Page 45 - Mechanical Engineers' Handbook (Volume 2)

34 Input and Output Characteristics

the energy exchange might be information transmission, but this is not considered here (we
would like information exchanges to take place at the lowest possible energy costs, but the
second law of thermodynamics rules out a free transmission).
As always, energy factors into two variables, such as voltage and current in electrical
systems, and we are concerned with the behavior of these in the energetic interaction. The
major difference in this perspective is that the system supplying energy cannot do so at a
fixed value. Neither the source nor the system receiving energy can fix its values for a
changing demand without a change in the value of a supply variable. The two subsystems
are in an equilibrium with each other and are forced by their connection to have the same
value of both of the appropriate energy variables. We concern ourselves with determining
and controlling the value of these energy variables at the interface where, obviously, only
one is determined by each of the interacting systems.

2 FAMILIAR EXAMPLES* OF INPUT–OUTPUT INTERACTIONS
2.1 Power Exchange
In the real world, pure sources and sinks are difficult to find. They are idealized, convenient
constructs or approximations that give our system analyses independent forcing functions.
We commonly think of an automobile storage battery as a source of 12.6 V independent of
the needed current, and yet we have all observed dimming headlights while starting an
engine. Clearly, the voltage of this battery is a function of the current demanded by its load.
Similarly, we cannot charge the battery unless our alternator provides more than 12.6 V, and
the charging rate depends on the overvoltage supplied. Thus, when the current demanded or
supplied to a battery approaches its limits, we must consider that the battery really looks
like an ideal 12.6-V source in series with a small resistance. The voltage at the battery
terminals is a function of the current demanded and is not independent of the system loading
or charging it in the interaction. This small internal resistance is termed the output impedance
(or input impedance or driving-point impedance) of the battery.
If we measure the voltage on this battery with a voltmeter, we should draw so little
current that the voltage we see is truly the source voltage without any loss in the internal
resistance. The power delivered from the battery to the voltmeter is negligible (but not zero)
because the current is so small. Alternatively, if we do a short-circuit test of the battery, its
terminal voltage should fall to zero while we measure the very large current that results.
Again, the power delivered to the ammeter is negligible because, although the current is very
large, the voltage is vanishingly small.
At these two extremes the power delivered is essentially zero, so clearly at some inter-
mediate load the power delivered will be a maximum. We will show later that this occurs
when the load resistance is equal to the internal resistance of the battery (a point at which
batteries are usually not designed to operate). The discussion above illustrates a simple
concept: Impedances should be matched to maximize power or energy transfer but should
be maximally mismatched for making a measurement without loading the system in which
the measurement is to be made. We will return to the details of this statement later.

*Many of the examples in this chapter are drawn from Chapter 6 of a manuscript of unpublished notes,
‘‘Dynamic Systems and Measurements,’’ by C. L. Nachtigal, used in the School of Mechanical Engi-
neering, Purdue University, 1978.