The role of eddy currents during a quench in quadrupole superconducting magnets
M. Sorbi, G. Bellomo
Ver. 1.0,
Milan 23 September 2013
Introduction
During the fast discharge of the LARP magnets HQ and LQ, it was observed that the experimental
current decay is faster than any simulated calculation. In the simulations QuenchPro, Roxie and
QLASA were used, with the nominal value of the magnet inductance (see Figs. 1-3)
Fig. 1 Measured (blue line) and QuenchPro calculated (red line) current decay during a quench of
LQ at 12.5 kA [1].
Fig. 2 Measured (blue line) and Roxie calculated (violet line) current decay during a quench of LQ
at 11.8 kA [2].
1
Fig. 3 Measured (blue line) and QLASA calculated (red line) current decay during a quench of HQ
at 13.6 kA. The green line is the calculated current decay with an artificially reduced value of
inductance of 30% [2].
The experimental faster current decay is very evident at the beginning of the discharge, i.e. before
the activation of the quench heater. At this stage the total resistance of the circuit is the value of the
dumping resistance, so an erroneous calculation of the current decay cannot be ascribed to an
erroneous evaluation of the total resistance (dump resistance + coils resistance) (see Fig. 4).
On the other hand the nominal inductance of the magnets has been measured independently
previously, and the measured value are also in agreement with the calculated data (Fig. 5).
In conclusion the simulation of the current decay during a quench cannot ever be fitted well using
the nominal value of the magnet inductance, even considering the about 10% reduced value of the
differential inductance at high current respect to the differential inductance at low current. In order
to reproduce the experimental current decay with quench code the only “practical” solution is to
reduce artificially the nominal inductance of the magnet of about 30%.
Fig. 4 Zoom of the first 10 ms of Fig. 1: measured (blue line) and QuenchPro calculated (red line)
current decay during a quench of LQ at 12.5 kA [1].
2
Fig. 5 Measured (red cross) and Roxie calculated (green line) inductance of HQ vs. the magnet
current. The blue rhombus represents the apparent inductance extrapolated by the current rate decay
during a quench fast discharge [2].
Eddy current calculation
A reasonable explanation of the faster decay of the magnet current can be found if we take in
account eddy currents in the magnet. In fact the rapid discharge of the current (of the order of 100
kA/s, i.e. up to 100 T/s at the beginning of the current decay) surely induces large currents in the
metallic components of the magnet: these eddy currents subtract energy from the coils, which
consequently reduces in current.
To estimate the eddy currents we can consider the following metallic parts coupled to the magnet:
1. The beam tube
2. The collars
3. The yoke
4. The external cylinder
5. The other components (keys, rods, etc.)
The yoke usually is realized with laminated iron for electrical transformer applications. Usually it is
supplied with superficial insulation in order to prevent eddy current from one lamination to the
other. So, at first order, their contribution for eddy current can be much lower respect to the other
magnet components.
In the following a 2D analysis is presented, so most of the extensive quantities (as resistance,
inductance, energies, etc) are intended per length unit. The role of the beam tube and collars has
been considered, first with an analytical approach, then with electromagnetic F.E. analysis.
Analytical analysis for beam tube and collars.
For eddy current analysis the beam tube and collars can be approximated to a thin, infinitely long
cylinder. From [3] an analytical solution can be found, which gives the following results:
- The thin cylinder inside or outside a varying quadrupolar field will be interested by surface
cos-2theta eddy currents, both during the transient phase and in the stationary condition.
- The transient phase has an exponential behavior, with a characteristic time :
 a  da
(1)
 0
4
where a is the cylinder average radius, da is the thickness and  the electric resistivity.
3
This exponential behavior of the transient phase of the superficial currents in the cylinder allows to
describe the coupling of the cylinder with the coils of the quadrupole as the coupling between a
primary circuit (the circuit of the coil, with its current I1, inductance L1 and resistance R1) with a
secondary circuit (the circuit of cylinder with its current I2, inductance L2 and resistance R2).
The current I2 of the secondary circuit represents the total current in cylinder, i.e.:


4
4


I 2  2  J s a  d 2 J s 0 a  cos2   d 2 J s 0 a


4
(2)
4
The associated inductance L2 to the cylinder can be calculated from an energetic equivalence
between the magnetic energy of the cylinder and the magnetic energy of the secondary circuit:
2
1
1
2
(3)
Az  J s ad  L2 I 2

20
2
which gives:

