Electrostatic-mechanical coupling involves coupling of forces produced by an electrostatic field with a mechanical device. Typically, this type of simulation is done on micro-electromechanical (MEMS) devices such as comb drives, switches, filters, accelerometers, and torsional mirrors. This section describes the direct-coupled electrostatic-structural coupling available in the TRANS126 transducer element.
TRANS126 is a reduced-order element intended for use as a transducer in structural finite element simulations or as a transducer in lumped electromechanical circuit simulation. Reduced-order means that the electrostatic characteristics of an electromechanical device are captured in terms of the device's capacitance over a range of displacements (or stroke of the device) and formulated in a simple coupled beam-like element.
This figure Figure 2.81: Extracting Capacitance shows a typical progression for calculating the devices capacitance in an electrostatic simulation, calculating the capacitance of the device over a range of motion (parameter d in Figure 2.81: Extracting Capacitance), and incorporating the results as the input characteristics for the transducer element:
The following related topics are available:
TRANS126 is a fully coupled element which relates the electrostatic response and the structural response of an electromechanical device. Because the element is fully coupled, you can use it effectively in static, harmonic, transient, and modal analyses. Nonlinear analysis can exploit the full system tangent stiffness matrix. Small signal harmonic sweep and natural frequencies reflect coupled full system behavior. For the case with motion in the x direction, the charge on the device is related to the voltage applied to the device as:
Q = C(x) (V)
where V is the voltage across the device electrodes, C(x) is the capacitance between electrodes (as a function of x), and Q is the charge on the electrode.
The current is related to the charge as:
I = dQ/dt = (dC(x)/dx) (dx/dt) (V) + C(x) (dV/dt)
where the term (dC(x)/dx) (dx/dt) (V) is the motion induced current and the term C(x) (dV/dt) is the voltage rate current.
The electrostatic force between the electrodes is given by:
F = (1/2) (dC(x)/dx) (V)2
As can be seen from the above equations, the capacitance of the device over a range of motion characterizes the electromechanical response of the device.
As shown in Figure 2.82: Reduced-Order Model, you can analyze MEMS devices using reduced-order models consisting of mechanical spring, damper, and mass elements (COMBIN14, COMBIN39, and MASS21), and the electromechanical transducer element (TRANS126). The transducer element converts energy from an electrostatic domain into a mechanical domain. It represents the capacitive response of a device to motion in one direction.
You can generate a distributed set of TRANS126 elements between the surface of a moving structure and a ground plane (EMTGEN). This arrangement allows for fully coupled electrostatic-structural simulations for cases where the gap is small compared to the overall area of the structure. Typical applications include accelerometers, switches, and micromirror devices.
The TRANS126 element supports motion in the nodal X, Y, and Z directions. You can combine multiple elements to represent a full 3D translational response of a device. Accordingly, you can model an electrostatic-driven structure by a reduced-order element that fully characterizes the coupled electromechanical response.
You can link the transducer element into 2D or 3D finite element structural models to perform complex simulations for large signal static and transient analysis as well as small signal harmonic and modal analysis. See Example: Electromechanical Analysis for a sample electromechanical analysis using the TRANS126 transducer element.
For a static analysis, an applied voltage to a transducer will produce a force which acts on the structure. For example, voltages applied (V1 > V2) to the electromechanical transducer elements (TRANS126) will produce an electrostatic force to rotate the torsional beam shown in Figure 2.83: Micromirror Model.
The static equilibrium of an electrostatic transducer may be unstable. With increasing voltage, the attraction force between the capacitor plates increases and the gap decreases. For a gap distance d, the spring restoring force is proportional to 1/d and the electrostatic force is proportional to 1/d2. When the capacitor gap decreases to a certain point, the electrostatic attraction force becomes larger than the spring restoring force and the capacitor plates snap together. Conversely, when the capacitor voltage decreases to a certain value, the electrostatic attraction force becomes smaller than the spring restoring force and the capacitor plates snap apart.
The transducer element can exhibit hysteresis as shown in Figure 2.84: Electromechanical Hysteresis. The voltage ramps up to the pull-in value and then back down to the release value.
The transducer element by nature has both stable and unstable solutions as shown in Figure 2.85: Static Stability Characteristics. The element will converge to either solution depending on the starting location (initial gap size).
System stiffness consists of structural stiffness and electrostatic stiffness and it can be negative. Structural stiffness is positive because the force increases when a spring is stretched. However, electrostatic stiffness of a parallel plate capacitor is negative. The attraction force between the plates decreases with an increasing gap.
