Phase noise analysis of micromechanical silicon resonant accelerometer

Sensors and Actuators A 197 (2013) 15–24

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Sensors and Actuators A: Physical journal homepage: www.elsevier.com/locate/sna

Phase noise analysis of micromechanical silicon resonant accelerometer Shi Ran, Jia Fang-xiu, Qiu An-ping, Su Yan ∗ MEMS Inertial Technology Research Center, Nanjing University of Science and Technology, Nanjing, 210094 Jiangsu, China

a r t i c l e

i n f o

Article history: Received 11 September 2012 Received in revised form 20 March 2013 Accepted 22 March 2013 Available online 6 April 2013 Keywords: Phase noise model Silicon resonant accelerometer Automatic gain control Resolution

a b s t r a c t The phase noise in micromechanical silicon resonant accelerometer (SRA) can perturb its frequency output and determines the minimum detectable change in acceleration (sensor resolution). In order to improve the resolution and optimize the sensor performance, the phase noise of SRA is studied in this paper. Theoretical model of phase noise of SRA is set up, especially the effect of the automatic gain control (AGC) circuit in oscillator to phase noise in low frequency range (1/f3 phase noise and 1/f5 phase noise) is analyzed, and the restriction of resolution performance arising from the 1/f3 phase noise caused by AGC is also illustrated. Phase noise experiments are then performed on a SRA prototype developed in MEMS Inertial Technology Research Center and validate the model. Compared to the previous work, the phase noise model in this paper matches the experiments results more precisely, and could provide guidance and reference for design in MEMS inertial device. © 2013 Elsevier B.V. All rights reserved.

1. Introduction Silicon resonant accelerometer (SRA) is a MEMS inertial device which is based on micro-machining process. Relative to traditional accelerometer, it is featured by wide dynamic range, high resolution, good stability, small size and light weight. And it has the potential to be fully integrated with IC into a single chip. Thus the SRA has become increasingly attractive in a wide range of applications. SRA is mainly composed by mechanical sensitive structure and closed-loop drive circuit [1]. The mechanical sensitive structure comprises seismic mass, microlever and resonator. When external acceleration is applied on the accelerometer, the inertial force acting on the seismic mass is amplified by microlever and applied to the resonator, which causes a change in stiffness of resonator and shift the resonant frequency. The resonator combines with closed-loop drive circuit to create an oscillator, which is self-oscillating in the resonant frequency of resonator. Therefore, the oscillator supplies a frequency output through which the acceleration could be detected. Closed-loop drive circuit and resonator noise perturbs both the amplitude and phase of the oscillator output. The amplitude noise is typically not relevant here as it does not impact the ability to measure oscillator frequency [2]. However, phase noise can still corrupt the output frequency of oscillator and determine the sensor resolution. Consequently, it is necessary to analyze the system composition and influence of the phase noise in SRA. There are already some articles conducting research on phase noise theory about SRA and oscillator [3–6]. In their model, the phase noise is composed of white phase noise, white frequency noise (1/f2 phase noise) and flicker phase noise (1/f3 phase noise): the brown noise of resonator and white noise of circuitry are modulated to phase domain through oscillator and result in white phase noise and white frequency noise; the mechanical or electrostatic nonlinearity of resonator could mix the flicker noise (1/f noise) of circuitry up to resonant frequency to perturb the resonator vibration, resulting in flicker phase noise. However, the experiment result shows that, the measured flicker phase noise in SRA is much larger than the theoretical value predicted by above model [5]. Therefore, the phase noise model in the previous work is not so accurate in the low frequency range, and the phase noise of SRA can also be caused by some other unidentified noise mechanisms. While the AGC which is commonly adopted in SRA to prevent the resonator from entering strong nonlinear region and keep the resonator vibration amplitude stable, will inevitably affect the SRA noise performance. In recent years, some attempts have been made to analyze the phase noise of AGC mechanism in other oscillator types [7–10]. In the voltage-controlled oscillator (VCO), with the use of AGC, the phase noise optimization can be accomplished independently without being constrained by startup consideration [7]. But the automatic amplitude control circuit also injects low-pass shaped noise into the VCO and affects phase output through indirect instability and the AM-PM conversion due to the varactors [8]. In the MEMS oscillator, Lee and

∗ Corresponding author. Tel.: +86 25 84313250; fax: +86 25 84303258 806. E-mail addresses: [email protected] (R. Shi), [email protected] (F.-x. Jia), [email protected] (A.-p. Qiu), [email protected] (Y. Su). 0924-4247/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.sna.2013.03.031

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R. Shi et al. / Sensors and Actuators A 197 (2013) 15–24

Fig. 1. Oscillator system in SRA.

