A Ledger Formulation of Impedance: Eliminating Imaginary Numbers Through Quantum Measurement Units

This paper examines electrical impedance from the perspective of Quantum Measurement Units (QMU) and proposes a ledger-based alternative to the conventional complex-number formulation used in alternating-current circuit theory.

Classical impedance theory represents impedance as

Z = R + jX,

where resistance and reactance are assigned the same dimensional unit and are distinguished through the imaginary operator j. The present work argues that this representation results from dimensional compression inherited from conventional SI/MKS unit systems.

Within the QMU framework, resistance and magnetic flux occupy distinct dimensional positions. Resistance is represented by the QMU quantity resn, while the stored-field component associated with conventional reactance is represented by the QMU quantity mflx. These quantities differ by magnetic-charge rank and therefore cannot be regarded as identical dimensional objects.

A ledger formulation of impedance is developed in which impedance is represented as the ordered pair

Z = (R, Φ),

where R denotes the dissipative component and Φ denotes the stored-field magnetic-flux component. Conventional reactance is recovered through dimensional projection, allowing standard impedance magnitude and phase relationships to be reproduced exactly while preserving dimensional separation.

The paper demonstrates recovery of conventional alternating-current results, including impedance magnitude, phase angle, and series RL behavior. It further argues that complex impedance functions as an efficient algebraic representation of two independent electrical components whose dimensional distinction has been compressed into a common unit assignment.

The resulting framework suggests that imaginary numbers are not fundamentally required for impedance theory. Instead, they may be interpreted as a mathematical bookkeeping device that preserves distinctions between quantities that remain explicitly separated within the QMU ledger.

The paper provides a conceptual foundation for subsequent work developing purely real-valued circuit methods, network analysis, and transmission-system calculations within the Quantum Measurement Units framework.