双曲复数拓扑群电子学：线性与非线性电路理论的统一公理框架

Classical circuit theory, based on real and complex numbers, has been immensely successful in analyzing linear time-invariant networks. However, it suffers from several limitations: • It treats series and parallel connections as ad-hoc formulas rather than algebraic operations. • It lacks a unified treatment of faults (short, open, etc.) as algebraic degenerations. • Nonlinear elements require separate treatment and break the linear algebraic structure. • Directional phenomena (e.g., reactive power, wave propagation) are handled with complex numbers but without a clear geometric interpretation. The introduction of hyperbolic complex numbers (split-complex numbers) a + bj with j2 = 1 offers a natural algebraic framework that overcomes these limitations. Their indefinite metric a2 −b2 captures the distinction between dissipative (real part) and reactive/rotational (imaginary part) components, while zero divisors a = ±b correspond to singularities like short and open circuits. Moreover, the algebra is equipped with two compatible group structures: addition (series) and a parallel operation derived from multiplication, which together form a bisemigroup structure that perfectly models circuit topology. In this paper, we build a complete axiomatic theory of circuits using: 1 1. Topological graphs to encode connectivity; 2. Hyperbolic complex numbers to model element characteristics; 3. Group theory to unify all operations, including faults and nonlinearities. The result is a self-contained, mathematically rigorous framework that not only reproduces all classical results (Kirchhoff’s laws, Thevenin/Norton theorems) but also extends them to nonlinear circuits and provides an algebraic foundation for fault diagnosis.