Politecnico di MilanoMSc Course - MEMS and Microsensors - 2020/2021

E09

Gyroscope Drive Design

Paolo Frigerio

29/10/2020

Problem

We have to design the electronic circuit needed to sustain the oscillation of the drive axisof a vibratory MEMS gyroscope. The system should guarantee a maximum sensitivityvariation of ±1.5%, worst case, within the automotive temperature range, i.e., from−45C to +125C.The drive resonator is both actuated and detected in single-ended mode.The primary loop of the drive oscillator is shown in Fig. 1. It is constituted by a TCA-based front-end, a dierentiator, and a hard-limiter with a dedicated supply voltage.The complete drive oscillator (with both the primary loop and the AGC loop) is shownin Fig. 2. The dierential gain of the INA has the following expression:

GINA = 1 +49.4 kΩ

RINA.

The variable-gain concept is implemented by means of acting on the supply voltage ofthe hard-limiter.The mechanical and electrical parameters of the gyroscope are listed in Table 1.

1

Mechanical

Symbol Value

Drive resonance frequency frd 20 000 Hz

Drive Q-factor @ 300 K Qd0 8000

Internal mass mi 1.5 nKg

External mass me 2.5 nKg

Process thickness h 24 µm

Gap g 1.8 µm

Number of drive comb ngers NCF 40

Target drive displacement amplitude xa0 5 µm

Electronics

Rotor bias voltage VDC 5 V

Supply voltage VDD ±5 V

Amplier voltage output swing VO ±4 V

Minimum capacitance Cmin 0.2 pF

Primary loop

Dierentiator feedback resistance RDIF,2 300 kΩ

Secondary (AGC) loop

Rectier gain GREC 1

LPF gain GLPF 1

LPF resistance RLPF 3 MΩ

Table 1: Parameters of the gyroscope.

We are asked to. . .

1. Dimension the parameters of the primary loop of the drive-mode oscillator (Fig.1), in order to obtain the target displacement amplitude, xa0, at the referencetemperature (300 K).

2. Calculate, considering the primary loop only (Fig. 1), the maximum percentagevariation of the drive displacement amplitude, and evaluate the required compen-sation 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 lter of the control loop.

Introduction

A vibratory MEMS gyroscope can be modeled as a resonator along the drive axis and asan accelerometer along the sense axis, that detects the sinusoidal motion induced by the

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Figure 1: Primary loop of the drive oscillator.

Coriolis force.Regarding the primary loop (the electronic circuit needed to start and sustain the os-cillation 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 thisexample, however, we will use a slightly dierent topology. Indeed, the primary loop iscomposed by a TCA-based front-end, that senses drive motion and provides an outputvoltage proportional to the drive displacement, followed by a dierentiator, needed toobtain 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 thede-gain stage, the voltage amplitude reduction is obtained by supplying the hard-limiteritself with a dedicated supply voltage, VHL.

Question 1

We want to calculate the actuation voltage amplitude, va, that has to be applied tothe actuation electrode in order to have the desired displacement amplitude, xa0, at thereference temperature. We know that, if a MEMS resonator is actuated at resonance,assuming a small-signal hypothesis on the actuation voltage, the displacement amplitudeis

xa =Q

kηdva.

In our case, Q is the quality factor of the drive resonator, k is the spring constant ofthe drive mode, ηd is the drive-actuation transduction coecient, and vda is the drive-

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Figure 2: Drive oscillator with AGC.

actuation sinusoidal voltage amplitude:

xa =Qd

kdηdavda.

We know that

ηda = VDC∂Cda

∂x,

where VDC is the DC voltage between the rotor and the stator electrode, and ∂Cda/∂x isthe drive-actuation capacitance variation per unit displacement. With the data providedin Table 1,

∂Cda

∂x=

2ε0hNCF

g= 9.44 fF/µm,

and

ηda = VDC∂Cda

∂x= 47.2 · 10−9 VF/m.

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Note that the resonator is symmetric. Hence,

ηd = ηda = ηdd = 47.2 · 10−9 VF/m.

We can then calculate the spring constant of the drive mode of the gyroscope:

kd = (2πfrd)2md = (2πfrd)2 (me +mi) = 63.2 N/m.

And we can nally evaluate the amplitude of drive-actuation sine-wave voltage requiredto obtain the target displacement amplitude:

vda =xaηdaQdkd

= 836 mV.

Remember that, with the circuit topology we are using, the voltage signal applied to thedrive 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 intoaccount the 4/π factor:

vda,sq =vda4/π

= 0.66 V.

