E09 Gyroscope Sense Design

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1 POLITECNICO DI MILANO MSC COURSE - MEMS AND MICROSENSORS - 07/08 E09 Gyroscope Sense Design Paolo Minotti 3/0/07 In this class we will learn how to design the readout electronics of a MEMS gyroscope, with particular focus on noise performance. We will calculate the sensitivity at different nodes of the readout chain, and we will learn how to input-refer the different noise sources as noise equivalent rate densities. PROBLEM We are asked to design the sense readout electronics of a mode-split capacitive MEMS gyroscope. The parameters of the device are reported in Table. C P is the parasitic capacitance between the front-end input and ground, due to chip pads, wire bonding, and interconnections. The topology of the circuit is shown in Fig..

2 Structure Process thickness h µm Gap g.9 µm External mass m e nkg Internal mass m i 8 nkg Sense axis Sense resonance frequency f r s 9800 Hz Sense damping coefficient b s µn/m/s) Parallel-plate length L PP 39 µm Parallel-plate cells N PP 5 Rotor-to-stator DC voltage V DC 0 V Drive axis Drive displacement amplitude x da 6 µm Drive resonance frequency f rd 9000 Hz Drive damping coefficient b s 00 nn/m/s) Electronics TCA feedback capacitance C F pf Parasitic capacitance C P 0 pf INA resistance R G AI N 4.94 kω INA gain G I N A kω / R G AI N TCA s op-amp voltage noise PSD s n,oav,v 4 nv/ Hz TCA s op-amp current PSD s n,oai,i fa/ Hz INA op-amp MOS overdrive voltage V OV 00 mv INA op-amp MOS γ coefficient γ /3 Table : Parameters of the gyroscope. We are asked to.... Calculate the sensitivity of the gyroscope, in terms of sense capacitance variation per unit angular rate, TCA output voltage variation per unit angular rate, and INA output voltage per unit angular rate.. Evaluate the intrinsic resolution of the sensor, due to thermo-mechanical noise only. 3. Size the feedback resistance of the TCA-based front-end, in order to obtain a wellbalanced system in terms of noise performance. 4. Calculate the voltage noise PSD of the INA required not to worsen the resolution of the sensor, and the needed bias current of the input op-amps of the INA, in order to fulfill this requirement.

3 Figure : Sense readout circuit. 3

INTRODUCTION What is a sensor? A sensor is a system whose input is a certain quantity, I, and whose output, O, is another quantity a voltage, a digital word, a frequency variation,... ): O = O I I.

What is the resolution of a sensor? The resolution of a sensor can be defined as the minimum detectable signal, i.e., as the input signal that produces an output signal equal to the output noise standard deviation Fig.

4 INTRODUCTION What is a sensor? A sensor is a system whose input is a certain quantity, I, and whose output, O, is another quantity a voltage, a digital word, a frequency variation,... ): O = O I I. We can define the sensitivity, S, as the output variation per unit input variation: S = O I. Figure : Resolution, minimum detectable signal, input-referred noise density. What is the resolution of a sensor? The resolution of a sensor can be defined as the minimum detectable signal, i.e., as the input signal that produces an output signal equal to the output noise standard deviation Fig. ). Said in other words, the minimum detectable signal is the input signal for which the output signal-to-noise ratio SNR) is equal to. We know that given a certain output noise power spectral density PSD), S n,o, the associated noise standard deviation, σ n,o, can be expressed as If S n,o is white, ) σ n,o = S n,o f d f. σ n,o = S n,o B, where B is the noise equivalent bandwidth of the system. To infer I min, we can equate the output SNR to : O SNR = O = I I O = I I σ n,o σ n,o Sn,o B =. 4

