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.

Example. Electrical: Resistor-Inductor-Capacitor (RLC) system.

R L

CNo power source, transient response depends on initial conditions

B1, B2 depend on initial conditions

i

Example. Mechanical: Spring-Mass-Dashpot system.

k

m

b

B1, B2 depend on initial conditions

No power source, transient response depends on initial conditions

x

Equations are the same if:

k

m

b

b m

1/k

R L

C1/C

L

R

or

x.

I <-> x.

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.

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 powere . q has units of energyp . f has units of energy

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).

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

• For the generalized capacitor (potential energy):

•For a linear electrical capacitor:

ε – permitivityA – areaG – Gap

•The mechanical equivalent is the linear spring.

(Check in table.)

Cspring = 1/k, F=kx

•Generalized Inductor or inertance (kinetic energy?)

Linear inertance: momentum

m – massv – velocityp – momentum

Electrical?

But what is this???

???

flow momentum?

p1

v

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

Reluctance

(Fmm in example!)

(Senturia, not necessary to approximate)