The field of the Hall D solenoid was calculated using the Poisson/Superfish
package (see the details here). Two ways were
used to calculate the inductance:
Full field energy
Poisson calculates the full energy E of the magnetic field. Using the
relation E=L*I²/2, where I is current, one can calculate
the inductance L. The standard
configuration was used.
The results are summarized in a table (the units are A and H):
Magnetic flux through the coils
Using the Poisson-produced field maps one can calculate the full
magnetic flux ∫BdS through any turn of the conductor and add them together
for any part of the coil obtaining the cumulative flux Φp. The full inductance
of this
part can be calculated as Lp=dΦp/dI, where I is the current through
the solenoid. A change of the current dI/dt would generate a voltage Lp·dI/dt.
The standard
configuration was used. In order to evaluate the dΦpdI≈ΔΦp/ΔI
the calculations were done at 1% different currents. The field maps had 1cm steps in Z and R.
The cubic spline was used to evaluate the field in the given point.
The conductor turns were assumed to be thin circular loops with the radius of the
center of the conductor.
The results are summarized in a table for all the readout taps (the units are A and H):
The same calculations for the pickup coils are presented in the next table:
Measurements
The inductance was calculated using the measured voltage on the taps and the pickup coils.
The voltage is generated during the dumps of the current on the dump resistor, and during
the regular ramping up or down the current by the power supply.
Magnet Dump
The dumps through the dump resistor RD provide the fastest change of the current
dI/dt=-RD/L·I. At 1300A dI/dt≈-3 A/s.
I used the dump from 1200A which happened at 2014/12/08:
Fast dump of 2014/12/08. The current is measured on the shunt. The time is in seconds.
A pdf picture is available.
The inductance was measured for a given current in the following way. The time dependence of the
current in the interval ±10A was fit with an exponential I·exp(-t/τ).
The time t was measured with respect to the center of the effective time interval.
The voltage on the coil was calculated as the sum of the voltages on all the taps, and
also was fit within the same interval with an exponential V·exp(-t/τ1).
The inductance was calculated as L=V/(I/τ). Also the effective dump
resistance (which includes the cables and the leads) was calculated as R=V/I.
The columns in the table show the current (A), the results for the inductance with the
error from the fit (H), and the effective dump resistance in Ohms.
The calculated value of the dump resistor grows from about 0.061Ω at the beginning
of the dump to about 0.069Ω at the end of the dump. It may be explained by
the heating of the dump resistor (0.061Ω nominal, made of stainless steel)
by about 140°C as the result of the dump. I do not know what type of stainless
steel the resistor is made of and used the thermal coefficient of 0.00094 I found
for the steel 18-8 (74% Fe, 18%Cr, 8% Ni).
Current ramping up or down
The power supply provides different ramping rates: from 0.2A/s at low currents to 0.06A/s at high currents.
The periods of stable voltages on the coils were selected, and the current dependence on time
was fit with a linear function, providing the ramp rate dI/dt.
It occurs that during such periods the next-order terms are negligible.
The voltage on the coils during these periods was averaged off and the inductance calculated as
L=V/(dI/dt).
In the following table different periods are presented. The dates and the ramp rates are specified in
separate lines. The first column shows the current, the second - the full inductance of the magnet,
while other 16 columns - the full inductances for the 16 taps.
A note: In order to measure the voltages correctly, especially at low ramp rates, one should
load a recent measurement of the offsets using the Yi's analyzer
The last period shows the results of a deliberately slow ramp up at low currents.
A similar table shows the results for the pickup coils. The columns contain the current
and the voltages on the 8 pickup coils for dI/dt=1A/s.
Comparison of the results
Comparison of the full inductance calculations and the measurements.
A pdf picture is available.
Results of the comparison:
The inductance measures at different ramp rates are consistent with each other.
The inductance measured at the dumps is very close to the one measured at the ramps,
however at I<600A the dump-derived values are systematically higher
by about a 1%. The dump resistor heating should not affect the result. The source
of this deviation is at the moment unclear.
The calculations using the flux integration describe well the measurements at the pickup coils,
but are systematically lower by 1-2% for the subcoils. The calculation uses a simplified geometry
of a uniform current through the whole coil, underestimating the field close
to a thin wire.
The calculations using the full energy are consistent with the flux-integration
ones at low currents, but deviate considerably to higher values at high currents.
The results for the individual taps and the pickup coils are shown below:
Comparison of the partial inductance calculations and the measurements.
A pdf picture is available.
Results:
The largest difference of about 5% between the calculations and the measurements
is observed for the pickup coils 2u and 1d. The well matched sum of all the pickup coils represent
to some extent the field integral along the magnet.
The subcoils match well, apart from the tap vtt11 (subcoil 1E). The measurements for the vtt11
are about 15% higher than the calculations. Scaling the values for the adjacent taps vtt10 and vtt12 to the ratio
of the turns (216/80 and 216/448) produce values closer to the calculated value for the vtt11
rather than to the measured value. At the moment there is no explanation for the discrepancy.