If the system stiffness is negative, convergence problems can occur near unstable solutions. If you encounter convergence problems while using TRANS126, use its built-in augmented stiffness method (KEYOPT(6) = 1). In this method, the electrostatic stiffness is set to zero to guarantee a positive system stiffness. After convergence is reached, the electrostatic stiffness is automatically reestablished for postprocessing and subsequent analyses.
You must completely specify the voltage across the transducer in a static analysis. You can also apply nodal displacements and forces. Applying initial displacements (IC) may help to converge the problem.
You can use TRANS126 to perform a prestressed modal analysis to determine the system eigenfrequencies. Of interest in many devices is the frequency shift when an applied DC voltage is placed on the electrodes of the transducer. You can simulate this effect by performing a static analysis of the device first with the DC voltage applied to the transducer, and then performing a prestressed modal analysis on the structure.
The TRANS126 element requires the unsymmetric eigenvalue solver (MODOPT,UNSYM) for modal analysis if a voltage is left unspecified at a transducer node. If the transducer element has a fully prescribed voltage (at both nodes), the problem becomes symmetric. In this case, set KEYOPT(3) = 1 for the transducer element and select a symmetric eigensolver (MODOPT,LANB). (MODOPT,LANB is the default.)
Linear perturbation is the preferred method for a prestressed modal analysis. Alternatively, activate prestress effects (PSTRES) when using TRANS126 with other element types that do not support linear perturbation.
You can simulate a prestressed full harmonic analysis on a structure, incorporating a transducer element TRANS126 to provide a small-signal AC voltage signal. Similarly, a mechanically excited structure will produce a voltage and current in the transducer. A static analysis must be performed prior to a small-signal harmonic analysis.
Typically, a device operates with a DC bias voltage and a small-signal AC voltage. The small-signal excitation simulation about a DC bias voltage is in essence a static analysis (with the applied DC voltage) followed by a prestressed full harmonic analysis (with the applied AC excitation). This capability is often required to tune a system's resonance frequency for such devices as filters, resonators, and accelerometers.
Linear perturbation is the preferred method for a prestressed harmonic analysis. The TRANS126 elements must use the full stiffness method (KEYOPT(6) = 0) in a linear perturbation harmonic analysis.
Alternatively, active prestress effects (PSTRES) when using TRANS126 with other element types that do not support linear perturbation.
You can run a full transient analysis incorporating TRANS126 attached to a complex finite element structure. You can apply any arbitrary large-signal time-varying excitation to the transducer or structure to produce a fully-coupled transient electromechanical response. You can apply both voltage and current as electrical loads, and displacement or force as mechanical loads.
Exercise care when specifying initial conditions for voltage and displacement
because you can specify both voltage VALUE1 and voltage
rate VALUE2 (IC), as well as
displacement and velocity.
You can specify convergence criteria for the voltage VOLT and/or current AMPS, and displacement U and/or force F (CNVTOL).
Linear and nonlinear effects can be included.
The TRANS126 element can be used to model reduced-order electromechanical devices in a coupled circuit simulation. The Mechanical APDL Circuit Builder provides a convenient tool for constructing a reduced-order model consisting of linear circuit elements (CIRCU124), mechanical spring, mass, and damper elements (COMBIN14, MASS21, and COMBIN39), and the electromechanical transducer element (TRANS126). TRANS126 links the electrical and mechanical models. Static, harmonic, and transient analysis of electromechanical circuit models may be performed.
This example is a direct coupled-field analysis of a MEMS structure.
The following topics are available:
Figure 2.86: Electrostatic Parallel Plate Drive Connected to a Silicon Beam
| Beam Properties | Parallel Plate Drive Properties |
|---|---|
| L = 150 µm | Ap = 100 ( µm)2 |
| b = 4 µm | gap = 1 µm |
| h = 2 µm | εr = 8.854e-6 pF/ µm |
| E = 1.69e5 µN/( µm)2 | |
| ρ = 2.332e-15 kg/( µm)3 |
A MEMS structure consists of an electrostatic parallel-plate drive connected to a silicon beam structure. The beam is pinned at both ends. The parallel-plate drive has a stationary component, and a moving component attached to the beam. Perform the following simulations:
Apply 150 Volts to the comb drive and compute the displacement of the beam.
For a DC voltage of 150 Volts, compute the first three eigenfrequencies of the beam (prestressed modal analysis).
For a DC bias voltage of 150 Volts, and a vertical force of 0.1 µN applied at the midspan of the beam, compute the beam displacement over a frequency range of 300 kHz to 400 kHz (prestressed harmonic analysis).
