Resistor box at the end of the Field Cage chain

 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.