A Physical Derivation of Wave Digital Elements

This section provides a ``physical'' derivation of Wave Digital Filters (WDF), which contrasts somewhat with the more formal derivation common in the literature. The derivation is presented as a numbered series of steps (some with rather long discussions):

1. To each element, such as a capacitor or inductor, attach a length of waveguide (electrical transmission line) having wave impedance $R_0$, and make it infinitesimally long. (Take the limit as its length goes to zero.) A schematic depiction of this is shown in Fig. N.1a. For consistency, all signals are Laplace transforms of their respective time-domain signals. The length must approach zero in order not to introduce propagation delays into the signal path.

   Figure N.1: a) Physical schematic for the derivation of a wave digital model of driving-point impedance $R(s)$. The inserted waveguide impedance $R_0$ is real and positive, but otherwise arbitrary. b) Expanded view of the interior of the infinitesimal waveguide section, also representing the termination impedance $R(s)$ as an impedance-step within the waveguide.

   

   Points to note:

   • The infinitesimal waveguide is terminated by the element. The element reflects waves as if it were a new waveguide section at impedance $R(s)$, as depicted in Fig. N.1b.

   • The interface to the element is recast as traveling-wave components $F^{+}(s)$ and $F^{-}(s)$ at impedance $R_0$. In terms of these components, the physical force on the element is obtained by adding them together: $F(s) = F^{+}(s)+F^{-}(s)$.

   • The waveguide impedance $R_0$ is arbitrary because it has been physically introduced. We will need to know it when we connect this element to other elements. The element's interface to other elements is now a waveguide (transmission line) at real impedance $R_0$.

   • The junction is ``parallel'' (cf. §J.2):
     • Force (voltage) must be continuous across the junction, since otherwise there would be a finite force across a zero mass, producing infinite acceleration.

     • The sum of velocities (currents) into the junction must be zero by conservation of mass (charge).