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Lumped Modeling with Circuit Elements, Ch. 5, Text

Learn how ideal elements represent real physical systems and how to model dynamic systems using circuit elements. Gain physical insight for timely solutions in electrical and mechanical systems.

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Lumped Modeling with Circuit Elements, Ch. 5, Text

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  1. Lumped Modeling with Circuit Elements, Ch. 5, Text • Ideal elements represent real physical systems. • Resistor, spring, capacitor, mass, dashpot, inductor… • To model a dynamic system, we must figure out how to put the elements from different domains together. • Alternatives include numerical modeling of the whole system. Lumped element modeling offers more physical insight and may be necessary for timely solutions.

  2. Example. Electrical: Resistor-Inductor-Capacitor (RLC) system. C No power source, transient response depends on initial conditions R L i B1, B2 depend on initial conditions

  3. Example. Mechanical: Spring-Mass-Dashpot system. x No power source, transient response depends on initial conditions k m B1, B2 depend on initial conditions b

  4. Equations are the same if: 1/k b m . k 1/C x . I <-> x L m or C R b R L

  5. Goal: Simulate the entire system. • Usual practice: • Write all elements as electrical circuit elements. • Represent the intradomain transducers (Ch. 6) • Use the powerful techniques developed for circuit analysis, linear systems (if linear), and feedback control on the whole MEMS system.

  6. Senturia generalizes these ideas. • Introduce conjugate power variables, effort, e(t), and flow, f(t). • Then, generalized displacement, q(t) • And generalized momentum, p(t) e . f has units of power e . q has units of energy p . f has units of energy

  7. Variable Assignment Conventions • Senturia uses e -> V, that is, effort is linked with voltage in the electrical equivalent circuit. He explains the reasons (for example potential energy is always associated with energy storage in capacitors).

  8. Following Senturia’s e -> V convention: • For effort source, e is independent of f • For flow source, f is independent of e • For the generalized resistor, e=e(f) or f=f(e) • Linear resistor e=Rf • Electrical, V=RI • Mechanical, F=bv

  9. For the generalized capacitor (potential energy): • For a linear electrical capacitor: ε – permitivity A – area G – Gap

  10. The mechanical equivalent is the linear spring. (Check in table.) Cspring = 1/k, F=kx

  11. Generalized Inductor or inertance (kinetic energy?) p1 Linear inertance: momentum flow momentum? m – mass v – velocity p – momentum Electrical? But what is this??? ???

  12. v

  13. Reluctance q=Ce, e=(1/C)q, Electrical Q=CV

  14. (Fmm in example!)

  15. (Senturia, not necessary to approximate)

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