Semiconductor devices - Micro-electromechanical devices - Part 32: Test method for the nonlinear vibration of MEMS resonators

IEC 62047-32:2019 specifies the test method and test condition for the nonlinear vibration of MEMS resonators. The statements made in this document apply to the development and manufacture for MEMS resonators.

Dispositifs à semiconducteurs - Dispositifs microélectromécaniques - Partie 32: Méthode d’essai pour la vibration non linéaire des résonateurs MEMS

L’IEC 62047-32:2019 spécifie la méthode d’essai et les conditions d’essai pour la vibration non linéaire des résonateurs MEMS. Les énoncés du présent document s’appliquent au développement et à la fabrication des résonateurs MEMS.

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Status
Published
Publication Date
23-Jan-2019
Current Stage
PPUB - Publication issued
Completion Date
24-Jan-2019
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IEC 62047-32
Edition 1.0 2019-01
INTERNATIONAL
STANDARD
NORME
INTERNATIONALE
colour
inside
Semiconductor devices – Micro-electromechanical devices –
Part 32: Test method for the nonlinear vibration of MEMS resonators
Dispositifs à semiconducteurs – Dispositifs microélectromécaniques –
Partie 32: Méthode d’essai pour la vibration non linéaire des résonateurs MEMS
IEC 62047-32:2019-01(en-fr)
---------------------- Page: 1 ----------------------
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---------------------- Page: 2 ----------------------
IEC 62047-32
Edition 1.0 2019-01
INTERNATIONAL
STANDARD
NORME
INTERNATIONALE
colour
inside
Semiconductor devices – Micro-electromechanical devices –
Part 32: Test method for the nonlinear vibration of MEMS resonators
Dispositifs à semiconducteurs – Dispositifs microélectromécaniques –
Partie 32: Méthode d’essai pour la vibration non linéaire des résonateurs MEMS
INTERNATIONAL
ELECTROTECHNICAL
COMMISSION
COMMISSION
ELECTROTECHNIQUE
INTERNATIONALE
ICS 31.080.99 ISBN 978-2-8322-6455-3

Warning! Make sure that you obtained this publication from an authorized distributor.

Attention! Veuillez vous assurer que vous avez obtenu cette publication via un distributeur agréé.

® Registered trademark of the International Electrotechnical Commission
Marque déposée de la Commission Electrotechnique Internationale
---------------------- Page: 3 ----------------------
– 2 – IEC 62047-32:2019 © IEC 2019
CONTENTS

FOREWORD ........................................................................................................................... 3

1 Scope .............................................................................................................................. 5

2 Normative references ...................................................................................................... 5

3 Terms and definitions ...................................................................................................... 5

4 Test parameters of nonlinear vibration of the resonators ................................................. 5

5 Test method for the amplitude-frequency response and phase-frequency response

of the nonlinear vibration ................................................................................................. 6

5.1 Test system ............................................................................................................ 6

5.2 Test conditions ....................................................................................................... 6

5.3 Test procedures ...................................................................................................... 7

6 Test method for the bending factor of the nonlinear vibrating frequency response ........... 7

7 Test method for the amplitude threshold for the nonlinear jump ....................................... 7

8 Test method for the frequency deviation as a result of the nonlinear vibration ................. 8

8.1 Frequency deviation of the self-excitation closed-loop system................................. 8

8.2 Frequency deviation of the phase-locked closed-loop system ................................. 8

8.3 Frequency deviation of the burst-excited system ..................................................... 9

Annex A (normative) Model of nonlinear vibration of MEMS resonators and bending

factor .................................................................................................................................... 10

A.1 Model of nonlinear vibration of MEMS resonators ................................................. 10

A.2 Solution of the nonlinear vibration model .............................................................. 11

A.3 Bending factor of the amplitude-frequency response ............................................. 11

Annex B (informative) Nonlinear jump of the frequency response of MEMS resonators ........ 14

Annex C (normative) Frequency deviation of MEMS resonators in the closed-loop

system .................................................................................................................................. 16

C.1 Effect of the frequency deviation of the MEMS resonator on the accuracy of

the resonant sensor .............................................................................................. 16

