MODELING THERMO-ELECTRO-MECHANICAL BEAMS IN SUGARINTRODUCTIONMODEL THEORYwhereMODEL VERIFICATIONDISCUSSIONCONCLUSIONACKNOWLEDGEMENTSREFERENCESMODELING THERMO-ELECTRO-MECHANICAL BEAMS IN SUGAR Colin Dewey, University of California, Berkeley INTRODUCTION QconvSUGAR is a simulation package for 3D MEMS devices that utilizes nodal analysis techniques [1]. It has been shown that nodal analysis is significantly faster than other simulation techniques and can be just as accurate. The latest version of SUGAR (v2.0) includes various models for 2D and 3D beams, electrical beams, and gap closing actuators [2]. In addition, SUGAR is easily extended through the use of user-defined models. Qrad Qcond w ∆x Qconv Qconv hQcond g QcondSUBSTRATE As the mathematics of heat transfer and thermal expansion are easily modeled by nodal analysis, I have implemented a SUGAR model for a simple thermo-electro-mechanical beam. Several groups have established the mathematics of thermo-electrical polysilicon beams [3, 4, 5]. Given the temperatures and voltages at the ends of a beam, the current passing through the beam, and the beam geometry, one can determine the temperature profile within the beam. From the temperature profile, the average temperature, resistance, and thermal expansion of the beam can be determined. Figure 1. A differential thermo-electric beamelement. In addition to the labeled heat flows,an electric current flows through the element,generating heat through joule heating. Thermo-electro-mechanical models are important in characterizing the properties of MEMS thermal actuators [6, 7, 8]. Hand analysis of thermal actuator designs is quite difficult and thus finite element analysis is often used. However, with large integrated systems, finite element analysis will be very computationally expensive. By using SUGAR, analysis of large systems involving thermal actuators will require more reasonable amounts of time and yield fairly accurate results. MODEL THEORY The equations used in the SUGAR model are based upon the model established by C. H. Mastrangelo in his PhD dissertation [3]. Figure 1 illustrates a differential beam element with the considered heat flows. The model takes into account several thermal effects: joule heating due to an electric current through the beam, thermal conduction through the ends of the beam and through the gap underneath the beam to the substrate, convection around the beam, and radiation from the beam. Following Mastrangelo’s derivation, but allowing for arbitrary temperatures (T1, T2) at the ends of the beam, the temperature profile in the steady-state within a beam is: φττττφ+−∆+−−=)sinh())21(sinh()cosh())21(cosh()()(LxTLxTxu where 1V1, T1 V2, T2 whAkAIhgwhshkTghkskwhkTwhkhTTlTTTTTTbbsbbgbgbbcsg==++=+=+++=+−=++===−=∆+=,112,141214112)(,,22/)(,2)(22332121ρδσγσβδβψδξγβεεψφετ I1, Q1 I2, Q2 I = V/R Figure 2. The SUGAR nodal model of athermo-electric beam. and where l, w, and h are the length, width, and height of the beam, g is the gap between the beam and the substrate, ξ is the temperature coefficient of resistance, Tg is the absolute temperature of the gas or fluid surrounding the beam, Ts is the absolute temperature of the substrate, hc is the convection heat-transfer coefficient, kb and kg are the thermal conductivities of the beam and surrounding gas, σb is the Stefan-Boltzmann constant for the beam, I is the current passing through the beam, and ρ is the resistivity of the beam at the temperature of the substrate. The parameter s is the shape factor in accounting for conduction from the beam to the substrate [5]. The temperatures T1 and T2 and the temperature profile are taken to be relative to the temperature of the substrate (the thermal ground). To adapt these equations for use within a SUGAR model, nodal equations must be derived. Figure 2 shows the nodal model used in SUGAR. In addition to the nodal state variables of displacement and rotation in a mechanical model, variables for temperature and voltage are assigned to the ends of the beam. The forcing functions corresponding to these new variables are, respectively, the heat flow (Q) and electric current (I) at the ends of the beam. To break out of the circular relationships of current determining temperature profile, temperature profile determining resistance, and resistance determining the current, a branch variable I is used to represent the electric current through the beam. A corresponding forcing function requires that Ohm’s law be satisfied. The forcing function equations are: [][]IRVVFIIIITTlAkQTTlAkQbranchbb−−=−==∆−−=∆+−=212121)coth()tanh()(2)coth()tanh()(2ττφτττφτ where R, the resistance of the beam, is given by: ()φττφξρ+−=+=)tanh()(1TuuAlR In the above equations, u is the average temperature along the beam. Once the average temperature in the beam has been calculated, the thermal stress and thermal force of the beam can be calculated: thththAFuEσασ=−= where α is the thermal coefficient of expansion of the beam and E is Young’s modulus. To complete the SUGAR model, the derivatives of the forcing functions with respect to the nodal state variables are determined to form the Jacobian matrix of the forcing function 2vector. SUGAR uses this matrix to converge on a solution for the state variables. MODEL VERIFICATION Verifying the implemented SUGAR model with experimental data was not an easy task, as the model is very sensitive to the values of the material properties and beam geometry. However, good correspondence was found between the experimental data of a simple thermal actuator by Allen et al. [8] and the simulated results of the actuator in SUGAR using standard material property values. A diagram of the thermal actuator at rest and with an applied DC voltage is shown in Figure 3. Figure 4. Temperature profile of simulatedthermal actuator with DC voltage of 4.36 V(current of 4.3 mA). The zero position istaken to be the point of contact between thelong “hot” arm and the anchor. As shown in Figure 4, the majority of the joule heating occurs in the “hot” arm of the actuator due to its large resistance. The “hot” arm expands more than the wide “cold” arm, which results a deflection of the tip. The correspondence of the experimental values with
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