5 The minimum detectable signal, I min, is thus I min = σ n,o O I Sn,o B =. O I Let s make a practical example, considering our gyroscope. In noiseless conditions, any angular rate would produce a certain Coriolis force that can be detected. When considering noise, however, the produced Coriolis force applied to the proof mass, in order to be detected, should be high enough to overcome, e.g., the thermo-mechanical noise which is unavoidably applied to the proof mass. Another example: the produced Coriolis force induces a sense displacement variation, that produces a current flowing through the sense port, that is integrated onto the TCA-based front-end, that outputs a sinusoidal voltage, whose amplitude, in order to be detected, should overcome the noise present at the output of the TCA which is due to both the thermo-mechanical noise of the resonator and the noise introduced by the electronics). From the previous analysis, it is evident, that, the lower the bandwidth, the lower is the minimum detectable signal, i.e., the better is the resolution. It is then more fair, when reporting the resolution, to evaluate it not considering the integrated rms noise, but in terms of noise density. In other words, the resolution is more often defined as the input equivalent noise PSD that produces an output equivalent noise PSD equal to the noise PSD that we effectively have at the output. Since the resolution can be easily calculated as ) O S n,o = S n,i, I S n,o S n,i = ) O. It is more intuitive and convenient to express noise PSDs reporting the square root of the PSD: s n,i = S n,i = S n,o ) O = s n,o. O I I Remember that, if we are, e.g., considering voltage noise, S n,v is expressed in V /Hz, while s n,v is expressed in V/ Hz. I We usually refer to resolution using adjectives such as better/worse, rather than higher/lower, as it might be misleading: one may refer to a low-resolution sensor as a low minimum detectable signal sensor; but, as we all know, generally speaking, high-resolution is commonly related with best performance think about highres TVs, digital cameras,... ). 5

We know that the PSD may show some frequency dependence white noise, flicker noise,... ). Hence, we need to know at which frequency we should evaluate noise PSD.

rate. With the help of Fig. 3, it is intuitive that we should evaluate the output noise PSD at f rd, i.e., at the frequency where we have our signal, independently from the fact the the noise PSD is white or not.

6 We know that the PSD may show some frequency dependence white noise, flicker noise,... ). Hence, we need to know at which frequency we should evaluate noise PSD. In the case of our gyroscope, for example, we know that the output signal is a sinusoidal voltage at the resonance frequency of the drive axis, f rd, whose amplitude is proportional to the angular rate. With the help of Fig. 3, it is intuitive that we should evaluate the output noise PSD at f rd, i.e., at the frequency where we have our signal, independently from the fact the the noise PSD is white or not. Figure 3: At which frequency should we evaluate the PSD? At the frequency where we have the signal to be detected. In case of our gyroscope, where the input signal is the angular rate, Ω, and the output signal is a sinusoidal voltage, V out, at f rd, given the sensitivity, S = V out, the resolution is or S n,ω = S n,v out frd ) S = S n,v out s n,ω = s n,v out frd ) V out Vout. frd ) ), Do we really need to calculate the overall output noise, i.e., to calculate how all the noise sources present within the whole readout chain reach the output node, V out, and then divide this output noise PSD for the sensitivity of the system? As we studied in other courses, we do not need to always compute the output voltage PSDs: it usually) happens that, once a noise source is introduced, both the signal and the noise face the same amplification and filtering). Is it thus possible to evaluate the resolution by just dividing the considered noise source for the sensitivity calculated from the input to the considered node, as graphically described in Fig. 4. There s a hidden assumption here: the bandwidth of the input signal should be much lower than the operation frequency. 6

Figure 4: How to calculate input-referred noise PSD. Few words about the notation used in the following. We define the sensitivity at a certain node, x, of the readout chain as S x : S x = x.

7 Figure 4: How to calculate input-referred noise PSD. Few words about the notation used in the following. We define the sensitivity at a certain node, x, of the readout chain as S x : S x = x. For example, the sensitivity as the sense displacement amplitude variation per unit angular rate is written as S ya = y a. Similarly, the sensitivity as the sense-electrode differential capacitance variation per unit angular rate is written as S Cs,di f f = C s,di f f. Regarding noise PSDs, we will use this notation: S n,x,y, referring to the noise PSD that we have at node Y or that we should equivalently apply at node Y ), to correctly model noise introduced by the noise source X. For example, let s consider a TCA-based front-end; the input-referred noise where input is the input of the frontend, i.e., the sense current, I s ) of the voltage noise of the op-amp will be written as which is expressed in A /Hz. S n,oav,is, QUESTION When a gyroscope is operated in mode-split, the resonance frequencies of the drive and sense modes are different: there is a certain, intended by design) frequency mismatch, f between them. In our case, f = f r s f rd = 800 Hz, 7

The mechanical sensitivity of a mode-split gyroscope can be expressed as S MS = x da x da =. ω ω r s ω rd We can observe that this value does not depend on the quality factor of the sense axis Q s.

