Trapped-Energy Vibratory Gyroscopes Using a Partially

Polarized Piezoelectric Ceramic Plate

Hiroshi Abe,1 Tetsuo Yoshida,

1 Toshihiro Ishikawa,

1 Noriko Miyazaki,

1 and Hiroshi Watanabe

2

12nd Technology Development Division, Research and Development Unit, Tokin Corporation, Sendai, 982-8510 Japan

2Department of Electrical Engineering, Fukushima National College of Technology, Iwaki, 970-8034 Japan

SUMMARY

Most conventional piezoelectric vibratory gyro-

scopes have been constructed by using the bending vibra-

tions as in a tuning-bar or a tuning-fork oscillator. However,

this kind of vibratory gyroscope faces substantial practical

problems because its vibration characteristics usually are

easily influenced by the support or lead installation of the

element. In contrast, if the trapped-energy vibrations are

used because they have a high degree of concentration of

the vibration energy and a nonvibrating region over a wide

range, these problems would be solved at once, and highly

precise and reliable piezoelectric vibratory gyroscopes

could be implemented. In this paper, we propose a trapped-

energy vibratory gyroscope with three electrodes, which

uses the thickness-shear vibrations excited by a parallel

electric field in a partially polarized piezoelectric ceramic

plate, as the structure for this type of trapped-energy vibra-

tory gyroscope and describe the principle of construction

for the gyroscope. First, the energy trapping of thickness-

shear vibrations in a partially polarized piezoelectric ce-

ramic plate, which is the basic theory of the vibratory

gyroscope, is theoretically analyzed and the mechanism

and characteristics of energy trapping are clarified. Next,

based on these results, the experimental results for the

piezoelectric vibratory gyroscope using a partially polar-

ized piezoelectric ceramic plate 1.5 mm thick are presented.

© 2001 Scripta Technica, Electron Comm Jpn Pt 2, 84(3):

44�52, 2001

Key words: Trapped-energy vibratory gyroscopes;

thickness-shear vibration; partially polarized piezoelectric

ceramic plate; parallel electric field excitation.

1. Introduction

A piezoelectric vibratory gyroscope is an angular

velocity sensor that uses the Coriolis force and is now being

implemented with substantial research centered on its ap-

plication to video camcorders and car navigation systems

[1�3]. Although high-precision vibratory gyroscopes of

low cost have recently been expected to become practical

in fields demanding high reliability such as automotive

airbags and vehicle stability control (VSC), conventional

piezoelectric vibratory gyroscopes which use bending vi-

brations in tuning bar or tuning fork have high sensitivity,

but are faced with the serious practical problems of the

degradation of the characteristics and the loss of reliability

caused by the mechanical support and lead wire installation.

In contrast, if the trapped-energy vibrations which have a

high degree of concentration of the vibration energy and a

wide nonvibrating region are used, the mechanical support

and electrical connections of the elements can have a solid

structure. As a result, highly precise and reliable piezoelec-

tric vibratory gyroscopes can be obtained [4, 5]. Previous

reports have discussed energy trapping of thickness-shear

vibrations excited by parallel electric field and this type of

trapped-energy gyroscope using the mass loading effect in

© 2001 Scripta Technica

Electronics and Communications in Japan, Part 2, Vol. 84, No. 3, 2001Translated from Denshi Joho Tsushin Gakkai Ronbunshi, Vol. J82-C-I, No. 12, December 1999, pp. 798�805

44

a LiTaO3 piezoelectric crystal plate or a fully polarized

piezoelectric ceramic plate [8, 15].

In this paper, we describe the principle of constitution

for a trapped-energy vibratory gyroscope that uses the

thickness-shear vibrations excited by parallel electric field

in a partially polarized piezoelectric ceramic plate. We then

propose a simple structure for the vibratory gyroscope with

three strip electrodes on one side of a piezoelectric ceramic

plate. We also analyze the energy trapping of thickness-

shear vibrations in a partially polarized piezoelectric ce-

ramic plate, which is the basic principle of this vibratory

gyroscope, by a one-dimensional theory, and verify the

validity of this analysis by experiments on the resonator.

