3.5: Thermodynamics of Discrete Electromechanical Coupling

In this section, the thermodynamic electric and magnetic energy storage subsystems are expanded to include the possibility of a finite number of discrete mechanical displacements of macroscopic material. Based on the notion of an energy function and a thermodynamic equilibrium, the force of electrical origin associated with each of these displacements is determined. Typically, the method exploits a knowledge of the electrical terminal relations to determine the forces. The approach is generalized in Secs. 3.7 and 3.8, where constitutive laws are the basis for finding the force density of electric origin. Except for mathematical manipulations, the derivations now reviewed draw upon all of the demanding issues confronted later in deriving force densities.

Electroquasistatic Coupling

An example of a lumped-parameter electroquasistatic system is given with Figure 2.11.1, including a schematic representation of a finite number of mechanical displacements.Associated with each of the displacements is an electromechanical force tending to displace a lumped element by an amount \(\delta \xi_{i}\).

Conservation of energy for the system with the geometry fixed is expressed by Equation 2.13.8. Now, an incremental increase in the total energy caused by placing an increment of charge \(\delta q_i\) on an electrode having the voltage \(v_i\) can be diminished by an amount equal to the work done on the external environment by the forces of electrical origin acting through the displacements of the associated mechanical entities. Thus, energy conservation requires that

\[ \delta w = \sum_{i = 1}^{n} v_i \delta q_i - \sum_{j = 1}^{m} f_j \delta \xi_j ; \quad w = w (q_1...q_n, \xi_1...\xi_m) \label{1} \]

Given the charges \(q_1...q_n\) and the displacement \(\xi_1...\xi_m\) as independent variables, the energy function is uniquely determined. The "displacements" should be recognized as generalized variables in that they could just as well be angular deflections, in which case the associated "forces" would be torques.

To determine \(w\), constitutive relations \(v_i(q_1...q_n, \xi_1...\xi_n)\) must be known so that Equation \ref{1} can be integrated. The integration is a line integral in a state-space composed of the independent variables.Because the \(f_j\)'s are not known, and are defined as equal to zero in the absence of electrical excitations, integration on the mechanical variables (j is carried out first. This gives no contribution because as the displacements are brought to their final values, \(f_j = 0\) (no work is required to assemble the system with the \(q_j\)'s \(= 0\)). Then, the integration on successive electrical variables is carried out, first on \(q_1\) with all other \(q_j\)'s \(= 0\), then on \(q_2\) with \(q_1\) at its final value and all others zero,etc. Formally, the integration of Equation \ref{1} gives

\[ w = \sum_{j=1}^{n} \int_o^{q_j} v_j (q_1...q_j^{'}, 0...0, \xi_1, \xi_2...\xi_m) \delta q_j^{'} \label{2} \]

Because the energy function is a state function specified by the independent variables, an incremental change in the total energy can also be written as

\[ \delta w = \sum_{i=1}^{n} \frac{\partial{w}}{\partial{q_i}} \delta q_i + \sum_{j=1}^m \frac{\partial{w}}{\partial{\xi_j}} \delta \xi_j \label{3} \]

If the \(q\)'s and the \(\xi\)'s are independent variables in the sense that Eqs. \ref{1} and \ref{3} hold for arbitrary combinations of incremental changes in these electrical and mechanical variables, then

\[ v_i = \frac{\partial{w}}{\partial{q_i}}; \quad f_j = - \frac{\partial{w}}{\partial{\xi_j}} \label{4} \]

Note that the \(q\)'s and \(\xi\)'s are not necessarily independent of each other unless the system is isolated from the total system in which it is imbedded. Given \(w\) from Equation \ref{2}, the electrical forces are determined.

consequence of the conservation of energy expressed by Equation \ref{1} is the reciprocity condition between pairs of terminal variables. For example, derivatives of Equation \ref{4} a, first with respect to \(q_j\) and then of the same equation but with \(i\) replaced by \(j\), and with respect to \(q_i\), are related by

\[ \frac{\partial{v_i}}{\partial{q_j}} = \frac{ \partial{^2 w}}{\partial{q_i}\partial{q_j}} = \frac{\partial{v_j}}{\partial{q_i}} \label{5} \]

Other reciprocity conditions follow from Equation \ref{4} by taking cross-derivatives to relate forces and voltages to each other.

In dealing with practical lumped-parameter systems, it is often convenient to use the voltages rather than the charges as independent variables. If all of the voltages are to be independent variables, it is appropriate to recognize that

\[ \sum_{i=1}^n v_i \delta q_i = \sum_{i=1}^n [ \delta (v_i q_i) - q_i \delta v_i ] \label{6} \]

so that substitution into Equation \ref{1} gives

\[ \delta w^{'} = \sum_{i=1}^n q_i \delta v_i + \sum_{j=1}^m f_j \delta \xi_j \label{7} \]

where \(a\) coenergy function has been defined in terms of the energy function as

\[ w^{'} (v_1...v_n, \xi_1,,,\xi_m) \equiv \sum_{i=1}^{n} v_i q_i - w \label{8} \]

The coenergy function is a particular case of an arbitrarily large number of functions that can be defined. Any combination of charges and voltages can be independent variables, and a hybrid energy function, appropriately defined as a state function of this combination. With the voltages as independent variables, an equation similar to Equation \ref{2} is found with the charges replaced by the voltages,and the voltages and displacements the independent variables:

\[ q_i = \frac{\partial{w^{'}}}{\partial{v_i}}; \quad f_j = \frac{\partial{w^{'}}}{\partial{\xi_j}} \label{9} \]

The coenergy function, like the energy function, is found from purely electrical considerations, as described in Sec. 2.13.

Magnetoquasistatic Coupling

Lumped-parameter electromechanical coupling in a magnetic field system, described schematically by Figure 2.12.1, can be given the same thermodynamic representation as that out-lined for electroquasistatic systems. The statement of conservation of energy for the system of discrete coils and mechanical displacements is the generalization of Equation 2.14.11, with the addition of the mechanical work done as an electrical force \(f_j\) causes an incremental displacement \( \delta \xi_j \) :

\[ \delta w = \sum_{i=1}^n i_{i} \delta \lambda_i - \sum_{j=1}^m f_j \delta \xi_j \label{10} \]

All of the arguments given for the electric systems follow for the magnetic field systems if variables are identified:

\[ \begin{align} q_i &\rightarrow \lambda_i, \quad v_i \rightarrow i_i \nonumber \\ w &= w(\lambda_1...\lambda_n, \xi_1...\xi_m); \quad w^{'} = w^{'}(i_1...i_n, \xi_1...\xi_m) \nonumber \end{align} \label{11} \]

The magnetic force is the negative partial derivative of the magnetic energy with respect to the appropriate associated displacement, with the other displacements and all of the flux linkages held constant. Similarly, the force can be found from the coenergy function by taking the derivative with respect to the associated displacement with the other displacements and the currents held constant.

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1. Electrons in vacuum can have a velocity approaching that of light. In that case an imposed magnetic field can have a crucial effect on the EQS dynamics (See Sec. 11.2).