0
(4)
32
Similarly, the resistance R2 of the secondary can be calculated from an equivalence between the
power dissipated by the surface current Js in the cylinder and the power dissipated in the circuit:
L2 
2

 da  J
2
s
ad R2 I 2
2
(5)
0
which gives:
R2 


(6)
8 a  da
Of course the time constant of the secondary circuit τ2=L2/R2 results equal to (1).
The coupling inductance M between the primary and the secondary circuit has been calculated from
direct integration [3], by supposing for the primary coil (the magnet coils) a perfect cos-2theta
distribution. For a thin cylinder inside the coils (like the beam tube):
c 
c 
ln  2 
2a 2 ln  2 
c

 c1  N
(7)
M  0 a 2 2 1 2 N1  L2 
1
2
2
16
c2  c1
c2  c1
where c1 is the aperture radius of the quadrupole (inner radius of the coils), c2 is the outer radius of
the coils and N1 is the total turn number of the quadrupole (turn number of each quadrant times 4).
For a thin cylinder outside to the quadrupole coils (as in the case of collars), the expression of M is:
2
2
2
2
c c
c c

M 
0 1 2 2 N 1  L2  1 2 2 N 1
(8)
64
a
2a
To calculate the eddy currents in the beam tube is then sufficient to write the Kirchoff equations
relative to the primary and secondary circuit:
 L1 I1  R1 I1  MI2  0
(9)
 

 L2 I 2  R2 I 2  MI1  0
As first approximation we can assume that the resistance R1 and R2 are constant in time (for
example R1 is equal to the dumping resistance of the coils).
The analytical solution of (9) can be easily found, assuming the boundary condition:
 I1 t  0  I 0
(10)

 I 2 t  0  0
which gives:
4
  t '    t ' 
e 1  1 e 2 


2
I 1 t   I 0 



1 1



2


t
  '   t '
e 1  1 e 2
2
M
I 2 t   I 0

R2 
1   2


(11)







(12)
where:
1 1
41  k 2 


      
1  2
 1 2
1
1  2 


2
1'
21  k 
1
2
1
1 1
41  k 2 
      
1  2
 1 2
1
1  2 


2'
21  k 2 
1
(13)
2
1

 1  1 
2 
 '
 1 '  1

  
 2  1  2  2 '
 2' 
L
1  1
R1
L
2  2
R2
M
k
L1  L2
If 2 < 1 (as in our case), a good first order approximation for (11-14) can be found:
t

  t '

 e 1  k 2  2 e 2 ' 