Finally, based on these results, we describe some experi-

mental results on the trapped-energy vibratory gyroscope

with three electrodes.

2. Structure of Trapped-Energy Vibratory

Gyroscope

2.1. Principle of constitution

A vibratory gyroscope is an angular velocity sensor

that uses the vibrations produced by the Coriolis force in

the direction perpendicular to the drive vibration direction

when the angular velocity acts. To achieve this, the two

vibration modes should be orthogonal (independent), and

it is desirable that they are doubly degenerated modes that

coincide in the resonant frequency. Trapped-energy modes

that satisfy this condition are the cross-sectional trapped-

energy mode in a piezoelectric bar and the thickness-shear

trapped-energy mode in a piezoelectric ceramic plate or

piezoelectric crystal plate [4, 6�8].

Previously, the authors have implemented the energy

trapping of thickness-shear vibrations without depositing

an insulator for mass loading on the excitation area between

the electrodes when the center part of the piezoelectric

ceramic plate was polarized in the thickness direction, and

a pair of strip electrodes was provided in the polarized part

as shown in Fig. 1. We showed experimentally how to

obtain a piezoelectric resonator with good characteristics

[7, 9]. This thickness-shear mode resonator excited by

parallel electric field has the driving electrodes installed on

one side of the plate. In addition, an important feature is its

ability to freely control the main displacement (shear dis-

placement) by changing the direction of driving electric

field (the direction of electrode arrangement). Thus, when

thickness-shear vibrations excited by this parallel electric

field are used, two groups of opposite electrodes are pro-

vided on a piezoelectric ceramic plate. If these groups are

arranged perpendicular to each other, the two orthogonal

thickness-shear vibrations can be independently driven and

detected. This is advantageous for constitution of the gyro-

scopes. Figure 2 is a schematic of a trapped-energy vibra-

tory gyroscope based on this principle. If a drive voltage is

applied between electrodes 1 and 1c, the thickness-shear

vibration in the x direction is excited between these elec-

trodes. If the angular velocity W is added in this state, the

thickness-shear vibration in the y direction, which is per-

pendicular to the angular velocity, is excited by the Coriolis

force. As a result, the vibration output for the y direction

that is proportional to this angular velocity can be detected

by electrodes 2 and 2c.

Fig. 1. Energy trapping of thickness-shear vibration

excited by parallel electric field.

Fig. 2. Trapped-energy vibratory gyroscope.

45

2.2. Trapped-energy vibratory gyroscope with

three electrodes

In the vibratory gyroscope with four electrodes

shown in Fig. 2, the driving electric field between electrodes

1 and 1c is affected by detection electrodes 2 and 2c. As a

result, the thickness-shear vibration (x-mode) cannot be

efficiently excited. Figure 3 shows the structure of the

trapped-energy vibratory gyroscope with three-electrodes

as a new design improvement. In the thickness-shear mode

resonator excited by parallel electric field shown in Fig. 1,

one of the two strip electrodes which are provided on the

plate surface is divided in two to make the three-electrode

structure of the DR electrode, PU1 electrode, and PU2

electrode, as shown in Fig. 3. If a driving electric field is

applied between the DR electrode and the (PU1 + PU2)

electrodes, the thickness-shear vibration in the x direction

is excited, and the thickness-shear vibration in the y direc-

tion, which is produced by the angular velocity W, can be

detected by the PU1 and PU2 electrodes. The two orthogo-

nal thickness-shear vibrations are driven and detected in the

three-electrode structure by using the current detection

circuit shown in Fig. 4 [10]. Since the input terminal of this

circuit is virtually connected by the function of the opera-

tional amplifier and the output voltage corresponding to the

input current is simultaneously obtained, the input terminal

can also serve as the detection terminal and the earth

terminal. The PU1 and PU2 terminals have the same phase

relationship with respect to the x direction vibration, and

the opposite phase relationship with respect to the y direc-

tion vibration produced by the Coriolis force. The output

voltage proportional to the angular velocity is detected as

the output difference between two current detection cir-

cuits. Figure 5 is a block diagram of the driving and detec-

tion circuit for the trapped-energy vibratory gyroscope with

three electrodes. After the outputs of two current detection

circuits are differentially detected, the detection circuit for

synchronized detection and rectification and an oscillation

circuit for self-excitation are provided.

