POLITECNICO DI MILANO MSC COURSE - MEMS AND MICROSENSORS - 207/208 E08 Gyroscope Drive Design Paolo Minotti 26/0/207 PROBLEM We have to design the electronic circuit needed to sustain the oscillation of the drive axis of a vibratory MEMS gyroscope. The system should guarantee a maximum sensitivity variation of ±.5%, worst case, within the automotive temperature range, i.e., from 45 C to +25 C. The drive resonator is both actuated and detected in single-ended mode. The primary loop of the drive oscillator is shown in Fig.. It is constituted by a TCA-based front-end, a differentiator, and a hard-limiter with a dedicated supply voltage. The complete drive oscillator (with both the primary loop and the AGC loop) is shown in Fig. 2. The differential gain of the INA has the following expression: G I N A = + 49.4 kω R I N A. The variable-gain concept is implemented by means of acting on the supply voltage of the hard-limiter. The mechanical and electrical parameters of the gyroscope are listed in Table.
Mechanical Symbol Value Drive resonance frequency f rd 20 000 Hz Drive Q-factor @ 300 K Q d0 8000 Internal mass m i.5 nkg External mass m e 2.5 nkg Process thickness h 24 µm Gap g.8 µm Number of drive comb fingers N CF 40 Target drive displacement amplitude x a0 5 µm Electronics Rotor bias voltage V DC 5 V Supply voltage V DD ±5 V Amplifier voltage output swing V O ±4 V Minimum capacitance C min 0.2 pf Primary loop Differentiator feedback resistance R DI F,2 300 kω Secondary (AGC) loop Rectifier gain G REC LPF gain G LPF LPF resistance R LPF 3 MΩ Table : Parameters of the gyroscope. We are asked to.... Dimension the parameters of the primary loop of the drive-mode oscillator (Fig. ), in order to obtain the target displacement amplitude, x a0, at the reference temperature (300 K). 2. Calculate, considering the primary loop only (Fig. ), the maximum percentage variation of the drive displacement amplitude, and evaluate the required compensation factor. 3. Dimension the parameters of the secondary (AGC) loop of the drive-mode oscillator (Fig. 2), considering the requirement on the maximum sensitivity variation. 4. Dimension the low-pass filter of the control loop. INTRODUCTION A vibratory MEMS gyroscope can be modeled as a resonator along the drive axis and as an accelerometer along the sense axis, that detects the sinusoidal motion induced by the Coriolis 2
Figure : Primary loop of the drive oscillator. force. Regarding the primary loop (the electronic circuit needed to start and sustain the oscillation of a resonator at a precise oscillation frequency) of the drive axis, in principle, it is possible to use the same circuit topology we studied for MEMS resonators. In this example, however, we will use a slightly different topology. Indeed, the primary loop is composed by a TCA-based front-end, that senses drive motion and provides an output voltage proportional to the drive displacement, followed by a differentiator, needed to obtain an overall 360 phase shift at the exact resonance frequency of the drive resonator; regarding the drive actuation circuitry, instead of having an hard-limiter followed by the de-gain stage, the voltage amplitude reduction is obtained by supplying the hard-limiter itself with a dedicated supply voltage, V HL. QUESTION We want to calculate the actuation voltage amplitude, v a, that has to be applied to the actuation electrode in order to have the desired displacement amplitude, x a0, at the reference temperature. We know that, if a MEMS resonator is actuated at resonance, assuming a smallsignal hypothesis on the actuation voltage, the displacement amplitude is x a = Q k η d v a. In our case, Q is the quality factor of the drive resonator, k is the spring constant of the drive mode, η d is the drive-actuation transduction coefficient, and v da is the drive-actuation sinusoidal voltage amplitude: x a = Q d k d η da v da. 3
Figure 2: Drive oscillator with AGC. We know that η da = V DC C da x, where V DC is the DC voltage between the rotor and the stator electrode, and C da / x is the drive-actuation capacitance variation per unit displacement. With the data provided in Table, and C da x = 2ɛ 0hN CF g = 9.44 ff/µm, C da η da = V DC x = 47.2 0 9 VF/m. Note that the resonator is symmetric. Hence, η d = η da = η dd = 47.2 0 9 VF/m. We can then calculate the spring constant of the drive mode of the gyroscope: k d = ( 2πf rd ) 2 md = ( 2πf rd ) 2 (me + m i ) = 63.2 N/m. 4