C.2 Frequency deviation of the self-exciting closed-loop system ................................. 16

C.3 Frequency deviation of the phase-locked closed-loop system ............................... 16

C.4 Oscillation frequency deviation of the burst-excited closed-loop system ................ 18

Figure 1 – Test system ........................................................................................................... 6

Figure 2 – 3 dB bandwidth of the linear amplitude-frequency response ................................... 8

Figure A.1 – Typical amplitude-frequency response of the nonlinear vibration of the

MEMS resonator ................................................................................................................... 12

Figure A.2 – Change of the amplitude-frequency response with the bending factor ............... 13

Figure B.1 – Jump phenomenon of the frequency response of a clamped-clamped

MEMS bridge resonator ........................................................................................................ 14

Figure B.2 – Three nonlinear amplitude-frequency responses with various amplitude level ... 15

---------------------- Page: 4 ----------------------
IEC 62047-32:2019 © IEC 2019 – 3 –
INTERNATIONAL ELECTROTECHNICAL COMMISSION
____________
SEMICONDUCTOR DEVICES –
MICRO-ELECTROMECHANICAL DEVICES –
Part 32: Test method for the nonlinear vibration of MEMS resonators
FOREWORD

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patent rights. IEC shall not be held responsible for identifying any or all such patent rights.

International Standard IEC 62047-32 has been prepared by subcommittee 47F: Micro-

electromechanical systems, of IEC technical committee 47: Semiconductor devices.
The text of this International Standard is based on the following documents:
FDIS Report on voting
47F/322/FDIS 47F/325/RVD

Full information on the voting for the approval of this International Standard can be found in

the report on voting indicated in the above table.

This document has been drafted in accordance with the ISO/IEC Directives, Part 2.

---------------------- Page: 5 ----------------------
– 4 – IEC 62047-32:2019 © IEC 2019

A list of all parts in the IEC 62047 series, published under the general title Semiconductor

devices – Micro-electromechanical devices, can be found on the IEC website.

The committee has decided that the contents of this document will remain unchanged until the

stability date indicated on the IEC website under "http://webstore.iec.ch" in the data related to

the specific document. At this date, the document will be
• reconfirmed,
• withdrawn,
• replaced by a revised edition, or
• amended.

IMPORTANT – The 'colour inside' logo on the cover page of this publication indicates

that it contains colours which are considered to be useful for the correct

understanding of its contents. Users should therefore print this document using a

colour printer.
---------------------- Page: 6 ----------------------
IEC 62047-32:2019 © IEC 2019 – 5 –
SEMICONDUCTOR DEVICES –
MICRO-ELECTROMECHANICAL DEVICES –
Part 32: Test method for the nonlinear vibration of MEMS resonators
1 Scope

This part of IEC 62047 specifies the test method and test condition for the nonlinear vibration

of MEMS resonators. The statements made in this document apply to the development and

manufacture for MEMS resonators.
2 Normative references

The following documents are referred to in the text in such a way that some or all of their

content constitutes requirements of this document. For dated references, only the edition

cited applies. For undated references, the latest edition of the referenced document (including

any amendments) applies.

IEC 62047-1, Semiconductor devices – Micro-electromechanical devices – Part 1: Terms and

definitions
3 Terms and definitions

For the purposes of this document, the terms and definitions given in IEC 62047-1 and the

following apply.
• IEC Electropedia: available at http://www.electropedia.org/
• ISO Online browsing platform: available at http://www.iso.org/obp
3.1
nonlinear vibration

vibration whose displacement has a nonlinear relationship with the elastic restoring force, with

the change of the vibration amplitude
3.2
nonlinear jump

jump phenomenon of the frequency response curve when the vibration amplitude exceeds a

certain threshold
3.3
frequency deviation

deviation of the vibration frequency of the resonator in a closed-loop system from the natural

