Page 56 - Mechanical Engineers' Handbook (Volume 2)

4 Operating Point of Static Systems 45
i I R 3 V
o
s
RR RR RR s (25)
12
23
13
and, as for the Thevenin equivalent, the circuit impedance is
´
RR RR RR
R Norton 12 23 13 (26)
R R
1 3
Note that R Thevenin R Norton , always.
´
4 OPERATING POINT OF STATIC SYSTEMS
A static system is a system without energy storage, a system in which there are only sources,
transducers, and dissipation elements. Such systems have no transient response: They respond
instantly to their inputs algebraically. The relationships among any of their variables are
proportionalities—simple static gains. If the inputs to a stable dynamic system are held
constant for long enough, it will become stationary; its variables will not change with time
provided only that there is sufficient dissipation in the system to damp out any oscillations.
Such a system is not static; it is at steady state. There is no exchange of energy among its
energy storage elements, and the dissipative elements completely determine the state of the
system.

4.1 Exchange of Real Power
If one system is supplying real power to another system in steady-state operation, then for
the purposes of a static analysis, energy storage elements can be ignored. Both the source
and the load can be considered to be purely resistive. If the source is separated from the
load at the point of interest (at least conceptually), then the characteristics of source and
load can be measured or computed. The load will be a power absorber—an electrical fluid
resistance or a mechanical damper—and its characteristics can be represented as a line in a
power plane coordinate system: voltage versus current, force versus velocity, torque versus
angular velocity, or pressure versus flow. There is no requirement that this line be straight,
and except that the measurement might be more difficult, there is no necessity that this line
be single valued or that it start at the origin. Figure 5 shows a selection of common dissipative
load characteristics.
Similarly, when the source portion of the system is loaded with a variable dissipation,
the line representing its output characteristics can be plotted on the same coordinates as the
load. Such characteristics are often given for pumps, servomotors, transistors, hydraulic
valves, electrical supplies, and fans. Again, there is no requirement that this line be straight
or that it be single valued, but it is very unusual for it to pass through the origin. If the
source or the load is not constant in time or can be controlled, then a family of these
characteristics will be required for variations in the parameters of the source or load. Figure
6 shows a selection of these sources. With very few exceptions, real sources cannot operate
at the maximum values of their power variables simultaneously: A battery cannot deliver its
maximum voltage and current at the same time. In more general terms, this means that in
spite of local variations, real source characteristics tend to droop from left to right in the
power plane; their average slope is negative.