1

ICMEAT2012-3028

A novel Design of a z-axis MEMS vibratory Gyroscope

Vahid Azizi1, Farshad Barzande

2, Amir Masoud Khodadadi Behtash

3

1M.Sc student, Amirkabir University of Technology; Email: vah.azizi@aut.ac.ir

2Assistant Professor, Amirkabir University of Technology; Email: fbarzande@aut.ac.ir 3M.Sc student, Amirkabir University of Technology; Email: amir.behtash@aut.ac.ir

Abstract

This paper presents a novel design and principle of

operation of a one-axis micro machined vibratory

gyroscope base on resonant sensing of Coriolis force.

Two layer of polysilicon wafer is developed and bonded

to create the actuating and sensing layers and electrodes.

A rotary fluctuation with electrostatic actuation is used

to vibrate the outer gimbal. The rotary motion is

detected by capacitive sensing between inner gimbal

and doped electrodes. The noise of system is considered

and the resolution of sensor is reported. The design has

several advantages over similar products include

simpler dynamic and mechanism, high resolution and

ability of fabrication via wet etching and RIE method.

With high quality factor, the gyroscope is able to work

on ambient pressure with high resolution about

0.5 / /s Hz . Finally the fabrication steps based on

available techniques are presented.

Keywords: Micro-machining gyroscope, electrostatic

actuation, capacitive sensing, resolution and scale factor

Introduction

In simplest terms, gyroscope is the sensor that measures

the rate of rotation of an object. The name “gyroscope”

originated from Leon Foucault, during his experiments

to measure the rotation of the Earth [1].

As the name implies, Microelectromechanical

Systems (MEMS) is the technology that combines

electrical and mechanical systems at a micro scale.

Practically, any device fabricated using photo-

lithography based techniques with micrometer scale

features that utilizes both electrical and mechanical

functions could be considered as MEMS.

Even though an extensive variety of micromachined

gyroscope designs and operation principles exist,

majority of the reported micromachined gyroscopes use

vibrating mechanical elements to sense angular rate.

The concept of utilizing vibrating elements to induce

and detect Coriolis force presents many advantages by

involving no rotating parts that require bearings and

eliminating friction and wear. That is the primary reason

why vibratory gyroscopes have been successfully

miniaturized by the use of micromachining processes,

and have become an attractive alternative to their

macro-scale counterparts.

Up to know many micromachined gyroscopes based

on different operation and fabrication process

techniques are introduced. In 1994, Draper laboratory

report the first micromachined gyroscope using single

crystal silicon structure [2]. J. Bernstein et all, improved

the design to achieve 1 / /s Hz resolution using

tuning fork gyroscope [3]. In 1995, Murata hired surface

micromachined polysilicon to develop a lateral axis

vibratory gyroscope using comb electrostatic driving

and capacitive sensing by electrode underneath the

structure. A resolution of 2 / /s Hz was reported

[4].

1997 was the year of rate sensors. Many researcher

and academics reported different configuration on

driving and sensing mode of gyro rate sensors and

techniques in their fabrication process. Berkeley Sensor

and Actuator Center (BSAC) introduced a 2μm

polysilicon dual axis (x and y) gyroscope with rotary

drive mode excitation and two torsional sensing modes

to achieve 0.24 / /s Hz resolution [5]. Robert Bosch

Gmbh reported a z-axis micromachined tuning fork

gyroscope for automotive application with 50μm drive

mode amplitude. The resolution of 0.4 / /s Hz is

related [6]. In the same year, Samsung presented two

gyroscopes similar to Murata’s sensor using low

pressure chemical vapor deposition. The devise shows

0.1 / /s Hz resolution with vacuum packing [7, 8].