This means that the hard-limiter should be biased (VHL) between −0.66 V and +0.66 V.Note that vda = 0.836 V is much lower than the rotor-stator DC voltage (5 V). The small-signal hypothesis is thus valid. What would happen if vda were too high1? There areat least two solutions: (i) one solution consists in adopting the push-pull conguration,that, as we learned in previous classes, does not require the small-signal hypothesis onthe ac voltage; (ii) another solution consists in re-designing the gyroscope, increasing thenumber of comb ngers used for the drive actuation: in this way, the required ac voltageis lower; of course VHL = vda,sq cannot be too low, otherwise the hard-limiter cannot beproperly biased.

We can evaluate the motional current amplitude owing through the drive-detection portof the drive resonator,

ima = xaηd (2πfrd) = 29.6 nA,

and the drive-detection capacitance variation amplitude,

Cdda = xa∂Cdd

∂x= 47.2 fF.

With a TCA-based front-end, the output voltage amplitude can be calculated as

VTCA,out,a = GTCACdda = |TTCA (frd)|Cdda =VDC

CFCdda,

or, equivalently,

VTCA,out,a =1

(2πfrd)CFima.

1Remember that the RLC model is valid if the ac voltage is much smaller than the DC voltage.

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A good design approach to choose the gain2 of the TCA is to maximize it: in this way,noise introduced by subsequent stages can be neglected. Which is the maximum gain ofan amplier? It's the highest value that prevents any saturation/clamping of the outputsignal. Since the amplier voltage output swing (VO) is ±4 V, given the previouslycalculated ima and Cdda, the desired feedback capacitance, CF , can be found as

CF =VDC

VOCdda = 59 fF,

which, unfortunately, is lower than the minimum one (200 fF). We will thus choose theminimum one,

CF = 200 fF.

This implies an output voltage amplitude of

VTCA,out,a =1

(2πfrd)CFCdda = 1.18 V.

For proper TCA behavior, the feedback pole frequency should be, at least, two decadesbefore the oscillation frequency,

1

2πRFCF<

1

100frd,

that corresponds to a (minimum) feedback resistor of 4 GΩ. With this choice, we guar-antee a small error (lower than 1 deg) on the ideal phase shift of the TCA (90).

Regarding the dierentiator stage, we can choose its gain,

GDIF = |TDIF (frd)| ,

following the same approach as before, i.e., we can force an output signal whose amplitudeis 4 V (VO). Hence,

GDIF =Vout,a,max

VTCA,out,a= 3.38.

If the poles are suitably designed, the dierentiator gain can be expressed as

GDIF = 2πfrdCDIF1RDIF2.

As the value of the feedback resistor, RDIF2, is given, 300 kΩ, the input capacitancevalue is

CDIF1 =GDIF

2πfrdRDIF2= 89 pF.

Regarding the dimensioning of the other two components, we can choose their values inorder to have the two poles at frequencies much higher than frd, namely two decades

2With gain we mean the magnitude of the transfer function at resonance.

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higher. Remember, the higher the pole frequency, the more precise is the phase shift ofthe stage. Assuming to place both poles at fp,DIF = 2 MHz:

CDIF2 =1

2πfp,DIFRDIF2= 265 fF,

and

RDIF1 =1

2πfp,DIFCDIF1= 885 Ω.

Question 2

The sensitivity of a MEMS gyroscope is proportional to the displacement amplitude ofthe drive-axis oscillation, xda:

S ∝ xda.

In absence of an amplitude-control loop, i.e., in presence of a xed drive-actuation voltageamplitude, vda, the drive displacement amplitude depends on the quality factor,

xda =Q

kηdavda.

In presence of temperature variations, the quality factor changes and, in turn, the drive-displacement amplitude changes. As seen in a previous class, the quality factor variationsin front of temperature variations can be roughly approximated with the following lin-earization,

∆Q

Q= −1

2

∆T

T.

Since∆T

T=

170 K

300 K' 56%,

the quality factor variation is about −28%. Since the drive displacement amplitude islinear with the quality factor, the sensitivity variation within the whole temperatureoperating range can be estimated as

∆S

S=

∆xdaxda

=∆Q

Q= −1

2

∆T

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 themaximum allowable variation is 1.5%, an open-loop-actuated drive resonator would notfulll the requirement. We need to introduce a compensation (control) loop, to reducethe displacement amplitude variations by a factor (at least) 28 / 3 ' 10.The displacement amplitude as a function of temperature variations within the wholetemperature range is reported in Fig. 3.

