This is our first example illustrating a
series connection of
wave digital elements. Figure
F.26 gives the physical scenario of
a simple
mass
dashpot system, and Fig.
F.27 shows the
equivalent circuit. Replacing element voltages and currents in the
equivalent circuit by
wave variables in an infinitesimal
waveguides
produces Fig.
F.28.
Figure F.26:
External force driving a mass which in turn
drives a dashpot terminated on the other end by a rigid wall.

Figure:
Electrical equivalent circuit of the mass and dashpot system of Fig.F.26.

The system can be described as an ideal force source
connected
in
parallel with the
series connection of mass
and
dashpot
.
Figure
F.29 illustrates the resulting wave
digital filter.
Note that the ports are now numbered for reference. Two more symbols
are introduced in this figure: (1) the horizontal line with a dot in
the middle indicating a series adaptor, and (2) the indication of a
reflectionfree port on input 1 of the series adaptor (
signal
). Recall that a reflectionfree port is always necessary
when connecting two adaptors together, to avoid creating a
delayfree
loop.
Let's first calculate the
impedance necessary to make input 1 of
the series adaptor reflection free. From Eq.
(
F.37), we require
That is, the impedance of the reflectionfree port must equal the
series combination of all other
port impedances meeting at the
junction.
The
parallel adaptor, viewed alone, is equivalent to a force source
driving impedance
. It is therefore realizable as in
Fig.
F.20 with the
wave digital spring replaced by the
massdashpot assembly in
Fig.
F.29. However, we can also carry out a quick analysis
to verify this: The alpha parameters are
Therefore, the
reflection coefficient seen at port 1 of the parallel
adaptor is
, and the
KellyLochbaum scattering
junction depicted in Fig.
F.20 is verified.
Let's now calculate the internals of the series adaptor in
Fig.
F.29. From Eq.
(
F.26), the beta parameters are
Following Eq.
(
F.30), the series adaptor computes
We do not need to explicitly compute
because it goes into a
purely resistive impedance
and produces no return wave. For the
same reason,
.
Figure
F.30 shows a wave flow diagram of the computations derived,
together with the result of elementary simplifications.
Figure F.30:
Wave flow diagram for the WDF modeling an ideal
force source in parallel with the series combination of a mass and
dashpot .

Because the difference of the two coefficients in Fig.
F.30 is 1,
we can easily derive the onemultiply form in Fig.
F.31.
Figure:
Onemultiply form of the
WDF in Fig.F.30.

Let's check our result by comparing the
transfer function from the
input
force to the force on the
mass in both the discrete and
continuoustime cases.
For the discretetime case, we have
where the last simplification comes from the mass
reflectance relation
. (Note that we are using the standard
travelingwave notation for the
adaptor, so that the
signs are swapped relative to elementcentric notation.)
We now need
.
To simplify notation, define the two coefficients as
From Figure
F.30, we can write
Thus, the desired transfer function is
We now wish to compare this result to the
bilinear transform of the
corresponding analog transfer function. From Figure
F.27, we
can recognize the mass and
dashpot as
voltage divider:
Applying the bilinear transform yields
Thus, we have verified that the force transferfunction from the
driving force to the mass is identical in the discrete and
continuoustime models, except for the bilinear transform
frequency
warping in the discretetime case.
Next Section: Wave Digital MassSpring OscillatorPrevious Section: Spring and Free Mass