COMSOL 6.4 - Reduced-Order Modeling and Equivalent Circuits

Reduced-order modeling seeks to reduce the number of degrees of freedom in a physical model, whilst still retaining the essential physics. It is frequently used in the design of resonant MEMS devices, which are usually designed to operate with the system vibrating in a particular mode. For a lightly damped resonant system driven at one of its resonant frequencies, it is reasonable to consider only the contributions to the system of a small number (m) of modes within the signal bandwidth. In some cases a single mode is sufficient. A system with n degrees of freedom has mass, stiffness, and damping matrices of size n-by-n. A reduced-order representation of the system considering m modes has size m-by-m. The reduction in complexity of the system, and the computational speed up is therefore significant when m « n. Determining a suitable reduced-order model of a complex system is a common simulation goal for MEMS designers. Frequently this is implemented as an equivalent circuit that can easily be integrated with SPICE models of the system as a whole. This section describes the theory of the reduced-order system and gives guidelines on how to obtain reduced-order models from a COMSOL Multiphysics model.

Consider a mechanical system, with n degrees of freedom, described by an equation of the form

where u is the displacement vector (size: n-by-1), K is the stiffness matrix (size: n-by-n), D is the damping matrix (size: n-by-n), and M is the mass matrix (size: n-by-n). In the frequency domain the problem takes the form

Initially consider the system in the absence of damping and forces. The undamped system has n eigenvalues ωi, which satisfy the equation

These eigenvectors are orthogonal with respect to M, as can be seen from the following construction, based on Equation 2-2:

using the results MT = M, and KT = K, which apply for physical reasons. Subtracting the first equation from the third gives:

so provided that the eigenfrequency are different:

Note that the above equations also imply

Next the following n-by-n matrix is constructed, with columns taken from the n eigenvectors:

Then consider the following matrix:

From Equation 2-3 it is clear that this is a diagonal matrix. Similarly from Equation 2-4 it is clear that UTKU is also diagonal.

From the properties of the eigenvectors it is possible to expand any function in terms of the eigenvectors. Thus the displacement u can be written as:

This equation can also be expressed in the form:

where a is a column vector containing the coefficients ai as rows. In general a is time dependent.

Now consider the original equation: Equation 2-1. First substitute for u using Equation 2-5. Then transform the equation to the modal coordinate system by premultiplying by UT. This gives:

It has already been established that the matrices UTMU and UTKU are diagonal, and frequently a damping model is chosen that results in a diagonal damping matrix. For example, in Rayleigh damping D = αM + βK, where α and β are constants. Alternatively, a damping ratio, ξi, is assigned to each mode.

The precise form of Equation 2-6 is determined by the normalization adopted for the eigenfunctions. In structural applications the eigenfunctions are often normalized such that UTMU = I. This is referred to as mass matrix scaling in the eigenvalue solver. In this case Equation 2-2 gives

where diag(ωi2) is the diagonal matrix with diagonal elements ωi2. Similarly, if damping ratios for each mode are defined, the damping matrix can be expressed in the form

Thus if mass matrix scaling is used Equation 2-6 takes the form

In MEMS applications it is common to scale the eigenvectors so that the point of maximum displacement has unit displacement magnitude. This is referred to as max scaling in the eigenvalue solver. For an individual mode this scaling has a simple physical interpretation — the corresponding component of a, ai, is the amplitude of the ith mode, measured at the point of maximum displacement, when the mode is driven by the force F. In this case Equation 2-6 takes the form

Here meff,i is the effective mass of the ith mode, ceff,i = 2meff,iξiωi is the effective damping parameter for the mode, and keff,i is the effective spring constant. Each element of the vector UTF gives the force component that acts on each of the respective modes.

For electromechanical systems it is common to express the force term, UTF, in terms of a driving voltage, V, applied to a terminal in the system. Given this:

here the vector η has components ηi which correspond to the electromechanical coupling factor for each of the modes in the system. Equation 2-8 now takes the form

To obtain the electromechanical coupling coefficients for a set of modes, it is possible to apply a 1 V signal to the system and then compute the induced forces: UTF = η. Reciprocally, energy conserving transducers also have the property that the current that flows into the transducer when it is operating in a particular mode (imot,i) is related to the displacement through the equation

So Equation 2-9 can be rewritten in the form

For each mode, the terms on the left of Equation 2-10 can be identified with a resistor capacitor and inductor, such that the mechanical system behaves like a set of parallel series LCR circuits. The equivalent modal resistances, capacitances, and inductances are given by:

For a given mode, these quantities are frequently referred to as the motional inductance, resistance, and capacitance.

The preceding discussion did not consider how to reduce the number of degrees of freedom in the system. For systems in which the vector UTF has only a few significant components (for example, components i = 1, …, m where m « n) the following approximation can be made:

The expression for u in matrix becomes:

where U' is now an m-by-n and a' is a vector with m components. The equation system in modal coordinates now takes the form

The matrices U'TMU', U'TDU', and U'TKU' now have dimensions m-by-m. Similarly the vector U'TF has m components. This results in a significant reduction in the system complexity.

If physically relevant damping is present in the system, the above theory must be modified as the damping matrix is no longer diagonal in the modal coordinate system. COMSOL can still handle this case as the modal solver does not make the assumption that any of the matrices are diagonal. In this case the eigenvalues become complex and the eigenvectors split into right and left eigenvectors. The right eigenvectors Ur are solutions of the equation:

As in the previous section, for a reduced set of modes, it is assumed that:

where U'r is the n-by-m matrix containing the right eigenvectors chosen for the modal analysis. Once again a' is a vector with m components. The system in modal coordinates takes the form

where U'l is the n-by-m matrix containing the left eigenvectors chosen for the modal analysis.

The matrices U'lTMU'r, U'lTDU'r, and U'lTKU'r are no longer necessarily diagonal. The modal solver accepts any linearly independent set of vectors to project the solution vector and equations onto and constructs the reduced-order system accordingly.

The and study steps have the property that they can assemble the modal matrices and make them available for output. In the node, the checkbox must be selected.

After the model has solved, right-click the node and select . In the output section choose the to display in the list. The mass matrix corresponds to the matrix U'lTMU'r the stiffness matrix corresponds to U'lTKU'r, and the damping matrix corresponds to U'lTDU'r. The vector U'lTF is available as the load vector. These matrices are given in a format that respects the normalization of the preceding . To change this select the node, and change the setting under the section. Use the setting if an equivalent Mass-Spring-Damper system is required, in which case the modal amplitude corresponds to the maximum displacement of the mode.