3. Energy Trapping in a Partially Polarized

Piezoelectric Ceramic Plate

In order to build the trapped-energy vibratory gyro-

scope employing thickness-shear vibrations excited by par-

allel electric field, the energy trapping of thickness-shear

vibrations in a partially polarized piezoelectric ceramic

plate, which is the basic principle, is analyzed based on

one-dimensional theory, and the mechanism and charac-

teristics of the energy trapping are revealed. In order to

rigorously analyze the energy trapping of the thickness-

shear vibrations excited by parallel electric field as shown

in Fig. 1, two-dimensional analysis that simultaneously

takes account of the energy trapping in the x and y directions

is needed [11, 12]. But to simplify the analysis here, we use

Fig. 3. Trapped-energy vibratory gyroscope with three

electrodes.

Fig. 4. Current detection circuit.

Fig. 5. Block diagram of driving and detection circuit for trapped-energy vibratory gyroscope.

46

the one-dimensional analysis of energy trapping where the

vibration distribution in the depth direction (y direction)

perpendicular to the displacement direction (x direction) is

always assumed to be uniform [13]. Some experimental

results of the resonator are presented and the applicability

of this analysis is verified.

3.1. Theoretical analysis

As shown in Fig. 1, the center region (2L u 2W ) of a

piezoelectric ceramic plate is polarized in the thickness

direction, and after forming a pair of strip electrodes (2W ule) face to face on that polarized region, the driving electric

field is applied to the region. If this kind of structure is used,

the driving electric field E and the polarization P are or-

thogonal to each other. The thickness-shear vibration that

has a shear displacement in the same direction as the applied

electric field E is excited in this region (2L u 2W ). In this

resonator structure, the region between the electrodes is an

electrically shortened state, that is, the state equivalent to

the poled and fully electroded plate. Consequently, the

cutoff frequency of the center region is lowered compared

to the surrounding unpoled region. As a result, the energy

trapping of thickness-shear vibrations may be realized in

the center region, as shown in Fig. 1(b). To explain this

trapping mechanism, the dispersion curves of the thickness-

shear vibrations were calculated for the cases where the

piezoelectric ceramic plate was fully electroded and was

unpoled. The material constants used in this calculation

were for the piezoelectric ceramics NEPEC-6 (Tokin

Corp.). These constants were measured by the procedure

shown in the Standard of Electronic Materials Manufactur-

ers Association of Japan EMAS-6100. The material con-

stant for the case where the plate is unpoled was determined

by measuring the resonance response just before the polari-

zation of the piezoelectric ceramic plate polarized before-

hand vanishes by a heat treatment.

Because two orthogonal thickness-shear vibrations

are used to construct the gyroscope as described above,

two-dimensional energy trapping of thickness-shear vibra-

tions propagating in the x direction and thickness-twist

vibrations propagating in the y direction, which have the

main displacement in the x direction, must be realized and

then the plate end in the width direction needs to be fixed.

Figure 6(b) shows the dispersion curves for thickness-

shear vibrations propagating in the x direction, which are

calculated under the condition of being uniform with no

variation in the y direction, in the piezoelectric ceramic

plate polarized in the thickness direction, as shown in Fig.