And we can finally evaluate the amplitude of drive-actuation sine-wave voltage required to obtain the target displacement amplitude: v da = x aη da Q d k d = 836 mv. Remember that, with the circuit topology we are using, the voltage signal applied to the drive electrode of the gyroscope is the output of the hard-limiter, which is a square-wave. As the voltage amplitude we just evaluated is referred to a sine-wave, we must take into account the 4/π factor: v da,sq = v da = 0.66 V. 4/π This means that the hard-limiter should be biased (V HL ) between 0.66 V and +0.66 V. Note that v da = 0.836 V is much lower than the rotor-stator DC voltage (5 V). The smallsignal hypothesis is thus valid. What would happen if v da were too high? There are at least two solutions: (i) one solution consists in adopting the push-pull configuration, that, as we learned in previous classes, does not require the small-signal hypothesis on the ac voltage; (ii) another solution consists in re-designing the gyroscope, increasing the number of comb fingers used for the drive actuation: in this way, the required ac voltage is lower; of course V HL = v da,sq cannot be too low, otherwise the hard-limiter cannot be properly biased. We can evaluate the motional current amplitude flowing through the drive-detection port of the drive resonator, i ma = x a η d ( 2πfrd ) = 29.6 na, and the drive-detection capacitance variation amplitude, C dda = x a C dd x = 47.2 ff. With a TCA-based front-end, the output voltage amplitude can be calculated as V TC A,out,a = G TC A C dda = TTC A ( frd ) Cdda = V DC C F C dda, or, equivalently, V TC A,out,a = ( 2πfrd ) CF i ma. A good design approach to choose the gain 2 of the TCA is to maximize it: in this way, noise introduced by subsequent stages can be neglected. Which is the maximum gain of an amplifier? It s the highest value that prevents any saturation/clamping of the output signal. Since the amplifier voltage output swing (V O ) is ±4 V, given the previously calculated i ma and C dda, the desired feedback capacitance, Ĉ F, can be found as Ĉ F = V DC V O C dda = 59 ff, Remember that the RLC model is valid if the ac voltage is much smaller than the DC voltage. 2 With gain we mean the magnitude of the transfer function at resonance. 5
which, unfortunately, is lower than the minimum one (200 ff). We will thus choose the minimum one, C F = 200 ff. This implies an output voltage amplitude of V TC A,out,a = ( 2πfrd ) CF C dda =.8 V. For proper TCA behavior, the feedback pole frequency should be, at least, two decades before the oscillation frequency, < 2πR F C F 00 f rd, that corresponds to a (minimum) feedback resistor of 4 GΩ. With this choice, we guarantee a small error (lower than deg) on the ideal phase shift of the TCA (90 ). Regarding the differentiator stage, we can choose its gain, G DI F = TDI F ( frd ), following the same approach as before, i.e., we can force an output signal whose amplitude is 4 V (V O ). Hence, G DI F = V out,a,max V TC A,out,a = 3.38. If the poles are suitably designed, the differentiator gain can be expressed as G DI F = 2πf rd C DI F R DI F 2. As the value of the feedback resistor, R DI F 2, is given, 300 kω, the input capacitance value is C DI F = G DI F 2πf rd R DI F 2 = 89 pf. Regarding the dimensioning of the other two components, we can choose their values in order to have the two poles at frequencies much higher than f rd, namely twice. Remember, the higher the pole frequency, the more precise is the phase shift of the stage. Assuming to place both poles at f p,di F = 2 MHz: C DI F 2 = 2πf p,di F R DI F 2 = 265 ff, and R DI F = 2πf p,di F C DI F = 885 Ω. 6
QUESTION 2 The sensitivity of a MEMS gyroscope is proportional to the displacement amplitude of the drive-axis oscillation, x da : S x da. In absence of an amplitude-control loop, i.e., in presence of a fixed drive-actuation voltage amplitude, v da, the drive displacement amplitude depends on the quality factor, x da = Q k η dav da. In presence of temperature variations, the quality factor changes and, in turn, the drivedisplacement amplitude changes. As seen in a previous class, the quality factor variations in front of temperature variations can be roughly approximated with the following linearization, Q Q = T 2 T. Since T T = 70 K 300 K 56%, the quality factor variation is about 28%. Since the drive displacement amplitude is linear with the quality factor, the sensitivity variation within the whole temperature operating range can be estimated as S S = x da = Q x da Q = T 2 T = 28%. A 28% sensitivity variation is too high for a high-performance product. For this reason, an AGC loop must be implemented in the drive oscillator of a gyroscope. As the the maximum allowable variation is.5%, an open-loop-actuated drive resonator would not fulfill the requirement. We need to introduce a