Spice Modeling of Magnet
C H Gough
Introduction
This Note is one of a series, documenting the development of the five Storage Ring Kicker systems for the SLS. The magnet impedances were measured up to 100MHz with a RF network analyser. In this Note, Pspice is used to find a reasonable equivalent circuit for fast transients, i.e frequency range from 100kHz up to 100MHz.
Circuit Approximations
Fig.1 Mechanical dimensions of magnet cross-section.
Terminal Impedances
The choice was made to view the conductors as ground-referenced, not bipolar floating above a ground plane. In this way, the input and output should have no coupling.
For the magnet connections, the first element is a unipolar line above a ground plane:
With h=40mm and d=10mm, Z=97.7, L=650nH/m, C=17pF/m and =3.3ns/m. The line length is 100mm go and 100mm return. For 100mm, this gives L=65nH and C=1.7pF.
By using the first section like a trombone between two test fixtures, the inductance and delay time could be found for different line lengths x:
(nH/m)
(ns/m)
So in spite of calibration with the test fixture, 45nH and -25ps offset remained. The inductance of L=43.3nH and the delay of
=451ps for 100mm do not agree well with the previous calculated value.
The second element is a strip line, unfortunately giving a coupling between the magnet input and output. The line length is 70mm.
The measured capacitance between lines is 10pF. The calculated inductance is around 1nH and even with unity magnetic coupling, the effect in simulations appears small, so this inductance is ignored.
Magnet Impedances
The magnet itself is not yet known. For a unipolar line between ground planes,
With h=104mm and d30mm, Z=89, L=297nH/m and C=37.5pF/m. This gives 529nH and 67pF for a length of 1.78m.
A model magnet was built using the conductor dimensions but with any dielectric material. Several different methods were tried to find the L and C values for the line.
The impedance and progation velocity can give the L and C values. With the measured impedance of 932 and an air velocity of 3e8 m/s,
This gives 55212nH and 63.81.4pF for a length of 1.78m.
Direct measurement of impedances with the network analyser gave L=58510nH for <6MHz with short circuit termination, and C=641pF for <6MHz with open circuit termination.
Measurement with an LCR meter gave 588nH at 10kHz (test fixture position on the magnet terminals is important !), and 60.0pF at 10kHz.
The line impedance Zc could be found by adjusting a carbon variable resistor at the line end for zero phase on S11 from low frequency up to 10MHz. The electrical time delay for zero phase shift on S12 from DC-10MHz. With these two values,
The measured impedance was 932. The total conductor length (including the terminations) is 1780mm. The time delay
measurement was re-checked. The through calibration was repeated using the terminal fixture instead of a simple SMA through piece; the calibration changed by 500ps, but the measurement by only 240ps. The measured time delay was 6.66.1ns. This gave L=62013nH and C=71.7.15pF.
With the metallised ceramic chamber in place, the magnetic flux is greatly reduced and the capacitance is greatly increased. As a guess
With a=6mm and b=40mm, Z=18, L=60nH/m and C=185pF/m. The line length is 600mm.
Open
G X
1k 1.8n 58.4p
10k 12n 58.1p
100k 100n 57.9p
100k 125u 62p
300k - 61p
1M 33m 64p
3M - 63p
10M 580m 68p
Short
R X
1k 7m 640n
10k 9m 580n
100k 16m 558n
100k 56m 581n
300k 44m 580n
1M 54m 581n
3M 80m 580n
10M 220m 620n
Zc 88.7
Fig.2 Magnet model.
Open
G X
1k 400n 336p
10k 1.3u 279p
100k 3.3u 269p
100k - 313p
300k - 268p
1M 77m 295p
3M 16m 650p
10M 3m 2.1uH
Short
R X
1k 2.2m 1.9u
10k 3.8m 1.89u
100k 10m 1.86u
100k 7m 1.89u
300k 72m 1.91u
1M 46m 1.92u
3M 2.35 2.08u
10M 24.5 2.42u
Zc 74
Fig.3 Magnet without chamber , with conductor strips separating the ferrites
Open
G X
1k 400n 335p
10k 1.3u 278p
100k 3.2u 267p
100k - 270p
300k 67m 267p
1M 83m 295p
3M 14m 697p
10M 3m 2.09uH
Short
R X
1k 2.3m 2.0u
10k 3.6m 1.89u
100k 8.5m 1.88u
100k 49m 1.88u
300k 34m 1.93u
1M 61m 1.94u
3M 2.44 2.08u
10M 28.2 2.51u
Zc 74
Fig.5 Magnet without chamber.
Open
G X
1k 390n 454p
10k 1.23u 399p
100k 3.2u 388p
100k 3.3m 401p
300k 210m 395p
1M 62m 455p
3M 13.5m 4.52nF
10M 2.9m 2.00uH
Short
R X
1k 3/16m 1.95u
10k 5.3m 1.91u
100k 98m 1.88u
100k 121m 1.91u
300k 998m 1.74u
1M 3.89 1.24u
3M 8.25 740n
10M 15 601n
Zc no definite value
Fig.6 Magnet with chamber.
Simulation Circuit
To maintain a constant low value of capacitance at low frequencies, there should be no series resistance with the shunt capacitance.
Real and Imaginary Components, Far End Terminated in 50
Real and Imaginary Components, Far End Open
Real and Imaginary Components, Far End Short
Measured Capacitance Conductor-Ground
Parallel Cp/Rp was used (much less variation than Cs/Rs).
100Hz 1kHz 10kHz 20kHz 100kHz
567pF 38M 462.6pF 2.66M 408.0pF 808k 402.7pF 594k 396.3pF 273k
References
"SR Kicker Inductance Measurements", C H Gough, 1 October 2000 Appendix
To a rough approximation, the magnet behaves like a transmission line. The inductance and capacitance is related to the phase constant by:
The first quarter-wave resonance is at 39MHz. Taking the transmission line length to be equal to the magnetic length of 0.6m,
The measured inductance is 3.2H/m, giving C=3.3mF/m, a silly value.
Noting that the centre of rotation is somewhere around 50,
gives C=1.3nF/m, again rather silly.
As simple LC resonator is defined by
With L=1.9H, C=8.8pF, a rather low value.
