150 JOURNAL OF MICROELECTROMECHANICAL SYSTEMS, VOL. 34, NO. 2, APRIL 2025
Modeling Tempe a u e E ec s in a MEMS Ring
Gy oscope: Towa d Physics-Awa e D i
Compensa ion
Meh an Hosseini-Pish oba , Membe , IEEE, and E dinc Ta a , Membe , IEEE
Abs ac — Tempe a u e plays an indispensable ole in he
long- e m pe o mance o MEMS gy oscopes, and despi e ex en-
si e s udies in he li e a u e, analy ical ea men o empe a u e
e ec s is s ill an open p oblem. This pape , o he bes o ou
knowledge, is he i s a emp o add ess his gap o ing
gy oscopes. We s a wi h a supe posi ion p inciple ha disen-
angles he mal displacemen ields om he gy oscope’s nominal
ib a ion. We se o h a geome ically nonlinea a ia ional
o mula ion o ob ain he empe a u e-induced s i ness ma ix.
We conduc empe a u e es s on ou 3.2 mm-diame e , 58 kHz
ing gy oscopes equipped wi h 16 capaci i e s ess senso s.
The expe imen al da a alida e ou analy ical modeling in he
ollowing key aspec s: 1) The model accoun s o no only changes
in ma e ial p ope ies bu also a less explo ed ac o , he mal
s esses. Thanks o a s ain in e pola ion module ha le e ages
he measu ed s esses, he model p edic s equency a ia ions
consis en ly and cap u es hys e esis loops a ising om esidual
s esses. No ably, we accu a ely es ima e he de ia ion o he
empe a u e coe icien o equency (TCF) om he expec ed
alue −30 ppm/◦C (based on he widely known −60 ppm/◦C
dependency o Young’s modulus o silicon). 2) The model is able
o cap u e s i ness couplings in he o de s o less han 0.1 N/m
(in a 7 kN/m de ice) and closely p edic s he quad a u e e o
and i s leakage in o he in-phase channel. Addi ionally, he model
inco po a es empe a u e a ia ions o mechanical scale ac o ,
d i e mode’s ampli ude, damping coupling, and sense mode’s
phase in e ms o hei con ibu ion o he in-phase e o . Based
on hese me i s, ou model se es as a building block owa d d i
compensa ion algo i hms encompassing he unde lying physics o
he empe a u e e ec s.
[2024-0163]
Index Te ms— Tempe a u e e ec s, ing gy oscope, s ess sens-
ing, quad a u e e o , in-phase e o .
I. INTRODUCTION
LONG-TERM pe o mance de e io a ion is pe asi e in
MEMS gy oscopes, hinde ing hei na iga ion-g ade
applicabili y. Tempe a u e has a p ominen ole in his p oblem
Recei ed 20 Sep embe 2024; e ised 13 Decembe 2024; accep ed
29 Decembe 2024. Da e o publica ion 15 Janua y 2025; da e o cu en
e sion 4 Ap il 2025. This wo k was suppo ed by he Eu opean Union’s
Eu opean Resea ch Council (ERC) unde G an 101116162-0-d i -ERC-2023-
STG. Views and opinions exp essed a e howe e hose o he au ho s only
and do no necessa ily e lec hose o he Eu opean Union o ERC. Subjec
Edi o V. Zega. (Co esponding au ho : Meh an Hosseini-Pish oba .)
Meh an Hosseini-Pish oba is wi h he Depa men o Elec ical and Elec-
onics Enginee ing, Bilken Uni e si y, 06800 Anka a, Tü kiye (e-mail:
[email p o ec ed]).
E dinc Ta a is wi h he Depa men o Elec ical and Elec onics Engi-
nee ing and he Na ional Nano echnology Resea ch Cen e (UNAM), Bilken
Uni e si y, 06800 Anka a, Tü kiye (e-mail: [email p o ec ed]).
Digi al Objec Iden i ie 10.1109/JMEMS.2024.3524796
due o i s co ela ion wi h scale ac o and bias d i [1],[2].