The parallel plate capacitance is given by the function Co/x where Co is equal to the free-space permittivity multiplied by the parallel plate area. The initial plate separation is 1 µm.
The modal and harmonic analyses must consider the effects of the DC voltage bias. The problem is set up to perform a prestressed modal and a prestressed harmonic analysis utilizing the static analysis results. The linear perturbation procedure is used for both the modal and harmonic analyses.
A consistent set of units are used (µMKSV). Since the voltage across TRANS126 is completely specified, the symmetric matrix option (KEYOPT(4) = 1) is set to allow for use of symmetric solvers.
The expected analytic results for this example problem are as follows.
| Frequency @ maximum displacement = 351.6 kHz |
| Maximum displacement = 22 µm (undamped) |
| Figure 2.87: Elements of MEMS Example Problem shows the transducer and beam finite elements. |
| Figure 2.88: Lowest Eigenvalue Mode Shape for MEMS Example Problem shows the mode shape at the lowest eigenvalue. |
| Figure 2.89: Mid Span Beam Deflection for MEMS Example Problem shows the harmonic response of the midspan beam deflection. |
The command text below demonstrates the problem input. All text prefaced with an exclamation point (!) is a comment.
/batch,list /title,Static, Modal, Harmonic response of a MEMS structure /prep7 /com L=150 ! beam length (micrometers) b=4 ! beam width h=2 ! beam height I=b*h**3/12 ! beam moment of inertia E=169e3 ! modulus ( micro newtons/micron**2) dens=2332e-18 ! density (kg/micron**3) per0=8.854e-6 ! free-space permittivity (pF/micron) plateA=100 ! capacitor plate area (micron**2) vlt=150 ! Applied capacitor plate voltage gapi=1 ! initial gap (microns) et,1,188,,,3 ! 3D beam element sectype,1,beam,asec secdata,b*h,b*h**3/12,,h*b**3/12,1,1 seccontrol,1e8,,1e8 mp,ex,1,E mp,gxy,1,E/2.6 mp,dens,1,dens et,2,126,,,,1 ! Transducer element, UX-VOLT degree of freedom, symmetric c0=per0*plateA ! C0/x constant for Capacitance equation r,2,0,0,gapi ! Initial gap distance rmore,c0 ! Real constant C0 n,1,-10 n,2,0 n,22,L fill n,999,,10 ! Orientation node for beam type,2 real,2 e,1,2 ! Transducer element (arbitrary length) type,1 real,1 e,2,3,999 ! Beam elements *repeat,20,1,1,0 nsel,s,loc,x,-10 nsel,a,loc,x,L d,all,ux,0,,,,uy ! Pin beam and TRANS126 element nsel,s,loc,x,0 d,all,uy,0 ! Allow only UX motion d,2,volt,vlt ! Apply voltage across capacitor plate nsel,s,loc,x,-10 d,all,volt,0 ! Ground other end of capacitor plate nsel,all d,all,uz,0 d,all,rotx,0 d,all,roty,0 fini /solu antype,static ! Static analysis rescontrol,define,last,last solve fini /post1 prnsol,dof ! print displacements and voltage prrsol ! Print reaction forces fini /solu antype,static,restart,last,last,perturb perturb,modal ! linear perturbation modal analysis solve,elform modopt,lanb,3 ! Block Lanczos; extract 3 modes mxpand ! Expand 3 modes solve finish /post1 file,,rstp set,1,1 ! Retrieve lowest eigenfrequency results pldisp,1 ! Plot mode shape for lowest eigenfrequency fini /solu antyp,static,restart,last,last,perturb perturb,harmonic ! linear perturbation harmonic analysis solve,elform hropt,full ! Full harmonic analysis option harfrq,300000,400000 ! Frequency range (Hz.) nsubs,500 ! Number of sampling points (substeps) outres,all,all ! Save all substeps ddele,2,volt ! delete applied DC voltage nsel,s,loc,x,L/2 ! Select node at beam midspan f,all,fy,.1 ! Apply vertical force (.1 N) nsel,all kbc,1 ! stepped load solve finish /post26 file,,rstp nsol,2,12,u,y, ! select node with applied force add,4,1,,,,,,1/1000 ! change to Kilohertz plcplx,0 ! magnitude /axlab,x,Frequency (KHz) ! set graphics options /axlab,y,Displacement /xrange,325,375 /gropt,divx,10 /gthk,axis,1.5 /device,text,1,100 xvar,4 plvar,2 ! Plot displacement vs. frequency prvar,2 ! Print displacement vs. frequency finish