compensation (control) loop, to reduce the displacement amplitude variations by a factor (at least) 28 / 3 0. The displacement amplitude as a function of temperature variations within the whole temperature range is reported in Fig. 3. QUESTION 3 To control the drive displacement amplitude, an AGC (automatic gain control) loop is introduced in most vibratory MEMS gyroscopes. The control circuit is designed as a negative feedback loop. Its raw schematic is reported in Fig. 4. We can consider the voltage difference V D V REF as the error signal of our negative feedback loop 3 : ε = V REF V D = V REF +G loop. If the loop gain is sufficiently high, the error is 0, hence V D V REF. In other words V D behaves as the virtual ground of our negative feedback loop, as depicted in Fig. 4, on the left. 3 For sake of simplicity the loop gain, G loop, will always be considered as a positive number, i.e., will be evaluated without considering the sign of the loop gain. 7
Figure 3: Open-loop drive displacement amplitude as a function of temperature variations. Figure 4: AGC as a negative feedback-loop. If the gain from the drive displacement amplitude, x da, to V D is known and constant, V D = K AGC x da, choosing V REF is equal to choosing a certain reference drive displacement amplitude, x da,re f : x da,re f = V REF K AGC, and the loop would force x da x da,re f, as shown in Fig. 4, on the right. Remembering that ε = V REF +G loop, and, equivalently, we can make the following observations. x da = x da,re f G loop +G loop, 8
Infinite loop gain: if the loop gain is infinite, the error signal is exactly zero, and x da = x da,re f, independently from temperature variations. Finite and constant (temperature-independent) loop gain: if the loop gain is finite, the error signal is finite... but constant: hence, the drive displacement amplitude will be different from the target one, but this new value will not depend from temperature variations. Finite and temperature-dependent loop gain: if the loop gain is finite, the error signal is finite (at 300 K) and, in addition, it will change if temperature changes: hence, the drive displacement amplitude will be different from the target one, and, in addition, it will depend from temperature variations. He must now evaluate the temperature dependence of our AGC loop gain. As one may intuitively guess, the loop gain, G loop, turns out to be directly proportional to Q d (we will demonstrate this later). We are thus in the case of a temperature-dependent loop gain: G loop Q d (T ). Given this linear proportionality, we can linearize the temperature dependance of the loop gain as G loop ( T ) = G loop0 2 G loop0 We can now express the temperature variations of x da with the AGC loop: which can be re-written as T 0 T. x da ( T ) = x da,re f G loop ( T ) +G loop ( T ), G loop0 2 x da ( T ) = x da,re f +G loop0 2 G loop0 T T 0 G loop0 T 0 T, As for x 0, we can write a x + a x a + a + x ( + a) 2, x da ( T ) = x da,re f G loop0 +G loop0 x da,re f G loop0 T 2 T 0 ( ) 2, +Gloop0 Defining x da0 = x da ( T = 0) = x da,re f G loop0 +G loop0, 9
we can re-write x da ( T ) as x da0 T 2 T x da ( T ) = x da0 0, +G loop0 As expected, the new (with the AGC) relative displacement amplitude variation is now reduced by a factor +G loop0 : ( ) x da T = = Q. x da0 2 +G loop0 Q 0 +G loop0 T 0 The effects of temperature variations on the drive displacement amplitude (hence on the sensitivity of the gyroscope) are reduced by a factor +G loop0, as expected from automation theory, when the AGC loop is introduced. As the required compensation factor was 0, we will design the AGC loop with a target loop gain of 20. The drive displacement behavior as a function of temperature with the control loop is reported in Fig. 5. Figure 5: Drive displacement amplitude as a function of temperature variations, for different situations. Note that the previously defined x da0 represents the drive displacement amplitude at the reference temperature. With a loop gain equal to 20, x da0 = 4.76 µm, i.e., 5% lower than the target value, 5 µm. We are now ready to evaluate the loop gain. Referring to Fig. 6, the loop gain we are going to calculate is the one marked as G loop,agc. How can we calculate the loop gain? As usual, we can (i) cut the loop (possibly at the output of a low-impedance voltage source, or at the input of an high-impedance amplifier), (ii) inject 0
an amplitude variation at the oscillation signal, and (iii) evaluate the quantity that returs to where the loop was cut. Figure 6: Complete drive-mode oscillator. highlighted. The signal path of the amplitude loop is Figure 7: Full-wave rectifier.