frequency of the resonator
4 Test parameters of nonlinear vibration of the resonators
Test parameters of nonlinear vibration of the resonators are:
a) amplitude-frequency response of the nonlinear vibration, ( ) ;
P ω
b) phase-frequency response of the nonlinear vibration, ( ) ;
---------------------- Page: 7 ----------------------
– 6 – IEC 62047-32:2019 © IEC 2019
c) bending factor of the amplitude-frequency response, b ;
d) amplitude threshold for the nonlinear jump, a ;
e) frequency deviation of the self-exciting closed-loop system, E ;
f) frequency deviation of the phase-locked closed-loop system, E ;
g) frequency deviation of the burst-exciting closed-loop system, E
5 Test method for the amplitude-frequency response and phase-frequency
response of the nonlinear vibration
5.1 Test system
A test system consists of the following equipments:
a) laser vibrometer;
b) micro-optical apparatus;
c) signal generator;
d) vacuum chamber;
e) mounting fixture;
f) vacuum pump;
g) angle valve.
The test system is illustrated in Figure 1.
Laser
Computer
vibrometer
Micro-optical
Vacuum pump
apparatus
Vacuum
chamber
Angle valve
Signal
generator
Mounting fixture
IEC
Figure 1 – Test system
5.2 Test conditions
a) Keep the ambient temperature within 23 °C ± 5 °C.

b) Maintain the vacuum degree of the vacuum chamber according to the actual operation of

the resonator.

c) Adjust the micro-optical apparatus to restrict the laser spot within the surface of the

resonator.

d) For transparent resonators, the laser beam should illuminate on the metal layer on the

resonator to enhance the reflected laser intensity.

e) The connection of the resonator, the installing base and the vacuum chamber should be

strong enough.
MEMS
resonator
---------------------- Page: 8 ----------------------
IEC 62047-32:2019 © IEC 2019 – 7 –

f) For out-of-plane vibrating resonators, the surface of the installing base should be parallel

to the horizontal plane. For in-plane vibrating resonators, the surface of the installing base

should be set to a certain angle about the horizon level, to ensure enough intensity of the

reflected laser into the detector.

g) The vacuum pump and the vacuum chamber should be connected with flexible bellows in

case of the vibration propagation from the pump to the chamber.

h) The angle valve should be tightly shut off to well maintain the vacuum level, and then the

pump turned off before operating the test procedure.
5.3 Test procedures

a) Set the frequency of the vibration excitation according to the estimated value of the

natural frequency of the resonator. And then implement the initial frequency scan around

the natural frequency within a wide range. The amplitude frequency response and the

phase frequency response can be measured according to the vibration displacement of

the resonators. And record the resonant frequency of the resonator.

b) Reset the vibration excitation parameters to implement the frequency scan for the second

time: reducing the interval of the frequency scan to half of that set in the initial frequency

scan and reducing the range of the frequency scan to ten times of the half-power

bandwidth of the amplitude frequency response. The amplitude frequency response and

the phase frequency response can be measured according to the vibration displacement

of the resonators. And record the resonant frequency of the resonator.

c) Compare the resonant frequencies obtained by the initial and the second tests. If the

discrepancy in the resonant frequencies is smaller than 1 ppm of the resonant frequency

measured in the second test, either the initial or the second test result can be deemed as

the accurate amplitude frequency response and the phase frequency response. If the

discrepancy in the resonant frequencies exceeds 1 ppm of the resonant frequency

measured in the second test, the third time frequency scan with further small frequency

interval should be implemented, until the discrepancy in last tested resonant frequency

and the previous one is smaller than that 1 ppm of the resonant frequency measured in

the last test.
6 Test method for the bending factor of the nonlinear vibrating frequency
response

a) Test the nonlinear vibrating amplitude frequency response of the MEMS resonator

according to the method presented in Clause 5. And obtain the resonant frequency ω and

the resonant amplitude a .

b) The bending factor can be calculated by substituting the resonant frequency ω and the

resonant amplitude a into Formula (A.11) as provided in Annex A. The value of ω can be

r n
determined by its design value.
ωω−
b= (1)
7 Test method for the amplitude threshold for the nonlinear jump

a) Obtain the bending factor of the amplitude frequency response of the MEMS resonator by

the method set out in Clause 6.

