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

Tensor Formulation of Electric Circuit Theory: A Real Geometry for Complex Phase, Impedance, and Power Se Kyun Kwon Department of Physics, Pohang University of Science and Technology, Pohang 37673, Republic of Korea (Correspondence: [email protected]) Abstract 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 twodimensional 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 𝐕=𝐙(𝐈), 𝑉𝛼=∑𝑍𝛼𝛽𝐼𝛽. Phase shift, active and reactive power, resonance, and impedance matching emerge naturally as geometric phenomena in ℝ2. Power flow is encoded in the power tensor 𝐓=𝐕⨂𝐈 , 𝑇𝛼𝛽 =𝑉𝛼𝐼𝛽, 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 complexvalued physics but a compressed representation of real two-dimensional geometry, offering a physically transparent and systematically extensible foundation for circuit theory. 1. Introduction Since the pioneering work of Heaviside and Steinmetz in the late nineteenth century, complex numbers have served as the dominant language of AC circuit analysis. The representation 𝑍 =𝑅+𝑗𝑋 has proven remarkably effective for computation, yet it obscures the underlying geometric and physical structure of sinusoidal systems. In standard formulations, the imaginary unit 𝑗 is treated as an abstract algebraic symbol rather than a physical operation, and the relations among resistance, reactance, phase shift, energy flow, and power decomposition remain embedded in algebraic shorthand rather than expressed as geometric entities. Complex numbers were adopted for their algebraic convenience—not because AC circuits are intrinsically complex-valued. In this work, we show that the complex representation is not fundamental to AC circuit theory. Instead, it emerges from a real, two-dimensional tensor geometry in which voltages and currents are vectors in ℝ2 and the imaginary unit is the canonical 90° rotation operator. Impedance is not a scalar but a genuine second-order tensor whose symmetric and antisymmetric parts describe dissipation and rotation. We construct an explicit algebra isomorphism between the field of complex numbers and a two-dimensional real matrix subalgebra, showing that the traditional phasor formalism is simply a compressed notation for this tensor algebra. Within this geometric framework, the fundamental circuit law assumes the coordinate-invariant tensor form 𝐕=𝐙(𝐈), 𝑉𝛼=∑𝑍𝛼𝛽𝐼𝛽. which holds in any orthonormal basis of ℝ2. This formulation restores the geometric content of AC circuit theory and provides transparent interpretation of phase, resonance, reactive energy, and power flow. Furthermore, the tensor formulation offers a natural platform for extending circuit theory to nonlinear elements, three-phase machines, distributed systems, and realgeometric physics. 