The added AGC circuitry is basically composed by three stages:. A full-wave rectifier (FWR), that, given a sine-wave at its input, outputs the fully-rectified waveform: as shown in Fig. 7. V FW R,in (t) = V a sin(ω 0 t) V FW R,out (t) = V a sin(ω 0 t), 2. A low-pass filter (LPF), that outputs a DC voltage proportional to the mean value of its input signal. In our case, if the input is the output would be V LPF,in (t) = V a sin(ω 0 t), V LPF,out (t) = mean[v a sin(ω 0 t) ] = 2 π V a. In this example, it is implemented with a single-pole active low-pass filter, but we are free to choose other topologies. 3. A differential gain stage, that is implemented with an instrumentation amplifier (INA), that compares the LPF output voltage with the AGC reference voltage. The difference between these two signals is the aforementioned error signal of the AGC loop, ε. The output of the INA is fed to the positive supply voltage of the hard-limiter and is inverted (multiplied for ) and fed to the negative supply of the hard-limiter. With this topology, given a certain INA output voltage, V out,i N A, the drive actuation signal is V da (t) = V I N A,out sq(ω 0 t), that describes a square-wave signal at ω 0, that toggles between +V I N A,out and V I N A,out. We are now ready to calculate the loop gain. Let s imagine to cut the loop at the INA output, and to apply a certain signal, v t. The power supply of the hard-limiter are thus ±v t. Hence, the drive actuation signal is a square wave between ±v t. The sinusoidal signal amplitude of the drive actuation voltage is thus 4/πv t. At resonance, drive displacement motion induces a motional current, whose amplitude is i ma = R md 4 π v t, where R md is the drive resonator motional resistance, R md = b ( ) d 2πfrd m η 2 = Q d d η 2 = 28.2 MΩ. d The TCA output voltage amplitude is thus v TC A,out,a = 4 ( ) i ma = ( ) 2πfrd CF 2πfrd CF R md π v t. 2
Note that this is the amplitude of the TCA output voltage, i.e., of the FWR input, which is a sinusoidal signal. The complete expression of the signal is v TC A,out (t) = v FW R,in (t) = v TC A,out,a sin(ω o t) = ( 2πfrd ) CF R md 4 π v t sin(ω o t). The LPF output, which is the virtual gound voltage, is 2/π times the amplitude of this signal, hence: V D = v LPF,out = 2 π v TC A,out,a = 2 4 ( ) π 2πfrd CF R md π v t. This signal is amplified by the gain of the INA, G I N A, and we are back at where we cut the loop: 2 4 v f = G I N A ( ) π 2πfrd CF R md π v t. As expected, the return signal is negative, i.e., we set up a negative feedback loop, as desired. The AGC loop gain is thus v G loop,agc = f = G 2 4 I N A ( ) π 2πfrd CF R md π. If we want a loop gain of 20, the INA gain, G I N A, should be v t G I N A = G loop,agc 2 4 ( ) π 2πfrd CF R md π = 7.49, that can be obtained with an INA gain resistance, R I N A, of 49.4 kω R I N A = = 3.00 kω. G I N A Note that as anticipated before. G loop,agc R md Q d, QUESTION 4 We have to dimension the low-pass filter of the AGC loop. As the gain of the LPF is unitary, the transfer function of the LPF can be expressed as V LPF,out (s) V LPF,in (s) = + sr LPF C LPF. From a spectral point of view, the signal at the input of the FWR is a pure tone at ω rd ; on the other hand, the rectified signal will have a DC tone, whose amplitude is 2/π times the amplitude of the input sine-wave, and other harmonics, as shown in Fig. 8. The only harmonic 3