Samsung demonstrated the benefits of mode decupling

using bulk machining technique on a 40μm single

crystal silicon wafer by achieving 0.013 / /s Hz

resolution in 1999-2000 [9]. At the same year, other

gyroscopes were reported using CMOS-MEMS

fabrication process [10, 11]. In 2002, Sandia National

Laboratory reported an integrated micromachined

gyroscope with resonant sensing similar to BSAC using

IMEMS process. A resolution of 0.3 / /s Hz was

demonstrated with one chip integrated electronics [12].

The Institute of Micromachining and Information

Technology (HSG-IMIT) reported a gyro sensor

according to the patent decoupling principle decoupled

angular velocity sensor (DAVED) using SOI

technology. The latest version showed the RMS noise

below /0.025 s (50 Hz bandwidth) and bias stability of

2

0.3 / s over the temperature range [13]. Bin Xiong et

al. introduced another bulk micromachined gyroscope

working at atmosphere in 2003. The device structure,

called “slots gyroscope”, consists of a proof mass with

slots linked up to substrate by suspend springs and is

fabricated by silicon–glass bonding and deep reactive

ion etching (DRIE) [14]. Cenk Acar et al. 2005;

introduced a novel micromachined gyroscope with 2

degrees-of-freedom (DOF) sense-mode oscillator that

provides inherent robustness against structural

parameter variations. The tested unit exhibited a

measured noise-floor of 0.64 / /s Hz at 50Hz

bandwidth in atmospheric pressure [15].

Along the progress in fabrication process and using

industry's longest-running MEMS multi-project wafer

(Poly-MUMPs), many other gyroscopes were developed

till 2011 [16-19].

In this paper, we review the principle of operation,

fundamental technical challenges in design and

fabrication technologies and application of one-axis

micro machined vibratory gyroscope.

Theory of operation

The Coriolis effect is a direct consequence of a body’s

motion in a rotating frame of reference. "Figure 1"

illustrates the Coriolis acceleration on an object moving

with a velocity vector v on the surface of Earth from

either pole towards the equator. The Coriolis

acceleration deflects the object in a counterclockwise

manner in the northern hemisphere and a clockwise

direction in the southern hemisphere. The vector Ω

represents the rotation of the planet [20].

Figure 1: Coriolis force on the surface of the earth

The gyroscope derives its precision from the large

angular momentum that is proportional to the heavy

mass of the flywheel, its substantial size, and its high

rate of spin ("Figure 2"). This, in itself, precludes the

use of miniature devices for useful gyroscopic action;

the angular momentum of a miniature flywheel is

miniscule. Instead, micromachined sensors that detect

angular rotation utilize the Coriolis effect.

Fundamentally, such devices are strictly angular-rate or

yaw-rate sensors, measuring angular velocity. However,

they are colloquially but incorrectly referred to as

gyroscopes [21].

Figure 2: Illustration of a conventional mechanical gyroscope

and the three rotational degrees of freedom it can measure

Suppose in a fixed frame of reference, a point oscillates

with a velocity vector v. If the frame of reference begins

to rotate at a rate Ω, this point is then subject to a

Coriolis force and a corresponding acceleration equal to

2Ω×v. "Figure 3" shows a tuning fork structure for

angular rate sensor. Because of coriolis force, the

energy transfers from primary flexural mode to a

secondary torsional mode [21].

Figure 3: Coriolis effect; energy transfers from primary

flexural mode to a secondary torsional mode

In "Figure 4", a lumped model of a simple gyroscope

suitable for a micro machined implementation is shown.

The proof mass is excited to oscillate along the x-axis

with a constant amplitude and frequency. Rotation about

the z-axis couples energy into an oscillation along the y-

axis whose amplitude is proportional to the rotational

velocity [22].

Figure 4: lumped model for vibratory rate gyroscope

Current gyroscope Micro structure

"Figure 5" shows the sense and actuating diaphragm

used in current gyroscope. The operation principle is

based on conservation of primarily angular momentum,

instead of linear momentum as in translational vibratory

gyroscopes (showed in "Figure 4").

