### 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).

^{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