 5: Simulated, closed loop frequency response for the analogue accelerometer for small deflections and for different settings of the proportional and integral gain.
Fig. 6: Measured frequency response of the analogue, closed loop accelerometer. The dashed line shows the simulated response for KP = 6 and KI = 0.6.
For verification the corresponding simulated, closed loop frequency responses are shown in fig. 5. It can be seen that the bandwidth of the accelerometer was increased considerably. As expected the higher the proportional gain setting, the higher the bandwidth and the flatter the frequency response. A proportional gain of 2 resulted in unsatisfactory performance, this was too low to counteract the integral gain required for accurate steady state performance. For a proportional gain of 6 the bandwidth was increased to 465 Hz and for gain of 10 the bandwidth was 357 Hz, this was due to a resonant peak. Obviously, derivative gain was not required here.
square of the frequency. Consequently, it is difficult to generate sufficient acceleration at low frequencies and at higher frequencies care has to be taken that the transducer does not saturate. The frequency response analyser used (Voltech TF 2000) related the signal from the accelerometer under test to the reference transducer to get a measured frequency response directly referring to acceleration .. An obvious discrepancy between the simulated and measured results is that in the simulation a slight decrease of gain is predicted at low frequencies due to the integral action of the controller where as the measured frequency response shows an increase of gain. Since the acceleration magnitudes are very small at low frequencies, the increase in gain at low frequencies is believed to originate from the way the measurement was taken. A mechanical resonance of the mounting of the sensing element and the reference accelerometer on the vibration table can also cause mid-range measurement errors.
3. MEASUREMENT RESULTS The micromachined accelerometer was tested dynamically by mounting it, together with a reference, Sundstrand Q-flex accelerometer on a vibration table. The reference accelerometer used had a flat frequency response up to 300 Hz. The measured, closed loop response is shown in fig .6 together with the simulated response for KP = 6 and KI = 0.6, the entire frequency range from 0 to 300 Hz lies within the 3 dB window, consequently the cut-off frequency is increased from 80 Hz in the open loop condition to above 300 Hz. Generally, the bandwidth of the device was satisfactory for the requirements specified in this research project.
Further experiments with different PID gain settings were carried out, these did not result in any improvement in performance. Another discrepancy between the hardware implementation and the mathematical model was that if the gain settings of the PI controller were increased considerably, the system started to exhibit ~n unstable oscillation; such behaviour is believed to originate from effects which are not considered in the mathematical model, e.g. cross-axis sensitivity, non-ideal parallel electrodes etc.
The measured frequency response agreed quite well with the theoretically predicted performance. With the test method used it is difficult to quote exact figures for the degree of agreement since the measured frequency response is only valid up to the bandwidth of the reference accelerometer. A further drawback this test method is the use of a vibration table without closed loop control of acceleration; the acceleration magnitude does not stay constant over the frequency range tested but increases with the
4. CONCLUSIONS In this paper a mathematical model for an accelerometer based upon a micromachined, capacitive sensing element is developed. In order to apply conventional control theory the assumption that the seismic mass stays close at the central position between
142
the electrodes has to be made. There was good agreement between measured and theoretical results. A closed loop control system, as expected, showed improved performance, the bandwidth was increased from 80 Hz to above 300 Hz. Under certain conditions the polarity of the feedback signal can change, this drawback can only be overcome by applying a different control strategy, e.g. use of oversampling conversion or sigma-delta modulation. Such a device not only results in an improved stability but also yields a direct digital output signal, (Lewis and Kraft, 1996, Smith et. ai., 1994, de Coulon et. al., 1993, Henrion et. ai., 1990, Lu et. aI., 1995). However, the design of the control system cannot be approached any longer with conventional control theory as these systems have an inherently nonlinear, discrete data structure.
Mathematical Model for a Micromachined Accelerometer. Trans. Inst. of Meas. and Control., Vol. 18, No.2, pp. 92-98, 1996. Lewis, c.P. and Kraft, M. Simulation of a Micromachined Digital Accelerometer in SIMULINK and PSPICE. UKACC 1nl. Con! on Control" Voll , pp. 205 - 209, Exeter UK, 1996. Lu, c., Lemkin, M. and Boser, B. A monolithic surface micromachined accelerometer with digital output. IEEE J. Solid-State Circuits, Vol. 30, No. 12, pp. 1367-1373, 1995. Marco, S., Samitier, J., Ruiz, 0., Herms, A. and Morante, J. Analysis of electrostatic damped piezoresistive silicon accelerometer. Sensors and Actuators, A 37-38, pp. 317-322, 1993.
REFERENCES Smith, T., Nys, 0., Chevroulet, M., de Coulon, Y. and Degrauwe, M. E1ectro-mechanical sigmadelta converter for acceleration measurements. IEEE International Solid-State Circuits Conference, pp. 160-161 , 1994.
Boxenhorn, B. and Greiff, P. Monolithic silicon accelerometer. Sensors and Actuators, A21-A23, pp. 273-277, 1990. Burstein, A. and Kaiser, W. J. Mixed analog-digital highly sensitive sensor interface circuit for lowcost microsensors. Sensors and Actuators, A52, pp. 193-197,1996. de
van Kampen, R. P., Vellekoop, M., Sarro, P. and Wolffenbuttel, R. F. Application of electrostatic feedback to critical damping of an integrated silicon accelerometer. Sensors and Actuators, A 43, pp. 100-106, 1994.
Coulon, Y., Smith, T., Hermann, J., Chevroulet, M. and Rudolf, F. Design and test of a precision servoaccelerometer with digital output. 7th Int. Corif. Solid-State Sensors and Actuators (Transducer '93), Yokohama, pp. 832835, 1993.
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