The polarization parameters determine depolarization through their product, (R_P C_P), and charging capacity through the resulting time at the selected current. In a first-order RC polarization model, the depolarization time is

[ t=R_P C_P\ln\left(1+\frac{T M_k(k)}{I_{\mathrm{Ch}}R_P C_P}\right), ]

and the required depolarization capacity is

[ Q_P=I_{\mathrm{Ch}}t. ]

Here, (R_P C_P) is the polarization time constant: larger values generally require longer relaxation or depolarizing phases.

Core takeaway: (R_P) and (C_P) do not independently set the time in this simplified model; their product, (\tau_P=R_P C_P), does. Once (\tau_P) and the test conditions are known, the required time follows from the logarithmic relation above, while (Q_P) is the charging current multiplied by that time.

How (R_P) and (C_P) Control Depolarization

The polarization time constant

The product

[ \tau_P=R_P C_P ]

is the characteristic time constant of the polarization branch.

A larger (R_P) indicates stronger voltage polarization for a given current, while a larger (C_P) represents greater stored polarization charge in the equivalent circuit. Increasing either parameter increases (\tau_P) when the other is held constant.

Depolarization-time relationship

Using (\tau_P), the time equation becomes

[ t=\tau_P\ln\left(1+\frac{T M_k(k)}{I_{\mathrm{Ch}}\tau_P}\right). ]

The variables (T) and (M_k(k)) represent the relevant discharge duration and cumulative discharge-current coefficient. The selected charging current is (I_{\mathrm{Ch}}).

For fixed test conditions, the depolarization time is a monotonically increasing function of (\tau_P). Therefore, cells with slower polarization dynamics require longer depolarization intervals.

Why the relationship is logarithmic

The logarithm reflects the exponential charging and discharging behavior of a first-order RC element. The polarization voltage changes rapidly at first, then approaches its equilibrium value more slowly.

Consequently, doubling (R_P C_P) does not necessarily double the calculated time exactly, but it does increase the time required to reach the same depolarization condition.

How Depolarization Capacity Is Calculated

Capacity required at a selected current

Once (t) has been calculated, the required depolarization capacity is

[ Q_P=I_{\mathrm{Ch}}t. ]

Substituting the time expression gives

[ Q_P

I_{\mathrm{Ch}}R_P C_P \ln\left(1+\frac{T M_k(k)} {I_{\mathrm{Ch}}R_P C_P}\right). ]

This is the additional charge that the tester must be able to deliver during the depolarizing phase at the selected current.

Current changes the time-capacity balance

Increasing (I_{\mathrm{Ch}}) generally reduces the time needed to apply the depolarizing charge, because the required current is delivered more quickly.

However, the capacity requirement is not determined by current alone. It depends on the interaction among (I_{\mathrm{Ch}}), (R_P C_P), and the prior discharge history represented by (T M_k(k)).

For protocol design, the tester must therefore be sized for both:

• Sufficient current, to complete depolarization within the desired time.
• Sufficient capacity, to deliver (Q_P) without reaching a hardware or software limit.

Applying the Two-Hour Relaxation Assumption

Establishing a practical upper bound

A common laboratory assumption is that full equilibrium is reached within approximately two hours. For a first-order model, this corresponds to limiting the time constant to

[ R_P C_P\leq \frac{2}{3}\ \text{h}, ]

because three time constants equal approximately two hours.

Under this assumption, the maximum depolarization time is

[ t\leq \frac{2}{3} \ln\left( 1+\frac{3T M_k(k)}{2I_{\mathrm{Ch}}} \right) \ \text{h}. ]

The corresponding capacity limit is

[ Q_P\leq \frac{2}{3}I_{\mathrm{Ch}} \ln\left( 1+\frac{3T M_k(k)}{2I_{\mathrm{Ch}}} \right) \ \text{Ah}. ]

These expressions provide conservative planning limits for automated test sequences when the cell is expected to reach equilibrium within two hours.

Interpreting the boundary condition

The two-hour condition is a modeling and test-protocol assumption, not a universal electrochemical law. A different definition of “fully rested”—for example, a voltage-recovery-rate threshold—can produce a different practical rest time.

The boundary should therefore be checked against measured relaxation behavior rather than applied blindly across chemistries, cell formats, temperatures, and aging states.

Measuring (R_P) and (C_P) in Cell Testing

Identifying the voltage response

A practical resting-identification method abruptly stops constant-current charging or discharging and records the voltage response.

The instantaneous voltage step helps identify internal resistance-related behavior, while the subsequent recovery curve provides information about the polarization dynamics and the effective (R_P) and (C_P) parameters.

Defining the end of rest

One practical criterion is to consider the cell fully rested when the voltage recovery speed falls below 10 mV per 180 seconds.

This criterion converts an abstract model parameter into an operational test decision: the tester can end the rest period when the measured recovery rate is sufficiently small.

Why parameter identification matters

The parameters can change with:

• State of charge
• Temperature
• Cell aging
• Discharge rate
• Electrode materials and fabrication processes

Using a single fixed (R_P) and (C_P) value for every condition may therefore understate the required rest time in some portions of the operating window.

Understanding the Trade-offs

Shorter tests versus more complete depolarization

Reducing the relaxation interval improves test throughput, but residual polarization can distort the measured voltage, capacity, energy efficiency, and apparent rate capability.

If the next cycle begins before the cell has sufficiently depolarized, the protocol may measure a mixture of true electrochemical behavior and memory from the previous cycle.

Higher current versus stress and safety

A higher (I_{\mathrm{Ch}}) can shorten the depolarization phase, but high current also increases voltage polarization, heat generation, and the risk of unwanted side reactions.

During fast-charge development, current should therefore be limited using voltage, temperature, SOC, and polarization feedback rather than selected solely to minimize test time.

First-order models versus real cells

The (R_P C_P) model is useful for protocol planning, but real lithium-ion cells often exhibit multiple polarization processes with different time constants.

A single time constant may not accurately represent long-tail relaxation, diffusion limitations, hysteresis, or behavior near high SOC. For high-accuracy evaluation, the model should be validated against measured recovery curves.

How to Apply This to Your Test Protocol

The practical workflow is to identify the polarization response, calculate the time and capacity, and then verify the result experimentally.

• If your primary focus is minimizing test duration: Estimate (R_P C_P) under the relevant SOC and temperature conditions, then select (I_{\mathrm{Ch}}) to meet the required depolarization time without exceeding voltage, thermal, or safety limits.
• If your primary focus is accurate cell characterization: Use the measured relaxation endpoint and recovery curve, rather than a fixed two-hour assumption, to define the rest interval.
• If your primary focus is tester sizing: Calculate (Q_P=I_{\mathrm{Ch}}t) for the worst-case polarization condition and provide sufficient current, energy, and compliance-voltage margin.
• If your primary focus is fast-charging protocol development: Track (R_P), (C_P), and polarization voltage across SOC, temperature, and aging so that charging current can be reduced when polarization becomes excessive.
• If your primary focus is comparing materials or cell designs: Compare the fitted time constant (R_P C_P) and the measured recovery behavior under identical test conditions.

Treating (R_P C_P) as a condition-dependent polarization time constant lets you convert cell behavior directly into defensible relaxation times and charging-capacity requirements.

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