The e o e, unde s anding he e ec s o empe a u e is c i ical
o imp o ing he pe o mance o hese senso s beyond he
cu en limi s. On his basis, we pe o med se e al es s on
ou ing gy oscopes by exposing hem o con olled empe -
a u e cycles. Se e al key obse a ions eme ged om hese
expe imen s:
1) Acco ding o he well-known ∼−60 ppm/◦C empe a u e
dependency o silicon’s Young’s modulus [3], we expec ed
o see a empe a u e coe icien o equency (TCF) o
∼−30 ppm/◦C; howe e , we measu ed e y di e en al-
ues such as −12 ppm/◦C and −14 ppm/◦C.
2) O e empe a u e cycles, hys e esis loops o med in he
equency- empe a u e plo s, which poin o some i e-
e sible esidual e ec s.
3) Quad a u e and in-phase channels exhibi ed ou pu s in
andem wi h he empe a u e cycles, indica ing nonuni o m
s ess p o iles ha a ec he s i ness symme y o he gy o-
scope. E en wi h a con inuous quad a u e cancella ion, he
in-phase e o pe sis ed, which signi ies a ia ion sou ces
o he han quad a u e leakage.
The abo e obse a ions poin o he mal s esses as a majo
mechanism o empe a u e e ec s. P e ious wo ks [2],[4]
ha e es ablished he po en ial o s ess calib a ion o d i
compensa ion, and [5] de eloped an analy ical model o
he e ec s o mechanical s ess. In p ac ice, he sou ce o
s esses in MEMS gy oscopes is usually he mal a he han
mechanical. Acco dingly, we aim o expand such analy ical
modeling’s ambi o encompass he empe a u e e ec s and
explain he expe imen al obse a ions. This a emp will se e
as a basis o unde s anding he physics behind he empe a u e
dependency o long- e m d i . P elimina y esul s o his
s udy– o only equency a ia ions–ha e been p esen ed in
he con e ence pape [6].
A. Li e a u e Re iew
In [1], a d i compensa ion was de eloped o a high
quali y ac o quad uple mass MEMS gy oscope, which p e-
en ed he need o a empe a u e senso by aking ad an age
o he d i e mode’s linea empe a u e dependency. In a
compa able app oach, he equency- empe a u e cha ac e -
is ics o an on-chip in eg a ed esona o we e le e aged o
calib a e a MEMS accele ome e [7]. Fo a NEMS-based
gy oscope, i was shown in [8] ha he d i e and sense
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HOSSEINI-PISHROBAT AND TATAR: MODELING TEMPERATURE EFFECTS IN A MEMS RING GYROSCOPE 151
Fig. 1. Ring gy oscope’s SEM (a), schema ics (b), and a angemen o elec odes (c).
modes’ TCF misma ch is esponsible o he scale ac o
d i . The empe a u e dependency o d i o a uning
o k gy oscope was in es iga ed in [9] based on empe a-
u e chambe es s, and ma e ial p ope ies and elec onics
pa ame e s a ia ions we e in oduced as he main ac o s.
Fo a JPL/Boeing Gy oscope, i was shown in [10] ha
d i e/sense equencies and quali y ac o s ha e an exponen-
ial esponse–wi h a conside able ime lag– o slow s ep-like
empe a u e changes. A pole-ze o cancella ion con olle was
de eloped in [11] o gua an ee abo e 90 Hz bandwid h o
a dual-mass MEMS gy oscope agains empe a u e-dependen
equency a ia ions. A mic o-o en was epo ed in [12] o
he mally isola e an ine ial measu emen uni (IMU) and
p e en empe a u e luc ua ions using a PID con olle . The
bias d i beha io o a g oup o consume -g ade MEMS
gy oscopes was s udied in [13] o e −40◦C o 85◦C, aking
in o accoun he empe a u e-dependency o quad a u e and
sense mode phase e o s. Mechanical s esses we e ci ed
as he sou ce o disc epancy be ween he expe imen al and
p edic ed quad a u e e o s. Mode e e sal has been p oposed
o empe a u e calib a ion o ci cula gy oscopes [14],[15].
