Useful Formulas For MEMS Resonators

S21 To/From Rx

Measurements at high frequency are usually made using S-parameters, but what typically matters more is Rx. Assuming that a series resonator is at resonance, its Lx cancels out its Cx and only the series Rx is left:

figures/s21.ps.svg

Circuit model of a series resonator at resonance, with S-parameter measurement.

S21 is defined as:

S21, db = 20log(V − 2)/(V + 1)|V + 2 = 0

V + 1 is called the incident wave which is really just vin ⁄ 2, as if the source were delivering power into a matched load, i.e.

V + 1 = (R0)/(R0 + R0)vin = (1)/(2)vin

V − 2 is the wave incident on the load. Again, it is just a voltage divider:

V − 2 = (R0)/(2R0 + Rx)vin

Divide the two and you get:

S21, db = 20log(2R0)/(2R0 + Rx)

Equivalently:

S21 = 10(S21, db ⁄ 20)
Rx = 2R0(1)/(S21) − 1

For more details, see Characterization of Crystal Resonators.