Nguyen [9] claimed that the 1/f3 noise from the automatic level control circuit could be superimposed on the DC voltage, and generate 1/f5 phase noise through electrical stiffness. He et al. [10] suggested that AGC circuit noise could be converted to phase noise by the mechanical nonlinearity of resonator in their new nonlinear MEMS oscillator model. Because of the high oscillating frequency of oscillator, these phase noise analyses in VCO and MEMS oscillator normally emphasize on the high frequency range, but short of the comprehensive analysis about low frequency range which we just concern in the SRA. In the research on SRA prototype developed by MEMS Inertial Technology Research Center [11–14], we found that, in addition to nonlinearity of the resonator, the multiplier or variable gain amplifier (VGA) in AGC circuit of the oscillator could up-convert the flicker noise to the resonant frequency to cause the flicker phase noise. From this point, this paper analyzes the main sources of the flicker phase noise, establishes a more accurate phase noise model, and validates the phase noise model by experiments. 2. Establishment of phase noise model 2.1. Oscillator of SRA As shown in Fig. 1, the oscillator in SRA consists of the resonator and the closed-loop circuit. Wherein the resonator utilizes doubleended tuning fork (DETF) structure, which is composed of drive combs, sense combs and vibrating beams. Sense combs detect the motion of vibrating beams and convert it into the current output Is . And Is would be converted to the drive voltage Vac by closed-loop drive circuit. Then the drive voltage Vac applies on the drive combs and produces an electrostatic force to drive the vibrating beams to vibrate in resonant frequency ωn . The transfer function from the drive voltage to the sense current can be found to be: H(s) =

1 s Is (s) = 2Kp KI Meff s2 + (ωn /Q )s + ωn 2 Vac (s)

(1)

where KP = KI = (∂C/∂x)Vdc , C is the capacitance of the drive electrode and sense electrode, Vdc is DC bias voltage of resonator, Keff is the effective stiffness of resonator, Meff is the effective mass of resonator, ωn is the resonant frequency, and Q is the quality factor. The resonator can be redefined as an equivalent LCR series circuit shown in Fig. 2, and the equivalent electrical components of this circuit are: Leq =

Meff 2Kp KI

,

Ceq =

2Kp KI Meff ωn2

,

Req =

Keff 2QKp KI ωn

(2)

Thus the transfer function of the equivalent is: H(s) =

(1/Leq )s

(3)

s2 + (Req /Leq )s + (1/Leq Ceq )

When the resonator works at the resonant frequency ωn , Eq. (3) becomes: H(ωn ) = Req =

Keff

(4)

2QKp KI ωn

Now the resonator can be treated as a resistance Req . The closed-loop circuit includes trans-impedance amplifier, VGA and AGC. The trans-impedance amplifier converts sense current Is to voltage output Voa with a gain of −Rf . Because of the nonlinearity of resonator, the resonant frequency ωn is a function of the resonator vibration amplitude |q|. Therefore, AGC and VGA are adopted to keep a stable gain and control the vibration amplitude |q| of resonator. And the VGA gain is −Avga .

Fig. 2. Equivalent electrical circuit of resonator.

R. Shi et al. / Sensors and Actuators A 197 (2013) 15–24

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In order to meet the self-exciting conditions, the loop gain of the oscillator should be equal to 1: Avga =

Req Rf

(5)

2.2. Linear phase noise Consider superimposing noise jamming on a ‘clean’ carrier signal of oscillator output with power of C and frequency of fn . At upper sideband fn + fb and lower sideband fn − fb , the power spectral density (PSD) of added noise is N0 (fn + fb ) and N0 (fn − fb ) respectively, and N0 (fn + fb ) = N0 (fn − fb ). According to the definition of phase noise [15,16], the phase noise PSD of the output signal of oscillator is: Sϕ (fb ) =

N0 (fn − fb ) C

(6)

