The resistor chains are different for the outer field cage and the inner field cage. The outer field cage has a "ground shield" attached to ring 182 and the shield is lined up with the "ground wires" in the anode sector. This prevents the shield from being located at the center of the ring and requires us to set the ring at the correct voltage for the ground shield rather than its own natural setting. Accomplishing this requires tuning both resistors.

The inner field cage does not have a "ground shield" and resistor 181 is just one more step in the resistor chain. It keeps its normal value of 2 M ohms while resistor 182 is trimmed to bring the field to ground potential. Ring 182 is not necessarily set to an integral multiple of ring-to-ring voltage steps above true ground because the gating grid, which determines the overall drift field, is set independently and its value is determined by the transparency of the grid and other external factors. (See the next page).

So in order to easily adjust resistors 181 and 182, they have been removed
from the TPC and installed in an external rack.

The resistor chain terminates in a scanning current meter so we can
monitor the currents running down each chain. This is a useful diagnostic
for helping to find short circuits in the voltage divider system.
A scanning voltmeter also monitors the voltage on ring 181 and 182 to ensure
that they are set properly.

To determine the setting on the variable resistors, let:

Z_{gg} - Z_{cm}
= the distance between the gating grid and the central membrane

V_{gg} - V_{cm }
= voltage on the gating grid minus the voltage on the central membrane

The drift field is then simply:

(1) E_{drift} = ( V_{gg
}- V_{cm} ) / ( Z_{gg} - Z_{cm} )

The field cage is built of rings with equal spacing between the rings.
This is true for all ring-to-ring gaps except for the gap between
the central membrane and the first ring. It is slightly wider.
In order to keep a uniform field gradient at the central membrane, the
first resistor (R_{0}) must be slightly larger to represent the
wider gap.

(2) R_{0} = Z_{01}
* ( R_{1 }/ Z_{12 })

As long as R_{0 } is chosen this way, the voltage and resistances
are well defined functions of Z in the region between the gating grid and
the central membrane.

(3) R(Z) = ( Z - Z_{cm}
) * ( R_{1 }/ Z_{12 })

(4) V(Z) = E_{drift }*
( Z - Z_{cm} ) + V_{cm}

Extending these equations to zero voltage will tell us the total resistance in the chain. Thus,

(5) R_{T} = ( V_{cm}
/ E_{drift } ) * ( R_{1 }/ Z_{12 })

This is enough information to calculate the settings on the variable
resistors R_{181} and R_{182}. There are several
known quantities:

E_{drift } = 146.5 V/cm
for P10 gas, chosen to be over the peak in the velocity curve
(eg. see previous page)

V_{gg } = -125 V
chosen to make the grid 100% transparent, see next page

R_{1 } = 2 M
Ohms

Z_{gg} = 208.7 cm

Z_{cm} = 0.0
cm

Z_{01} = 1.225 cm

Z_{12 }=
1.15 cm

Z_{gs} = 209.3 cm

From (1) V_{cm}
= -30,700 Volts

From (2) R_{0}
= 2.130 M Ohms

From (5) R_{T}
= 364.440 M Ohms

Since there are 180 identical resistors in the chain plus R_{0},
R_{181}, and R_{182} this means that

R_{181} + R_{182} = R_{T}
- R_{0} - 180 * 2 M Ohms = 2.310
M Ohms

In the special case of the inner field cage where there is no ground shield

R_{181 } = 2
M Ohms

R_{182} = 310
K Ohms

R_{T} is the same for the outer field cage but it is split
differently between R_{181} and R_{182} because
the ground shield is partway between the center of rings 181 and 182 and
we want to bias ring 182 so that the ground shield is at the correct voltage
rather than the ring. We can easily calculate these values using
equations (3) and (4).

R_{gs} = ( Z_{gs} - Z_{cm} ) * (
R_{1 }/ Z_{12 })
= 364.000 M Ohms

V_{gs} = E_{drift }* ( Z_{gs}
- Z_{cm} ) + V_{cm }
= -37.5 Volts

R_{gs} is the total resistance between the central membrane
and the ground shield and to make better sense of this number we should
subract off the fixed resistor values. Thus in the special case of
the outer field cage

R_{181 } = R_{gs}
- R_{0} - 180 * 2 M Ohms
=
1.870 M Ohms

R_{182} =
R_{T} - R_{0} - 180 * 2 M
Ohms
- R_{181}
= 440 K Ohms

Page created by Jim Thomas, send comments to jhthomas@lbl.gov.