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Verilog-A RF MEMS Model Library

Large-signal compact models of electrostatically actuated RF MEMS devices — version 2.0.0

Overview

The library provides Verilog-A compact models of an electrostatically actuated fixed-fixed beam capacitive switch, a cantilever ohmic switch and a fixed-fixed beam resonator for DC, small-signal and large-signal circuit simulation [1]. Each device is represented by a single-degree-of-freedom (1-DOF) damped nonlinear mass–spring system coupled to one (switches) or two (resonator) nonlinear capacitive transducers. Since the electrostatic force is evaluated from the instantaneous terminal voltage, RF self-actuation and hold-down require no separate bias path.

Equations of motion

Newton's law is written as a force balance on a kinematic node, m dv/dt + b(g) v + Fs(z) = Fe(z, V) + Fc(g), with v = dz/dt and gap g = g0 − z, using time derivatives only. In DC analysis the operating point is therefore the static equilibrium at the applied bias.

Electrostatics

The capacitance C(z), including an optional Palmer fringing correction, is the single source of both the transducer current d(CV)/dt and the force Fe = ½V² dC/dz; charge and energy are therefore conserved. Because Fe scales approximately as 1/g² against a nearly linear restoring force, the default capacitive switch pulls in at 35.7 V, at 0.644 of the effective gap, and is held down down to 2.10 V. The up- and down-state capacitances are 38.2 fF and 1.54 pF.

Mechanics

The linear stiffness of the fixed-fixed beam is the sum of a bending term, 192EI/l³/(1 − r² + r³/2), and a tension term, 4σ(1 − ν)tw/(l − W/2), with r = W/l the electrode coverage. The cubic coefficient k3 = π4Ewt/(8l³) represents mid-plane stretching and is positive (hardening). Electrostatic spring softening results from Fe and is not included in k3; the net Duffing behaviour of the resonator is therefore bias-dependent.

Damping

The damping coefficient is the sum of a gap-independent anchor-loss term, √(k1m)/Qanchor, and a squeeze-film term for a rigid rectangular plate, βμeffa³b/g³, where μeff = μ/(1 + 9.638 Kn1.159) accounts for gas rarefaction [3]. The mean free path and viscosity scale with ambient pressure and temperature. For the default capacitive switch, Q = 1.12 at rest, and the switching time at 1.4 VP is 14.3 µs at 1 atm and 3.8 µs at 10 mbar, within the range reported for this class of device [4].

Contact

The contact force comprises van der Waals attraction, derived from a Hamaker constant with the (−3, −9) exponent pair of the flat/flat Lennard-Jones interaction [2], and a Hertzian asperity wall at an effective roughness standoff gr, which determines the down-state capacitance. The contact resistance of the ohmic switch scales as F−1/2 (elastic) or F−1/3 (plastic) [5].

Noise and drift

A Brownian force source, SF = 4kBTb, follows the instantaneous damping; the ohmic switch adds Johnson–Nyquist noise of the contact resistance. Dielectric charging, the dominant cause of long-term drift of capacitive switches, is available as an optional trapped-charge state.

Limitations. The 1-DOF representation holds the beam at a single gap and cannot represent progressive peeling; the release voltage is therefore underestimated (VR/VP ≈ 0.06, against 0.3–0.7 measured). Fringing is a first-order analytic correction, squeeze-film damping is taken in the incompressible limit without the gas-spring term, and the beam is rigid.

Download

Verilog-A_RF_MEMS_Models_v2.0.0.zip

27 files, 24 302 145 bytes.
SHA-256 505927f94ab6fbadeab6afa5ce781d3077bc1c967b02e4d1c542037a499cd007

Single DOF Models/The three 1-DOF models, their _DEBUG variants, ADS component definitions (.ael), mems_disciplines.vams, a Spectre oscillator example and an ADS workspace
Multiple DOF Models/Multi-DOF library for Verilog-A and SpectreHDL with user guide [8]; unchanged
tests/Python reference model and regression checks
CHANGELOG.mdChanges with respect to v1.4.1
INSTALL.htmlInstallation notes per simulator

Example (Spectre):

ahdl_include "RF_MEMS_RESONATOR.va"

Compatibility with v1.4.1. Module names and port lists are unchanged; several parameters are renamed or redefined. Results obtained with v1.4.1 and parameter sets extracted against it should be regenerated (see CHANGELOG.md).

Models

RF_MEMS_CAPACITOR
(in, out)		 Fixed-fixed beam capacitive switch, switched capacitor or varactor, with series resistance and inductance and optional dielectric charging.
OHMIC_CANTILEVER_RF_MEMS_SWITCH
(s, d, g)		 Cantilever ohmic switch with separate actuation electrode and force-dependent contact resistance at a dimple.
RF_MEMS_RESONATOR
(in, out, beam)		 Fixed-fixed beam resonator with two gap-closing transducers and stiffness coefficients k1…k5.
*_DEBUG As above, with internal quantities (displacement, velocity, acceleration, damping, capacitance, forces) available on *_CHECK outputs, 1 V per SI unit.