Therefore, it can be concluded that the key of building the theoretical model of phase noise lies on calculating the noise superimposed on the sideband of the carrier signal. In Fig. 1, three noise sources are injected into the oscillator system: v2nres is the noise due to the resonator, v2nT represents all noises in closed-loop drive circuit referred to the output node of the trans-impedance amplifier, v2ndc is the noise due to DC voltage reference. These noise sources result in output noise of the oscillator:

v2nout

   2  H(s)Rf Avga 2   Avga 2 2     v2 = (v + vndc ) +  1 − H(s)Rf Avga  nres 1 − H(s)Rf Avga  nT

(7)

Correlating (1), (4), (5) and (7), the noise voltage at an offset frequency fb is written as:

      2  fn  fn 2 2 vnres (fn − fb ) + v2ndc (fn − fb ) + A2vga 1 + v2nT (fn − fb )  2Qf 2Qf

v2nout (fn − fb ) = 

b

(8)

b

where v2nres is brown noise of the resonator.

v2nres (fn − fb ) = 4kTReq

(9)

The noise of the closed-loop circuit v2nT consists of four terms: thermal noise v2nRf of Rf , the noise of trans-impedance amplifier v2na (1 + (fce1 /f )) (including white noise and flicker noise), the noise of VGA v2nvga (1 + (fce2 /f )) (including white noise and flicker noise), and the AGC noise v2nagc . Wherein v2nagc will be discussed later. Now consider the first three terms that:

v2nT (fn − fb ) =

v2nres (fn − fb ) Avga

+ v2nvga



fce2 1+ fn − fb



+ v2na



fce1 1+ fn − fb





1+

Req (ωn − ωb )Cin Avga

2 (10)

where Cin is the input capacitance of trans-impedance amplifier, fce1 and fce2 are the corner frequencies between white noise and flicker noise of trans-impedance amplifier and VGA. The resonant frequency fn of SRA, is much larger than the corner frequency fce1 , fce2 and accelerator Bandwidth BW. Considering offset frequency fb < BW, Eq. (10) can be rewritten as:

v2nT (fn − fb ) = where F =

v2nres (fn − fb ) Avga





+ v2na 1 +

Req ωn Cin Avga

2

+ v2nvga =

4kTReq + v2na F 2 + v2nvga Avga

(11)

2

1 + (Req ωn Cin /Avga ) . The flicker noise from trans-impedance amplifier and VGA can be ignored while calculating noise

v2nT (fn − fb ) in the vicinity of resonant frequency.

As shown in Fig. 3, AGC is composed of the rectifier, the lowpass filter and the proportional-integral controller (PI). The frequency response of PI is:

 |HPI (f )| =

R  P

RI

2

+

1/2RI CI f



2 =

(P)2 +

 I 2 f

(12)

where P = RP /RI is the proportional parameter, and I = 1/2RI CI is the integral parameter. And the AGC noise v2nagc can be written as:



v2nagc (f ) = v2nw 1 +

fce3 f





1 2

  I 2   P2 +

(f/flpf ) + 1

f

(13)

where v2nw is electrical white noise of AGC, fce3 is the corner frequency between white noise and flicker noise, flpf is the cutting off frequency of the lowpass filter.

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R. Shi et al. / Sensors and Actuators A 197 (2013) 15–24

Fig. 3. Block diagram of AGC.

Fig. 4. Upconversion of low frequency noise from AGC around resonant frequency.

flpf is normally chosen to satisfy flpf ≤ fce3 , flpf ≤ I/P and flpf  fn . Thus Eq. (13) can be rewritten as:

 εagc v2nagc (f )

f3

=

f < flpf

(14)

f ≥ flpf

0 where εagc = v2nw fce3 I 2 .

Considering that the cutting off frequency of lowpass filter satisfies flpf  fn , AGC noise v2nagc (fn − fb ) is small enough to be ignored just as the flicker noise from trans-impedance amplifier and VGA, if it is calculated in the manner of Eq. (11). Thus, in the previous articles [3–6], the AGC noise is not taken into account in calculating phase noise.