2. Real-Geometric Representation of Phasors A complex voltage phasor 𝑉 =𝑉𝑥+𝑗𝑉𝑦 is identified with a real two-dimensional vector 𝐕=(𝑉𝑥, 𝑉𝑦)𝑇. Similarly, a current phasor is 𝐈=(𝐼𝑥, 𝐼𝑦)𝑇. We introduce two fundamental rank-2 tensors on ℝ2. The Euclidean metric tensor in ℝ2 is 𝕀=𝛿𝛼𝛽=(1 0 0 1), and the canonical generator of 90° rotation is 𝕁=𝐽𝛼𝛽=(0 −1 1 0 ), 𝕁2=−𝕀, 𝑒𝜃𝕁 =(cos𝜃 −sin𝜃 sin𝜃 cos𝜃). Multiplication by the imaginary unit corresponds to the action of 𝕁, 𝑗𝐼 ⟷ 𝕁(𝐈), ∑𝐽𝛼𝛽𝐼𝛽. Thus, the imaginary unit is not an algebraic symbol but a concrete linear transformation of the vector space. The complex plane is simply the real vector space ℝ2 equipped with the rotation tensor 𝕁. 3. Impedance as a Second-Order Tensor The classical scalar impedance 𝑍 =𝑅+𝑗𝑋 is naturally lifted to the impedance tensor 𝐙=𝑅𝕀+𝑋𝕁, 𝑍𝛼𝛽=𝑅𝛿𝛼𝛽+𝑋𝐽𝛼𝛽. Explicitly, 𝐙=𝑍𝛼𝛽=(𝑅 −𝑋 𝑋 𝑅 ). This representation has a clear geometric interpretation. The term 𝑅𝕀 is a symmetric isotropic scaling tensor representing dissipation, whereas the term 𝑋𝕁 is an antisymmetric rotation tensor generating a vector orthogonal to the input. Because 𝕁 generates the Lie algebra 𝔰𝔬(2), the reactance is fundamentally a rotation rate in the voltage-current plane. The fundamental circuit law becomes the tensor equation 𝐕=𝐙(𝐈)=𝑅𝐈+𝑋𝕁(𝐈), 𝑉𝛼=∑𝑍𝛼𝛽𝐼𝛽=𝑅 𝐼𝛼+𝑋∑𝐽𝛼𝛽𝐼𝛽. Voltage is therefore the vector sum of a component parallel to the current 𝑅𝐈, which transfers dissipative power, and a component orthogonal to the current 𝑋𝕁(𝐈), which participates only in reactive energy exchange. 4. Algebraic Isomorphism between Complex Numbers and Tensor Algebra We now formalize the statement that the complex algebra ℂ is isomorphic to a real matrix subalgebra generated by 𝕀 and 𝕁. Define the mapping Φ: ℂ→ℝ2×2, Φ(𝑥+ 𝑗𝑦)=𝑥𝕀+𝑦𝕁, where 𝕀=(1 0 0 1), 𝕁=(0 −1 1 0 ). 4.1 Linearity For complex numbers 𝑧1=𝑥1+𝑗𝑦1 and 𝑧2=𝑥2+𝑗𝑦2, Φ(𝑧1+𝑧2)=(𝑥1+𝑥2)𝕀+(𝑦1+𝑦2)𝕁=Φ(𝑧1)+Φ(𝑧2). Thus, Φ is additive. Homogeneity with respect to real scalars follows immediately, Φ(𝑐𝑧)=𝑐Φ(𝑧), 𝑐 ∈ℝ. 4.2 Multiplicativity The product in ℂ is 𝑧1𝑧2=(𝑥1+𝑗𝑦1)(𝑥2+𝑗𝑦2)=(𝑥1𝑥2−𝑦1𝑦2)+𝑗(𝑥1𝑦2+𝑦1𝑥2). On the matrix side, Φ(𝑧1)Φ(𝑧2)=(𝑥1𝕀+𝑦1𝕁)(𝑥2𝕀+𝑦2𝕁) =(𝑥1𝑥2−𝑦1𝑦2)𝕀+(𝑥1𝑦2+𝑦1𝑥2)𝕁, because 𝕁2=−𝕀 and 𝕀 commutes with 𝕁. Therefore Φ(𝑧1𝑧2)=Φ(𝑧1)Φ(𝑧2), and Φ is an algebra homomorphism. 4.3 Isomorphism onto a Subalgebra The image of Φ is exactly the two-dimensional real subspace 𝒜 ={𝑥𝕀+𝑦𝕁∣𝑥,𝑦 ∈ℝ}⊂ℝ𝟐×𝟐. 𝒜 is closed under matrix addition and multiplication, and Φ is clearly injective. Hence Φ is an algebra isomorphism ℂ≅𝒜. This proves that classical complex AC theory is a special case of real tensor algebra on ℝ𝟐. Complex numbers are simply a convenient notation for matrices of the form 𝑥𝕀+𝑦𝕁. 5. Coordinate Invariance and Physical Fundamentality In the tensor framework, voltages and currents are vectors, and impedance is a secondorder tensor. Under a orthonormal transformation of basis represented by a rotation matrix 𝑅𝛼𝛽∈ SO(2), 𝑉 𝛼=∑𝑅𝛼𝛽𝑉𝛽, 𝐼𝛼=∑𝑅𝛼𝛽𝐼𝛽, and the impedance tensor transforms as 𝑍𝛼𝛽=∑𝑅𝛼𝛾𝑅𝛽𝛿𝑍𝛾𝛿 The circuit law, 𝐕=𝐙(𝐈), 𝑉𝛼=∑𝑍𝛼𝛽𝐼𝛽, retains its form in the new basis as 𝐕 =𝐙 (𝐈), 𝑉 𝛼=∑𝑍𝛼𝛽𝐼𝛽. Thus, the tensor equation is coordinate-invariant establishing that the fundamental circuit law is a physical law independent of the particular axis orientation in ℝ𝟐. By contrast, the complex equation 𝑉 = 𝑍𝐼 implicitly assumes a specific identification of the real and imaginary axes with the chosen coordinate axes in ℝ𝟐. Changing the basis corresponds to a nontrivial transformation of the complex representation. The tensor framework is therefore more fundamental; it encodes the geometry and physics in a basis-independent manner, while the complex notation corresponds to a particular coordinate choice. 