3

Figure 5: Sense and actuate diaphragm of current gyro sensor

The gyroscope structures consist of a rotational drive

oscillator that generates and maintains a constant

angular momentum, and a sense-mode angular

accelerometer that measures the sinusoidal Coriolis

moment.

In the drive-mode, the outer drive gimbal is excited

about the x-axis ("Figure 6"). In the presence of an

angular rate input about z-axis, the sinusoidal Coriolis

torque is induced about the y-axis, which causes the

sense-mode response of the inner mass ("Figure 7").

Gimbals are commonly used in torsional gyroscope

suspension systems to decouple the drive and sense

modes, and to suppress undesired modes.

Many suspension system and gimbal configurations

are possible in torsional vibratory gyroscopes. Similar to

linear gyroscope systems, the suspension system that

supports the masses and gimbals usually consists of thin

flexible beams, formed in the same structural layer as

the proof-mass.

Figure 6: schematic diaphragm of current gyroscope; (a) static

mode; (b) actuating mode

Figure 7: Transferring energy to the sense mode; (a) static

mode; (b) sensing mode

Design and equations

(A) Fundamentals

Using electrostatic actuation on electrode underneath

the structure, rotary fluctuation on torsional beam is

created. Based on conservation of angular momentum,

any rotation about z axis transfers energy of actuating

gimbal to the sense mode. The final movement of sense

electrodes is measured by capacitive sensing.

To found out the basically modeling of sensor, let

define the angular deviation of drive and sense mode

with θd and ϕ respectively ("Figure 8").

Figure 8: Angular deviation direction of drive and sense mode;

θd drive axis, ϕ sense axis.

The rotational equation of sense and drive mass could

be extracted by considering the vibration equation

around drive and sense axis. By assumption that the

angular rate input is constant, i.e. 0z , for small angle

oscillation, the rotational vibration equation of sense

and drive proof masses could be extracted respectively

as follows:

( )s s s s s s

y y y z y x d zI D K I I I (2)

( ) ( )d s d s d

x x d x x d x d dI I D D K M (3)

The only force on drive gimbal is electrostatic force.

When gyro rotates around z axis with constant velocity

Ω, the energy transfers to sense mode and force it to

vibration with angle ϕ. The term ( )s s s

z y x d zI I I is

coriolis force on the sense mode. The equation (2)

shows the fact that the movement of sense mode is

related to the rotation of diaphragm. This is the basic of

operation of torsional gyroscopes.

By solving the second rotational vibration equation,

drive angle could be extracted to: 0 sin( )d d d dt (4)

In which the amplitude and phase are defined as

follows:

0

22 2

11

d

d

x

d d d

M

KQ

(5)

1

2

1

tan

1

d dd

d

Q

(6)

In the resent equation, ωd is the natural frequency and

Qd is the quality factor of drive gimbal.

4

d deq x

d d d s

eq x x

K K

m I I

(7)

1

2

d d d seq eq x x

d dd d s

eq x x

K m I IQ

C D D

(8)

The response of sense mass to coriolis force is;

sin( )o

dt (9)

Similar to drive angular vibration response, the

amplitude and phase of sense response are extracted as

follows:

2

0

22 2

( )

11

s s s

z y x do

z s

y s

d

d d

s s s

I I I

I

Q

(10)

1

2

1

tan

1

d

s sd

d

s

Q

(11)

ωs and Qs are natural frequency and quality factor of

sense mode respectively.

s

y

s s

y

K

I

(12)

s

y s

s s

y

IQ

D

(13)

(B) Resonance characteristics

The amplitude response of one forced vibration system

is deeply depends on the ratio of input frequency to

natural frequency of the system. "Figure 9" shows the

amplification ratio of a second order vibration system

versus frequency ratio for different damping ratio (ζ).

For under damped system, large amplification is

possible if the frequency of input force is around the

natural frequency of the system.