However, in our research, it is discovered that, when v2nagc is multiplied by the output voltage of trans-impedance amplifier Voa  cos(ωn t) in the VGA, the low frequency could be modulated to the vicinity of resonant frequency fn (Fig. 4). And then the modulated AGC noise can be written as:

v2nT (fn − fb ) = v2nagc (f )|Voa |2 =

εagc |Vac |2 A2vga

fb3

=

εagc 2Pres Req fb3

(15)

Av2ga

where Pres is the power of the resonator. Pres =

ωn Keff |q|2 1 |Vac ||Is | = 2 Q

(16)

where |Vac | is the amplitude of drive voltage of the resonator, |Is | is the amplitude of the sense current. From Eqs. (15) and (11), we get:

v2nT (fn − fb ) =

4kTReq εagc 2Pres Req + v2na F 2 + v2nvga + 3 Avga A2vga fb

(17)

Substituting (17) and (8) into (6), we can get the phase noise of oscillator of SRA (the full derivation can be seen in supplementary document): Sϕ (fb ) =

A2vga Px

+



Q2 4kTQ + Avga Rx

4kT (1 + Avga ) A2vga Q



+



v2na v2ndc Rx A2vga



1+

Rx ωn Cin QAvga

 1 + Rx

v2na

2 

 1+



+ v2nvga



Rx ωn Cin QAvga

2 

 + v2nvga

 f 2 n 2fb

 + 2εagc

1 fb3

+ εagc

fn2 2Q 2 fb5

(18)

R. Shi et al. / Sensors and Actuators A 197 (2013) 15–24

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where Px = QPres = ωn Keff |q|2

Rx = QReq =

(19)

Keff

(20)

2ωn Kp KI

The first term of Eq. (18) is white phase noise ϕNF . The second term is 1/f2 phase noise (white frequency noise). The 3rd and 4th terms are 1/f3 phase noise (flicker phase noise) and 1/f5 phase noise respectively which are both caused by AGC noise v2nagc . 2.3. Nonlinear phase noise Because of the nonlinearity of resonator, the kinetic equation can be written as: Meff q¨ +

Keff ωn Q

q˙ + Keff q + K3 q3 = Peff

(21)

where q is the vibration displacement of the resonator, Peff is the electrostatic drive force, K3 is the 3rd order nonlinear stiffness coefficient. Thus the resonant frequency is dependent on the vibration amplitude.



ωr = ωn

1+

3K3 3K3 |q|2 ≈ ωn + ωn |q|2 4Keff 8Keff

(22)

where ωr is the natural frequency of the resonator, |q| is the vibration amplitude of the resonator. Considering the noise voltage v2nout the electrostatic drive force can be written as: Peff

1 ∂C = 2 ∂x



   Vac  cos(ωn t) + v2nout + Vdc

2

(23)

Eq. (23) consists of DC term, the fundamental harmonic term and the second harmonic term. And the fundamental harmonic term plays the most significant role in driving the resonator. Hence the other two terms can be removed, and the vibration amplitude is given by |q| =

Q Q ∂C |P | = |Vac | Keff eff Keff ∂x





v2nout + Vdc

(24)

Substitute (24) into (22), and neglect the high order infinitesimal, the equation may be rewritten as: ωr = ωn +

3K3 3 8Keff

ωn Q

2

∂C ∂x

2 2

|Vac |

2 Vdc

+

3K3 3 4Keff

ωn Q

2

∂C ∂x

2

 |Vac |2 Vdc

v2nout

(25)

The second term is a fixed frequency shift caused by the nonlinearity of the resonator, while the 3rd term is the frequency noise caused by v2nout which is modulated to the resonant frequency by the nonlinearity of the resonator. Substitute s = j2fb into Eq. (7). Considering that fn  fb , it can be written as:

v2nout (fb ) = A2vga v2nT (fb )

(26)

Because AGC noise v2nagc is up-converted to the resonant frequency in VGA (Fig. 3), it will not appear in v2nT (fb ). Moreover, considering 1 2Cin fb

 Rf , v2nT (fb ) can be given by:

v2nT (fb ) =

v2nres Avga



+ v2na 1 +

fce1 fb





+ v2nvga 1 +

fce2 fb



=



4KTRx + v2na + v2nvga + v2na fce1 + v2nvga fce2 QAvga

1 fb

(27)

Substituting (26) and (27) into Eq. (25), the respective phase noise can be derived: Sϕ (fb ) =

Sf (fb ) fb2

=

9 2 K32 |q|4 f 16 n K 2 V 2 eff dc

 4KTR A x vga Q

+ A2vga v2na + Av2ga v2nvga

 1 fb2

+

9 2 K32 |q|4 2 1 A (v2 fce1 + v2nvga fce2 ) 3 f 16 n K 2 V 2 vga na f eff dc b

(28)