6. Geometric Interpretation of Circuit Phenomena 6.1 Phase shift The phase angle 𝜙 between voltage and current is simply the geometric angle between the vectors 𝐕 and 𝐈 in ℝ𝟐. For a single impedance, 𝐙=𝑅𝕀+𝑋𝕁=(𝑅 −𝑋 𝑋 𝑅 ), we may factor 𝐙 into its magnitude and rotation components, 𝐙=√𝑅2+𝑋2(cos𝜙 −sin𝜙 sin𝜙 cos𝜙)=√𝑅2+𝑋2𝑒𝜙𝕁. Here, 𝜙 =tan−1(𝑋 𝑅 ⁄ ),cos𝜙 = 𝑅 √𝑅2+𝑋2, sin𝜙 = 𝑋 √𝑅2+𝑋2 . This decomposition shows that the phase angle is determined by the ratio of the antisymmetric and symmetric components of the impedance tensor. Thus, the phase shift in AC circuits is not an abstract complex-number operation, but the physical rotation generated by the antisymmetric tensor 𝑋𝕁 relative to the symmetric part 𝑅𝕀. The familiar phase lag or phase lead arises from the geometric action of the rotation generator 𝕁 acting on the current vector. 6.2 Resonance as vanishing antisymmetric part For a series 𝑅𝐿𝐶 circuit, the frequency-dependent reactance is 𝑋(𝜔)=𝜔𝐿−1 𝜔𝐶 , and the impedance tensor is 𝐙(𝜔)=𝑅𝕀+𝑋(𝜔)𝕁, 𝑍𝛼𝛽(𝜔)=𝑅𝛿𝛼𝛽+𝑋(𝜔) 𝐽𝛼𝛽. Resonance occurs when 𝑋(𝜔0)=0, so that 𝐙(𝜔0)=𝑅𝕀, 𝑍𝛼𝛽(𝜔0)=𝑅𝛿𝛼𝛽. Geometrically, the rotation component vanishes, and the voltage becomes collinear with the current. Resonance is therefore the condition that the antisymmetric part of the impedance tensor vanishes, leaving a purely symmetric scaling operator. 6.3 Power tensor and energy interpretation Define the power tensor 𝐓=𝐕⨂𝐈 , 𝑇𝛼𝛽 =𝑉𝛼𝐼𝛽, Decompose it into symmetric and antisymmetric parts: 𝐓𝑠=1 2(𝐓+𝐓𝑇), 𝑇(𝛼𝛽)=1 2(𝑇𝛼𝛽 +𝑇𝛽𝛼), 𝐓𝑎=1 2(𝐓−𝐓𝑇), 𝑇[𝛼𝛽]=1 2(𝑇𝛼𝛽 −𝑇𝛽𝛼). The active (real) power is 𝑃 =⟨𝐕,𝐈⟩= ∑𝑉𝛼𝐼𝛼=𝑉𝑥𝐼𝑥+𝑉𝑦𝐼𝑦. This can be expressed as the trace of the symmetric part of the power tensor, 𝑃 =tr(𝐓)=∑𝑇(𝛼𝛼). The reactive power is 𝑄 =−𝐕×𝐈=∑𝐽𝛼𝛽𝑉𝛼𝐼𝛽=−(𝑉𝑥𝐼𝑦−𝑉𝑦𝐼𝑥). Using the Levi-Civita symbol 𝜀𝛼𝛽 (with 𝜀12 =1), we have 𝑄 =−∑𝜀𝛼𝛽𝑇𝛼𝛽. Since 𝜀𝛼𝛽 is antisymmetric, this contraction selects the antisymmetric part of 𝐓, 𝑄 =−∑𝜀𝛼𝛽𝑇[𝛼𝛽]. Thus, 𝑃 arises from the symmetric part of 𝐓 and represents net energy transfer and dissipation. On the contrary, 𝑄 arises from the antisymmetric part of 𝐓 and represents oscillatory energy exchange between electric and magnetic fields, or between storage elements, with no net-work over a cycle. The geometric orthogonality ⟨𝕁(𝐈),𝐈⟩=0 explains why reactive power does not contribute to net energy transfer. 7. Composition Laws and Network Theory 7.1 Series connection For series elements, 𝐙(series)=𝐙1+𝐙2. This is the direct tensor analogue of complex impedance addition. 7.2 Parallel connection and admittance tensor Introduce the admittance tensor, 𝐘=𝐙−1. In the single-element case, 𝐘=𝐶𝕀+𝐵𝕁, 𝑌𝛼𝛽=𝐶 𝛿𝛼𝛽+𝐵 𝐽𝛼𝛽, with 𝐶 = 𝑅 𝑅2+𝑋2, 𝐵 = −𝑋 𝑅2+𝑋2 . Parallel elements combine via 𝐘(𝑝𝑎𝑟𝑎𝑙𝑙𝑒𝑙)=𝐘1+𝐘2.