Principal parameters

g_0, l, t, w, W, t_d Geometry; stiffness and effective mass are derived from it unless overridden (k_1_ovr, k_3_ovr, m_ovr).
E, nu, rho, sigma, epsilon_r Beam and dielectric properties; defaults: electroplated gold on Si3N4.
p_amb, lambda, mu Ambient pressure; mean free path and viscosity at the reference conditions (p_ref, T_ref).
Q_anchor Anchor-loss quality factor (gap-independent).
A_H, z_vdw, g_r, p_c Hamaker constant, van der Waals equilibrium separation, roughness standoff, asperity contact pressure.
alpha_f, R_s, L_s Fringing correction (1 on, 0 off) and series parasitics.
en_charge, V_c_sat, tau_c, tau_d Dielectric charging; disabled by default.
R_c, F_ref, n_c Ohmic switch: contact resistance at the reference force and Holm exponent.

Examples

Requirements

A simulator supporting Verilog-A [7]: Keysight ADS [9], Cadence Spectre/SpectreRF, Synopsys HSPICE, or ngspice ≥ 39 with OpenVAF. Supported analyses are DC, AC, noise, S-parameter, harmonic balance, periodic and quasi-periodic steady state, and transient. The mechanical state is carried on kinematic nodes and is therefore visible to the simulator, which is a prerequisite for harmonic-balance and PSS analysis [6]. Dielectric charging, with a time constant of the order of 103 s, should be disabled in these analyses.

OpenVAF and ngspice

All six modules of release 2.0.0, including the _DEBUG variants, compile with OpenVAF. The models of v1.4.1 do not: their equations of motion use the implicit contribution Pos(velocity): ddt(Pos(z)) == …, which OpenVAF rejects, whereas release 2.0.0 expresses the same equations as a force balance using ddt only. For use with ngspice ≥ 39, each model is compiled to an OSDI library, which is loaded in the netlist; mems_disciplines.vams must reside in the directory of the source file.

openvaf RF_MEMS_CAPACITOR.va        # produces RF_MEMS_CAPACITOR.osdi

.control
pre_osdi RF_MEMS_CAPACITOR.osdi
.endc

Solver tolerances. The default absolute tolerance of position, velocity and force in disciplines.vams (10−6) equals the full travel of the beam. mems_disciplines.vams reduces these tolerances through the standard override macros and must be included before disciplines.vams.

Verification. All six modules compile with OpenVAF (through VerilogAE 1.0.0), and the equations are cross-checked against the reference model. Release 2.0.0 has not been simulated in Spectre, ADS, HSPICE or ngspice.

Changelog

2.0.0 — 19 September 2026

Version 2.0.0 is a correctness release of the 1-DOF models. It corrects the van der Waals coefficient, which was 59 orders of magnitude too small, the Lennard-Jones exponent, the rarefaction coefficient and the mean free path; it replaces the squeeze-film model, which overestimated the damping at rest by a factor of 22, adds a gap-independent anchor-loss term, removes a discontinuous current in the ohmic switch and a non-conserving current in the resonator, and corrects the hold-down voltage expressions. Results obtained with v1.4.1 should be regenerated.

Module names and port lists are unchanged; parameter names and meanings are not. The Multiple DOF Models are unchanged. The complete record is given in CHANGELOG.md in the archive.

References

  1. K. Van Caekenberghe, “Modeling RF MEMS devices,” IEEE Microw. Mag., vol. 13, no. 1, pp. 83–110, Jan.–Feb. 2012. [Online]. Available: https://ieeexplore.ieee.org/document/6132308
  2. J. N. Israelachvili, Intermolecular and Surface Forces, 3rd ed. Academic Press, 2011.
  3. T. Veijola, H. Kuisma, J. Lahdenperä, and T. Ryhänen, “Equivalent-circuit model of the squeezed gas film in a silicon accelerometer,” Sens. Actuators A, Phys., vol. 48, pp. 239–248, 1995. [Online]. Available: https://www.sciencedirect.com/science/article/pii/0924424795009957
  4. G. M. Rebeiz, RF MEMS: Theory, Design, and Technology. Wiley, 2003.
  5. R. Holm, Electric Contacts: Theory and Application, 4th ed. Springer, 1967.
  6. K. Kundert, “Hidden state in SpectreRF,” Designer's Guide Consulting, ver. 2, May 2019. [Online]. Available: https://designers-guide.org/analysis/hidden-state.pdf
  7. Verilog-AMS Language Reference Manual, ver. 2.4, Accellera Systems Initiative, 2014. [Online]. Available: https://www.accellera.org/images/downloads/standards/v-ams/VAMS-LRM-2-4.pdf
  8. J. Iannacci, “Mixed-domain fast simulation of RF and microwave MEMS-based complex networks within standard IC development frameworks,” in Advanced Microwave Circuits and Systems, V. Zhurbenko, Ed. InTech, 2010. [Online]. Available: https://www.intechopen.com/books/advanced-microwave-circuits-and-systems
  9. Using Verilog-A and Verilog-AMS in Advanced Design System, ADS 2011.01, Agilent Technologies (now Keysight Technologies), 2011. [Online]. Available: https://edadownload.software.keysight.com/eedl/ads/2011_01/pdf/veriloga.pdf