In Eq. (28), the first term is 1/f 2 phase noise ϕ1/f 2 , the second term is 1/f3 phase noise ϕ1/f 3 . 2.4. The overall phase noise of SRA Combine Eqs. (18) and (28), then we get the overall phase noise of SRA which consists of white phase noise ϕNF , 1/f2 phase noise ϕ1/f 2 , 1/f 3 phase noise ϕ1/f 3 and 1/f5 phase noise ϕ1/f 5 . Sϕ (fb ) = ϕNF +

ˇ1/f 2 fb2

+

ˇ1/f 3 fb3

+

ˇ1/f 5 fb5

(29)

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R. Shi et al. / Sensors and Actuators A 197 (2013) 15–24

Fig. 5. Phase noise power regions.

where ˇ1/f 2 is coefficient of 1/f2 phase noise, ˇ1/f 3 is coefficient of 1/f 3 phase noise, ˇ1/f 5 is coefficient of 1/f 5 phase noise. ϕNF =



A2vga 2fn Keff |q|2

ˇ1/f 2 =

ˇ1/f 3 =

9K32 A2vga |q|4

Q2 4kTQ + Avga Rx



2 V2 16fn2 Meff dc

9K32 A2vga |q|4



2 V2 16fn2 Meff dc

ˇ1/f 5 = εagc





v2na (1 +

4KTRx + v2na + v2nvga QAvga

2fn Rx Cin QAvga

 +

2 

A2vga 8fn Meff |q|2

+ v2nvga



4kT (1 + Avga ) A2vga Q

(30)

+

v2ndc Rx A2vga

+

v2na Rx





1+

2fn Rx Cin QAvga

2  +

v2nvga Rx

(31)



v2na fce1 + v2nvga fce2 + 2εagc

(32)

fn2 2Q 2

(33)

According to (29)–(33), we plot the overall phase noise of SRA in Fig. 5. Therefore, we should determine the dominant kind of phase noise based on the operating bandwidth of SRA, and then study the methods to reduce the phase noise. (1) When BW > fc20 , white phase noise dominates, Sϕ (fb ) ≈ ϕNF . According to Eq. (30), reducing quality factor Q or increasing vibration amplitude |q| can both reduce the phase noise, thus improve the resolution. (2) When fc20 > BW > fc32 , 1/f 2 phase noise dominates, Sϕ (fb ) ≈ ˇ1/f 2 /fb2 . According to Eq. (31), increasing quality factor Q can reduce the phase noise and improve the resolution. And there is an optimal value |q|opt for the vibration amplitude of the resonator which obtains the optimal resolution performance.

 |q|opt =

2

2 Keff Vdc

(4kTRx (1 + Avga )/A2vga Q ) + (v2ndc /A2vga ) + v2na (1 + (2fn Rx Cin /QAvga ) ) + v2nvga

9K32 fn Rx

(4KTRx /QAvga ) + v2na + v2nvga

 16 (34)

(3) When fc32 > BW > fc53 , 1/f 3 phase noise dominates, Sϕ (fb ) ≈ ˇ1/f 3 /fb3 . According to Eq. (32), reducing the vibration amplitude |q| can reduce the phase noise. (4) When BW < fc53 , 1/f 5 phase noise dominates, Sϕ (fb ) ≈ ˇ1/f 5 /fb5 According to Eq. (33), increasing quality factor Q can reduce the phase noise and improve the resolution. According to the analyses above, the phase noise can be effectively reduced by accurately adjusting the quality factor Q and vibration amplitude |q| under the bandwidth condition. But the 1/f3 phase noise ϕ1/f 3 can be divided to two parts: one is ϕ1/f 3 1 =



2 V2 9K32 A2vga |q|4 /16fn2 Meff dc





v2na fce1 + v2nvga fce2 (1/fb3 ) which is caused by the nonlinearity of the resonator; the other is ϕ1/f 3 2 = (2εagc /fb3 )

which is caused by up-conversion of the AGC noise. And 2εagc /fb3 is independent on Q or |q|. In the design of silicon resonant accelerometer, if we choose relatively small amplitude |q| to satisfy ϕ1/f 3 1  (2εagc /fb3 ), (2εagc /fb3 ) will become a bottleneck constraining the 1/f3 phase noise, and then ϕ1/f 3 min = (2εagc /fb3 ).

R. Shi et al. / Sensors and Actuators A 197 (2013) 15–24

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Fig. 6. The SRA developed by MEMS Inertial Technology Research Center: (a) the SEM image of mechanical sensitive structure of SRA; (b) vacuum packaged mechanical sensitive structure of SRA; and (c) silicon resonant accelerometer prototype.