In his me hod, he de ice’s ope a ion pe iodically swi ches
be ween he wo wineglass modes o gene a e wo di e en
ou pu s whose combina ion cancels ou he bias. In [16],
ecu en neu al ne wo ks wi h a blend o machine lea ning
algo i hms we e used o model and compensa e o he d i .
Simila ly, neu al ne wo ks we e u ilized in [17] o empe a-
u e calib a ion o imp o e upon he con en ional polynomial
i ing.
B. Rele ance
Analy ical models o empe a u e e ec s in MEMS gy o-
scopes a e absen in he li e a u e. In ha espec , he exigency
o ou wo k lies in add essing he ollowing:
•In he exis ing s udies, he emphasis is o en on he empe a-
u e dependency o he ma e ial p ope ies, and he ole and
signi icance o he mal s esses ha e no been me iculously
explo ed.
•Cu en d i compensa ion me hods ocus on ex ac ing
pa e ns om inpu -ou pu da a (using, e.g., linea i ing o
neu al ne wo ks), whe e he physics unde pinning he d i
phenomenon is ea ed as a black box. The emphasis o such
TABLE I
RING GYROSCOPE’SPARAMETERS
me hods is, in e ec , on co ela ion a he han causa ion.
Ou goal is o lay he g oundwo k o models ha shed ligh
on he go e ning physics and he oo cause o he d i .
C. Con ibu ion
Ou gy oscope, shown in Fig. 1, has a double- ing s uc u e
designed o ope a e in he n=2 wineglass modes a ∼58 kHz.
The de ice was ab ica ed by a silicon-on-glass (SOG) p ocess
om (111) silicon wi h wa e -le el acuum packaging [18],
and i s dimensions a e gi en in Table I. Six een elec ode
pai s a ound he ou e ing a e used o di e en ial d i e,
sense, quad a u e cancella ion, and equency uning. Mo e-
o e , we ha e 16 capaci i e s ess senso s (eigh inne , eigh
ou e , 45◦sepa a ed) o moni o ex e nal s esses ac oss he
subs a e. We se ou an analy ical model o he e ec s o
empe a u e on he s i ness dis ibu ion o his ing gy oscope.
The depa u e poin o ou analysis is ha we can delinea e
wo mechanisms o such e ec s:
1) Change in ma e ial p ope ies, especially he
∼−60 ppm/◦C dependency o Young’s modulus;
2) The mal s esses ha a ise om 1) con lic be ween he mal
de o ma ions and s uc u al cons ain s, 2) coe icien o
he mal expansion (CTE) misma ches a he silicon-glass
and die a achmen in e aces.
Ou modeling adhe es o he linea heo y o he moelas-
ici y whe e s ains a e he linea sum o a he mal and a
mechanical pa [19]. The inno a i e ea u es o ou app oach
a e as ollows.
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152 JOURNAL OF MICROELECTROMECHANICAL SYSTEMS, VOL. 34, NO. 2, APRIL 2025
Fig. 2. Supe posi ion o h ee loading s a es o modeling empe a u e e ec s.
•We p opose a supe posi ion amewo k by dis inguishing
he mal componen s o he o al displacemen ield om he
nominal ib a ion o he gy oscope. Conside ing geome ic
nonlinea i y, we p esen a po en ial unc ion o calcula e
he s i ness ma ix induced by such he mal displacemen s
wi hin a a ia ional amewo k.
•The on-chip s ess senso s measu e he mechanical s ains
ha enable us o in e pola e he subs a e’s s ain ield in
combina ion wi h he mal s ains. We use his in e pola ion
o calcula e he displacemen s o he suppo ing beams and
changes in he elec os a ic gap. Fu he mo e, we show he
mechanical s ains con ain esidual e ec s o e empe a u e
cycles ha accoun o equency- empe a u e hys e esis.
