Dynamic Impedance Phasor Helix

This paper demonstrates that the Bode and Nyquist diagrams — traditionally treated as separate analytical tools in control theory — are 2D projections of a single 3D geometric object: a helical curve traced by the complex-valued frequency response G(jω) parameterized over frequency. The framework, termed the Dynamic Impedance Phasor Helix, unifies magnitude, phase, and frequency into one continuous structure, revealing geometric relationships that are lost in conventional 2D representations. Interactive visualizations for LFC and RLC system variants are included.本文证明，传统上在控制理论（control theory）中被视为独立分析工具的波特图（Bode diagrams）与奈奎斯特图（Nyquist diagrams），实为单一三维几何对象的二维投影：该对象是由按频率参数化的复值频率响应G(jω)所描绘的螺旋曲线。该框架被命名为动态阻抗相量螺旋（Dynamic Impedance Phasor Helix），可将幅值、相位与频率统合为一个连续结构，揭示出传统二维表征中丢失的几何关联。本文还包含针对LFC与RLC系统变体的交互式可视化内容。