Tensor Formulation of Electric Circuit Theory: A Real Geometry for Complex Phase, Impedance, and Power

Classical electric circuit theory represents voltages, currents, and impedances using complex numbers, a convention adopted historically for algebraic convenience rather than physical necessity. Here we develop a complete real-tensor formulation of circuit theory in which voltages and currents are genuine vectors in a two-dimensional real space, while impedances are second-order tensors composed of an isotropic scaling operator and the antisymmetric generator of the rotation group SO(2). The traditional complex impedance 𝑍 = 𝑅 + 𝑗𝑋 is replaced by the tensor 𝐙 = 𝑅𝕀 + 𝑋𝕁, 𝑍𝛼𝛽 = 𝑅𝛿𝛼𝛽 + 𝑋𝐽𝛼𝛽, (𝛼, 𝛽 = 1,2) where 𝕀 is the identity and 𝕁 is the canonical 90° rotation tensor. We define a mapping Φ: ℂ → ℝ2×2, Φ(𝑥 + 𝑗𝑦) = 𝑥𝕀 + 𝑦𝕁 that establishes an algebra isomorphism between complex numbers and the two-dimensional real subalgebra spanned by 𝕀 and 𝕁, demonstrating that the complex formalism is merely the algebraic projection of a richer real-geometric structure. The fundamental circuit law becomes the coordinate-invariant tensor equation, a genuine geometric physical law: 𝐕 = 𝐙(𝐈), 𝑉^𝛼 = Σ[𝑍𝛼𝛽 𝐼_𝛽]. Phase shift, active and reactive power, resonance, and impedance matching emerge naturally as geometric phenomena in ℝ^2. Power flow is encoded in the power tensor 𝐓 = 𝐕⨂𝐈, 𝑇𝛼𝛽 = 𝑉^𝛼 𝐼^𝛽, whose symmetric part describes dissipative transfer of energy and antisymmetric part encodes reversible oscillatory exchange. This tensor formulation reveals that complex AC analysis is not intrinsically complex-valued physics but a compressed representation of real two-dimensional geometry, offering a physically transparent and systematically extensible foundation for circuit theory.