To maximize the sensitivity of a vibration system, it

is generally desirable to operate at or near the peak of

the sense-mode response curve. This is typically

achieved by matching drive and sense resonant

frequencies. In other words, both drive and actuate

gimbal in platform of gyroscope should vibrate in

resonance situation.

The quality factor of vibration system is also directly

proportional to its sensitivity or scale factor. For an

under damped second order system, the amplification

ratio could be written as:

0

0

1

2

resx

QF k

(14)

Large quality factor results big amplification ratio and,

simultaneously, maximize the sensitivity or scale factor

of system.

Accordingly, the maximum sensitivity is possible if:

d

s d n

s d

(15)

Figure 9: Amplification of second order vibration system in

various frequencies and different value of damping ratio

In resonance situation, the phase of drive and sense

mode (respect to input) are -90° and -180° respectively.

The amplitude of responses could be extracted as

follows:

0

0

0 0

2

0

( )

( )

res

res d d

x

s s s

z y x n s

res z ress

y n

s s s

z y x s

res z ds d

y n x

MQ

K

I I I Q

I

I I I Q MQ

I K

0( )s s s

z y x s d

res z d s

x y n

I I I MQ Q

K I

(16)

Scale factor of system is product of drive and sense

quality factor:

2.

s d sy x x

s d ns d s

y x x

I I IQ Q

D D D

(17)

(C) Electrostatic actuation

The letter Md in (3) is harmonic torque which works as

input force on diaphragm of gyroscope. This moment is

produces by input voltage and electrostatic driving. The

electrostatic perpendicular force between two plates of

parallel capacitor ("Figure 10") could be found as [20]:

20

22en d

WLF V

d

(18)

5

Figure 10: Electrostatic force on parallel plate of capacitor

In current gyroscope, two couple of electrodes in each

side of driving gimbal is implemented. The total

moment of electrostatic normal force should be

harmonic around x axis. "Figure 11" shows the

actuating gimbal and drive electrodes underneath the

structure.

Figure 11: Drive electrodes

The driving voltage on the left side, V1, and the right

side, V2, are given as:

1 0

2 0

sin

sin

dc

dc

V V V t

V V V t

(19)

According to Equation (18), the driving forces towards

left side, F1, and towards right side, F2, are:

201 12

202 22

2

2

WLF V

d

WLF V

d

(20)

The moment of F1 and F2 around x axis is:

1 2dM rF rF (21)

From "Figure 11", r is the distance of electrostatic force

vector from x axis.

By substitution of equation (20) into (21), the harmonic

driving moment around x axis could be extracted as:

002

2sin

sin

M

d dc

d

r WLM V V t

d

M M t

(22)

In contrast of DC gain in driving voltage, the final

driving moment is completely harmonies with no DC

gain.

(D) Squeeze film characteristics

Squeeze film damping occurs when two parallel plates

move toward each other and squeeze the fluid film in

between. Squeeze film damping effects are quite

complicated, and can exhibit both damping and stiffness

effects depending on the compressibility of the fluid.

"Figure 12" show the distribution of air pressure due

to movement of parallel plate towards each other. This

effect damps the vibration of plates, especially in

microsystems scale.

Figure 12: Energy lost in due to squeeze film damping

Modeling of squeeze film is quite complicated and it is

not described here. Just the microsystem designer

should take care about damping and elastic constant in

modeling process due to squeeze file effect. The

squeeze file damping constant and squeeze film elastic

constant are reported in [23, 24] as: 2

0

8 2,0 2 2 2 2 4

64 1

( ) ( ) /

e

m n odd

P AK

h nmn m

(23)

2

2

0

6 2,0 2 2 2 2 4

64

( ) ( ) /

d

m n odd

nm

P AC

h nmn m

(24)

In which, A is the area of parallel plates, ho is the initial

gap, Po is the air pressure, m and n are odd numbers and

𝜔 is the frequency of vibration. η is the length to width

ratio (l/w) of plates and σ is the squeeze number: 2

2

0

12

a

l

P h

(25)

Where, μ is the dynamic viscosity of the air.