b) The nonlinear jump phenomenon is presented in Annex B. Figure B.1. The amplitude

threshold for the nonlinear jump can be calculated by substituting the bending factor b

into Formula (2).
__________
ppm = part per million
---------------------- Page: 9 ----------------------
– 8 – IEC 62047-32:2019 © IEC 2019
a = (2)
33Qb
where
a is the amplitude threshold for the nonlinear jump;
is the quality factor of the resonator with linear vibration.
The value of can be obtained by the following formula:
(3)
Q=
ωω−
where

ω and ω are the boundary points of the -3 dB bandwidth of the linear amplitude-frequency

1 2
response, which is shown in Figure 2.
0 dB
–3 dB
ѡ ѡ ѡ
1 n
Frequency
IEC
Figure 2 – 3 dB bandwidth of the linear amplitude-frequency response
8 Test method for the frequency deviation as a result of the nonlinear vibration
8.1 Frequency deviation of the self-excitation closed-loop system

a) Obtain the bending factor of the amplitude frequency response of the MEMS resonator by

the method provided in Clause 6.

b) Frequency deviation of the self-excitation closed-loop system can be calculated by

substituting the bending factor into Formula (C.3).
E ×100 % (4)
8.2 Frequency deviation of the phase-locked closed-loop system

a) Obtain the bending factor of the amplitude frequency response of the MEMS resonator by

the method provided in Clause 6.
Amplitude
---------------------- Page: 10 ----------------------
IEC 62047-32:2019 © IEC 2019 – 9 –

b) Frequency deviation of the phase-locked closed-loop system can be calculated by

substituting the bending factor b into Formula (C.12).
π 2
(5)
E ×100 %
8.3 Frequency deviation of the burst-excited system

a) Obtain the bending factor of the amplitude frequency response of the MEMS resonator by

the method provided in Clause 6.

b) Frequency deviation of the burst-excited system can be calculated by substituting the

bending factor b into Formula (6)
− ωt
Ee= (6)
where
t is the time with the definition of t= 0 when disconnecting the excitation;
e is the natural constant.
---------------------- Page: 11 ----------------------
– 10 – IEC 62047-32:2019 © IEC 2019
Annex A
(normative)
Model of nonlinear vibration of MEMS
resonators and bending factor
A.1 Model of nonlinear vibration of MEMS resonators

The vibration behaviour of the MEMS resonators can be illustrated by the Duffing equation

which is shown in Formula (A.1)
3 *
  (A.1)
mx++cxk xk+ x =F cosωt
( )
where
m is the equivalent mass of the resonator;
c is the coefficient of the damping;
k is the linear stiffness coefficient;
k is the nonlinear stiffness coefficient;
is the amplitude of the driving force;
ω is the frequency of the driving force;
x is the vibration displacement;
t is the time.
Then transform Formula (A.1) by dividing it by the equivalent mass, m :
c k F
  (A.2)
x+ω x=−−x x + cos(ωt)
mm m

For an actual resonator, the nonlinear term− kx m in Formula (A.2) is small. Therefore a

small parameter ε is introduced to implement the multiple scales algorithm. Formula (A.2) is

transformed to:
  (A.3)
x+ω x=−−2εμx εαx +Fcosωt
( )
where
(A.4)
(A.5)
2mε
F= (A.6)
---------------------- Page: 12 ----------------------
IEC 62047-32:2019 © IEC 2019 – 11 –
A.2 Solution of the nonlinear vibration model

The solution of the nonlinear vibration model is derived by the multiple scales algorithm,

which is shown in Formula (A.7).
xacosωt−+φ Ο ε (A.7)
( ) ( )
where
a is the vibration amplitude of the MEMS resonators;

is the phase delay between the vibration displacement of the MEMS resonator and the

force;
Ο ε is the symbol of the same order infinitesimal of ε .
( )

The amplitude-frequency response of the MEMS resonator is shown in Formula (A.8) in an

implicit expression.
2 22
ω=ω+±ba −ε μ (A.8)
4ωa
where
3αε
b= (A.9)

The phase-frequency response of the MEMS resonator is shown in Formula (A.10) in an

implicit expression.
(A.10)
ω=ω+−b sinφ μεcotφ
2 22
4εω μ
A.3 Bending factor of the amplitude-frequency response

