# How to Model Warburg Impedance in Circuit Simulators

October 15, 2020

### Key Takeaways

• Electrochemical cells such as batteries can be described in terms of their Warburg impedance.

• This is used in an equivalent circuit model for an electrochemical cell to describe its electrical behavior, particularly in terms of its poles and zeros in the Laplace domain.

• If you can extract the Warburg impedance for your electrochemical system, you can include the cell in a circuit simulation. Batteries can be described using an equivalent circuit model with Warburg impedance

Circuit models are the basis for all electronics, but even they have their limits. The electrical behavior of LTI systems can be described in terms of the fundamental passive elements (resistors, capacitors, and inductors), even though a system may not be made up of these elements in reality. This means that circuit models are a very useful language for describing electrical behavior in many electrical systems, ranging from transmission lines to electrochemical systems.

In the realm of electrochemistry, the electrochemical behavior of the system can be summarized in terms of its Warburg impedance. This distributed element model provides some important conceptual insight into how charge is accumulated, stored, and released from a battery or electrochemical cell. Because this model formulates impedance in terms of a distributed ladder network, it can be easily included in a SPICE simulation alongside other circuits. Here’s how Warburg impedance is described and how you can use it in your circuit simulations. ## Warburg Impedance Equivalent Circuit

Developing the Warburg impedance of an electrochemical cell requires making a few basic assumptions that turn out to be extremely accurate. The Warburg impedance model for electrochemical cells can be derived from the distributed element model used for transmission lines with RLCG elements. When we consider the construction of an electrochemical cell and migration of charge in the cell, we can note the following observations:

1. Slow dynamics. Charge transfer is entirely diffusion-limited, which is equivalent to operating with very slow circuit dynamics. Therefore, Z → 0 for any inductive elements, and the L elements in the distributed element transmission line model can be ignored.

2. Resistive transport. Charge transport within a cell is driven by an  electrochemical interface reaction, which has some resistance. Therefore, we keep the resistive element in the distributed element transmission line model.

3. Capacitive charging. The electric double layer in an electrochemical cell acts like an imperfect conductor (R > 0) separated by a nearly perfect insulator (G = 0), which has capacitive impedance according to Maxwell’s equations. Therefore, we keep the distributed capacitance in the standard transmission line model (C > 0).

Ignoring L and G in the distributed element transmission line model while keeping R and C gives the following circuit model describing an electrochemical cell. 5-element model for an electrochemical system

In this RC ladder circuit model, we can see that this has the same form as a transmission line’s distributed element circuit with L = G = 0 in the standard transmission line impedance equation. We can then use this condition to derive the Warburg impedance: Warburg impedance equation in terms of a distributed element circuit model.

Note that R and C may be different depending on the state of charge and the dominant electrochemical reaction in the cell. Therefore, it’s common to have different Warburg impedance values representing different states of charge in the system. For this reason, we can define a Warburg coefficient for a particular electrochemical cell, which may be a function of total stored charge, frequency, and whether the cell is charging or discharging: Warburg impedance in terms of a frequency-dependent Warburg coefficient.

We now have a phenomenological model that can be used to describe the dynamics of an electrochemical system. Note that the Warburg coefficient is generally a complex function. The Warburg coefficient can be determined from an electrochemical impedance spectroscopy data.

## Transient Behavior

For battery designers, perhaps the most important point is the behavior in the Laplace domain, which tells you how fast the system can discharge when connected to a short circuit. In other words, it’s important to know the transient behavior of the system. Since the above system is being modeled as a distributed element model, we can define a transfer function for this system as a function of the load impedance. From this, the transient behavior can be determined in terms of the system’s poles and zeros in the Laplace domain, or directly in the time domain.

The maximum discharge rate occurs when the system is short circuited. In this case, we can calculate the poles and zeros of the system in terms of the input impedance. It turns out that there is a spectrum of values that define the charge and discharge rate in the system: Poles and zeros of the input Warburg impedance.

These values can be used to analytically describe the transient behavior of the system using exponential functions. In general, since Warburg systems can be quite large, it’s often easier to run a circuit simulation for an electrochemical system. To run a circuit simulation for a system with an electrochemical element, you simply need to create a Warburg model for the system in terms of its R and C values. You can use the following procedure:

1. Create the distributed RC ladder circuit shown above in your circuit design program. Select a large number of sections; usually N = 50-100 is fine.

2. Connect the input to a voltage source with the electrochemical systems output voltage, and connect the output to the rest of the system you’re designing.

3. Start running simulations. Normally a transient simulation or pole-zero analysis is used for these systems.

The front-end design features from Cadence can be used to build models for a range of systems, including electrochemical systems with a defined Warburg impedance. You can use the modeling applications in the PSpice Simulator to extract important parameters in an electrochemical system and use these in a range of circuit simulations. The set of analysis tools in PSpice can also help you optimize your design’s interactions with an electrochemical system.

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