--- title: "What can one impedance arc tell us?" description: "Introduce complex impedance through sinusoidal response, derive an interfacial arc, and separate parameter uncertainty from non-identifiability." canonical_url: "https://sandboxphysics.com/stories/electrochemistry/impedance-identifiability/" language: "en" page_type: "story" alternate_languages: zh-CN: "https://sandboxphysics.com/zh/stories/electrochemistry/impedance-identifiability/" last_modified: "2026-10-28" --- [Electrochemistry and ion transport](https://sandboxphysics.com/stories/electrochemistry/) L3 · Chapter 7 # What can one impedance arc tell us? Sandbox Physics · 2026-10-28 A complete impedance arc can determine solution resistance, effective reaction resistance and capacitance. It may still leave microscopic reaction pathways unresolved. First ask what each frequency measures, then fit a saved spectrum to distinguish uncertain parameters from parameters these data cannot determine at all. ## Before you begin Use chapter six’s small-signal charge-transfer resistance. Complex numbers are introduced through sinusoidal voltage and phase lag. No prior knowledge of phasors or frequency-domain analysis is assumed. [Review the preceding chapter](https://sandboxphysics.com/stories/electrochemistry/butler-volmer/) - Interpret complex impedance as an amplitude ratio and phase difference. - Recover effective parameters from arc intercepts and peak frequency. - Distinguish limited frequency coverage from structurally indistinguishable pathways. ## Rock the voltage slowly, then quickly Apply a small sinusoidal voltage to a stable electrode at frequency $f$, completing $f$ cycles per second. Angular frequency is $\omega=2\pi f$ in radians per second. If response is linear at this amplitude, current oscillates at the same frequency with a different amplitude and possibly a shifted peak time. Impedance records both the voltage-to-current amplitude ratio and their phase difference. Amplitude alone misses essential information: resistor current peaks with voltage, whereas capacitor current leads by a quarter cycle. At high frequency, charging the same capacitor with a given current requires a smaller voltage excursion. Reading: MIT 10.626 ## Use a complex number to keep magnitude and phase Real and imaginary parts record two components a quarter cycle apart. The imaginary unit $\mathrm i$ obeys $\mathrm i^2=-1$; physical current remains $I$. Factor out the shared time dependence $e^{\mathrm i\omega\tau}$. The remaining complex amplitude is a phasor. Actual voltage or current is the real part of the complete expression. A resistor has in-phase voltage and current, giving real impedance. Capacitor current is capacitance multiplied by the voltage derivative. Differentiating the shared exponential multiplies by $\mathrm i\omega$, giving negative imaginary capacitive impedance. Phasors turn the time derivative into algebra. $$ Z=\frac{\widehat V}{\widehat I},\qquad Z_R=R,\qquad Z_C=\frac{1}{\mathrm i\omega C}=-\frac{\mathrm i}{\omega C} $$ Hats denote complex amplitudes and impedance is in ohms. We use positive-exponential time dependence. Nyquist plots show negative imaginary part upward for the capacitive arc. Reading: MIT 10.626 ## Combine reaction and charging in parallel Reaction current and double-layer charging feel the same small interfacial overpotential, so they form parallel branches. Solution resistance $R_s$ carries their summed current and is in series. Add the parallel admittances, or reciprocal impedances, then invert to obtain this simplified Randles-type model. Capacitance $C_{\rm dl}$ and charge-transfer resistance $R_{\rm ct}$ describe the whole modelled electrode. Their product $\tau_r=R_{\rm ct}C_{\rm dl}$ sets the time scale for current partition. The model holds temperature and surface concentrations fixed and includes neither diffusion impedance nor distributed capacitance. $$ Z(\omega)=R_s+\left(R_{\rm ct}^{-1}+\mathrm i\omega C_{\rm dl}\right)^{-1} $$ At low frequency the capacitor is effectively open, so impedance approaches the sum of resistances. At high frequency it effectively shorts the reaction branch, leaving solution resistance. $$ Z'=R_s+\frac{R_{\rm ct}}{1+(\omega\tau_r)^2},\qquad -Z''=\frac{R_{\rm ct}\omega\tau_r}{1+(\omega\tau_r)^2} $$ Primes $Z\prime$ and $Z^{\prime\prime}$ denote real and imaginary parts. Eliminating frequency gives a semicircle, whose top occurs at $\omega\tau_r=1$. Reading: MIT 10.626 ## Read three parameters from three features Suppose the high-frequency intercept is $35.0\;\Omega$, the low-frequency intercept $215\;\Omega$, and the top of the arc occurs at $49.1\;\mathrm{Hz}$. The first gives solution resistance. Their difference gives $R_{\rm ct}=180\;\Omega$. The peak condition then gives $C_{\rm dl}\approx18.0\;\mathrm{\mu F}$. The relaxation time is $3.24\;\mathrm{ms}$. The table gives noiseless circuit truth with frequency scaled by the peak frequency $f_c$. The default synthetic record covers $0.1\;\mathrm{Hz}$ to $10^5\;\mathrm{Hz}$. Reproducible seeded Gaussian noise has standard deviation $0.8\;\Omega$ independently in each real and imaginary component. $$ C_{\rm dl}=\frac{1}{2\pi f_cR_{\rm ct}},\qquad f_c=\frac{1}{2\pi\tau_r} $$ Use hertz and ohms to obtain farads. The horizontal position of the arc top is halfway between its intercepts. Noiseless default circuit; synthetic recorded points separately include labelled noise | $f/f_c$ | $Z\prime$ / $\Omega$ | $-Z^{\prime\prime}$ / $\Omega$ | | --- | --- | --- | | 0.1 | 213.2 | 17.82 | | 1 | 125.0 | 90.00 | | 10 | 36.78 | 17.82 | ## Save a spectrum before estimating its parameters Acquisition saves frequency, real part, imaginary part and noise settings. Fitting reads that record and varies positive resistances and capacitance to compare predictions with the saved points. Changing resistance, capacitance or noise acquires a new record. Changing the pathway fraction or fitted frequency band repeatedly analyses the same saved points. For equal real and imaginary noise scales, add the squared residuals of both components and minimize their sum. Dividing by a positive common noise variance gives the expression below with the same minimizing parameters. A noiseless reference uses unweighted residuals. Residuals should fluctuate around zero. A coherent frequency-dependent trend can indicate missing physics. A small residual also needs adequate coverage of both intercepts and the arc top. At high frequencies alone, much of the reaction information lies in small corrections. Many reaction resistances can produce nearly coincident spectra within the noise. Broader frequency coverage can improve this precision problem. $$ \chi^2=\sum_k\frac{(Z'_{k,{\rm rec}}-Z'_{k,{\rm model}})^2+(Z''_{k,{\rm rec}}-Z''_{k,{\rm model}})^2}{\sigma^2} $$ Index $k$ labels frequency and $\sigma$ is the known noise standard deviation of each component. Correlated errors require a covariance instead of this independent equal-variance assumption. ## Some ambiguity survives perfect measurements Split the effective reaction branch into two parallel pathways. For any $0