Geometry Design

Geometrical parameters of drive-sense mode of gyro

diaphragm and suspension beams are shown in "Figure

13" and "Figure 14" respectively. To find out the

amplitude of sense mode vibration diagram and

consequently the rate of gyroscope, one critical step is

the right modeling of coefficients in relation (2) and (3),

it means the equivalent torsional stiffness and damping

constant for both drive and sense mode.

6

Figure 13: Geometrical parameters of gyroscope diaphragm

Figure 14: Suspension beam parameters

Equivalent torsional stiffness

For torsional gyroscope shown in the "Figure 13", total

stiffness is sum of elastic stiffness in suspension beams

and stiffness comes from squeeze film phenomena. In

this section, extraction of driving mode stiffness is

presented. For sense mode stiffness, the conditions are

the same. d

x elastic beams eqK k k (26)

Where, keq is the equivalent squeeze film stiffness.

Elastic beam stiffness is basically originated from

elastic deformation accrued in angular vibration. For

drive mode suspension system with two beams the

elastic stiffness is:

2( )elastic beams

SGk

L

(27)

Where;

43

4

2(1 )

10.21 1

3 12

d

d dd d

d d

L L

EG

w wS t w

t t

(28)

For squeeze film elastic stiffness, derivation method is

quite different. "Figure 15" shows the section-cut of

gyroscope diaphragm. When the diaphragm is actuated

about x axis, gap distance between different potion of

diaphragm and substrate is changed. For small angle

deviation, we can assume four linear springs for

different portion of diaphragm, originated form squeeze

film phenomena. To find the equivalent squeeze film

stiffness about x axis, the linear springs could be

replaced with a torsional spring shown in "Figure 16".

Figure 15: linear spring modeling of squeeze film phenomena

Figure 16: equivalent squeeze film stiffness

According to potential energy conservation theory, the

potential energy stored in four linear springs ("Figure

15") is equal to potential energy stored in equivalent

torsional spring ("Figure 16").

2 2 2 2 2

1 1 2 2 3 3 4 4

1 1 1 1 1

2 2 2 2 2eqk k x k x k x k x

(29)

Where, x1 to x4 are the displacements of linear springs.

1 4 1

2 3 2

x x r

x x r

(30)

By equation (29) and (30), the equivalent squeeze film

stiffness is: 2 2 2 2

1 1 2 1 3 2 4 2eqk k r k r k r k r (31)

According to "Figure 15", the gyroscope diaphragm is

symmetrical about z axis, so:

1 2

3 4

k k

k k

(32)

With combination of (27), (31) and (32), driving mode

stiffness is:

2 2

1 1 3 22( ) 2 2d

x

SGK k r k r

L

(33)

Where, k1 and k3 are squeezing film stiffness and are

extracted from relation (23):

3 4

4

2

0

8

0

10.21 1

(1 ) 3 12

4. .

d d d d dx

d d d

t w E w wK

L t t

PA B C D

h

(34)

7

Where: 2( ) ( )ox oy iy oy iyA D D D D D

2,

2 2 2 2 4

1

( )( ) ( ) /

2

m n oddoy iy

ox

Bn D D

mn mD

3

x yC S S

2,

2 2 2 2 4

1

( ) ( ) /2

m n oddy

x

DnS

mn mS

Same method is hired to find out sense mode stiffness.

