On the Application of the Principle of Energy Equipartition to Linear Time-Invariant <i>RLC</i> Networks

The energy equipartition principle, rooted in classical statistical mechanics, has been formerly tailored to predict the total integrated thermal noise in <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">RLC</i> circuits, assigning an average energy 1/2<italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">kT</i> to every degree of freedom of the network. Although the implications of this result are primarily theoretical, it can still be used in some practical cases and as verification, based on first principles, of the noise estimate. In this brief, we show that the definition of degrees of freedom must be substantially revised to properly evaluate the total average thermal noise energy in <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">RLC</i> networks with multiple reactive components. By exploiting the state-space representation of a system and the property of controllability, we highlight the impact of network degeneracies such as capacitor-only cut-sets and inductor-only loops on the energy equipartition. As a key result, we show that a correct application of the principle of energy equipartition is achieved by assigning an average energy 1/2<italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">kT</i> to each nonzero natural frequency of the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">RLC</i> network. Some representative examples are illustrated that fully support our findings.