6(a). The vertical axis shows the normalized frequency

W� 2SfH /v�, and the horizontal axis shows the normalized

propagation constant kxH. Here, v �c44E /U�1 / 2 and kx indi-

cate the wave numbers of the thickness-shear vibrations

propagating in the x direction. In the figure, the broken and

solid lines show the dispersion curves for the cases of fully

electroded and unpoled piezoelectric ceramic plate, respec-

tively. The symbols TS1 and TS3 represent the fundamental

thickness-shear and thickness-shear modes respectively.

Next, we examine the dispersion curves for the thick-

ness-twist mode which has the main displacement in the x

direction and propagates in the y direction. Figure 6(c)

shows the dispersion curves calculated under the assump-

tion of being uniform in the x direction. The horizontal axis

in the figure represents the normalized propagation con-

stant kyH in the y direction, where ky is the wave number of

the thickness-twist mode in the y direction. The symbols

TT1 and TT3 represent the fundamental and third thickness-

twist modes, respectively. As seen from Figs. 6(b) and 6(c),

the dispersion curves for modes TS1 and TT1 become the

low cut type for both cases of x direction propagation and

y direction propagation. Therefore, in the case of the fun-

damental thickness-shear mode, the two-dimensional en-

ergy trapping, which is of the forward-wave type both in

the x and y directions can be realized in the region between

the cutoff frequencies WU1, WU1g , and WE1 for the unpoled

and poled plates shown on the vertical axis in Fig. 6.

Figure 7 shows the resonant frequency spectrum of

the trapped-energy modes S�0, S�1, and S�2. This represents

the relationship between the dimensions (2L, 2W) of the

exciting region (polarized region) and the normalized reso-

nant frequency WR� 2SfRH /v� of the trapped mode, that is,

Here, D1 and D2 are given as follows:

Fig. 6. Dispersion curves for thickness-shear (TS) and

thickness-twist (TT) modes in the piezoelectric ceramic

plate poled in thickness direction.

47

where '1 and '2 are the frequency decreases represented by

Here, E1 and E2 are the constants when the characteristics

near the cutoff frequencies of the dispersion curves shown

in Figs. 6(b) and 6(c) are approximated by

As seen in Fig. 7, if the ratios of the dimensions of the

exciting region satisfy the conditions of

the single resonant response of the fundamental trapped

mode S�0 can be obtained. Substituting the material con-

stants of the piezoelectric ceramic plate into Eq. (1), the

dimension ratios L/H and W/H for obtaining the single

resonance response are given as follows:

3.2. Experiments

In order to verify the validity of the analysis based on

one-dimensional energy trapping theory, we constructed

the thickness-shear mode resonator shown in Fig. 1(a) on a

piezoelectric ceramic plate (NEPEC-6, Tokin Co.) having

contour dimension l0 = 25 mm and thickness 2H = 1 mm,

and then measured its impedance characteristics. In the

experiment, the plate end of the resonator was glued to a

plastic board for suppression of spurious responses. Figures

8(a) and 8(b) show the impedance characteristics of Reso-

nator 1 (L/H = 5, W/H = 5) and Resonator 2 (L/H = 7, W/H =

5) with the dimension ratios which satisfy Eq. (2). As seen,

a single resonance response of the fundamental trapped

mode S�0 was clearly observed near the frequency of 970

kHz in both cases. These resonant frequencies were normal-

ized by the material constants, and were plotted on the

frequency axis of the dispersion curves in Fig. 6. As a result,

the normalized resonant frequencies WR0 became 1.678 and

1.665 and were located between the two cutoff frequencies

WU1 (= 1.691) and WE1� S / 2 N 1.571�. Therefore, it was

experimentally verified that the energy trapping of this

(1)

(2)

Fig. 7. Frequency spectrum of trapped-energy modes as functions of the electrode dimensions L/H and W/H.

Fig. 8. Impedance characteristics of thickness-shear-mode trapped-energy resonators for case of D2 � W/H � �CC2.

48

thickness-shear vibration occurred in the frequency range

between these two cutoff frequencies.

