Distribution of Relaxation Times (DRT): Separating What EIS Alone Cannot

Electrochemical Impedance Spectroscopy (EIS) is one of the most widely used techniques for characterizing battery cells. A single EIS measurement captures how a cell responds across a range of frequencies, producing the familiar Nyquist plot of real versus imaginary impedance. But Nyquist plots have a fundamental limitation: when multiple electrochemical processes have similar time constants, their arcs overlap and become impossible to separate visually.

This is where the Distribution of Relaxation Times (DRT) becomes valuable.

What Is DRT?

DRT is a mathematical technique that deconvolves an impedance spectrum into its constituent relaxation processes. Instead of viewing impedance as a single curve, DRT produces a plot of impedance contribution (gamma) versus time constant (tau). Each peak in the DRT corresponds to a distinct electrochemical process occurring in the cell.

In practical terms, DRT answers the question: what physical processes are contributing to this cell's impedance, and how much does each one contribute?

Common processes that DRT can separate include:

• Solid Electrolyte Interphase (SEI) layer responses at high frequencies
• Charge transfer at the electrode-electrolyte interface
• Solid-state diffusion within electrode particles at lower frequencies
• Contact resistance and current collector contributions

When these processes have overlapping time constants, they appear as a single broad arc on a Nyquist plot. DRT resolves them into distinct peaks, each with a characteristic time constant and magnitude.

Why DRT Is Difficult

Computing the DRT from EIS data is an inverse problem. The measured impedance spectrum is related to the underlying DRT through a Fredholm integral equation of the first kind. Inverting this integral is inherently ill-posed: small noise in the measurement can produce large, spurious oscillations in the result. Without careful treatment, the output is dominated by noise artifacts rather than real electrochemical processes.

This is why most teams that use DRT today do so through custom MATLAB or Python scripts that require significant expertise to configure correctly. The critical challenge is selecting the right amount of regularization — a mathematical constraint that suppresses noise without erasing real features. Too much regularization smooths away genuine peaks. Too little lets noise dominate.

Historically, this has required manual tuning and expert judgment, which limits DRT adoption to research groups with dedicated expertise.

How Micantis Implements DRT

The Micantis platform includes a one-click DRT analysis that automates the entire process. Here is what happens under the hood.

Tikhonov Regularization with Radial Basis Functions

The Micantis DRT engine uses Tikhonov regularization with a Radial Basis Function (RBF) discretization approach. This method was established by Wan, Saccoccio, Chen, and Ciucci in their 2015 work on DRT deconvolution (Electrochimica Acta, 184, 483-499). RBF discretization has advantages over piecewise linear methods, particularly when EIS data does not cover the full frequency range of interest.

The algorithm works by:

1. Constructing kernel matrices that relate the measured real and imaginary impedance to the underlying DRT through Gaussian RBF basis functions
2. Building a regularization matrix using a first-order smoothness constraint (penalizing large derivatives in the DRT)
3. Solving a constrained quadratic program that minimizes the combined fit error across both real and imaginary impedance components, subject to the regularization penalty and a non-negativity constraint

The non-negativity constraint is physically motivated: the DRT represents resistive contributions, which cannot be negative.

Automatic Lambda Optimization

The regularization parameter (lambda) controls the trade-off between fitting the data closely and keeping the DRT smooth. Micantis uses Generalized Cross-Validation (GCV) to find the optimal lambda automatically. GCV selects the lambda that would best predict left-out data points, without requiring the user to make any judgment calls. Users can also override this and specify a fixed lambda if they prefer.

Gaussian Peak Fitting

Once the DRT is computed, Micantis can optionally decompose it into a sum of Gaussian peaks. Each peak is characterized by:

• Height — the magnitude of the impedance contribution
• Center — the characteristic time constant (in seconds)
• Width — how spread out the process is in the time-constant domain

The peak fitting uses nonlinear optimization with multiple random initializations to avoid local minima. A separation penalty encourages the optimizer to find distinct, well-separated peaks rather than duplicates. This decomposition directly maps to physical processes: a peak at a short time constant may correspond to charge transfer, while a peak at a longer time constant may indicate diffusion.