The nonlinear amplitude-frequency response of the MEMS resonator can be derived from

Formula (A.8), which is shown in Figure A.1. The curve between the left and the right

branches is the bone curve.
---------------------- Page: 13 ----------------------
– 12 – IEC 62047-32:2019 © IEC 2019
0,0
ѡ r
归一f11化频率
Frequency
IEC
Key
1 left branch
2 bone curve
3 right branch
Figure A.1 – Typical amplitude-frequency response
of the nonlinear vibration of the MEMS resonator
The bone curve is a parabolic form, which can be presented by Formula (A.11).
ωω−
(A.11)
where
ω is the natural frequency of the resonator.
The parameter in Formula (A.11) is the bending factor, which can be obtained by
substituting the resonant point (ω , a ) into Formula (A.11).
r r
ωω−
(A.12)

The intensity of the nonlinear vibration is evaluated by the bending factor . Figure A.2 shows

the change of the nonlinear amplitude-frequency response with the bending factor. The

amplitude-frequency response with negative bending factor bends towards the lower

frequency, which is noted as the softening spring effect. And the one with the positive bending

factor bends towards the higher frequency, which is known as the hardening spring effect.

The amplitude-frequency response with b = 0 indicates the linear vibration.
A11
Nor归m一ali化zed振 幅amplitude
---------------------- Page: 14 ----------------------
IEC 62047-32:2019 © IEC 2019 – 13 –
1 2 4
1,0
0,8
0,6
0,4
0,2
0,0
0,9 999 996 0,9 999 998 1,0 000 000 1,0 000 002 1,0 000 004
Norma归一lized化频率 ffr1n1equency
IEC
Key
1 b < 0
2 b = 0
3 b > 0
4 b > 0, and b >b
4 4 3
Figure A.2 – Change of the amplitude-frequency response with the bending factor
Normalized amplitude
归一A化1n1振幅
---------------------- Page: 15 ----------------------
– 14 – IEC 62047-32:2019 © IEC 2019
Annex B
(informative)
Nonlinear jump of the frequency response of MEMS resonators

The nonlinear jump occurs when the vibration amplitude exceeds a certain value, leading to

the jump of the frequency response, which is shown in Figures B.1 a) and B.1 b). Amplitude

and phase jumps would cause the breakdown of the resonator. Therefore, it is necessary to

evaluate the amplitude threshold for the nonlinear jump to keep the vibration amplitude of the

resonator in a resonant sensor smaller than the amplitude threshold.
0,14
0,12
0,10
0,08
0,06
Jump
Jump
0,04
0,02
0,00
41,6 41,8 42,0 42,2 42,4 42,6
频率 /kHz
FreFrqequueencnyc, yk /kHzHz
IEC
a) Amplitude jump
-20
-40
-60
-80
-100
Jump
-120
-140
Jump
-160
-180
-200
41,6 41,8 42,0 42,2 42,4 42,6
Frequ频率ency/,k HzkHz
Frequency /
IEC
b) Phase jump
Key
1 forward scan
2 backward scan
Figure B.1 – Jump phenomenon of the frequency
response of a clamped-clamped MEMS bridge resonator

There are three amplitude-frequency response curves in Figure B.2. The one with the

amplitude peak of ar does not take on the nonlinear jump. With the increase of the amplitude

peak, the one with the amplitude peak of ar jumps around its peak. Between the peaks of

ar and ar3, a given amplitude peak of ar demarcates the boundary between the nonlinear

1 2

jump and the non-nonlinear jump. This given amplitude peak is referred to as the amplitude

threshold for the nonlinear jump.
AIMpli t/udµeA, μm
振幅 /
Phase, deg out
Phase /°
相位 /deg
---------------------- Page: 16 ----------------------
IEC 62047-32:2019 © IEC 2019 – 15 –
ar (=ac)
jump
Frequency
forward scan
backward scan
IEC
Figure B.2 – Three nonlinear amplitude-frequency responses
with various amplitude level

For a resonator with natural frequency of 40 kHz, factor of 30 000, and measured bending

factor of 1,228 E+15 rad/(s·m ), according to Formula (2), the amplitude threshold i

...

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