3. Phase noise experiments 3.1. SRA prototype A SRA prototype (Fig. 6) developed by MEMS Inertial Technology Research Center is adopted for the phase noise experiments. The SRA prototype structure is mainly composed of two identical resonators, microlever and seismic mass (Fig. 6a). When external acceleration is applied on the accelerometer, the inertial force acting on the seismic mass will be amplified by microlever and applied to the resonators. One resonator is under tension with resonant frequency increasing; the other is under pressure with resonant frequency decreasing. And the difference between the two resonant frequencies is just the output of SRA. This differential design not only helps restrict the common mode errors, but also effectively improves the scale factor of SRA. The mechanical sensitive structure is fabricated by Deep Dry Silicon on Glass (DDSOG) process, and installed in a LCC (leadless chip carrier) surface mounted ceramic tube (Fig. 6b) with device-level vacuum packaging technology.

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Table 1 Silicon resonant accelerometer parameters. Sensor parameter

Symbol

Value

Effective stiffness 3rd order nonlinear stiffness coefficient Resonant frequency Quality factor DC bias voltage of resonator Equivalent resistance Input capacitance Vibration amplitude of resonator VGA gain

Keff K3 fn Q Vdc Req Cm |q| Avga

108.6 N/m 1 × 101 N/m3 26,906 Hz 120,000 10 V 306 k 1 pF 1.8 ␮m 0.153

2

vndc

White noise of DC Voltage reference White noise of amplifier Corner frequency of amplifier

fce1

White noise of VGA Corner frequency of VGA Noise coefficient of AGC

√ 5.1 × 10−7 V/ Hz √ −8 2.75 × 10 V/ Hz 100 Hz √ 5.22 × 10−6 V/ Hz 100 Hz 1.3 × 10−7 Hz2

v2na

v2nvga

fce2 εagc

Fig. 7. Calculated phase noise of SRA.

Table 2 Comparison of phase noise coefficients of SRA between calculated and measured. Phase noise coefficient

Calculated

ϕNF ˇ1/f 2 ˇ1/f 3 ˇ1/f 5

4.27 × 10−9 7.61 × 10−11 2.60 × 10−7 3.27 × 10−9

Measured Hz−1 Hz Hz2 Hz4

6.31 × 10−10 Hz−1 Not measured 1.00 × 10−7 Hz2 1.26 × 10−10 Hz4

3.2. Comparison of the phase noise model to previous work The test bandwidth of the SRA prototype is 100 Hz, and white phase noise dominates in this bandwidth. According to section 2.4, reducing quality factor Q or increasing vibration amplitude |q| may be suggested to reduce the white phase noise. However, in order to overcome the deleterious effects of parasitic feed through capacitance of the resonator, high Q is adopted in SRA by vacuum packaging technology to ensure start-up of the oscillator. Thus increasing vibration amplitude |q| is chosen to reduce the white phase noise in the optimal design of SRA. Substitute the parameters shown in Table 1 into the phase noise models, the analytical value for the phase noise in SRA is calculated (see in Fig. 7 and Table 2). For comparison, the parameters in Table 1 are also calculated according to the model in previous work [4–6] neglecting the AGC influence, and the results are illustrated in Fig. 7 with dotted line. Fig. 7 shows that, the model of the white phase noise established in this article is the same as the model from previous work. While there is a large difference in the low frequency range. Because of considering the 1/f3 phase noise and 1/f5 phase noise caused by the AGC noise v2nagc , the phase noise in low frequency range of this article’s model is larger than that of the previous model [4–6].

R. Shi et al. / Sensors and Actuators A 197 (2013) 15–24

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Fig. 8. Measured phase noise of SRA.

