Page 79 - Mechanical Engineers' Handbook (Volume 2)

68 Input and Output Characteristics

Admittances are the reciprocals of impedances, and both are unique properties of a
system. The Laplace operator (s) expresses these properties as a polynomial ratio. The de-
nominator polynomial (when set to zero, it becomes the characteristic equation of the system)
is always the same, and the numerator polynomial is a function of the location of the point
considered in the system. It was also shown that driving-point impedances are not a function
of the controlled variables on any ideal sources the system contains. Instead, all effort sources
may be replaced by solid connections, and all flow sources may be removed before the
driving-point impedance is computed.
When two systems are connected together at a driving point, port, or pair of terminals,
usually so that one can pass energy or information to the other, then there is a favorable
relationship between the impedances of the two systems that depends on the objective of the
connection. When it is desired to pass energy or power from one system to the other, then
the output impedance of the driving system should match the input impedance of the driven
system. If neither the driver nor the driven are adjustable, then a transducer, gyrator, or
transformer is used to match them by selecting the modulus to achieve the match. Any
impedance seen through a transformer, for example, appears to be increased or diminished
by the square of the transformer ratio. In a chain of subsystems, it is not necessary to install
the transformer at the driving point under consideration; the correct ratio can be determined
no matter where it is placed within the chain because the square of the modulus will always
appear in one of the driving-point impedances.
If the interconnection represents a measurement interface, then the most favorable re-
lationship between the driving-point impedances is the largest possible mismatch consistent
with obtaining the measurement. The ideal instruments for measuring efforts have infinite
input impedance and the ideal instruments for measuring flows have infinite input admit-
tances. Instruments that measure the integral of flows, such things as volume, charge, and
displacements, should have very low compliance (should displace easily, have low volume
themselves, or have small capacitances), while instruments that measure the integral of ef-
forts, such things as flux linkage or momentum, must have low mass or inductance.
The operating point of a pair of coupled systems is at the intersection of their input and
output characteristics in the power or energy plane. If one of these, for example the source
or output characteristic, exists in the power plane, that is, is static, but the other is energetic
(i.e., dynamic: massive, inductive, capacitive, etc.), then the source characteristic must en-
close the trajectory of the load characteristic at the highest frequency of interest, and ideally,
the source characteristic and load trajectory should be tangent at the maximum power point
or should be made tangent there by suitable choice of system parameters.
The key issue in this chapter is this: Whenever two dynamic systems are connected, an
interaction occurs. If the connection is to meet its objectives, the nature of this interaction
must be explored and controlled.

REFERENCES
1. H. M. Paynter, Analysis and Design of Engineering Systems, MIT Press, Cambridge, MA, 1960.
2. F. A. Firestone, ‘‘A New Analogy between Mechanical and Electrical Systems,’’ Journal of the Acous-
tical Society of American, 4, 249–267, (1932/33).
3. A. C. Bell and S. Ramalingam, ‘‘Design and Application of a Tensile Testing Stage for the SEM,’’
Journal of Engineering Materials and Technology, 96, 157–162 (July 1974).
4. L. C. Nachtigal, ‘‘Dynamic Systems and Measurements.’’ Unpublished notes, Purdue University, 1978.