  We generalize the two-dimensional scalar differential equations for the
thickness-extensional operating modes of thin film bulk acoustic wave
resonators (FBARs) to include the effects of surface impedance so that the
equations can be used to model sensors based on FBARs. A mass sensor is
analyzed as an example to show the effectiveness of the equations obtained. The
mass layer inducer frequency shifts are calculated. It is also found that the
vibration may be confined under the driving electrode or the mass layer. These
are fundamental to mass sensor design.