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Figure 3: Open-loop drive displacement amplitude as a function of temperature varia-tions.

Question 3

To control the drive displacement amplitude, an AGC (automatic gain control) loop isintroduced in most vibratory MEMS gyroscopes. The control circuit is designed as anegative feedback loop. Its raw schematic is reported in Fig. 4. We can consider thevoltage dierence VD − VREF as the error signal of our negative feedback loop3:

ε = VREF − VD = VREF1

1 +Gloop.

If the loop gain is suciently high, the error is ' 0, hence VD ' VREF . In other wordsVD behaves as the virtual ground of our negative feedback loop, as depicted in Fig. 4, onthe left.

Figure 4: AGC as a negative feedback-loop.

3For the sake of simplicity the loop gain, Gloop, will always be considered as a positive number, i.e.,

will be evaluated without considering the − sign of the loop gain.

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If the gain from the drive displacement amplitude, xda, to VD is known and constant,

VD = KAGCxda,

choosing VREF is equal to choosing a certain reference drive displacement amplitude,xda,ref :

xda,ref =VREF

KAGC,

and the loop would force xda ' xda,ref , as shown in Fig. 4, on the right.Remembering that

ε = VREF1

1 +Gloop,

and, equivalently,

xda = xda,refGloop

1 +Gloop,

we can make the following observations.

• Innite loop gain: if the loop gain is innite, the error signal is exactly zero, andxda = xda,ref , independently from temperature variations.

• Finite and constant (temperature-independent) loop gain: if the loop gain is nite,the error signal is nite. . . but constant: hence, the drive displacement amplitudewill be dierent from the target one, but this new value will not depend fromtemperature variations.

• Finite and temperature-dependent loop gain: if the loop gain is nite, the errorsignal is nite (at 300 K) and, in addition, it will change if temperature changes:hence, the drive displacement amplitude will be dierent from the target one, and,in addition, it will depend from temperature variations.

We must now evaluate the temperature dependence of our AGC loop gain. As one mayintuitively guess, the loop gain, Gloop, turns out to be directly proportional to Qd (wewill demonstrate this later). We are thus in the case of a temperature-dependent loopgain:

Gloop ∝ Qd (T ) .

Given this linear proportionality, we can linearize the temperature dependence of theloop gain as

Gloop (∆T ) = Gloop0 −1

2

Gloop0

T0∆T.

We can now express the temperature variations of xda with the AGC loop:

xda (∆T ) = xda,refGloop (∆T )

1 +Gloop (∆T ),

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which can be re-written as

xda (∆T ) = xda,ref

Gloop0 −1

2

Gloop0

T0∆T

1 +Gloop0 −1

2

Gloop0

T0∆T

= xda,ref ·Gloop0 ·

(1− 1

2

∆T

T0

)(1 +Gloop0)

(1− 1

2

Gloop0

1 +Gloop0

∆T

T0

) .Considering small relative temperature variations, we can linearize the denominator byexploiting the well-known linearization 1/(1 + ε) ≈ 1− ε, applicable when ε 1:

xda(∆T ) ≈ xda,ref ·Gloop0

1 +Gloop0

(1− 1

2

∆T

T0

)·(

1 +1

2

Gloop0

1 +Gloop0

∆T

T0

).

We can now multiply the two terms in parentheses and neglect the term proportional to(∆T/T0)

2, arriving to the nal result:

xda(∆T ) ≈

nominal displacement xda0︷ ︸︸ ︷xda,ref ·

Gloop0

1 +Gloop0−

closed-loop drift︷ ︸︸ ︷xda,ref

Gloop0

1 +Gloop0· 1

2

∆T

T0︸ ︷︷ ︸open-loop drift

· 1

1 +Gloop0.

As expected, the new (with the AGC) relative displacement amplitude variation is nowreduced by a factor 1 +Gloop0:

∆xdaxda0

=

(1

2

∆T

T0

)1

1 +Gloop0=

∆Q

Q0

1

1 +Gloop0.

The eects of temperature variations on the drive displacement amplitude (hence onthe sensitivity of the gyroscope) are reduced by a factor 1 + Gloop0, as expected fromautomation theory, when the AGC loop is introduced.As the required compensation factor was 10, we will design the AGC loop with a targetloop gain of 20.The drive displacement behavior as a function of temperature with the control loop isreported in Fig. 5.Note that the previously dened xda0 represents the drive displacement amplitude at thereference temperature. With a loop gain equal to 20, xda0 = 4.76 µm, i.e., ' 5% lowerthan the target value, 5 µm.