We demons a e he e ec i eness o ou model h ough
compa ison wi h he expe imen al da a. The model
add esses he a o emen ioned expe imen al obse a ions
by success ully p edic ing he equency a ia ions, o ma ion
o equency- empe a u e hys e esis loops, TCF, and
quad a u e/in-phase e o s. Addi ionally, he model p o ides
insigh in o he e olu ion o he gy oscope’s s uc u al
s i ness o e empe a u e. This app oach allows us o
iden i y di e en ypes o s i ness based on how hey
mani es in he gy oscope’s pe o mance.
D. O ganiza ion
Sec ion II explains he solid mechanical basis o he
modeling. Sec ion III elabo a es on he p inciples o s ess
sensing in connec ion wi h he analy ical model. Sec ion IV
links he p e ious wo sec ions o he gy oscope’s equency
a ia ions and quad a u e/in-phase e o s. Sec ion Vp o ides
he de ails o empe a u e es s and discusses he analy ical-
e sus-expe imen al esul s. Sec ion VI concludes he pape
wi h a iew on u u e di ec ion.
II. ANALYTICAL MODELING
The mechanical s uc u e o he gy oscope is composed o
Ring#1 (ou e ing), Ring#2 (inne ing), eigh connec ing
beams be ween Rings#1 and 2, and eigh suppo ing beams
joining Ring#2 o he ancho ed inne s uc u e (see Fig. 1-(b)).
Ou objec i e, in a nu shell, is o ma hema ically desc ibe how
his s uc u e’s s i ness a ies as empe a u e changes. Le
T0be he ini ial s ess- ee empe a u e, and he gy oscope
is subjec o he he mal load 1T=T−T0. As he c ux
o ou modeling, we decompose he displacemen ield o he
gy oscope’s mo ing s uc u e in o he supe posi ion o h ee
dis inc s a es (see Fig. 2):
1) S a e I. The mal expansion (o con ac ion) o he mo -
ing s uc u e while i is ixed a he suppo ing beams
(i.e., homogeneous bounda y condi ions). The esul ing
s esses a e due o con lic s be ween he he mal de o -
ma ions o he beams and ings.
2) S a e II. The in e nal ancho ed s uc u e unde goes he mal
expansion (con ac ion) as well and pushes (pulls) he
mo ing pa s as a esul (amoun ing o nonhomogeneous
bounda y condi ions). The ensuing s esses a e mos ly
exe ed on Ring#2.
3) S a e III. Nominal n=2 wineglass ib a ion o he
gy oscope in which, Ring#1 displacemen ield is go e ned
by he mode shapes cos(2θ) (d i e mode) and sin(2θ)
(sense mode).
Despi e hei sha ed he mal o igin, we dis inguish S a e I om
S a e II since, o s i ness calcula ion, we a e conce ned wi h
he gy oscope’s mo ing s uc u e. S a e I desc ibes he he mal
e ec s endogenous o he mo ing s uc u e, while S a e II
accoun s o he exogenous e ec s in he o m o bounda y
loads a he suppo ing beams. S a es I and II ake place in
a slow ime scale de e mined by he empe a u e cycles (e.g.,
in he o de o minu es o hou s) while S a e III occu s in
a as ime scale de ined by he ope a ion equency (in he
o de o µs). This slow- as ime scale sepa a ion indica es
ha S a es I and II can be concei ed as he equilib ium
displacemen ield o a quasis a ic he mal loading. We hen
ob ain he esul ing s ess ield σ, which we can ea as a p e-
s ess in conjunc ion wi h he gy oscope’s nominal ib a ion.
In a a ia ional amewo k, we conside he modi ied po en ial
unc ion Ude ined as [20]
dU
d−V=1
2EE2
n+σEn(1)
(d−Vis he di e en ial olume elemen ) o desc ibe he in e -
ac ion be ween S a es I-III and o calcula e he s i ness ma ix
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HOSSEINI-PISHROBAT AND TATAR: MODELING TEMPERATURE EFFECTS IN A MEMS RING GYROSCOPE 153
[kσ
i j ] ∈ R2×2induced by σ:
dkσ
i j
d−V=∂2
∂Qi∂QjdU
d−V−dU
d−Vσ=0Q1,Q2=0
.(2)
He e, Enis he G een-Lag ange s ain [21] associa ed wi h he
nominal n=2 ib a ion o he gy oscope ( ha is, S a e III)
and Q1and Q2a e he displacemen s o he cos(2θ) and
sin(2θ) modes, espec i ely. In Equa ion (1), i is essen ial o
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154 JOURNAL OF MICROELECTROMECHANICAL SYSTEMS, VOL. 34, NO. 2, APRIL 2025
Fig. 3. Tempe a u e a ia ions o silicon and glass CTEs.
