After ring 181, the potentials are determined by a box of resistors which sit outside the TPC. This is well documented, but at the time of this writing is not complete. This was particularly relevant during Run 9 when an electrical short developed inside the TPC between rings 181 and 182 of the outer field cage on the west end (OFCW). Shown here is a plot of the resistors:
Note that the ammeter is essentially a short to ground, while the voltmeters are documented in the Keithley 2001 Manual to express over 10 GOhm of resistance (essentially infinite resistance). The latter only occurs when the input voltage is below 20 V. The voltages at rings 181 and 182 are above 20 V and below 200 V (though actually at negative voltage), so their voltage is scaled by the shown 1.11 and 10.0 MOhm resistors to be stepped down by a factor of x10. The readings are then multiplied by x10 before being recorded in the Slow Controls database.
After Run 9, this box was disconnected and resistances were measured for the OFCW portion. Because the resistors were not separated from each other, equivalent resistances were actually measured. In the below math, R182eq refers to the resistance measured across resistor R_182, while Rfull is the resistance measured between the input to the box from ring 181 to the output for the ammeter. R111 is the combined 10 + 1.11 MOhm pair.
double R111 = 10.0 + 1./(1./1.11 + 1./10000.) 11.11 double R182eq = 0.533 // measured double Rfull = 2.31 // measured double pa = R111 - Rfull double pb = R111*(R111 - 2*Rfull) double pc = (R182eq - Rfull)*R111*R111 double R181 = (-pb + sqrt(pb*pb - 4*pa*pc))/(2*pa) 2.3614 double R182 = 1./(1./R182eq - 1./R111 - 1./(R181+R111)) 0.5841 double R182b = 1./(1./R182 + 1./R111) double Vcm = 27960. double Rtot = 364.44 // full chain double Inorm = Vcm/Rtot 76.720 double V181_norm = Rfull * Inorm 177.22 double V182_norm = V181_norm * R182b/(R181 + R182b) 33.724 double Rshorted = 1./(1./R111 + 1./R182 + 1./R111) 0.5286 double Rmiss = Rfull - Rshorted 1.7814 double Ishorted = Vcm/(Rtot - Rmiss) 77.097 double V181_short = Rshorted * Ishorted 40.750 double Iexcess = Ishorted - Inorm 0.377
Note that many of these numbers would be different for the inner field cages.
External resistors to make up for missing resistance can also be added to the chain here.