Problem $2$: An electro$-$thermo$-$mechanical actuator $(30$ points$)$

Thermal actuators are widely used MEMS components. Here we will design an actuator that can act as an optical shutter. The actuator takes the shape of a bent beam connected to a voltage source. As current flows through the beam, it heats up$,$ expands, and deflects, moving a square plate attached to the beam.

a$)$ Assuming steady state, determine analytically the temperature profile $\backslash ($ T$(\backslash$mathrm$\{$x$\})\backslash )$ as a function of the beam coordinate $\backslash ($ x $\backslash )$ when voltage $\backslash ($ V $\backslash )$ is applied. Assume $1-$D uniform power dissipation, ignore changes to the electrical resistance $\backslash ($ R $\backslash )$ as the beam heats up$,$ and assume that the ends of the beam are thermally grounded.

b$)$ Determine analytically the change in length $\backslash (\backslash$Delta l $\backslash )$ as a function of $\backslash ($ V $\backslash ).$

c$)$ For an arbitrary angle $\backslash (\backslash$phi $\backslash ),$ determine an analytical expression for $\backslash ($ y$\_\{0\}\backslash )$ as a function of $\backslash (\backslash$Delta l $\backslash ).$

d$)$ As a designer, you are presented with the following three material choices for this actuator. Which material will provide the greatest elongation for a given voltage stimulus?

e$)$ Assume the actuator is made of polysilicon, the total length $\backslash ($ l $\backslash )$ of the actuator beams is $1$ mm $,$ the actuator has a $\backslash (2\backslash$mu $\backslash$mathrm$\{$~m$\}\backslash$times $2\backslash$mu $\backslash$mathrm$\{$~m$\}\backslash )$ cross$-$section, and it is angled at $\backslash (\backslash$phi$=5\backslash )$ degrees. How much voltage is needed to move the plate $\backslash (5\backslash$mu $\backslash$mathrm$\{$~m$\}\backslash )?$ How much power will be dissipated in the actuator at this voltage? What will be the max temperature in the beam?

f$)$ To determine the response time of this structure, we can lump the thermal system. First, determine analytically the equivalent thermal resistance $\backslash ($ R$\_\{$T$\}\backslash )$ relating the heat current $\backslash ($ I$\_\{$Q$\}\backslash )$ and the maximum temperature $\backslash ($ T$($l $/2)\backslash ).$ You can ignore any beam thermal expansion in this calculation.

g$)$ The thermal capacitance of the structure will be dominated by the plate, so determine analytically the thermal capacitance $\backslash ($ C$\_\{$T$\}\backslash )$ of the plate.

h$)$ Finally, determine analytically the overall thermal time constant of the system. For the structure described in part e$,$ what is the numerical time constant? Assume that the square plate is $\backslash (150\backslash$mu $\backslash$mathrm$\{$~m$\}\backslash$times $150\backslash$mu $\backslash$mathrm$\{$~m$\}\backslash )$ and is $\backslash (2\backslash$mu $\backslash$mathrm$\{$~m$\}\backslash )$ thick, and assume a specific heat of $\backslash (700\backslash$mathrm$\{$~J$\}/\backslash$mathrm$\{$kg$\}-\backslash$mathrm$\{$K$\}\backslash )$ and a density of $\backslash (2300\backslash$mathrm$\{$~kg$\}/\backslash$mathrm$\{$m$\}^\{3\}\backslash )$ for polysilicon. Problem $2$: An electro$-$thermo$-$mechanical actuator $(30$ points$)$

Thermal actuators are widely used MEMS components. Here we will design an actuator that can act as an optical shutter. The actuator takes the shape of a bent beam connected to a voltage source. As current flows through the beam, it heats up$,$ expands, and deflects, moving a square plate attached to the beam.

a$)$ Assuming steady state, determine analytically the temperature profile $\backslash ($ T$(\backslash$mathrm$\{$x$\})\backslash )$ as a function of the beam coordinate $\backslash ($ x $\backslash )$ when voltage $\backslash ($ V $\backslash )$ is applied. Assume $1-$D uniform power dissipation, ignore changes to the electrical resistance $\backslash ($ R $\backslash )$ as the beam heats up$,$ and assume that the ends of the beam are thermally grounded.

b$)$ Determine analytically the change in length $\backslash (\backslash$Delta l $\backslash )$ as a function of $\backslash ($ V $\backslash ).$

c$)$ For an arbitrary angle $\backslash (\backslash$phi $\backslash ),$ determine an analytical expression for $\backslash ($ y$\_\{0\}\backslash )$ as a function of $\backslash (\backslash$Delta l $\backslash ).$

d$)$ As a designer, you are presented with the following three material choices for this actuator. Which material will provide the greatest elongation for a given voltage stimulus?

e$)$ Assume the actuator is made of polysilicon, the total length $\backslash ($ l $\backslash )$ of the actuator beams is $1$ mm $,$ the actuator has a $\backslash (2\backslash$mu $\backslash$mathrm$\{$~m$\}\backslash$times $2\backslash$mu $\backslash$mathrm$\{$~m$\}\backslash )$ cross$-$section, and it is angled at $\backslash (\backslash$phi$=5\backslash )$ degrees. How much voltage is needed to move the plate $\backslash (5\backslash$mu $\backslash$mathrm$\{$~m$\}\backslash )?$ How much power will be dissipated in the actuator at this voltage? What will be the max temperature in the beam?

f$)$ To determine the response time of this structure, we can lump the thermal system. First, determine analytically the equivalent thermal resistance $\backslash ($ R$\_\{$T$\}\backslash )$ relating the heat current $\backslash ($ I$\_\{$Q$\}\backslash )$ and the maximum temperature $\backslash ($ T$($l $/2)\backslash ).$ You can ignore any beam thermal expansion in this calculation.

g$)$ The thermal capacitance of the structure will be dominated by the plate, so determine analytically the thermal capacitance $\backslash ($ C$\_\{$T$\}\backslash )$ of the plate.

h$)$ Finally, determine analytically the overall thermal time constant of the system. For the structure described in part e$,$ what is the numerical time constant? Assume that the square plate is $\backslash (150\backslash$mu $\backslash$mathrm$\{$~m$\}\backslash$times $150\backslash$mu $\backslash$mathrm$\{$~m$\}\backslash )$ and is $\backslash (2\backslash$mu $\backslash$mathrm$\{$~m$\}\backslash )$ thick, and assume a specific heat of $\backslash (700\backslash$mathrm$\{$~J$\}/\backslash$mathrm$\{$kg$\}-\backslash$mathrm$\{$K$\}\backslash )$ and a density of $\backslash (2300\backslash$mathrm$\{$~kg$\}/\backslash$mathrm$\{$m$\}^\{3\}\backslash )$ for polysilicon.