use he G een-Lag ange s ain o cap u e all he second-o de
e ms ha p oduce s i ness ( e e o he discussion o geo-
me ic nonlinea i y in [5]). Following he a o emen ioned
supe posi ion p inciple and Equa ion (2), he o al s i ness
ma ix o he gy oscope
K=Km+KT+KS−Ke−Kβ,(3)
comp ises he ollowing pa s:
•Km= [km
i j ] ∈ R2×2is he nominal mechanical s i ness
ma ix o he wineglass mode shapes in he absence o
he mal s esses (S a e III). I he alue o his ma ix a
he ini ial empe a u e T0is Km,0, hen i should be scaled
by he a ia ions o Young’s modulus, Km=E(T)
E(T0)Km,0.
•KT= [kT
i j ] ∈ R2×2is he he mal s i ness ma ix
esul ing om S a e I o he mal loading, cap u ing he
s i ness p oduced by he mal de o ma ions o he mo -
ing s uc u e while no ex e nal bounda y condi ions a e
applied.
•KS= [kS
i j ] ∈ R2×2is he s ess s i ness ma ix co -
esponding o S a e II o he mal loading; his s i ness
ma ix e lec s he e ec s o s esses ha expansion
(con ac ion) o he inne s uc u e imposes on he mo ing
pa s.
•Ke= [ke
i j ] ∈ R2×2is he s i ness ma ix associa ed wi h
he linea elec os a ic o ces.
•Kβ= [kβ
i j ] ∈ R2×2is he equi alen s i ness ma ix o
he Du ing- ype nonlinea elec os a ic o ces.
No e ha Keand Kβappea wi h he nega i e sign in (3)
because o hei elec os a ic o igin. The calcula ion de ails
o Kmand Kecan be ound in [22] and [23], espec i ely.
We app oxima e Kβusing He’s a ia ional me hod, which has
been shown o p o ide good accu acy in p edic ing esonance
equencies o oscilla o s wi h polynomial nonlinea i ies [24].
Ou app oach o compu e KTand KSconsis s o
1) calcula ing he displacemen ields o S a es I and II using
he Ri z me hod [25];
2) ob aining he s ess ield σin Equa ion (1) based on he
s ain-displacemen ela ion and he cons i u i e law;
3) de e mining he induced s i ness using Equa ions (2).
An exhaus i e ea men o his p ocedu e based on a ia-
ional p inciples o solid mechanics is epo ed in [26], and
we he e only quo e he inal esul s o he s i ness ma ices
in Box 1. I is impo an o no e ha he equa ions o
KTand KS ep esen an ex ensional- ype s i ness, which is
p opo ional o E A/Ras opposed o bending- ype ha ing
he E I/R3 ac o . These wo s i ness o igina e om he
in e ac ion o he ex ensional s esses ac oss he ings and he
bending mo ions o he n=2 mode shapes and could be
ha dening o so ening depending on he sign o he s esses.
As a esul , o cap u e his ype o s i ness in he modeling,
he assump ion o cen e line inex ensibili y–ubiqui ous in he
ing gy oscope li e a u e–should be e oked.
