A silicon micro$-$frame is shown in the figure, with a doped

piezoresistor layer having a Gauge factor G$.$ Derive an analytical

expression for the relative change of resistance under F$.$ The resistor is

doped uniformly throughout the length. Each arm of the cantilever has

a length of L $,$ a width W $,$ and a thickness t $.$ The uniform doping length

of the piezoresistor region is t$\_($p$),$ and t$\_($p$)$ is less than half of t$.$

Hints:

The mechanical deformation and resistance associated with the horizontal bar connecting the two arms should be

ignored

The strain on the piezoresistor varies over the length and depth of the piezoresistor. We expect a linear change in

strain on the beam surface, from the anchor to the tip.