Page 327 - The Mechatronics Handbook

a capacitor); the required voltage is determined at equilibrium, defined when the time rate of change of
charge motion is zero (i.e., all the charges are at rest relative to each other).

Extensive and Intensive Variables
In the formalism of thermodynamics, the amount of stored energy and the displacement that determines
it are extensive variables. That is, they vary with the spatial extent (i.e., size or volume) of the object
storing the energy. The total elastic energy stored in a uniform rod of constant cross-sectional area in an
idealized uniform state of stress is proportional to the length (and hence volume) of the rod; so is the
total relative displacement of its ends; both are extensive variables. The total electrostatic energy stored
in an idealized parallel-plate capacitor (i.e., one with no fringe fields) is proportional to the area of the
plates (and hence, for constant gap, the volume they enclose); so is the total separated charge on the
plates; both are extensive variables (cf., Breedveld, 1984).
Equilibrium of these storage elements is established by an intensive variable that does not change with
the size of the object. This variable is the gradient (partial derivative) of the stored energy with respect to
the corresponding displacement. Thus, at equilibrium, the force on each cross-section of the rod is the
same regardless of the length or volume of the rod; force is an intensive variable. If the total charge
separated in the capacitor is proportional to area, the voltage across the plates is independent of area;
voltage is an intensive variable.
Dynamics is not solely due to the storage of energy but arises from the transmission and deployment
of power. The instantaneous power into an equilibrium storage element is the product of the (intensive)
gradient variable (force, voltage) with the time rate of change of the (extensive) displacement variable
(velocity, current). Using this thermodynamics-based approach, all intensive variables are considered
analogous, as are all extensive variables and their time rates of change, and so on.
Drawing analogies from a thermodynamic classification into extensive and intensive variables may
readily be applied to fluid systems. Consider the potential energy stored in an open container of incom-
pressible fluid: The pressure at any specified depth is independent of the area at that depth and the
volume of fluid above it; pressure is an intensive variable analogous to force and voltage, as our common
physical intuition suggests it should be. Conversely, the energy stored in the fluid above that depth is
determined by the volume of fluid; energy and volume are extensive variables, volume playing the role
of displacement analogous to electrical charge and mechanical displacement. Pressure is the partial
derivative of stored energy with respect to volume and the instantaneous power into storage is the product
of pressure with volumetric flow rate, the time rate of change of volume flowing past the specified depth.
An important advantage of drawing analogies from a classification into extensive and intensive vari-
ables is that it may readily be generalized to domains to which the ‘‘system-of-particles’’ image may be
less applicable. For example, most mechatronic designs require careful consideration of heating and
cooling but there is no obvious flow of particles associated with heat flux. Nevertheless, extensive and
intensive variables associated with equilibrium thermal energy storage can readily be identified. Drawing
on classical thermodynamics, it can be seen that (total) entropy is an extensive variable and plays the
role of a displacement. The gradient of energy with respect to energy is temperature, an intensive variable,
which should be considered analogous to force, voltage, and pressure. Equality of temperature establishes
thermal equilibrium between two bodies that may store heat (energy) and communicate it to one another.
A word of caution is appropriate here as a classification into extensive and intensive variables properly
applies only to scalar quantities such as pressure, volume, etc. As outlined below, the classification can
be generalized in a rigorous way to nonscalar quantities, but care is required (cf., Breedveld, 1984).

Equilibrium and Steady State
In some (though not all) domains energy storage may also be based on motion. Kinetic energy storage

may be associated with rigid body motion or fluid motion; magnetic energy storage requires motion of
charges. The thermodynamics-based classification properly groups these different kinds of energy stor-
age as analogous to one another and generically they may be termed kinetic energy storage elements.

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