Page 78 - Mechanical Engineers' Handbook (Volume 2)

8 Concluding Remarks 67

consider the tank to be a simple lumped capacitor, at least in the time scale of the spark
event.

7.1 Impedance of a Distributed System
Imagine a long, slender tank of water, a trough, open at the top, and perform the following
thought experiment. If one end wall were moved inward suddenly, the level of water at that
end would rise higher up than was required by the change in tank volume because the
remaining water further along the tank would not change level instantaneously. Then, this
wave coming down off the tank wall would travel to the other end where it would slosh up
the far wall and return. This wave would continue to slosh back and forth at decreasing
amplitude as viscosity took its toll, until finally the surface would be calm again at a new,
slightly higher level. Each time the wave reached an end, it would be returned in kind, that
is, with the same sign.
Now suppose that the far end of the tank opened into a large lake, so large that no level
changes would take place when an end wall was moved, and again move the remaining end
wall inward quickly. Then, when the first wave reached the opening to the lake, it would
leave the tank and be lost in the lake. But that would involve water leaving the tank in
excess of the volume change in the tank, and a negative wave would return from the open
end to signal the new, lower level required by the loss. The closed end would reflect this
wave as a rarefaction, and when that returned to the open end, lake water would spill back
in as an upward wave. Eventually these alternating processes would return the level in the
tank to that of the lake.
A closed end returns a wave of like kind, and an open end returns a wave of opposite
kind. If there are no losses as the waves hit the ends of the tank, a wave of strength 1is
reflected with strength 1 from a closed end and with a strength 1 from an open end. This
implies that there is an end condition somewhere between closed and open from which a
wave will not be reflected at all. A suitably constructed porous wall, in this example, would
simply absorb the wave completely by accepting and dissipating all of its energy. The im-
pedance of this wave-matched wall is the wave impedance of the channel and is a charac-
teristic of it, depending on the inductive and capacitive properties of the medium.
The 75- and 300- markings on the antenna connections of a television receiver imply
two things: First the input impedances of those terminals are resistive at 75 and 300 ,
respectively, and, second, the coaxial cable and flat-lead antenna wiring are really waveguides
whose wave impedances are 75 and 300 . By matching the impedance of the cable at the
receiver terminals, we are assured that all the incoming wave energy will be absorbed by
the receiver and none will be reflected back up the cable to the antenna and thus lost.

8 CONCLUDING REMARKS

This chapter has demonstrated an alternative viewpoint for the interaction of systems with
each other. Control engineers are quite accustomed to transfer functions: relationships in the
frequency (s or Laplace) domain between a variable at one point in a system and another at
some other point, most often between inputs to and outputs from a controlled system. This
chapter has dealt with a special class of these relationships between the complementary
variables of power at a single point in a system. These special transfer functions are called
driving-point impedances or admittances, and they determine how one subsystem will load
or be loaded by another.