Figures 9(a) and 9(b) show the impedance charac-

teristics of Resonator 3 (L/H = 5, W/H = 7) and Resonator

4 (L/H = 7, W/H = 7) in which the dimension ratios L/H and

W/H are set in the range not satisfying Eq. (2). In this case,

the inharmonic overtone responses of trapped mode S�1

appear besides the fundamental trapped mode S�0. This fact

corresponds well to the predictions described above.

4. Control of Resonant Frequencies by

Electrode Dimensions

For obtaining a highly sensitive piezoelectric vibratory

gyroscope, it is usually desirable to make the x-mode reso-

nant frequency fRx for the drive coincide with the y-mode

resonant frequency fRy for the detection vibrations. Hence,

we proposed a technique that made consciously a minute

resonant frequency difference between the two modes and

reduced the process variations without completely degen-

erating the two modes beforehand [14]. In either case, a

critical practical problem faced in designing a vibratory

gyroscope is the control of the resonant frequency differ-

ence between the two modes given by

We investigated experimentally the dependence of 'fon the electrode dimensions in the electrode configuration

shown in Fig. 10. The x-mode is excited when the driving

voltage is applied to the electrodes (PU1 + PU2) and DR.

Meanwhile, the y-mode is excited between the electrodes

Fig. 9. Impedance characteristics of thickness-shear mode trapped-energy resonators for case of D2 � W/H ! �CC2.

Table 1. Electrode dimensions

2L1 2L2 a1 a2 b d1 d2

(a) 5.0 5.0 0.5 Var iable 2.0 4.5-a2 1.0

(b) Variable 5.0 0.5 0.5 2.0 2L1-1.0 1.0

(c) 5.0 5.0 0.5 1.5 2.5-d2/2 3.0 Variable

Unit: mm

(3)

Fig. 10. Electrode configuration for gyroscope.

49

PU1 and PU2. In this experiment, the Pb(Zr, Ti)O3 piezo-

electric ceramic plate (NEPEC-6, Tokin Corp.) has thick-

ness 2H = 1 mm and contour length l0 = 25 mm. The

polarized region is the center area (2L1 u 2L2) of the plate.

Table 1 shows the electrode dimensions of the gyroscope.

Figure 11 shows the changes in 'f with respect to the

electrode dimensions a2, 2L1, and d2. It is seen from this

result that 'f can be freely controlled by changing the shape

dimensions of the electrode.

5. Experiments on Gyroscope

Based on the experimental results described in Sec-

tion 4, the electrode dimensions were designed to make the

frequency difference 'f extremely small, and a prototype

of a piezoelectric vibratory gyroscope with the structure of

three electrodes shown in Figs. 3 and 10. The contour

dimension of the plate used is 25 u 25 u 1.5 mm. The

electrode dimensions are 2L1 = 2L2 = 7 mm, a1 = 0.75 mm,

a2 = 4.9 mm, and b = 2.1 mm. The plate end about 1 mm

wide is glued with an epoxy resin to the glass epoxy

substrate for suppression of unwanted spurious modes and

mechanical support of the device. As shown in Fig. 12, the

experimental results reveal that the resonant frequencies for

x mode and y mode almost agree at 647 kHz, and the

resonance Q is around 900. The variations in Q before and

after fixing are less than 10%. The changes in the resonant

frequency caused by the fixing were not measured because

the effects of the supporting and fixing are very small.

Figure 13 shows the measured output voltage as a function

of the angular velocity. As seen, a detection sensitivity of

2.8 mV/deg/s, which is proportional to the angular velocity,

is obtained. Here, the driving voltage is 2 VP-P, and the total

circuit-gain for the differential amplifier and dc amplifier is

about 4000.

Fig. 11. Resonant frequency difference between x- and

y-modes.

Fig. 12. Impedance characteristics for x- and y-modes.

Fig. 13. Relationship between angular velocity and

gyro output voltage.