8 ω = πf = 506 rad/hz. Mode-split gyroscopes have several advantages with respect to mode-matched ones. First, we do not need to guarantee that the two resonance frequencies are exactly the same as in mode-matched gyroscopes. The mechanical sensitivity of a mode-split gyroscope can be expressed as S MS = x da x da =. ω ω r s ω rd We can observe that this value does not depend on the quality factor of the sense axis Q s. Hence, as shown in Fig. 5, Q-factor variations have no effects on the sensitivity. Mode-matched gyroscope, on the other hand, must face this issue, as their sensitivity, that can be expressed as S M M = x da = x da x da Q s ω ME MS ω =, r s ω r s Q s depends on Q s. Figure 5: Mechanical transfer functions of drive- and sense-axis of a mode-split gyroscope. Another pro relates with the bandwidth-vs-resolution trade-off. Mode-matched gyroscope do have a bandwidth-vs-resolution trade-off: in facts, both the bandwidth, which is limited by the bandwidth of the mechanical transfer function, and the resolution depend on the sense axis damping coefficient, b s. This means that acting in order to increase the resolution would reduce the bandwidth of the sensor, and vice-versa. As we can see from Fig. 5, the bandwidth of our mode-split gyroscope is no more dependent on the bandwidth of the 8

9 resonant peak; indeed, it is much higher, and can be easily limited with an electronic filtering placed at the end of the sense readout chain. With mode-split operation, the bandwidth-vsresolution trade-off is broken, and it is possible, e.g., to lower the pressure inside the package as much as possible, thus increasing the noise performance of the device, without affecting its sensing bandwidth. For these reasons, most state-of-the-art MEMS gyroscope are mode-split operated. The sense displacement amplitude sensitivity of our mode-split gyroscope, defined as the sense displacement amplitude variation per unit angular rate, is S ya = y a = x da ω = x da πf =.9 nm/rad/s). Note the unit of measurement! Remember that angular rate is expressed in rad/s: given the Coriolis force expression F c = mvω, with a kg mass and a m/s speed, in order to have a N force, the object should rotate at rad/s! Not Hz, neither dps! As rad/s = π Hz, one can re-write the previously calculated sensitivity as.9 nm S ya = rad/s Note that, numerically speaking, this is exactly x da f =.9 nm π Hz = 7.5 nm/hz, = 7.5 nm/hz. where f is 800 Hz. As Hz = 360 dps, the sensitivity can be more conveniently expressed in terms of m/dps, as S ya = 0 nm = 0.8 pm/dps. 360 dps Hz Hz We can now express the sensitivity in terms of single-ended sense capacitance variation per unit angular rate: S Cs = C s = C s y s y = C s y S y a. 9

10 Here, C s / y is the single-ended capacitance variation of the sense port per unit displacement along the y-axis, C s y = C s0 g, were C s0 = ɛ 0hL PP N PP = 506 ff, g is the single-ended capacitance of one sense electrode, and g is the rotor-to-stator gap. Hence, C s y = C s0 = 66 ff/µm. g The sense capacitance variation sensitivity, S Cs, can be thus evaluated as S Cs = C s = C s0 g S y a = 66 ff/µm 0.8 pm/dps = 5.54 af/dps. As mentioned, C s / is the single ended capacitance variation per unit angular rate. In presence of a certain angular rate, Ω, the capacitance of one sense port increases of a factor S Cs Ω, while the other capacitance decreases of the same factor, i.e., S Cs Ω. The complete expression of the two capacitance variations at the two sense ports is C s t) = C s0 + S Cs Ωsinω rd t), and C s t) = C s0 S Cs Ωsinω rd t). We can define the differential capacitance variation per unit angular rate, as S Cs,di f f = C s,di f f = C s = S C s =. af/dps. We know that, if the sense-port is biased with a DC voltage applied between the rotor and the stator, V DC, the sense-port capacitance variation induces a current flowing through the electrode that flows in the feedback path of the front-end used to detect the sense displacement. In our case, the sense current is integrated in the feedback capacitance of the TCA. With a sense capacitance variation expressed as the TCA output voltage can be expressed as C s t) = C sa sinω rd t), V TC A t) = G TC A C s t), where G TC A = ) ) V TTC A j ωrd TC A j ωrd = ) C s j ωrd = V DC C F = 5 0 V/F 0

