阻抗与界面响应实验室
科学说明与审核记录

本工具计算理想 Randles 电路的解析响应，不做实验拟合，不从电路参数唯一推断真实微观机制。AI 审核不等同于电化学专家认证。

1. 电路拓扑

溶液电阻 Rₛ 与并联网络串联；并联的两条支路分别是双电层电容 Cdl，以及电荷转移电阻 Rct 串联半无限扩散 Warburg 元件 W。它不是 Rct 与 Cdl 串联，也不是 W 置于整个并联电路之外。

Impedance & Interface Response Lab
Scientific Notes & Review

This tool calculates an ideal Randles circuit analytically. It does not fit experiments or uniquely infer microscopic mechanisms from circuit parameters. AI review is not certification by an electrochemistry expert.

1. Circuit topology

Solution resistance Rₛ is in series with a parallel network. One branch is the double-layer capacitor Cdl; the other is charge-transfer resistance Rct in series with semi-infinite Warburg diffusion W. Rct and Cdl are not in series, and W is not placed outside the entire parallel combination.

2. 符号、单位与显示

采用 eʲωᵗ 约定，电容和本 Warburg 元件的虚部为负。Nyquist 纵轴为 −Z″；Bode 相位显示 arg Z，通常为负；电流相对电压超前 −arg Z。Nyquist 横纵轴每欧姆的显示像素相同，以保持半圆与 45° 尾部的几何含义。频率范围 100 kHz–1 mHz，161 个对数等距频率；当前频率连续独立计算，不强制对齐采样点。

Rₛ、Rct 单位 Ω；Cdl 内部单位 F，输入 µF；σ 单位 Ω·s⁻¹ᐟ²，使 σ/√ω 为 Ω。交流电压为峰值 V̂，不是 RMS。波形各除以自身峰值，所以相位可直接比较，曲线高度不能用于比较 V 与 A 的绝对大小；峰值分别显示在图下。横轴 t/T，演示周期放慢为 3 秒，与物理频率解耦。

3. 参数与边界

σ = 0 去掉 Warburg 贡献，得到理想 Rₛ + (Rct ∥ Cdl)。其高频截距为 Rₛ，低频截距 Rₛ + Rct，弧顶角频率为 1/(Rct Cdl)，弧顶 −Z″ = Rct/2。σ > 0 时，1/(2πRct Cdl) 只作 RC 尺度参考，不能当作实际阻抗峰。半无限 Warburg 适用于扩散长度未碰到有限边界的近似，极低频时真实实验可偏离。

界面电荷球随归一化电压相位示意变化，数量、大小和位置不代表定量电荷密度、电势或原子电子轨迹。三电极硬件为通用示意。电路视图与示意拓扑互相对应；相机移动不改变阻抗。

4. 验证与审核

自动测试独立核对无 Warburg 时的高低频极限、RC 弧顶、Warburg 的等实部/负虚部、非负被动电路的阻抗符号，以及由相量反演的电流超前。数值基准：Rₛ = 20 Ω、Rct = 250 Ω、Cdl = 40 µF、σ = 0 时，f = 15.9154943 Hz 对应 Z = 145 − 125j Ω。Rₛ = 10 Ω、Rct = 100 Ω、Cdl = 0、σ = 30 Ω·s⁻¹ᐟ²、ω = 1 rad/s 时，Z = 140 − 30j Ω。

独立 AI 交叉审核已用六个频率（0.001、0.01、1、15.91549、1000、100000 Hz）下的独立闭式 RC 解核对，并重构 I(0) = V̂ Re(Z)/|Z|²；数值误差小于 10⁻¹¹。独立审核未发现实质科学阻塞；不等同于人工电化学专家认证。

5. 未包含的物理过程

模型假设小信号、线性、时不变和理想元件。不包含 CPE、感抗、有限长度扩散、电极非均匀性、非线性动力学、漂移、噪声或真实仪器传递函数。界面没有求解 Poisson–Nernst–Planck，也没有进行原子尺度电荷输运。1–10 mV 峰值是演示输入；模型自动保持线性，不构成真实材料在线性范围内的证明。

6. 公开依据

2. Sign, units and display

The eʲωᵗ convention gives negative imaginary impedance for the capacitor and this Warburg. Nyquist displays −Z″; Bode displays arg Z, usually negative. Current leads voltage by −arg Z. Nyquist uses equal pixels per ohm on both axes to preserve semicircle and 45°-tail geometry. The spectrum has 161 logarithmically spaced frequencies from 100 kHz to 1 mHz. The selected frequency is evaluated continuously rather than snapped to a sampled point.

Rₛ and Rct are in Ω; Cdl is stored in F with µF input; σ is in Ω·s⁻¹ᐟ² so σ/√ω is in Ω. AC voltage is peak amplitude V̂, not RMS. Each waveform is divided by its own peak, so phase can be compared directly but trace heights do not compare absolute volts and amperes. Both physical amplitudes are displayed below. The horizontal coordinate is t/T; playback is slowed to 3 seconds per illustrated cycle and is separate from physical frequency.

3. Parameters and limits

σ = 0 removes Warburg, giving ideal Rₛ + (Rct ∥ Cdl). Its high-frequency intercept is Rₛ and low-frequency intercept Rₛ + Rct. The apex has angular frequency 1/(Rct Cdl) and −Z″ = Rct/2. For σ > 0, 1/(2πRct Cdl) is only an RC reference scale, not necessarily an impedance peak. Semi-infinite Warburg approximates diffusion before a finite boundary is encountered; real experiments can depart from it at low frequency.

Charge markers illustrate normalized voltage phase; their number, size and locations are not quantitative charge densities, potentials or electron trajectories. Three-electrode hardware is generic. The 3D circuit and schematic share the same topology. Camera motion does not alter impedance.

4. Validation and review

Automated tests independently check high/low-frequency limits without Warburg, the RC apex, equal Warburg real and negative-imaginary components, passive impedance signs and phasor-derived current lead. Benchmark: with Rₛ = 20 Ω, Rct = 250 Ω, Cdl = 40 µF and σ = 0, f = 15.9154943 Hz gives Z = 145 − 125j Ω. With Rₛ = 10 Ω, Rct = 100 Ω, Cdl = 0, σ = 30 Ω·s⁻¹ᐟ² and ω = 1 rad/s, Z = 140 − 30j Ω.

Independent AI cross-review compared a separately derived closed-form RC solution at six frequencies (0.001, 0.01, 1, 15.91549, 1000 and 100000 Hz) and reconstructed I(0) = V̂ Re(Z)/|Z|², agreeing within 10⁻¹¹. No material scientific blocker was identified. This is not human electrochemistry expert certification.

5. Excluded physics

Small-signal, linear, time-invariant response and ideal elements are assumed. CPEs, inductance, finite-length diffusion, electrode heterogeneity, nonlinear kinetics, drift, noise and real instrument transfer functions are excluded. No Poisson–Nernst–Planck or atomistic charge-transport model is solved. The 1–10 mV peak voltage is an illustrative input; automatically linear model behavior does not prove a real material remains linear.

6. Sources