Bayesian Uncertainty Quantification

For users who need confidence bounds on their DRT, Micantis offers an optional Bayesian analysis using Hamiltonian Monte Carlo (HMC) sampling for truncated multivariate Gaussians. This follows the method of Pakman and Paninski (Journal of Computational and Graphical Statistics, 23(2), 518-542, 2014).

The Bayesian approach generates many plausible DRT solutions consistent with the data and the regularization prior, then reports the 0.5th and 99.5th percentile bounds. This tells you not just what the best-fit DRT looks like, but how certain you can be about each feature. Peaks that appear consistently across samples are robust; features that vary widely may be artifacts or poorly constrained by the data.

When to Use DRT

DRT is most valuable when you need to understand why a cell's impedance looks the way it does, not just what it is. Common use cases include:

• Degradation studies: Track how individual loss mechanisms (SEI growth, loss of active material, lithium plating) evolve over cycling. DRT can reveal which process is driving degradation before it shows up clearly in capacity data.
• Material characterization: Compare electrode formulations, electrolyte compositions, or separator treatments by examining which relaxation processes change and by how much.
• Quality control: Identify cells with anomalous internal resistance distributions that may indicate manufacturing defects, even when the total impedance looks acceptable.
• Failure analysis: When a cell fails, DRT can help identify which electrochemical process was responsible by comparing the failed cell's DRT to a healthy reference.

DRT Versus Equivalent Circuit Modeling

DRT and Equivalent Circuit Models (ECMs) are complementary, not competing approaches. An ECM requires you to propose a circuit topology in advance — you need a hypothesis about what processes are present. DRT does not require this assumption. It lets the data tell you how many processes exist and where their time constants fall.

In practice, DRT is often used first to identify the number and approximate time constants of the relevant processes. This information then guides the construction of an appropriate equivalent circuit model.

What Micantis DRT Produces

A complete DRT analysis in Micantis generates three result views:

1. DRT plot — gamma (impedance contribution in ohms) versus tau (time constant in seconds), showing the distribution of relaxation processes
2. Fitted Nyquist plot — the original EIS data overlaid with the fit from the DRT model, so you can assess fit quality
3. Frequency-domain plots — real and imaginary impedance versus frequency, with residuals showing where the fit deviates from the data

With optional peak analysis enabled, individual Gaussian peak fits are overlaid on the DRT plot. With Bayesian analysis, confidence bounds appear as shaded regions.

All results support multi-cell analysis — run DRT on several cells simultaneously and compare their relaxation time distributions directly.

References

• Wang, Z., Wang, Y., Py, B., Maradesa, A., Liu, J., Wan, T. H., Saccoccio, M., & Ciucci, F. (2025). "DRTtools: Freely Accessible Distribution of Relaxation Times Analysis for Electrochemical Impedance Spectroscopy." ACS Electrochemistry, 1(12), 2680-2689.

• Wan, T. H., Saccoccio, M., Chen, C., & Ciucci, F. (2015). "Influence of the Discretization Methods on the Distribution of Relaxation Times Deconvolution: Implementing Radial Basis Functions with DRTtools." Electrochimica Acta, 184, 483-499. Establishes the RBF discretization approach.

• Maradesa, A., Py, B., Wan, T. H., Effat, M. B., & Ciucci, F. (2023). "Selecting the Regularization Parameter in the Distribution of Relaxation Times." Journal of The Electrochemical Society, 170, 030502. Covers regularization parameter selection for ridge regression in DRT.

• Ciucci, F., & Chen, C. (2015). "Analysis of Electrochemical Impedance Spectroscopy Data Using the Distribution of Relaxation Times: A Bayesian and Hierarchical Bayesian Approach." Electrochimica Acta, 167, 439-454. Introduces Bayesian methods for DRT.

• Effat, M. B., & Ciucci, F. (2017). "Bayesian and Hierarchical Bayesian Based Regularization for Deconvolving the Distribution of Relaxation Times from Electrochemical Impedance Spectroscopy Data." Electrochimica Acta, 247, 1117-1129. Extends the Bayesian approach with credible intervals.

Try It

DRT analysis is available in the Micantis WorkBook platform as a built-in analysis workflow. Select your EIS data, click DRT, and get results. No scripting, no parameter tuning, no MATLAB license required.