50

6. Conclusions

In this paper, we have proposed a new trapped-energy

vibratory gyroscope with three electrodes that uses the

thickness-shear vibrations excited by parallel electric field

in a partially polarized piezoelectric ceramic plate, and have

presented its principle of constitution and some experimen-

tal results. The analysis used was based on the one-

dimensional energy trapping theory for thickness-shear

vibrations in a partially polarized piezoelectric ceramic

plate. For the case of the fundamental thickness-shear

modes, we have shown that the energy trapping of the

modes, which become the forward-wave type in both the x

and y directions, can be realized in the frequency range

between the two cutoff frequencies for the fully electroded

and unpoled plates. In addition, we have verified this fact

experimentally and have derived the dimension conditions

where a clean single resonance response of the trapped

mode can be obtained. We demonstrated experimentally

that the resonant frequency difference between the two

orthogonal thickness-shear vibrations can be freely control-

led by changing the shape dimensions of the three elec-

trodes for drive and detection placed in the polarized region.

The experimental results of the gyroscope using a piezo-

electric ceramic plate with a thickness of 2H = 1.5 mm have

shown that a detection sensitivity of 2.8 mV/deg/s could be

obtained. The trapped-energy vibratory gyroscope with

three electrodes proposed here has two-dimensional and

solid structure and the resonant frequencies of the modes

can easily be adjusted by the electrode dimensions. Hence,

inexpensive and highly reliable vibratory gyroscopes can

be obtained. These facts indicate that new uses are expected

in the application fields of piezoelectric vibratory gyro-

scopes. A trapped-energy vibratory gyroscope using the

thickness-shear vibrations usually has the drawbacks of

high resonant frequency and low detection sensitivity. In

the future, we would like to design more compact gyro-

scopes with lower frequency by analysis of the vibration

distribution and by adoption of the plano-mesa structure,

and work to improve its performance.

Acknowledgment. We are very grateful to Profes-

sor Kiyoshi Nakamura of the Graduate School of Engineer-

ing of Tohoku University for valuable advice.

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51

AUTHORS (from left to right)

Hiroshi Abe (member) received his bachelor�s degree in engineering specializing in electronics from Iwate University in

1989 and joined Tokin Corp. He performs R&D on sensors, primarily piezoelectric vibratory gyroscopes. He is now with the

2nd Technology Research Lab of the Research and Development Unit.

Tetsuo Yoshida (member) received his bachelor�s degree in engineering specializing in communications from Tohoku

University in 1968 and joined Tohoku Metal Industries (now Tokin Corp.). He received a doctorate from Tohoku University in

1995. He has conducted R&D primarily on ceramic filters, ultrasonic motors, and piezoelectric devices such as piezoelectric

vibratory gyroscopes. He is now with the 2nd Technology Research Lab of the Research and Development Unit.

Toshihiro Ishikawa (member) received his graduate degree from Yamagata University in 1994 and joined Tokin Corp.

He is engaged in development of sensors such as ceramic filters and piezoelectric vibratory gyroscopes. He is now with the 2nd

Technology Research Lab of the Research and Development Unit.

Noriko Miyazaki (member) received her bachelor�s degree in engineering specializing in electronic information from

Yamagata University in 1996 and joined Tokin Corp. She is engaged in R&D on piezoelectric vibratory gyroscopes. She is now

with the 2nd Technology Research Lab of the Research and Development Unit.

Hiroshi Watanabe (member) received his bachelor�s degree in electrical engineering from Fukushima National College

of Technology in 1967. He became a member of the technical staff on the Faculty of Engineering of Tohoku University in 1969

and a research associate in 1984. He became a lecturer in the Department of Electrical Engineering, Fukushima National College

of Technology, in 1984, an associate professor in 1985, and a professor in 1993. In 1995, he was a visiting researcher at Princeton

University (Monbusho overseas research fellow). His research interests are bulk acoustic wave devices such as piezoelectric

resonators and filters, piezoelectric vibratory gyroscopes, and piezoelectric tactile sensors. He was the recipient of the 1979

Ohm Technical Award. He is a member of the Acoustical Society of Japan and IEEE.

52