Page 55 - Mechanical Engineers' Handbook (Volume 2)

44 Input and Output Characteristics

´
3.6 Source Equivalents: Thevenin and Norton
´
Thevenin’s and Norton’s theorems were both originally developed for electric circuits. They
are duals; that is, while Thevenin uses a series voltage source, Norton uses a parallel current
´
source to construct an equivalent to a real subsystem being considered as a source. Both of
these theorems are completely generalizable to any system, provided that the appropriate
´
sources are used. In this chapter, the mobility analogy has been adapted so that Thevenin
equivalents are constructed using sources of voltage, force, pressure, or torque, and Norton
equivalents are constructed using sources of current, velocity, flow, or angular velocity. In
the development below, these two classes of sources will be referred to as effort and flow
sources.

´
Thevenin Equivalent
Assume that a subsystem being considered as a source contains within its structure one or
more ideal sources. Insofar as can be measured externally at the driving point (the point of
connection between any two systems considered to be source and load), any active subsys-
tem, no matter what its load, can be replaced by a new effort source added in series with
the original subsystem; all the original internal sources are set to zero. The value of the new
source effort variable is the output that would appear at the driving point if the load were
disconnected from the original subsystem. Setting an effort source to zero is equivalent to
connecting its nodes together; setting a flow source to zero is equivalent to removing it from
the system.
´
The output impedance of this Thevenin equivalent is clearly the impedance looking in
at the driving point; it is the derivative of the driving-point effort variable with respect to
´
the driving-point flow variable with the load disconnected. Thevenin’s theorem simplifies
this calculation because it tells us that the internal sources can be set to zero before the
calculation is made, and the system topology is often substantially simplified by this move.
´
We have already seen an example of this [see Eqs. (15)–(17)]. The Thevenin equivalent of
this circuit is simply a new source determined with the current i 0 (see Fig. 2):
o
R 3 V R R RR RR 3
3
1
1
2
2
V Th´evenin s I s (22)
R R 3 R R 3
1
1
in series with a resistance determined with sources V and I zeroed:
s
s
RR RR RR
R 12 23 13 (23)
Th´evenin
R R 3
1
Norton Equivalent
Assume the same subsystem considered above. Insofar as can be measured externally at the
driving point, any active subsystem, no matter what its load, can be replaced by a new flow
source added in parallel with the original subsystem, all the original sources set to zero. The
value of the new source flow variable is the flow that would pass through a short circuit
substituted at the driving point for the original load.
Referring again to Eq. (15), with V set to zero, we obtain the following:
o
R V R R RR RR
0 3 1 2 2 3 1 3 (I i ) (24)
R R 3 s R R 3 s o
1
1
we see that i for a short circuit would be
o