# Interdisciplinary Applied Mathematics

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The Modified Nodal Analysis (MNA) technique for circuit simulation is based on Kirchhoff’s current law, which states that the sum of all outgoing currents at each node is equal to zero, i.e.,

У~Ак = 0.

k

Kirchhoff’s current law, when considered at each node, generates a set of algebraic differential equations, which can be transformed into a set of algebraic equations by applying a suitable numerical integration scheme. The nonlinear algebraic equations can be solved by numerical methods (e.g., a Newton’s method), which require repeated solution of a set of linear equations of the form

Av = b.

The observation that each element in the circuit contributes to the above equation leads to the element stamp concept. Figure 17.6 shows a simple resistor (of conductance G) connected between nodes t and o. The branch current, i,    given    by    i =    G(vt    — v0),    appears    with    a    positive    sign in the

current equation    at    node    t and    with    a negative    sign    at    node    o. Hence,

the resistor contributes the term +G(vtv0) to the tth equation and the term —G(vt — v0) to the oth equation. An element stamp (which is the coefficient matrix A having rows corresponding to each node, and columns corresponding to each variable) for the resistor is shown in Figure 17.6. If vt    and    vo    are the    tth    and    the    oth    elements    of    the    voltage    vector,    v,

respectively, the resistor’s contribution to the coefficient matrix adds a quantity +G    to positions    (t, t)    and    (o, o)    and a    —G    to the    positions (t, o)

and (o, t). The lumped-constant models for MEMS is based on the fact that    all    the    MEMS    structures are    built    from    a    common    set    of basic    (or

atomic) elements, such as beams, anchors, and plates. Though these are less accurate than distributed constant models, they reduce the computational effort by a great deal. For many applications the accuracy obtained from a lumped-constant model could be sufficient. Just as the conventional circuit simulation relies on the lumped-constant models of the electronic devices, microsystem simulation can be achieved by extending the concept of element stamps to microdevices and their constitutive elements. In this

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