With no detail of extraction steps, the sense mode

stiffness is:

3 4

4

2 3

0

8

0

10.21 1

(1 ) 3 12

4.

s s s s sy

s s s

x y

t w E w wK

L t t

P S SE

h

(35)

Where:

2,

2 2 2 2 4

1

( ) ( ) /2

m n oddx

y

E

nSmn m

S

Equivalent damping ratio

Drive and sense damping is basically consisting of

viscous and thermoelastic damping. Thermoelastic

damping is the intrinsic material damping that occurs as

a result of thermal energy dissipation due to elastic

deformation. In a vibrating beam, alternating tensile and

compressive strains across the width cause irreversible

heat flow, which in turn results in an effective damping

due to dissipation of vibration energy. Under vacuum

conditions, thermoelastic damping is one of the primary

damping mechanisms [25]. However when system

works under ambient pressure, the magnitude of

thermoelastic damping is negligible against viscous

damping.

For current gyroscope, working at ambient pressure,

only viscous damping (squeeze film damping) is

considered. The extraction methods for both drive and

sense mode damping constants are quite similar to those

of stiffness. "Figure 17" shows the linear damper

modeling of squeeze film damping for drive mode

actuation. To find out the equivalent damping, linear

damper are replaced with a rotary damper shown in

"Figure 18".

Figure 17: linear damper models of squeeze film damping

Figure 18: equivalent rotary damper for drive mode actuation

According to dissipation energy equality between rotary

damper and linear dampers, we could write: 2 2 2 2

1 1 2 1 3 2 4 2eqdC C r C r C r C r (36)

Where, Ceqd is equivalent viscose damping of both drive

and sense mode. d s

eqd x xC D D (37)

By combination of (36) and (37), and consideration of

symmetry properties about z axis, the drive equivalent

damping ratio is:

2 2

1 1 3 22 2

sdxx

DD

d s

x xD D C r C r

(38)

Where C1 and C3 are squeeze file damping and are

computed from (24). By replacing the equivalent

amount of parameters in (38) we have:

30

6

0

4. .d s

x x x y

PD D F G S S H

h

(39)

Where;

2( ) ( )ox oy iy oy iyF D D D D D

2

2

2,

2 2 2 2 4

( )

2

( )( ) ( ) /

2

oy iy

ox

m n oddoy iy

ox

n D Dm

DG

n D Dmn m

D

2

2

2,

2 2 2 2 4

2

( ) ( ) /2

y

x

m n oddy

x

nSm

SH

nSmn m

S

8

Same method is hired to find out sense mode equivalent

damping ratio. With no detail of extraction steps, the

sense mode damping ratio is: 3

0

6

0

4.

x ys

y

P S SD I

h

(40)

Where; 2

2

2,

2 2 2 2 4

2

( ) ( ) /2

x

y

m n oddx

y

nSm

SI

nSmn m

S

Mechanical-thermal noise1

Noise is an area of science and technology that poses

practical problems but also has deep intellectual

attractions. Noise in components such as MEMS, and in

systems containing them, has two fundamental origins,

one external (extrinsic) and the other internal (intrinsic)

[26]. The magnitude of external noise signals coupled

into a MEMS device varies with the local conditions.

Our focus is on the fundamental noise sources within

MEMS devices because they provide the hard limits on

the device performance. The reduction of intrinsic noise

from various basic mechanisms is an important

challenge to the MEMS designer. One of the most

effective sorts of intrinsic noise is mechanical-thermal

noise which usually limits the accuracy of MEMS

devices [27-31].

The fundamental mechanism of mechanical-thermal

noise is Brownian motion of molecules or small

particles around MEMS devices. In other words, a

random arrival and collision of atoms and molecules

with surface of MEMS structures, produces an output

signal, called mechanical-thermal noise.

The equivalent rate of mechanical-thermal noise is

(detail extractions are described in [32]) :

,

0

2 B yy

z n

x x x

k TD

A m

(1)

Where, Ωz,n is noise equivalent rate in rad s Hz , Dyy is

sense mode damping ratio, kB is Boltzmann’s constant

( 231.38066 10 J / K ), Ax is amplitude of drive vibration

diagram, ωox is drive natural frequency and mx is driving

mass.

