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):

- To each element, such as a capacitor or inductor, attach a
length of waveguide (electrical transmission line) having wave
impedance
, and make it infinitesimally long. (Take the limit as
its length goes to zero.) A schematic depiction of this is shown in
Fig.F.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 F.1:**a) Physical schematic for the derivation of a wave digital model of driving-point impedance . The inserted waveguide impedance is real and positive, but otherwise arbitrary. b) Expanded view of the interior of the infinitesimal waveguide section, also representing the termination impedance 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 , as depicted in Fig.F.1b. - The interface to the element is recast as traveling-wave
*components*and at impedance . In terms of these components, the physical force on the element is obtained by adding them together: . - The waveguide impedance
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 . - The junction is ``parallel'' (cf. §7.2):
- 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).

- Force (voltage) must be continuous across the junction, since
otherwise there would be a finite force across a zero mass, producing
infinite acceleration.

- The infinitesimal waveguide is

- Reflectance of a General Lumped Waveguide Termination
- Reflectances of Elementary Impedances
- Capacitor Reflectance
- Inductor Reflectance
- Resistor Reflectance
- Choosing Impedance to Simplify Element Reflectance
- Digitizing Elementary Reflectances by Bilinear Transform

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