This deposit contains a self-contained research article and a compact Engineer’s Reference Card presenting a purely real framework for impedance in Quantum Measurement Units (QMU). The approach avoids complex numbers, keeps units explicit, and expresses phase/magnitude using real rotation–scaling matrices. The work develops three equivalent, engineer-friendly methods: Two-component method (cos/sin). Steady-state signals are represented by cosine and sine components so that impedance acts as a real 22 matrix. A simple detuning function b () =/Fq - Fq/ captures both magnitude and phase. The normalized series impedance is Zₒ₄ₑ₈₄ₒ () =Z₀bmatrix1&-b\ b&1bmatrix, where Z₀ is the chosen base impedance unit. Magnitude and angle follow from G () =1+b () ² and () =! (b () ), so the channel is a rotation by () and a scaling by Z₀, G (). Time-domain operator method (no phasors). Normalized differentiation/integration operators reproduce the same in-quadrature behavior without introducing complex numbers. For a series channel driven by current i (t), the voltage is v (t) =Z₀ (i+Dₜ i-Iₜ i) with Dₜ= (1/Fq), d/dt and Iₜ=Dₜ^-1. The cycle average power remains purely real: p=12 (Vc Ic+Vₛ Iₛ). Quadrature-operator method. A symbolic 90^ shift operator encodes phase within a real algebra equivalent to the two-component rotation, yielding the compact series form Z () =, 1+Q, b (), Z₀ (with Q representing a +90^ rotation in the cos/sin plane). The article shows how a ledger identity aligns inductive and capacitive channels on a single base impedance unit so series/parallel combinations add like dimensions. It derives resonance, magnitude/angle, average power, one-port reductions, and two-port cascades. Real-valued forms are given for transmission sections and coupled inductors (mutual inductance), and small-signal active templates (e. g. , transconductance and negative differential resistance). A purely real formulation of S-parameters is provided. With a reference Z₀ per port, the power-wave vectors are a=12Z₀ (V+Z₀ I) and b=12Z₀ (V-Z₀ I), giving the reflection matrix= (Z-Z₀) (Z+Z₀) ^-1= (Z-Z₀), adj (Z+Z₀) / (Z+Z₀). Two-port scattering follows by stacking port waves into a real 44 block matrix whose 22 blocks directly encode rotation and scaling. Engineer’s Reference Card. The companion card distills the results into printable boxes: normalized RLC entries, composition rules (series Z-sum and parallel Y-sum), rotation–scaling identities, S-parameter definitions, transmission/ladder recipes, and quick procedures for matching and detuning. It is intended for RF/microwave and analog engineers who want dimensionally clean, real-valued network methods that avoid phasors while preserving the usual design intuition.
David W. Thomson (Fri,) studied this question.