A. Tempe a u e-Dependen Ma e ial P ope ies
Based on he widely used app oxima ion dE/EdT≈
−60 ppm/◦C[3], we conside he ollowing model o he
Young’s modulus:
E(T)≈170 1−60 ×10−6(T−20)[GPa],(4)
whe e Tis in ◦C. Acco ding o he CTE measu emen s in [27]
and [28] o silicon and bo osilica e glass, espec i ely,
we use he ollowing polynomial app oxima ions alid o
T∈ [20◦C,100◦C]:
Si: α(T)≈2.3332 +9.2618 ×10−3T
−1.8901 ×10−5T2[ppm/◦C]; (5)
Subs a e: ˜α(T)≈3.1137 +1.5215 ×10−3T
−1.0524 ×10−5T2+1.3980 ×10−8T3[ppm/◦C].(6)
As plo ed in Fig. 3, he silicon’s CTE has a 22% a ia ion
when he empe a u e ises om 20 ◦C o 100 ◦C while his
alue o he subs a e is %1.
III. STRESS SENSING
A o al o 16 s ess senso s measu e he mal s esses ac oss
he gy oscope (Fig. 4-(a)). Eigh senso s a e placed in he
in e nal suspension egion, and he es a e loca ed ou side
he ou e elec odes. This con igu a ion helps us cap u e he
s ess e ec s on he suppo ing beams and elec os a ic gap.
Each s ess senso has an unbalanced b idge- ype mechanical
s uc u e ha con e s and ampli ies x-displacemen s in he
y-di ec ion ( ed lines in Fig. 4-(b)), which hen is picked up
by AC-modula ed di e en ial capaci ance eading [4],[29].
I is impo an o no e ha he s ess senso s do no espond
o pu e uncons ained he mal expansion (e.g., in a ma ched
CTEs scena io). The FEM simula ions isualized in Fig. 4-(b)
demons a e ha pu e he mal load (1T=50◦C) gene a es
a negligible displacemen g adien ac oss he s ess senso ’s
in e digi a ed inge s and has no app eciable e ec on hei
gaps. On he con a y, he pu e mechanical load εxx =
+42 µm/m (1x=50 nm applied o he unde nea h ancho s)
p oduces a signi ican ly mo e p onounced displacemen g a-
dien ha he s ess senso can esol e and pick up. In ou
gy oscope, hese mechanical s ains p ima ily esul om he
CTE misma ches o he di e en ma e ials, including PCB,
solde , ce amic package, and die-a ach epoxy. The la e is
pa icula ly impo an due o i s la ge CTE (∼140 ppm/◦C) and
p opensi y o de eloping esidual s esses. On his basis, he
ou pu ˜εi j ( )o he s ess senso loca ed a pola coo dina es
(˜
Ri,˜
θj)is gi en by
˜εi j ( )=∂˜u
∂ ( , θ, )(˜
Ri,˜
θj)− ˜εT( ), (7)
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HOSSEINI-PISHROBAT AND TATAR: MODELING TEMPERATURE EFFECTS IN A MEMS RING GYROSCOPE 155
Fig. 4. S ess senso s’ dis ibu ion (a); FEM esul s o y-displacemen (nm) unde pu e he mal and mechanical loads (b) (wo king p inciple is explained
in ed lines).
Fig. 5. In e pola ion o subs a e’s s ain and displacemen ields; ˜εTand
εTa e he he mal s ains a he subs a e and silicon laye , espec i ely.
whe e ˜u and ˜εTa e he adial displacemen and he mal s ain
in he subs a e le el. We hen p oceed by he in e pola ion
algo i hm explained in [5] o ob ain he adial displacemen
(see Fig. 5 o isualiza ion):
˜u ( , θ, )≈Z
0
φ⊤(ϱ, θ)a∗( )dϱ+ ˜εT( ) ,(8)
whe e a∗is he op imal s ain in e pola ion coe icien wi h
he ec o o Fou ie - ype basis unc ions φ⊤(ϱ, θ):
a∗( ):=a g min
a
1
2X
i,j˜εi j ( )−φ⊤(˜
Ri,˜
θj)a2
,
φ( , θ) := [1, ]⊤⊗ [1,cos(θ), sin(θ), . . . ,
cos(Nhθ), sin(Nhθ)]⊤.(9)
He e, ⊗s ands o he enso p oduc , and Nhis he selec ed
numbe o ha monics. This in e pola ion p ocess enables us o
calcula e he displacemen s 1i( )o suppo ing beams and
a ia ions o elec os a ic gaps a he silicon laye . As illus-
a ed in Fig. 5, hese calcula ed displacemen s comp ise wo
pa s:
1) The displacemen imposed by he subs a e ia ancho s.
