### Equivalent Circuits

The concepts of ``circuits'' and ``ports'' from classical
circuit/network theory [35] are very useful for
*partitioning* complex systems into self-contained sections
having well-defined (small) interfaces. For example, it is typical in
analog electric circuit design to drive a high-input-impedance stage
from a low-output-impedance stage (a so-called ``voltage transfer''
connection). This large impedance ratio allows us to neglect
``loading effects'' so that the circuit sections (stages) can be
analyzed separately.

The name ``analog circuit'' refers to the fact
that electrical capacitors (denoted ) are analogous to physical
springs, inductors () are analogous to physical masses, and
resistors () are analogous to ``dashpots'' (which are idealized
physical devices for which compression velocity is proportional to
applied force--much like a shock-absorber (``damper'') in an
automobile suspension). These are all called
*lumped elements*
to distinguish them from *distributed parameters* such as the
capacitance and inductance per unit length in an electrical
transmission line. Lumped elements are described by ODEs while
distributed-parameter systems are described by PDEs. Thus, RLC analog
circuits can be constructed as *equivalent circuits* for lumped
dashpot-mass-spring systems. These equivalent circuits can then be
digitized by *finite difference* or *wave digital*
methods. PDEs describing distributed-parameter systems can be
digitized via finite difference methods as well, or, when wave
propagation is the dominant effect, *digital waveguide* methods.

As discussed in Chapter 7 (§7.2), the *equivalent
circuit* for a force-driven mass is shown in Fig.F.10. The
mass is represented by an *inductor* . The driving
force is supplied via a *voltage source*, and the mass
velocity is the *loop current*.

As also discussed in Chapter 7 (§7.2), if two physical
elements are connected in such a way that they share a *common
velocity*, then they are said to be formally connected *in
series*. The ``series'' nature of the connection becomes more clear
when the *equivalent circuit* is considered.

For example, Fig.1.9 shows a mass connected to one end of a spring, with the other end of the spring attached to a rigid wall. The driving force is applied to the mass on the left so that a positive force results in a positive mass displacement and positive spring displacement (compression) . Since the mass and spring displacements are physically the same, we can define . Their velocities are similarly equal so that . The equivalent circuit has their electrical analogs connected in series, as shown in Fig.1.10. The common mass and spring velocity appear as a single current running through the inductor (mass) and capacitor (spring).

By Kirchoff's loop law for circuit analysis, the sum of all voltages
around a loop equals zero.^{2.13} Thus, following
the direction for current in Fig.1.10, we have
(where the minus sign for
occurs because the current enters its minus sign),
or

*i.e.*,

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Impedance Networks

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Modal Representation