We are now ready to evaluate the loop gain. Referring to Fig. 6, the loop gain we aregoing to calculate is the one marked as Gloop,AGC .How can we calculate the loop gain? As usual, we can (i) cut the loop (possibly atthe output of a low-impedance voltage source, or at the input of an high-impedanceamplier), (ii) inject an amplitude variation at the oscillation signal, and (iii) evaluatethe quantity that returns to where the loop was cut.The added AGC circuitry is basically composed by three stages:

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Figure 5: Drive displacement amplitude as a function of temperature variations, for dif-ferent situations.

1. A full-wave rectier (FWR), that, given a sine-wave at its input, outputs the fully-rectied waveform:

VFWR,in (t) = Va sin (ω0t) → VFWR,out (t) = Va |sin (ω0t)| ,

as shown in Fig. 7.

2. A low-pass lter (LPF), that outputs a DC voltage proportional to the mean valueof its input signal. In our case, if the input is

VLPF,in (t) = Va |sin (ω0t)| ,

the output would be

VLPF,out (t) = mean [Va |sin (ω0t)|] =2

πVa.

In this example, it is implemented with a single-pole active low-pass lter, but weare free to choose other topologies.

3. A dierential gain stage, that is implemented with an instrumentation amplier(INA), that compares the LPF output voltage with the AGC reference voltage.The dierence between these two signals is the aforementioned error signal of theAGC loop, ε. The output of the INA is fed to the positive supply voltage of thehard-limiter and is inverted (multiplied for −1) and fed to the negative supply ofthe hard-limiter. With this topology, given a certain INA output voltage, Vout,INA,the drive actuation signal is

Vda (t) = VINA,outsq (ω0t) ,

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Figure 6: Complete drive-mode oscillator. The signal path of the amplitude loop is high-lighted.

Figure 7: Full-wave rectier.

that describes a square-wave signal at ω0, that toggles between +VINA,out and−VINA,out.

We are now ready to calculate the loop gain. Let's imagine to cut the loop at the

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INA output, and to apply a certain signal, vt. The power supply of the hard-limiterare thus ±vt. Hence, the drive actuation signal is a square wave between ±vt. Thesinusoidal signal amplitude of the drive actuation voltage is thus 4/πvt. At resonance,drive displacement motion induces a motional current, whose amplitude is

ima =1

Rmd

4

πvt,

where Rmd is the drive resonator motional resistance,

Rmd =bdη2d

=(2πfrd)m

Qdη2d

= 28.2 MΩ.

The TCA output voltage amplitude is thus

vTCA,out,a =1

(2πfrd)CFima =

1

(2πfrd)CF

1

Rmd

4

πvt.

Note that this is the amplitude of the TCA output voltage, i.e., of the FWR input, whichis a sinusoidal signal. The complete expression of the signal is

vTCA,out (t) = vFWR,in (t) = vTCA,out,a sin (ωot) =1

(2πfrd)CF

1

Rmd

4

πvt sin (ωot) .

The LPF output, which is the virtual gound voltage, is 2/π times the amplitude of thissignal, hence:

VD = vLPF,out =2

πvTCA,out,a =

2

π

1

(2πfrd)CF

1

Rmd

4

πvt.

This signal is amplied by the gain of the INA, GINA, and we are back at where we cutthe loop:

vf = −GINA2

π

1

(2πfrd)CF

1

Rmd

4

πvt.

As expected, the return signal is negative, i.e., we set up a negative feedback loop, asdesired. The AGC loop gain is thus

Gloop,AGC =

∣∣∣∣vfvt∣∣∣∣ = GINA

2

π

1

(2πfrd)CF

1

Rmd

4

π.

If we want a loop gain of 20, the INA gain, GINA, should be

GINA =Gloop,AGC

2

π

1

(2πfrd)CF

1

Rmd

4

π

= 17.49,

that can be obtained with an INA gain resistance, RINA, of

RINA =49.4 kΩ

GINA − 1= 3.00 kΩ.

Note that

Gloop,AGC ∝1

Rmd∝ Qd,

as anticipated before.

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Question 4

We have to dimension the low-pass lter of the AGC loop. As the gain of the LPF isunitary, the transfer function of the LPF can be expressed as

VLPF,out (s)

VLPF,in (s)=

1

1 + sRLPFCLPF.