3.3. Model validation In order to verify the two phase noise models in Fig. 7, SRA prototype is placed in an incubator to work in a constant temperature, and a high precision frequency measurement circuit is used to detect its frequency output, and then the phase noise data can be derived by power spectrum transformation of measured frequency (see in Fig. 8 and Table 2). From Figs. 7 and 8, we can see that the phase noise model in this paper matches the experiment results more precisely compared to the previous model in [4–6]. It is worth noting that the 1/f2 phase noise is not found in neither calculated data nor measured data of the SRA prototype in Figs. 7 and 8. This is because the 1/f2 phase noise ϕ1/f 2 is inversely proportional to quality factor Q, while the 1/f3 phase noise is limited by the AGC noise, ϕ1/f 3 min = (2εagc /fb3 ), and the minimum value is independent of |q| or Q. Therefore the 1/f2 phase noise will be covered by 1/f3 phase noise and cannot be measured, when Q is increased in SRA to make ϕ1/f 3 min = (2εagc /fb3 )  ϕ1/f 2 . Thus 1/f3 phase noise 2εagc /fb3 generated by AGC is the noise limit of the low frequency range of SRA. Especially under the condition of BW < fc20 , 1/f 3 noise 2εagc /fb3 directly limits the accelerometer resolution. 3.4. Discussion From Table 2, it can be noted that the measured data for white phase noise and 1/f5 phase noise are approximately 10 dB lower than the model prediction. This can be caused by some non-ideal factors in the SRA prototype. According to Eq. (30), we think there are two reasons for the lower white phase noise in the measurement: (1) under the control of a fixed AGC circuit, the resonator vibration amplitude |q| can be increased while tans-impedance Rf is decreased by the parasitic capacitances in drive circuit; (2) the quality factor Q in the closed loop is actually lower than the measured value by ring-down test method in an open loop. The increased |q| and lower Q can both result in a lower measured data for whit phase noise. As for the lower 1/f5 phase noise, the white noise v2nw in Eq. (13) is the equivalent white noise of all the electronic devices in AGC. While the parasitic capacitances in AGC can result in low pass filters which will restrict the white noise, therefore the equivalent value can be overstated in the model. Meanwhile, the increased capacitance C1 in PI circuit by parasitic capacitance can also result in lower 1/f5 phase noise. 4. Summary In this paper, the phase noise model of SRA is set up. Especially the effect of the AGC circuit to the phase noise in low frequency range (1/f3 phase noise and 1/f5 phase noise) is analyzed. The restriction of resolution performance arising from 1/f3 phase noise generated by AGC is illustrated, and experiments are performed on SRA prototype developed by our research group to validate the phase noise model. Compared to the previous model in [4–6], the phase noise model in this paper matches the experiment results more precisely, and could provide guidance and reference for design in MEMS inertial device. From Table 2, the calculated phase noise of theoretical model is close to the measured phase noise. We will further improve the phase noise model. Research about reducing the effect of the AGC noise to the phase noise and further improving the resolution will also be performed. Appendix A. Supplementary data Supplementary data associated with this article can be found, in the online version, at http://dx.doi.org/10.1016/j.sna.2013.03.031. References [1] [2] [3] [4] [5] [6]

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Biographies Shi Ran was born in China in December 1986. He received the B.E. and M.E. degrees in Instrumentation Science and Technology from Nanjing University of Science and Technology in 2008 and 2011, respectively. In 2011 he joined the laboratory to pursue his Ph.D. degree in Instrumentation Science and Technology. His present field of interest is MEMS inertial sensors. Jia Fang-xiu was born in China in June 1981. She received the B.S. degree in Mechanical and Electrical Engineering from Henan university of Science and technology in 2003, and received the M.D. and Ph.D. degrees in Instrumentation Science and Technology from Harbin Institute of Technology, in 2005 and 2010, respectively. She is currently a lecturer in Nanjing University of Science and Technology. Her research interests include laser range finding, temperature control and MEMS circuits design. Qiu An-ping was born in China in December 1971. She received the B.S. degree from Hefei University of Science and Technology in 1993, and received the M.E. and Ph.D. degrees in Instrumentation Science and Technology from Southeast University in 1998 and 2001, respectively. During 2001and 2005, she served in Southeast University as an associate professor. Since 2005, she has been a professor in School of Mechanical Engineering, Nanjing University of Science and Technology. She has been involved in the research of micromechanical inertial sensors, currently focusing on design and development of micromechanical gyroscopes and resonant accelerometers. She holds 12 patents. Su Yan was born in China in December 1967. He received the B.E., M.E. and Ph.D. degrees in Instrumentation Science and Technology from Southeast University in 1989, 1996 and 2001, respectively. During 2001 and 2005, he served in Southeast University as an associate professor. Since 2005, he has been a professor in School of Mechanical Engineering, Nanjing University of Science and Technology. His research interests include MEMS sensors, biosensors, inertial navigation systems, and infrared sensors.