# Voltage Response of Equivalent 1-RC Model for a Lithium Ion Cell

Context: While studying Equivalent Circuit Modelling of Li-Ion cells, I came across a rough approach to approximate the model parameters. My source is this (Refer Book Page 39 in sample) The model is as follows:

The discussion under consideration is that a Li-Ion cell is supplied with a discharge pulse and voltage response is recorded as follows:

The intent is to relate the parameters to different parts of this voltage response. I understand well that at t=20min, the sudden rise in voltage shall be due to $$\R_{0}\$$ alone but I am not able to understand the $$\\Delta v_{\infty}=(R_{0}+R_{1})\Delta i\$$. There is an equation (2.8) in discrete form, again from the source, which goes as: $$\v[k]=OCV(z[k])-R_{1}i_{R_{1}}[k]-R_{0}i[k]\$$ which calculates cell voltage at a given time stamp. The source says:

The overall steady-state voltage change can be found from Eq. (2.8) to be $$\\Delta v_{\infty}=(R_{0}+R_{1})\Delta i\$$, again with signs computed so that $$\R_{0}\$$ and $$\R_{1}\$$ are both positive, knowing that the capacitor voltage will converge to zero in steady state.

I think this equation works under the assumption that at t=20min, the capacitor was fully charged and hence acted as an open circuit basically. But the source doesn't mention it and ideally also, during pulse response an experimentalist won't wait for the time around for the capacitor to get charged. My clear question is that while the discharge current was present and while the voltage through capacitor was gradually increasing against OCV but did not enter a steady state, does the $$\\Delta v_{\infty}\$$ equation still hold? I tried, but I could not analytically derive it as well. I hope I have put this across well.

• In the steady state, the capacitor current becomes zero, its voltage becomes whatever the voltage across R1 is due to battery current R1 x I_steady Commented Aug 20 at 5:51