For current gyroscope the noise equivalent rate

could be rewritten as:

, 0

2

( )

s

B y

z n d s

d d x x

k TD

I I

(41)

Where;

1 Also called as “white noise”

0 0

d res d d

x

MQ

K

The mechanical-thermal noise (also called white noise)

defines the resolution of the sensor and is expressed in

/ /s Hz or / /hr Hz , which means the standard

deviation of equivalent rotation rate per square root of

bandwidth of detection [33]. According to relation (41),

one efficient way to minimize the effects of noise in

sensors, is maximize the quality factor Qd or driving

mode inertia ( d s

x xI I ). Although there were extensive

efforts from the researchers, the performance of the

micro gyroscope is not enough to apply for some

applications which require high precision. "Table 1"

shows the range of resolution required for typical high

performance application [33].

Table 1: Resolution requirement of gyroscope for typical high

performance application

Application Resolution (deg/hr) Current

capability

Inertial

navigation 0.01-0.001 Impossible

Tactical weapon

guidance 0.1-1.0 Impossible

Heading and

altitude reference 0.1-10 Challenging

For current gyroscope, using genetic algorithm (GA), as

an optimization tools, the resolution of sensors has been

minimized to 0.5 / /s Hz . In genetic algorithm,

available in MatLab toolboxes, the geometry parameters

of gyroscope diaphragm and resolution of system has

been defined as variable vectors and fitness function

respectively. The fact of resonance in both drive and

sense vibration was implemented using nonlinear

constraints. By different types of combination,

recombination and mutation, finally the resolution has

been minimized and the fitness value for resolution of

sensor was achieved.

Modeling results and Simulation

The mathematical model of gyroscope has been

simulated using MatLab code and Simulink software. In

rotational vibration equations of sense and drive mode

of gyroscope (Equation (2) and (3)), Md works as input

signal and it is the result of electrostatic actuation, and

Ωz is the input rate of sensor which is aimed to be

estimated using relation (16). The implementation of

modeled gyroscope is shown on "Figure 19" using an

open loop system. To summarize the simulation blocks,

the estimation loops for θ and ϕ are implemented using

two subsystems. The output port of "Divide 3" is the

estimated magnitude of input rate of gyroscope, which

is the input port of "Divide 1".

9

Figure 19: implementation of modeled gyroscope using an

open loop system

"Figure 20" shows the estimated rate of gyroscope for 8

deg/s rotational speed. A step in 0.05s resets the

MinMax running resettable block and estimation is

started. The final value is about 7.85 deg/s. This

estimation shows the modeling equations are able to

estimate input rate with an error about 2 %. This error

has been verified for other magnitude of input rate. One

of the most affective factors of error in estimation of

input rate is the way of modeling of stiffness and

damping in both sense and drive modes of system.

Figure 20: Estimated rate of modeled gyroscope for 8 deg/s of

input rate

Fabrications

The current gyroscope consists of two main layers; the

middle layer (oscillator masses) fabricated using surface

micro machining and lower layer fabricated using

surface-bulk micromachining. "Figure 21" shows two

layer of current gyroscope. These layers are fabricated

separately and bonded together using anodic bonding

technology.

The fabrication process of middle layer is shown in

"Figure 22". At the first step, the base silicon layer is

thinned to 100μm using chemical-mechanical polishing

methods (a). 200nm of Cr as sacrificial layer is

deposited on one side of silicon layer using electron

beam and vapor deposition (b). 1-2μm of photoresist

covers the Cr layer and is developed using lithography

process (c and d). At the next step, the Cr is patterned

and gets the mask layout used in the previous step and

photoresist is removed (e and f). Silicon is etched with

Reactive Ion Etching (RIE) process (h). Finally, Cr is

etched away and Au is deposited on one side layer as

upper electrode for both drive and sense modes (g).

