ADS implementation of a compact model for circuit simulation of electromechanical components

A micro-electromechanical system can be often conveniently described by a monodimensional harmonic oscillator:

Fig.1

where for electrostatically controlled systems the external force stems from the applied control voltage V_In. This shows the implementation of a non-linear electromechanical capacitor, where the capacitance C depends on the mechanical vertical displacement z as induced by the applied control voltage signal V_In.

Fig. 2

Because C is not constant over time (due to z variation) the relationship between current, capacitance and voltage is:

(1)

where V₁ and I₁ are the voltage and current of the (externally seen as) 1-Port non linear component. The non-linearity is provided by the capacitance expression:

(2)

where eps0 and epsr are the air and dielectric permittivity, t_d is the dielectric thickness and g₀ is the initial gap between suspended beam and bottom electrode.

The beam has mass m, and is surrounded by air with friction coefficient b . Then from the principle of virtual work force can be expressed as:

(3)

(see also Rebeiz 2.32). Considering Fig.1 the dynamic equation is:

(4)

which is actually the D'Alembert equation:

(5)

and can be transduced to the electrical domain if the mechanical displacement is mapped into the charge in a capacitor (see Feynman 23.5):

(6)

thus:

(7)

which, because I=dQ/dt , can also be written as:

(8)

(6) shows that in the electrical domain F_ext has the dimension of a electromotive force or a voltage, and every term on the left-hand side represents the voltage drop at the respective component. However, for implementation purposes it is convenient to deal with F_extk rather than F_ext:

(9)

This is equivalent to scaling the whole circuit such that v is mapped into v'=Cv which can still be regarded as a voltage even though psychically the product provides a charge. Then the displacement can be directly seen as the voltage across the capacitor of the scaled equivalent electric circuit and defined as a port voltage in the ADS Symbolically Defined Device (SDD) with no need to explicitly integrate I to get Q (or z).Therefore the non-linear capacitor can be implemented with additional equations at all ports of a 6-ports SDD:

(P1.1)

(P3.1)

(P3.2)

(P4.1)

(P4.2)

(P6)

(P5)

(P1.2)

with the last one properly defining the non linear capacitor.