**Figure 14.**

*<sup>k</sup>*<sup>1</sup> <sup>¼</sup> *<sup>G</sup> <sup>m</sup> dKT*

where *B* and *k*<sup>1</sup> represent the bias and scale factor, respectively; *KT* is the total stiffness of the folded beams; *m* is the total mass of proof mass and moving fingers; *e* represents the asymmetry of capacitive gap induced by the fabrication error; *d* is

From the equation of bias, it can be seen that if the asymmetry of capacitive gap

*dKT*

*α<sup>s</sup>* � *αeq*

*<sup>α</sup>eq* � *<sup>α</sup><sup>s</sup>*

where *La* expresses the distance from the anchor to the midline; *Lf* denotes the half length of an anchor for fixed fingers; *lf* defines the locations of first fixed finger, as shown in **Figure 13b**; *α<sup>s</sup>* indicates the CTE of silicon; *αeq* is called as equivalent CTE describing the thermal deformation of the top surface of the substrate and calculated by the analytical model for the MEMS die attaching proposed in litera-

Deriving the differential of the bias to the temperature, the TDB is expressed as

*<sup>m</sup>*Δ*<sup>T</sup>* <sup>¼</sup> *KA* � *KB*

Deriving the differential of the scale factor to the temperature, the thermal drift

Due to the asymmetry induced by the fabrication error, *KA* and *KB* are different from each other, and TDB is proportional to the difference between *KA* and *KB*. Therefore, the consideration on the imperfection is very important for discussing

Based on the discussion on TDB and TDSF, the factors affecting thermal drift

1. The model shows that TDB is only caused by thermal deformation, while TDSF consists of two parts caused by stiffness temperature dependence and thermal deformation, respectively. The two parts of TDSF are positive and negative,

2. The first part of TDSF can be reduced by high doping. TDB and the second part of TDSF can be both reduced by soft adhesive die attaching or increasing

respectively. However, the second part has a greater absolute value.

*dT* <sup>¼</sup> *<sup>α</sup><sup>E</sup>* <sup>þ</sup> *<sup>α</sup><sup>s</sup>* (7)

Δ*TLa* (8)

*La* (10)

(11)

*d* 

Δ*T* (9)

*<sup>m</sup> <sup>α</sup>eq* � *<sup>α</sup><sup>s</sup>*

*lf* þ *Lf <sup>α</sup>eq* � *<sup>α</sup><sup>s</sup>*

is not considered, then the bias nulls. In equations of bias and scale factor, the parameters varying with temperature include *KT*, *e*, and *d*. Variation of the stiffness

> *TCS* <sup>¼</sup> <sup>1</sup> *KT*

<sup>Δ</sup>*<sup>e</sup>* <sup>¼</sup> *KA* � *KB KA* þ *KB*

Δ*d* ¼ *dα<sup>s</sup>* þ *lf* þ *Lf*

ture; *KA* and *KB* stand for the spring stiffness connecting proof mass.

<sup>Δ</sup>*<sup>T</sup>* ¼ � *KT*Δ*<sup>e</sup>*

*<sup>k</sup>*1jΔ*T*¼<sup>0</sup>Δ*<sup>T</sup>* <sup>¼</sup> *TCS* � *<sup>α</sup><sup>s</sup>* <sup>þ</sup>

and method suppressing the thermal drift can be obtained:

TDB <sup>¼</sup> <sup>Δ</sup>*<sup>B</sup>*

of scale factor (TDSF) is expressed as

the thermal drift.

substrate thickness.

**102**

TDSF <sup>¼</sup> <sup>Δ</sup>*k*<sup>1</sup>

The variations of *e* and *d* are calculated by analytical method [44]:

the capacitive gap.

is induced by TCEM and CTE [7]:

*Reliability and Maintenance - An Overview of Cases*

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*Accelerometer with optimization for TDSF.*

3. In silicon structure, TDB can be reduced by middle-locating anchors for moving electrodes in sensitive direction or decreasing the stiffness asymmetry of springs, while the second part of TDSF can be reduced by middle-locating anchors for fixed electrodes in sensitive direction.

The TDSF of the MEMS capacitive accelerometer can both be induced by the TCEM and the thermal deformation, so the structure of the accelerometer is optimized to make the TCEM and thermal deformation compensate each other, as shown in **Figure 14** [62]. As such, TDSF is suppressed significantly.