11 is the gain of the TCA, i.e., the transfer function from capacitance variation to output voltage of a TCA evaluated at resonance, ω rd. Hence, the TCA sensitivity of the gyro, defined as the single-ended voltage amplitude variation of the TCA output per unit angular rate, can be expressed as S VTC A = V TC A The differential TCA sensitivity is thus = G TC AS Cs = 3.3 mv/hz = 7.7 µv/dps. S VTC A,di f f = S VTC A = 55.5 µv/dps. The two outputs of the two TCAs are connected with the inputs of an INA, whose gain is G I N A = kΩ R G AI N =. The total sensitivity of the gyroscope, defined as the voltage amplitude variation of the INA output per unit angular rate, can be thus expressed as S VOUT = S VTC A G I N A = S VTC A,di f f G I N A = 60 µv/dps. QUESTION We can define the intrinsic resolution of a sensor as the one we have assuming noiseless readout electronics, i.e., considering the intrinsic noise of the sensing element only. In the case of our gyroscope, intrinsic resolution is due to the unavoidable thermo-mechanical noise associated with the MEMS structure. Note that, for gyroscopes, the resolution is usually referred to as the noise equivalent rate density, or NERD. Thermo-mechanical noise PSD of the sense resonator can be easily modeled in terms of force noise PSD applied to the proof mass: S n,me MS,F = 4k b T b s, expressed in N /Hz; this noise source is white, i.e., it is constant with frequency. s n,me MS,F = S n,me MS,F = 4k b T b s = 9 fn/ Hz. Remembering what we learned in the Introduction, if we evaluate the F / parameter, i.e., the Coriolis force variation per unit angular rate, we can easily infer the intrinsic resolution of our gyroscope, by simply dividing the force noise PSD for F /. As the Coriolis force amplitude can be expressed as F = m s v da Ω = m s ω rd x da Ω,

12 linearly proportional to the angular rate, F = F Ω = 4πm s f rd x da =.5 nn/rad/s) = 00 pn/dps, where m s = m i = 8 nkg. The intrinsic resolution NERD), due to MEMS noise only, is thus s n,me MS,Ω = s nf F = 9 fn/ Hz 00 pn/dps = 643 µdps/ Hz. Note the units of measurement: if we correctly evaluate the sensitivity at node X in terms of X/dps, we can divide the noise PSD, expressed in X/ Hz, for the sensitivity, thus obtaining the input-referred resolution directly expressed in dps/ Hz. We can write the complete expression of the NERD, by expliciting previous equations: s n,me MS,Ω = kb T b s πm s f rd x da. Be careful about the units of measurement. Remember that, if no conversion is considered, the sensitivity is expressed in N/rad/s), hence, the obtained NERD would be expressed in rad/s)/ Hz! If one wanted to evaluate the resolution in terms of dps/ Hz, a conversion factor should be included in the equation, and one might write: s n,me MS,Ω = kb T b s πm s f rd x da 80 dps π rad/s = 643 µdps/ Hz, which is correctly expressed in dps/ Hz. This is the ultimate resolution limit of our mode-split gyroscope. QUESTION 3 A well-balanced sensor from a noise performance point-of-view is a sensor where noise introduced by the readout electronics is equal to the intrinsic noise introduced by the sensor. It is silly to design the readout electronics that introduces higher noise with respect to the intrinsic one: we would be, with no reason, worsening the resolution of the sensor 3. On the other hand, it is silly, as well, to design a too-low-noise readout electronic circuit, whose noise is negligible with respect to the intrinsic one: we are wasting design time, power consumption,..., just to increase the resolution of a factor with respect to the well-balanced case where intrinsic and electronic noise are equal. 3 There might be, however, some situation where it is not possible to achieve such target; it may happen that electronics noise, due to some circumstances, might be only but higher than the intrinsic one.