We ob ain his pa based on he subs a e’s in e pola ed
displacemen ield (8) and glass CTE.
2) The displacemen due o he silicon’s he mal expan-
sion/con ac ion, calcula ed based on he silicon CTE and
he dis ance om ancho s o he poin s o in e es (bases
o suppo ing beams and edges o elec odes).
Elec onic noise is he dominan ac o in he pe o mance o
he s ess senso s gi en ha hey unc ion a 10 kHz, which
is conside ably below hei ∼300 kHz mechanical esonance.
We ound he esolu ion o hese senso s o be 5.3 nano-S ain,
which p o ed o be su icien in cap u ing he he mal s esses
du ing he expe imen s.
IV. TEMPERATURE EFFECTS ON GYROSCOPE’S
PERFORMANCE
The s i ness ma ix in Equa ion (3) is he key o quan i-
ying empe a u e e ec s on he gy oscope’s equencies and
ou pu e o s. The ma ix Kcan be concei ed as he pe u -
ba ion o he ini ial s i ness K0a he e e ence empe a u e
T0:
K0=Km,0−Ke,0−Kβ,07→ K=K0+δK,
δK=E(T)
E(T0)−1Km,0+KT+KS−(Ke−Ke,0)
−(Kβ−Kβ,0)(10)
whe e Km,0,Ke,0, and Kβ,0a e he ini ial nominal, elec o-
s a ic, and Du ing s i ness ma ices, espec i ely, and δKis
he pe u bing s i ness. Based on (10), a pe u ba ion analysis
can be ca ied ou o he a ia ions o eigen alues ( equen-
cies) and eigen ec o s (mode shapes’ o ien a ion). We e e
o [5],[26] o he ma hema ical de ails o such an analysis and
only ci e he pe inen esul s he e. Assume ha ini ially ( ha
is, co esponding o K0) he gy oscope has he mode shapes
cos(2θ−ψ1,0)and sin(2θ−ψ1,0)wi h he equencies ω1and
ω2and modal masses m∗
1and m∗
2, espec i ely. The o a ion
angles ψ1,0, ψ2,0a e due o ab ica ion impe ec ions and can
be de e mined om he ini ial quad a u e e o (see Box 2 o
an illus a ion). Unde he e ec s o empe a u e, hese modal
cha ac e is ics will be pe u bed such ha ωi,07→ ωi,0+δωi
and ψi,07→ ψi,0+δψi:
δωi≈δk∗
ii
2m∗
iωi,0
, δψi≈δk∗
12
m∗
i(ω2
1,0−ω2
2,0),i=1,2,(11)
whe e δk∗
i j a e he componen s o he s i ness pe u ba ion
δKas obse ed om he (ini ial) o a ed mode shapes:
δK∗=T⊤δK T,T=cos(ψ1,0)−sin(ψ2,0)
sin(ψ1,0)cos(ψ2,0).(12)
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156 JOURNAL OF MICROELECTROMECHANICAL SYSTEMS, VOL. 34, NO. 2, APRIL 2025
The a ia ions o equencies, mode shape o ien a ion,
and quali y ac o changes o e empe a u e gi e ise o
e o s in he gy oscope’s quad a u e and in-phase channels.
We ha e p o ided he de ailed exp essions in Box 2 o
a mode-misma ched gy oscope. We ha e assumed ha he
d i e mode is exci ed in o esonance by he AC o ce
F( )=Fdcos(ω1 )(using PLL and AGC loops), and he
equency spli is much la ge han he bandwid h. Acco ding
o he equa ions in Box 2, he ollowing ema ks a e in
o de :
•Tempe a u e-dependen a ia ions o mechanical scale ac o
(SF, see Box 2 o he de ini ion), d i e o ce/displacemen
(Fd,Ad), and sense mode’s phase (β2) cause a ia ions o
he o e all scale ac o .