Figure 21: layers of current gyroscope

Figure 22: fabrication of oscillator layer

The next layer of gyroscope is fabricated using bulk and

surface micromachining. The fabrication step is shown

in the "Figure 23". 500μm silicon wafer is hired as base

substrate (a). Same the oscillator layer process, 200nm

of Cr is deposited on one side of silicon layer (b). After

spinning developing photoresist, Cr layer is patterned

using HCI to get the mask layout (c-f). Then wafer is

etched about 25μm (which is the gap between

electrodes) using RIE method (h). Finally Cr is etched

away and Au is deposited using another mask to create

the underneath electrodes (g).

These layers finally are attached together using anodic

bonding process.

Figure 23: Fabrication of base layer

10

Conclusions

A novel micro machined vibratory gyroscope with

simple structure was designed and fabricated using RIE

technology. The design has novelty in layout and

structure and it is the first micro gyroscope with ability

of fabrication using available technologies in Iran. The

challenges in modeling of torsional gyroscopes with

decouple structures were presented. The survey also

shows principle of operations, methods of modeling

stiffness and damping in torsional planer rate sensors.

Thermal –mechanical noise of system was considered

and for current gyroscope it has been minimized to

0.5 / /s Hz which puts the sensor in category of

Heading and altitude reference as other challenging

gyroscopes. Surface micro machining steps of both

diaphragm layer and frame was described. The design

process based on fabrication technology restrictions,

proved the fact that planer torsional rate sensor are more

convenient in fabrication process and with optimized

resolution, they could works in the field of resolution of

comb drive challenging gyroscope.

Acknowledgment

The authors gratefully acknowledge Mr. N.sayyaf, M.Sc

engineer on manufacturing and mechatronics, for his

great efforts and cooperation in the field of microsystem

technology.

List of Symbols

s

xI x-moment of inertia of sensing plate

s

yI y-moment of inertia of sensing plate

s

zI z-moment of inertia of sensing plate

s

yD Sense-direction damping ratio of the

sensing mass s

yK Torsional stiffness of the suspension beam

connecting the sensing plate to the drive

gimbal d

xI x-moment of inertia of drive plate

d

xD Drive-direction damping ratio of the drive

mass s

xD Drive-direction damping ratio of the

sensing mass d

xK Torsional stiffness of the suspension beam

connecting the drive gimbal to the

substrate.

dM Harmonic moment acted of drive gimbal

d Drive direction deflection angle of the drive

gimbal 0

d Amplitude of drive deflection angle of the

drive gimbal

d Phase of drive deflection angle of the drive

gimbal

d Natural frequency of drive gimbal

Frequency of harmonic moment acted on

drive gimbal (Input frequency)

M Amplitude of harmonic moment acted on

drive gimbal

dQ Drive gimbal quality factor

Sense direction deflection angle of the

sensing mass o Amplitude of deflection angle of the

sensing mass

Phase of deflection angle of the sensing

mass

z Input rate

s Sensing mass natural frequency

sQ Sensing mass quality factor

dcV Direct portion of input voltage

0V Amplitude of sinusoidal portion of input

voltage

n, m Odd Number

0P Ambient pressure

A Area of electrode plates

Squeeze number

0h Initial gap between electrodes

eK Equivalent stiffness in due to squeeze film

effect

dC Equivalent damping ratio in due to squeeze

film effect Dynamic viscosity of air (1.983e-5 Kg/m.s)

l Length of electrode plate

w Width of electrode plate

E Module of elasticity (for Silicon is about 98

Gpa)

G Shear modulus

Poisson ratio

A0 y-outer dimension of frame

B0 x-outer dimension of frame

Ai y-inner dimension of frame

Bi x-inner dimension of frame

Dox x-outer dimension of drive gimbal

Doy y-outer dimension of drive gimbal

Dix x-inner dimension of drive gimbal

Diy y-inner dimension of drive gimbal

Sx x-dimension of sensing mass

Sy y-dimension of sensing mass

t Wafer thickness

kB Boltzmann’s constant ( 231.38066 10 J / K )

T Temperature in Kelvin

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