Fix responsivity in iv

[satoru99]

There are two issues. 1. Calculation of R_L The load resistance (R_L) should be shunt resistance (R_sh) + parasitic resistance (R_par) as written in comment. It is calculated as follows https://github.com/simonsobs/sodetlib/blob/1b384e513d3d4a3d4bfd8282f378b29e065c591b/sodetlib/operations/iv.py#L395-L397 However, from the equation of voltage: R_sh * I_bias = (R_tes + R_sh + R_par) * I_tes. the R evaluated here should be R_tes + R_par, and, the R_L only contains R_par and not R_sh. (Also, R_n might include contribution from R_par.) To give the expected R_L, we should add + R_sh.

2. Calculation of dV/dI The responsivity is calculated as follows. Note that the rL comes from R_L above and thus underestimated by R_sh. https://github.com/simonsobs/sodetlib/blob/1b384e513d3d4a3d4bfd8282f378b29e065c591b/sodetlib/operations/iv.py#L299 Here, dv is calculated from v_tes. But from the theoretical model, the correct value seems to be the derivative of v_bias. Since, v_tes = v_bias * R / (R + R_L), this difference affects at small R.

These two issues affect the responsivity calculation. It's necessary to be fixed.

[satoru99]

Oh, there is another v_bias. Here, I mean v_bias = i_bias * R_sh and not following i_bias * bias_line_resistance. https://github.com/simonsobs/sodetlib/blob/1b384e513d3d4a3d4bfd8282f378b29e065c591b/sodetlib/operations/iv.py#L342

[msilvafe]

Thanks @satoru99, yes we (I) seem to have dropped R_par completely from all of these equations and we did not use the thevenin equivalent voltage (i_bias*R_sh) as you pointed out here. I follow your math/logic (from Irwin-Hilton) and agree that we should fix this. These seem like the are pretty simple changes to make, and I would love to see things tightening up at small R from this. Are you planning to show some example re-analyzed IVs and submit a PR with these fixes or work with @RemingtonGerras to do this?

[RemingtonGerras]

Confluence page of notes + summary of the PR