•The misalignmen be ween he nominal d i e-sense axes
and he ac ual mode shapes ( he angles ψ1,2) leads o a
p ojec ion o he misma ch be ween he modes on o he
gy oscope’s ou pu ( ep esen ed by he di e ence be ween
he Aand B e ms). This misalignmen esul s in Type I
and II e o s, and, acco ding o (11), depends on s i ness
coupling.
•Type I e o is he main componen o he quad a u e, which
can be conside ed as he equi alen a e o he s i ness
coupling (since k12 ∝ |1ω|sin(2ψ1)). The demodula ion
angle ϕdem is used o compensa e o he nonideal phase
angle o he sense mode. Type I quad a u e e o can leak
in o he in-phase due o his phase e o (compa e he
cos(ϕdem)and sin(ϕdem)componen s). Since quad a u e is
o en o de s o magni ude la ge , his ype o leakage could
p oduce signi ican in-phase e o s. Type II mainly a ec s
he in-phase because he bandwid h is smalle han he
equency spli . We no e ha scale ac o a ia ion is p esen
in Type I and II h ough d i e o ce/displacemen ampli ude
changes.
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HOSSEINI-PISHROBAT AND TATAR: MODELING TEMPERATURE EFFECTS IN A MEMS RING GYROSCOPE 157
Fig. 6. Flowcha o modeling empe a u e e ec s.
Fig. 7. Tempe a u e es se up; he empe a u e senso (PTAT) and he
gy oscope on -ends a e benea h he daugh e boa d; he aluminum lid
p o ec s he wi e bonds.
•Type III e o a ises om a di ec in-phase e m, d,
which we assume o igina es om he damping coupling
be ween he wo modes. In ha espec , no only scale
ac o a ia ions, bu also empe a u e luc ua ions o d
impac his ype o e o . In e ec , Type III ep esen s
empe a u e-p one componen s o he in-phase ha would
pe sis e en a e elimina ing Type I and II wi h a quad a u e
con ol loop.
Fig. 8. Tempe a u e p o iles and d i e mode’s bandwid h a ia ions (ini ial
alue: 1.8 Hz) du ing Tes s#1 and 2.
•Type IV e lec s he e ec o scale ac o a ia ions on
ex ac ing he a e.
The lowcha in Fig. 6illus a es he algo i hmic p ocedu e
o he model om expe imen al inpu da a o he gy oscope’s
pe o mance ou pu s.
V. RESULTS AND DISCUSSION
A. Gy oscope Speci ica ions
We will epo wo da ase s–labeled as Tes s#1 and 2– om
wo di e en de ices. Fo he de ice in Tes #1, we measu ed
he equencies and quali y ac o s o he d i e (cos(2θ))
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158 JOURNAL OF MICROELECTROMECHANICAL SYSTEMS, VOL. 34, NO. 2, APRIL 2025
Fig. 9. S ess senso s’ measu ed da a e sus empe a u e o Tes #1 (i: inne , o: ou e , and numbe s indica e he angula posi ion).
Fig. 10. D i e mode’s equency a ia ions o e ime and empe a u e; expe imen al e sus analy ical p edic ions.
and sense (sin(2θ)) modes as 57.637 kHz, 31.93 ×103and
57.723 kHz, 30.13 ×103, espec i ely. The ini ial uncompen-
sa ed quad a u e e o is -3225◦/s, which amoun s o a mode
shape o a ion angle o −3.3◦. De ails o ex ac ing he mode
shape o a ion angle om he equency esponse can be ound
in [22]. A e co ec ing o elec os a ic so ening and he
Du ing e ec , we ob ained he ini ial mechanical equency
o he d i e mode as 58.045 kHz. Simila speci ica ions wi h
a sligh a iance hold o he de ice in Tes #2.
B. Expe imen al Se up
Fig. 7shows he se up used o he empe a u e expe -
imen s. The daugh e boa d con ains he MEMS die wi h
an on-PCB hea e unde nea h i . The gy oscope’s on -ends
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