1
I 1 t   I 0 



1 k2 2



1


 
1
1 

   1  k 2 2 
1'
1 
1 

1 k2 2
1
1
1

 
2
2'
2 1 k
5
(14)
(15)
(16)
(17)
(18)
(19)
(20)
(21)
(22)
A simplified HQ model
Beam tube:
In the case of a realistic quadrupole 1 m long like HQ, we can set:
N1  46  4  184
(23)
L1  N 1 L2  4.16 mH
R1  32 mΩ
 1  130 ms
c1  60 mm
c2  90 mm
a  55 mm
da  5 mm
L2  0.123 H
(24)
(25)
(26)
(27)
(28)
(29)
(30)
(31)
 AISI  5  10 7 Ωm
(32)
R2  7.14  10 4 Ω
 2  0.172 ms
M  0.146 H
k  0.545
(33)
(34)
(35)
(36)
2
 4  10 4
1
(37)
2
This gives:
k2
so with a very good approximation:
2
(38)
1
1
This means that the magnet discharge will follow a behavior very similar to a pure exponential
discharge with the intrinsic constant time 1: the beam tube has a negligible effect in the coil current
discharge.
1'  1
1 k2
Collars:
For the aluminum collars a similar approach can be used. The above relations should be used with
more caution because the approximation of infinitely thin cylinder may not be valid.
We can assume for the collars:
a  90 mm
da  30 mm
L2  0.123 H
 AL  3  10 8 Ωm
R2  4.36  106 Ω
 2  28.2 ms
M  17.7 H
k  0.78
This gives:
6
(39)
(40)
(11)
(42)
(43)
(44)
(45)
(46)
k2
so with the first order approximation:
2
 0.17
1
(47)
2
 1.17
1
 2 '  0.334   1  9.4 ms
 1 '  1.17   1
1 k2
(48)
(49)
The Fig. 6 and 7 show the results. The current decay of the coil coupled with the collars is faster
than a pure exponential decay with constant time1, in a similar way as the experimental data. Of
course the comparison with the experimental data is valid in the first 5-10 ms, when the heaters are
not yet activated and the resistance R1 can be considered constant.
16000
Pure expontial
14000
Coupling
Current (A)
12000
Coupling first
order approx
10000
8000
6000
4000
2000
0
0.000
0.100
0.200
Time (s)
0.300
0.400
Fig. 6 Current decay of the magnet with a pure exponential time (blue line) and with the coupling of
the collars (red line and green line). The first order approximation can be considered valid for the
solution.
15000
14500
Current (A)
14000
13500
Pure expontial
13000
Coupling
12500
Coupling first order approx
12000
0.000
0.002
0.004
0.006
0.008
Time (s)
Fig. 7 Zoom of Fig. 6 for the first 10 ms.
7
0.010
Electromagnetic F.E. analysis of collars.
The above results for collars have been obtained in the approximation of very thin cylinder. A
solution considering thick collars is hard to be obtained analytically. However the calculation can
be performed with the transient analysis of OPERA 2D. The coil current decay must not be guided
by a pre-fixed function, otherwise the effect of the coupling on the coil current is cancelled. Instead,
the option “external circuit” in the analysis can be used, to simulate the discharge of the magnet in
an external dumping resistance.
A model has been created, with 3 block for the coils to cancel the b6, b10 and b14 component of the
field. In order to obtain a gradient of 168 T/m (as HQ) the total current resulted 695 kA/quadrant
(without iron contribution). For a 1 m long magnet, the corresponding magnetic energy resulted
761.4 kJ. With 47 turns/quadrant the inductance of the magnet is then:
(50)
L1  6.96 mH
The chosen external resistance is:
(51)
R1  54.1 mΩ
in order to obtain a constant time:
(52)
 1  129 ms
The Figs. 8-9 show the model and the current density in collars after 5 ms and 10 ms from the
beginning of the coil discharge.
Fig. 8 Eddy current calculation with OPERA-2D transient analysis, with collars coupled to coils.
The coils are schematized to 3 blocks/octant. This pictures shows the currents density in collars
after 5 ms from the magnet discharge in the dumping resistance.
8
Fig. 9: As Fig 8, eddy current calculation with OPERA-2D transient analysis, with collars coupled
to coils. This pictures shows the currents density in collars after 10 ms from the magnet discharge in
the dumping resistance. The eddy currents in the collars have fully penetrated and the decrease
towards the pole is similar to a cos-2theta function.
The Figs. 10-11 reports the coil current discharge vs. the time: the behavior is quite similar to the
analytical solution with the approximation of thin cylinder for the collars. By fitting the current
discharge of OPERA with the analytical formula of 1st order approximation (20), it resulted:
1 k2
2
 1.13
1
(53)
 2 '  10 ms
(54)
which again are similar to the values (48-49) obtained in the analytical case.
16000
Pure Exponential
14000
Opera
Current (A)
12000
10000
8000
6000
4000
2000
0
0.000
0.050
0.100
0.150
0.200
0.250
0.300
0.350
Time (s)
Fig. 10 Current decay of the magnet with a pure exponential time (blue line) and with the coupling
of the collars (red line), as calculated by transient analysis OPERA 2D with the “external circuit”
for the coils.
9
15000
Current (A)
14500
14000
13500
13000
Pure Exponential
12500
Opera
12000
0
0.002
0.004
0.006
0.008
0.01
Time (s)
Fig. 11 Zoom of Fig. 8 for the first 10 ms.
Effect on the MIITs
The energy transfer from the coils to the metallic parts surely has benefic effects for the hot spot
temperature of the magnet during a quench. In the approximation of pure exponential decay, the
total MIITs balance results:
1 2
(55)
I0
2
With a current decay given by (20), the MITTs at the first order are:
1 2 
 
(56)
I 0 1  k 2 1 
2
2 

So, for the above example, the reduction of the MIITs is about 15%. Of course this is a very rough
estimation, because during the discharge the total resistance of the coil increases due to the
temperature variation. However the benefit could be larger, because the ratio of the two time
constant increases and so larger energy is transferred to the coupled circuits.
Simulation behavior of eddy currents in quench codes
The real benefic effect of the coupled metallic part with the coils can be simulated in quench codes
by inserting in the model some fictitious coils coupled with the main coils. The inductance, mutual
inductance and resistance of this fictitious coil have to be chosen in order to reproduce the intrinsic
characteristic of the real coupled metallic part, as calculated from an analysis with OPERA 2D.
Conclusion
This analysis demonstrates that the eddy currents have an important impact in the current decay of
the magnet, as observed in the experimental data. The 2-D study shows that the beam tube has
marginal effect, whereas the collars have an important role. A future more specific and detailed
model with OPERA 2D should include also the other metallic parts and consider the iron yoke for a
more correct evaluation of the field.
A critical aspect is the evaluation of the average electrical resistivity of the components, as the pack
of collars, which can be sensibly larger than the resistivity of bulk material.
10
The study require a 3-D analysis too, because in the coil-end region the z-directed field could
induce not marginal additional currents on the magnet components.
The above analytical models can be applied also for dipole magnets: in this case the current
distribution in collars a beam tube would be a cos-theta function; the parameter of the secondary
circuit can be calculated in a similar way.
References
[1]
Lidia Rossi, Study of the superconductive-to-resistive transition for US-LARP high field
quadrupoles for the LHC upgrade, Bachelor degree thesis, Università degli Studi di Milano,
2012
[2]
V. Marinozzi, Study of the quench propagation in the high-field superconducting
quadrupoles for the LHC luminosity ugrade, Master degree thesis, Università degli Studi di
Milano, 2013
[3]
W.R.Smythe, Static and Dynamic Electricity, McGraw-Hill, 1950, p.411
11