Figure 8: Loop gain of the AGC.

From a spectral point of view, the signal at the input of the FWR is a pure tone atωrd; on the other hand, the rectied signal will have a DC tone, whose amplitude is 2/πtimes the amplitude of the input sine-wave, and other harmonics, as shown in Fig. 8.The only harmonic that we want is the one at DC, proportional to the drive displace-ment amplitude. The role of the lter is thus to lter out all the other harmonics. Astemperature variations have a very narrow bandwidth (temperature variations are veryslow), we could place the pole at a very-low frequency. How low? As in every negativefeedback loop, the choice of the poles is related with stability issues. Before, we evaluatedthe DC loop gain, Gloop0, i.e., the loop gain at DC. What about the loop gain at otherfrequencies?Regarding the frequency behavior of the AGC loop, one can demonstrate that all thestages can be modeled as frequency-independent-gain stages. . . except the MEMS res-onator 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

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transfer function isIa (s)

Va (s)=

1

Rm

1

1 + s2Q

ωo

.

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 voltageamplitude, when actuated at resonance.The new frequency-dependent AGC loop gain can be thus written as

Gloop,AGC = GINA2

π

1

(2πfrd)CF

1

Rmd

4

π

1

1 + s2Q

ωrd

1

1 + sRLPFCLPF.

This is a two-poles system, where the rst pole frequency, the one due to the MEMS, isgiven,

fp1 =1

2π2Qd

ωrd

=frd2Qd

= 1.25 Hz,

while the second one, fp2 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 poleof the loop gain should be equal (or higher) to the gain-bandwidth product (GBWP) ofthe loop. In other words,

fp2 ≥ Gloop0 · fp1 = 25 Hz.

This is the minimum pole frequency of the LPF. With

CLPF =1

2π · (Gloop0 · fp1) ·RLPF= 2.12 nF,

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a 45 phase margin is ensured.The whole situation is graphically represented in Fig. 9.Choosing a value of 2.12 nF for the capacitance, all the components of the circuit aredimensioned. The primary loop is designed in such a way that the circuit oscillatesat the resonance frequency of the drive resonator, while the secondary (AGC) loop isdesigned to suitably reject the eects of temperature variation on the drive displacementamplitude (thus on the sensitivity) according to the given specication.

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Appendix: base-band model of a mass-spring-damper

system for actuation amplitude variation effects

A mass-spring-damper system is modeled by the dierential equation

mx+ bx+ kx = F.

which can be rewritten in Laplace domain as

X (s)

F (s)=

1

k

1

1 + sQ

ωo+s2

ω2o

.

Let's assume an external force, F , in the form

F (t) = Fa (t) sin (ωot) ,

where Fa is the slowly varying amplitude of the force and ωo is the resonance frequencyof the structure. As the resonator is actuated at resonance, the displacement will have asinusoidal waveform, as well, that can be expressed as

x (t) = −xa (t) cos (ωot) .

We can write,x (t) = −xa (t) cos (ωot) + xa (t)ωo sin (ωot) ,

andx (t) = −xa (t) cos (ωot) + 2xa (t)ωo sin (ωot) + xa (t)ω2

o cos (ωot) .

Hence,m(−xa (t) cos (ωot) + 2xa (t)ωo sin (ωot) + xa (t)ω2

o cos (ωot))

+

+b (−xa (t) cos (ωot) + xa (t)ωo sin (ωot)) +

+k (−xa (t) cos (ωot)) = Fa (t) sin (ωot) ,

which can be re-written as

[2xa (t)mωo + xa (t) bωo − Fa (t)] sin (ωot) = 0,[−mxa (t) + xa (t)mω2

o − bxa (t)− kxa (t)]

cos (ωot) = 0

We can re-write the rst equation as

2xa (t)mωo + xa (t) bωo = Fa (t) ,

which can be rewritten in Laplace domain:

2smωoXa (s) + bωoXa (s) = Fa (t) ,

Xa (s) [2smωo + bωo] = Fa (t) ,

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Xa (s)

Fa (s)=

1

2smωo + bωo=

Q

k

1 + s2Q

ωo

.

This is the base-band model of a MEMS resonator when dealing with slow variations ofthe actuation signal amplitude.Following the same approach, one can demonstrate that the baseband-equivalent mo-tional impedance of the resonator can be modeled as:

Ia (s)

Va (s)=

1

Rm

1

1 + s2Q

ωo

.

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.

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