IEC 62047-32:2019

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

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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

INTERNATIONAL
ELECTROTECHNICAL
COMMISSION
COMMISSION
ELECTROTECHNIQUE
INTERNATIONALE
ICS 31.080.99 ISBN 978-2-8322-6455-3

– 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

INTERNATIONAL ELECTROTECHNICAL COMMISSION
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SEMICONDUCTOR DEVICES –
MICRO-ELECTROMECHANICAL DEVICES –

Part 32: Test method for the nonlinear vibration of MEMS resonators

FOREWORD
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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.

– 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.
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:
Aω
a) amplitude-frequency response of the nonlinear vibration, ( ) ;
P ω
b) phase-frequency response of the nonlinear vibration, ( ) ;

– 6 – IEC 62047-32:2019 © IEC 2019
c) bending factor of the amplitude-frequency response, b ;
d) amplitude threshold for the nonlinear jump, a ;
c
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
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
r
the resonant amplitude a .
r
b) The bending factor can be calculated by substituting the resonant frequency ω and the
r
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−
rn
b= (1)
a
r
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
– 8 – IEC 62047-32:2019 © IEC 2019
4ω
n
a = (2)
c
33Qb
where
a is the amplitude threshold for the nonlinear jump;
c
is the quality factor of the resonator with linear vibration.
Q
The value of can be obtained by the following formula:
Q
ω
n
(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).
b
ba
s
E ×100 % (4)
ω
n
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
=
b) Frequency deviation of the phase-locked closed-loop system can be calculated by
substituting the bending factor b into Formula (C.12).
ba
π 2
(5)
E ×100 %
ω
n
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
n
ba
Q
Ee= (6)
ω
n
where
t is the time with the definition of t= 0 when disconnecting the excitation;

e is the natural constant.
=
– 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;
F
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)
n
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
( )
n
where
k
(A.4)
α=
mε
c
(A.5)
μ=
2mε
*
F
F= (A.6)
m
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
φ
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.
F
2 22
ω=ω+±ba −ε μ (A.8)
n
4ωa
n
where
3αε
b=  (A.9)
8ω
n
The phase-frequency response of the MEMS resonator is shown in Formula (A.10) in an
implicit expression.
F
(A.10)
ω=ω+−b sinφ μεcotφ
n
2 22
4εω μ
n
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.
=
– 12 – IEC 62047-32:2019 © IEC 2019
a
r
0,0
ѡ
ѡ r
n
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−
n
a=
(A.11)
b
where
ω is the natural frequency of the resonator.
n
The parameter in Formula (A.11) is the bending factor, which can be obtained by
b
substituting the resonant point (ω , a ) into Formula (A.11).
r r
ωω−
rn
b=
(A.12)
a
r
The intensity of the nonlinear vibration is evaluated by the bending factor . Figure A.2 shows
b
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
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
– 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
ar
ar (=ac)
jump
ar
ѡ
n
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
Q
factor of 1,228 E+15 rad/(s·m ), according to Formula (2), the amplitude threshold is 75,6 nm.

Amplitude
– 16 – IEC 62047-32:2019 © IEC 2019
Annex C
(normative)
Frequency deviation of MEMS resonators in the closed-loop system
C.1 Effect of the frequency deviation of the MEMS resonator on the accuracy
of the resonant sensor
There are three kinds of closed-loop systems in the resonant sensors: the self-excited closed-
loop, the phase-locked closed loop and the burst-excited closed loop. All these closed-loop
systems take advantage of the linear frequency response characteristics to track the natural
frequency of the resonator. However, under the nonlinear vibration, the frequency response
would be changed. Consequently, the oscillation frequency of the closed-loop system drifts
from the natural frequency of the resonator, which cause the measurement error of the
resonant sensor.
C.2 Frequency deviation of the self-exciting closed-loop system
In a self-exciting closed-loop system, the driving force and the damping force nearly balance
each other. The vibration behaviour of the resonator is shown in Formula (C.1).
x+ω x=−εαx (C.1)
n
By the Lindstedt-Poincaré algorithm, the solution of Formula (C.1) is obtained. And the
vibration frequency is:
3αa
s 2
ωω=++ε Ο ε
sn ( )
8ω
(C.2)
n
=ω ++ba Ο ε
ns ( )
where
is the vibration frequency of a resonator in the self-exciting closed-loop system;
ω
s
a is the vibration amplitude of a resonator in the self-exciting closed-loop system;
s
Ο ε ε .
is the symbol of the same order infinitesimal of
( )
The frequency deviation of the self-exciting closed-loop system is:
ω −ω ba
sn s
(C.3)
E ≈×100 %
ωω
nn
C.3 Frequency deviation of the phase-locked closed-loop system
In the phase-locked loop, the vibration displacement maintains π 2 delay relative to the

driving force. Substitute φ=π 2 into Formula (A.9):
F
ω ωb+ (C.4)
π 2n
2 22
4εω μ
n
=
=
where
ω is the vibration frequency of a resonator in the phase-locked closed-loop system.
π 2
The frequency deviation of the phase-locked closed-loop system is:
ωω−
π 2n F
Eb ×100 % (C.5)
2 32
ω
4εω μ
n
n
F , in Formula (C.5), could be obtained from Formula (A.8):

F= 2ωaεμ (C.6)
nr
Therefore, Formula (
...