Figure 8: Loop gain of the AGC. that we want is the one at DC, proportional to the drive displacement amplitude. The role of the filter is thus to filter out all the other harmonics. As temperature variations have a very narrow bandwidth (temperature variations are very slow), we could place the pole at a verylow frequency. How much low? As in every negative feedback loop, the choice of the poles is related with stability issues. Before, we evaluated the DC loop gain, G loop0, i.e., the loop gain at DC. What about the loop gain at other frequencies? Regarding the frequency behavior of the AGC loop, one can demonstrate that all the stages can be modeled as frequency-independent-gain stages... except the MEMS resonator and the LPF. In fact, when dealing with actuation-voltage amplitude variations, the resonator can be modeled (see Appendix) as a base-band equivalent system whose transfer function is I a (s) V a (s) = R m + s 2Q. Be careful! This is not the ratio between the motional current and the actuation voltage! This is the ratio between the motional current amplitude and the actuation voltage amplitude, when actuated at resonance. The new frequency-dependent AGC loop gain can be thus written as 2 4 G loop,agc = G I N A ( ) π 2πfrd CF R md π ω o + s 2Q ω rd + sr LPF C LPF. 4
This is a two-poles system, where the first pole frequency, the one due to the MEMS, is given, f p = 2π 2Q d ω rd = f rd 2Q d =.25 Hz, while the second one, f p2 depends on the LPF design. Figure 9: Loop gain of the AGC. As in every negative feedback loop, to ensure the stability of the loop, the second pole of the loop gain should be equal (or higher) to the gain-bandwidth product (GBWP) of the loop. In other words, f p2 G loop0 f p = 25 Hz. This is the minimum pole frequency of the LPF. With C LPF = 2π (G loop0 f p ) RLPF = 2.2 nf, a 45 phase margin is ensured. The whole situation is graphically represented in Fig. 9. Choosing a value of 2.2 nf for the capacitance, all the components of the circuit are dimensioned. The primary loop is designed in such a way that the circuit oscillates at the resonance frequency of the drive resonator, while the secondary (AGC) loop is designed to suitably reject the effects of temperature variation on the drive displacement amplitude (thus on the sensitivity) according to the given specification. 5
APPENDIX: BASE-BAND MODEL OF A MASS-SPRING-DAMPER SYSTEM FOR ACTUATION AMPLITUDE VARIATION EFFECTS A mass-spring-damper system is modeled by the differential equation which can be rewritten in Laplace domain as X (s) mẍ + bẋ + kx = F. F (s) = k Let s assume an external force, F, in the form + s Q ω o + s2 ω 2 o F (t) = F a (t)sin(ω o t), where F a is the slowly varying amplitude of the force and ω o is the resonance frequency of the structure. As the resonator is actuated at resonance, the displacement will have a sinusoidal waveform, as well, that can be expressed as x (t) = x a (t)cos(ω o t).. We can write, and Hence, ẋ (t) = ẋ a (t)cos(ω o t) + x a (t)ω o sin(ω o t), ẍ (t) = ẍ a (t)cos(ω o t) + 2ẋ a (t)ω o sin(ω o t) + x a (t)ω 2 o cos(ω ot). m ( ẍ a (t)cos(ω o t) + 2ẋ a (t)ω o sin(ω o t) + x a (t)ω 2 o cos(ω ot) ) + +b ( ẋ a (t)cos(ω o t) + x a (t)ω o sin(ω o t))+ +k ( x a (t)cos(ω o t)) = F a (t)sin(ω o t), which can be re-written as [2ẋ a (t)mω o + x a (t)bω o F a (t)]sin(ω o t) = 0, [ mẍa (t) + x a (t)mω 2 o bẋ a (t) kx a (t) ] cos(ω o t) = 0 We can re-write the first equation as 2ẋ a (t)mω o + x a (t)bω o = F a (t), which can be rewritten in Laplace domain: 2smω o X a (s) + bω o X a (s) = F a (t), X a (s)[2smω o + bω o ] = F a (t), 6
Q X a (s) F a (s) = = k 2smω o + bω o + s 2Q. ω o This is the base-band model of a MEMS resonator when dealing with slow variations of the actuation signal amplitude. Following the same approach, one can demonstrate that the baseband-equivalent motional impedance of the resonator can be modeled as: I a (s) V a (s) = R m + s 2Q. This frequency dependent behavior should be taken into account when dealing, e.g., with the AGC stability, i.e., when the transfer function of the amplitude of the signal is of concern. ω o 7