13 Figure 6: Electrical model for noise calculations on the TCA. It is possible to identify three main noise sources introduced by the front-end electronics: i) the op-amp voltage noise, ii) the op-amp current noise, iii) the feedback resistance thermal noise. As shown in Fig. 6, it is convenient to model both the op-amp current noise and the feedback resistance thermal noise as current noise sources at the front-end input, while opamp voltage noise is better modeled as a voltage noise source applied at the positive input of the op-amp. Let s first evaluate the input-referred or input-equivalent) noise of the TCA front-end, as sense capacitance noise. In this way, we will be familiar with typical numbers related to capacitive sensors: remember that, in MEMS world, the interface between the mechanical domain and the electronic domain is usually indicated as the sense capacitance variation; hence, if one wanted to compare two electronic front-ends in terms of resolution, he may compare them in terms of input-equivalent capacitance noise density, which is expressed in F/ Hz. Remember that our signal is a sinusoidal tone at f rd. Hence, if needed, we will evaluate the transfer functions at f rd, neglecting any frequency dependence of noise spectra. We will here consider only one front-end. Regarding the voltage noise of the op-amp, it is convenient to calculate the voltage noise at the output of the TCA, and then bring it back to the capacitance: ) TVout /V + frd S n,oav,cs = S n,oav,v ) TVout /C s frd, 3

14 hence, ) TVout /V + frd s n,oav,cs = s n,oav,v ). TVout /C s frd As and we get: s n,oav,cs = s n,oav,v TVout /V + frd ) = + C P C F, TVout /C s frd ) = V DC C F, + C P C F V DC C F ) s n,oav,v C P V DC = 4 zf/ Hz, Note that, as long as C P C F, s n,oav,cs does not depend on the feedback capacitance value. This is a common property of TCA based front-ends. The current noise of the op-amp can be expressed as an equivalent sense capacitance variation noise PSD, as well: ) TVout /I frd s n,oai,cs = s n,oai,i ) = s TVout /C s frd n,oai,i πf rd C F V DC C F s n,oai,cs = s n,oai,i πf rd V DC = zf/ Hz. Note that op-amp current noise contribution is much lower than the voltage noise one. Note that also s n,oai,cs does not depend on C F. The same approach can be used for the thermal noise of the feedback resistance: 4k b T s n,rf,i =. R F Hence ) TVout /I frd s n,rf,cs = s n,rf,i ) = s TVout /C s frd n,rf,i πf rd C F V DC C F = 4k b T R F πf rd V DC. To infer the noise equivalent rate densities of the considered noise sources, one should take into account the differential readout. Let s assume a certain noise PSD at the output of both TCAs due to the electronics only); remember that the two noise contributions are not correlated, as they are produced by uncorrelated noise sources. The differential readout doubles 4

15 the signal in amplitude) and doubles the noise PSD in power). Hence, when considering the SNR, the differential readout improves the resolution of a factor. As a general rule, if c = a b, S n,c = S n,a + S n,b. In our case, and i.e., V di f f = +V s.e. ) V s.e. ) = V s.e., S n,di f f = S n,s.e., s n,di f f = s n,s.e.. When dealing with a sensor, the input-referred resolution is s n,i = s n,di f f S di f f, where S di f f is the differential sensitivity, which is twice the single-ended one, S di f f = S s.e.. Hence: s n,i = s n,di f f S di f f = s n,s.e. S s.e., which is better with respect to the one calculated when considering one front-end alone! Let s make a simple numerical example, to better clarify this. Let s imagine a TCA whose output voltage PSD is 00 nv/ Hz, and a gyroscope whose single-ended sensitivity is 4 70 nv/dps. One may input-refer the resolution by directly considering the factor, as s n,i = s n,s.e. S s.e. = 00 nv/ Hz 70 nv/dps dps/ Hz, or, equivalently, may sum the noise sources in power!) and divide for the differential sensitivity, as s n,i = s n,di f f S di f f = 00 nv/ Hz ) + 00 nv/ Hz ) 70 nv/dps = 40 nv/ Hz 40 nv/dps dps/ Hz. Given these concepts, we can go back to our gyroscope. As the sense capacitance variation per unit angular rate is known, it is possible to input-refer the previously calculated capacitance noises, as s n,oav,ω = s n,oav,cs S Cs = s n,oav,cs C s = 50 µdps/ Hz, 4 Numbers provided here are on purpose not realistic, but allow easy calculations. 5

