Introduction: System Efficiency, COP, and Work

Engineers often arbitrarily interchange the terms efficiency (ε) and coefficient of performance (COP) as if they were synonyms — which they are not. An EM system’s or EM process’s thermodynamic efficiency is defined with respect to all its energy inputs, whereas the system’s COP is defined with respect to only a subset of the inputs (those furnished by the operator). Thus, it would be incorrect to refer to a system that outputs more energy than the operator had input as one whose efficiency exceeds 100%.
While both COP and ε are energy-output versus energy-input comparisons, they compare quite different things. For a real system with losses, the system efficiency is always ε < 100%. Yet under proper conditions the system can still exhibit COP > 1.0.
Another common misconception of the definition of work compounds the confusion. Many people think of work as a change of magnitude of energy. However, rigorously, work is only a change of form of energy.
The basic system diagram of primary concern is given in Figure 1 below.

Figure 1. Basic system for efficiency and COP determinations.

1.1 Efficiency ε indicates how much useful work or useful energy output the system produces, in comparison to the total energy input to the system from all sources (e.g., from both the operator and the active environment). In Figure 1, the system efficiency is the ratio of the useful system output divided by the sum of (i) the operator’s input and (ii) the input from the environment. Conventionally, efficiency ε is this ratio expressed as a percentage.

1.2 The Coefficient of Performance, COP, indicates how much useful work or useful energy output the system produces in comparison to the operator’s energy input only. In Figure 1, the system COP is given by the useful output divided by the operator’s input only. The COP ratio is conventionally expressed as a decimal fraction.

1.3 Work is rigorously a change of form of some energy. A useful work process changes input energy to a different form in a way that is of use to the operator. An example is an electric motor receiving EM energy input and outputting mechanical energy that rotates a shaft. System losses produce non-useful work.

1.4 All EM energy occurs as continuous energy flows from source charges. All EM fields and potentials mathematically decompose into ongoing sets of EM energy flows, as shown by Whittaker {1}. All EM systems collect input energy by potentialization, but first the energy must be in the required suitable form.

1.5 No work is done if the input energy can potentialize without changing its form. Recall the definition of work as a change in energy’s form, not its magnitude. Power is the rate of doing work. Some systems can potentialize using the input energy directly, in the exact form in which it was supplied. Because no conversion of the form of the energy is involved, such potentialization does not expend work. Consider the example of a receiving circuit that is potentialized from voltage in a separate circuit, in the absence of a direct dq/dt current flow between the two circuits. If current is then “pushed” simultaneously and separately (asymmetrically) through the load and losses, without being rammed back up through the back emf of the original energy source, then this free potentialization also usefully accomplishes free dissipative output work simultaneously. This class of EM system is called asymmetric.

1.6 Work is done if the input energy’s form must be changed in order to potentialize. Some systems must convert their input energy to a different form before using it. In converting the energy, they expend power and work. This conversion, and the associated work, occurs before using the energy to potentialize. As an example, electrical engineers have been taught to build only the symmetric class of Maxwellian circuits, those which use half their collected free potentialization energy to kill the source dipole — the source of their own energy flow — faster than they power their loads. Hence, constant operation of these symmetric systems requires the operator to keep paying for input energy, to continuously keep restoring their self-destroyed source dipole. This class of Maxwellian system is called symmetric.

1.7 Of interest in this paper are working EM systems, in other words those that receive input energy and process it by changing its form to produce useful output energy or useful work in a load.

1.8 This paper is primarily concerned with systems that utilize positive EM energy. Negative EM energy can indeed be evoked and used in circuits and systems, producing effects and phenomenology that are startlingly different and highly useful; however, that topic is out of scope for this paper.

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