16 We can calculate s n,oai,ω = s n,oai,cs = s n,oai,cs S Cs C s s n,rf,ω = s n,rf,cs S Cs = = 06 µdps/ Hz, 4k b T R F πf rd V P C s s n,oa,ω = s n,oav,ω + s n,oai,ω = 5 µdps/ Hz.. Op-amp noise, 5 µdps/ Hz, is lower than the intrinsic one, 643 µdps/ Hz. We can then calculate R F for which s n,oa,ω + s n,rf,ω = s n,me MS,Ω. We can then write thus, s n,rf,cs S Cs = 4k b T R F πf rd V P C s = s n,me MS,Ω s n,oa,ω = 377 µdps/ Hz, R F = 4k b T ) C ) =.3 GΩ. s πf rd V P s n,me MS,Ω s n,oa,ω With this value of feedback resistance, the total noise equivalent rate density is ) s n,tot,ω = s s n,me MS,Ω + sn,eln,ω = n,me MS,Ω + s n,oav,ω + s n,oai,ω + s n,rf,ω s n,tot,ω = mdps/ Hz. The main contributions are given by the MEMS thermo-mechanical noise and the voltage noise of the op-amp, which are almost equal. Note that with this value of feedback resistance, the feedback pole is f p,tc A = πr F C F = 59 Hz, which is well below the operation frequency, f rd. The TCA is thus correctly designed. QUESTION 4 Up to now we considered only the noise introduced by the front-end. What about the noise introduced by the second stage, i.e., by the INA? We can calculate the differential voltage noise PSD at the input of the INA, S n,tot,vtc A,di f f, due to all the previously considered noise sources MEMS + front-end), and then compare 6

17 this with the input-referred noise of the INA, S n,i N A,VTC A,di f f. The differential voltage noise PSD at the output of the TCA due to both MEMS and front-end noise, can be easily calculated by multiplying the previously calculated input-referred rate noise density for the TCA voltage differential sensitivity, S VTC A,di f f : s n,tot,vtc A,di f f = s n,tot,ω S VTC A,di f f = 50.5 nv/ Hz. In order to have a negligible INA noise, its equivalent input noise, S n,i N A,V, should be lower than 50.5 nv/ Hz. As a rule of thumb, we can dimension it to be 0. times in power i.e., /3 in voltage): s n,i N A,V,max = 6.8 nv/ Hz Which is the expression of S n,i N A,V? Input-referred INA noise is due to the two op-amps voltage noise sources, S n,oai N Av,V, and the thermal noise of the gain resistance, S n,ri N A,V : where Hence, S n,oai N Av,V should be equal to S n,i N A,V = S n,oai N Av,V + S n,ri N A,V, S n,ri N A,V = 4k B T R G AI N = 9.04 nv/ Hz. S n,oai N Av,V = S n,i N A,V,max S n,ri N A,V 6.8 ) ) nv/ Hz 9.04 nv/ Hz S n,oai N Av,V = = 0.0 nv/ Hz. The expression of the input voltage noise of an op-amp is S n,oa,v = 4k bt γ g m, where the factor takes into account both input transistors, γ = /3, and g m is the transconductance of the input differential pair: g m = I D V OV. where I D is the bias current of the input transistors, and V OV is their overdrive voltage. The bias current required to have an op-amp with an input voltage noise S n,oai N Av,V is thus: I D = 4k bt γv OV S n,oai N Av,V =.0 µa. The tail current source, that biases the differential pair, should thus be equal to I t ail = I D = 43.9 µa. Fig. 7 reports all noise contribution, as a pie chart. As expected, MEMS thermo-mechanical noise and voltage op-amp noise contributions are equivalent and constitute the 84% of the overall resolution. 7

18 Figure 7: Pie chart showing all noise contributions. 8

E08 Gyroscope Drive Design

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