USPAS - Superconducting Magnet Theory and Measurements

83 Pages·2001·0.531 MB·English

by  A. Jain

Measurements of Field Quality Using Harmonic Coils
Animesh K. Jain
Brookhaven National Laboratory, Upton, New York 11973-5000
[US Paticle Accelerator School on Superconducting Accelerator Magnets, January
22-26, 2001, Houston, Texas, USA.]
1. Radial and Tangential Coils
2. A Dipole Coil
3. General Case of a 2m-pole Coil
4. Flux Through a Coil of an Arbitrary Shape
5. Rotating Coil of an Arbitrary Shape
5.1 Tangential Coil as a special case
5.2 Radial Coil as a special case
6. An Array of Rotating Coils
7. Imperfect Motion of Rotating Coils
7.1 Transverse Vibrations
7.2 Torsional Vibrations
8. Examples of Practical Coils – HERA and RHIC
9. Analog and Digital Bucking
10. Magnet Dependent Bucking Algorithm
11. Coil Construction Errors
11.1 Finite Size of Coil Windings
11.2 Random Variation of Radius
11.3 Random Variation of Angular Position
11.4 Random Variation of Opening Angle (tangential coil)
11.5 Unequal Radii of the two grooves (tangential coil)
11.6 Offset in Rotation Axis
US Particle Accelerator School on Superconducting Accelerator Magnets, Jan. 22-26, 2001, Houston, TX.
12. Systematic Errors in Coil Parameters
12.1 Error in Radius
12.2 Error in Angular Position
12.3 Error in Opening Angle
13. Effect of a Finite Averaging Time
14. Errors in Placing the Coil in the Magnet
14.1 Coil Axis different from the Magnet Axis
14.2 Sag of the Measuring Coil
14.3 Coil Axis Tilted relative to the Magnet Axis
15. Calibration of a Five-Winding Tangential Coil
15.1 Radii and the Opening Angle
15.2 Angular Positions
15.3 “Tilt” of Tangential Winding (unequal radii of the two grooves)
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US Particle Accelerator School on Superconducting Accelerator Magnets, Jan. 22-26, 2001, Houston, TX.
A Radial Coil (or B Coil)
q
Y
B
q
R
2
q
Number of Turns = N
X
Length of the coil = L
R
1
w
A radial coil has a flat loop of wire whose plane coincides with the radial
plane of the rotating cylinder. The two sides of the loop are located at radii R
1
and R , as shown above. The flux through the coil at an angular orientation q
2
is:
R
2
R (cid:243) ¥ (cid:230) (cid:246) n- 1
F (q ) = NL (cid:242)2Bq (r,q )dr = NL(cid:244)(cid:244) (cid:229) C(n)(cid:231)(cid:231) r (cid:247)(cid:247) cos(nq - na n)dr
Ł R ł
R1 ı n=1 ref
R
1
Ø n nø
¥ (cid:230) (cid:246) (cid:230) (cid:246)
F (q ) = (cid:229) NLRref Œ (cid:231)(cid:231) R2 (cid:247)(cid:247) - (cid:231)(cid:231) R1 (cid:247)(cid:247) œ C(n)cos(nq - na )
Œ œ n
n Ł R ł Ł R ł
n=1 º ref ref ß
w q = d
If the coil rotates with an angular velocity and is the angular position
q w d.
at time t = 0, then = t + The flux as a function of time is then:
Ø n nø
¥ (cid:230) (cid:246) (cid:230) (cid:246)
F (t) = (cid:229) NLRref Œ (cid:231)(cid:231) R2 (cid:247)(cid:247) - (cid:231)(cid:231) R1 (cid:247)(cid:247) œ C(n)cos(nw t + nd - na )
Œ œ n
n Ł R ł Ł R ł
n=1 º ref ref ß
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US Particle Accelerator School on Superconducting Accelerator Magnets, Jan. 22-26, 2001, Houston, TX.
A Radial Coil (or B Coil)
q
Y
B
q
R
2
q
Number of Turns = N
X
Length of the coil = L
R
1
d
Angular position at time t=0:
w
Ø n nø
¥ (cid:230) (cid:246) (cid:230) (cid:246)
F (q ) = (cid:229) NLRref Œ (cid:231)(cid:231) R2 (cid:247)(cid:247) - (cid:231)(cid:231) R1 (cid:247)(cid:247) œ C(n)cos(nq - na )
Œ œ n
n Ł R ł Ł R ł
n=1 º ref ref ß
Ø n nø
¥ (cid:230) (cid:246) (cid:230) (cid:246)
F (t) = (cid:229) NLRref Œ (cid:231)(cid:231) R2 (cid:247)(cid:247) - (cid:231)(cid:231) R1 (cid:247)(cid:247) œ C(n)cos(nw t + nd - na )
Œ œ n
n Ł R ł Ł R ł
n=1 º ref ref ß
The voltage signal induced in the radial coil is:
V(t) = - (cid:230)(cid:231) F¶ (cid:246)(cid:247) = (cid:229) ¥ NLR w ØŒ (cid:230)(cid:231)(cid:231) R2 (cid:246)(cid:247)(cid:247) n - (cid:230)(cid:231)(cid:231) R1 (cid:246)(cid:247)(cid:247) nøœ C(n)sin(nw t + nd - na )
Ł ¶ t ł ref Œ Ł R ł Ł R ł œ n
n=1 º ref ref ß
The amplitude of the voltage signal is proportional to the angular velocity.
For analysis based on voltage signals, it is essential to control the angular
velocity and make corrections for any speed fluctuations. The integrated
voltage signal gives the flux, which is independent of angular velocity.
The above expressions assume that the two sides of the coil loop are located
on the same side of the origin, as shown in the figure. If the two sides are
located on opposite sides of the origin, as is true for many practical coils, then
one should replace R by –R in the above equations.
1 1
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US Particle Accelerator School on Superconducting Accelerator Magnets, Jan. 22-26, 2001, Houston, TX.
A Tangential Coil (or B Coil)
r
Y
D
B
r
Number of Turns = N
R
c
Length of the coil = L
q
D
Opening Angle =
X
w
A tangential coil has a loop of wire whose plane is at right angles to the radius
vector through the center of the loop. The two sides of the loop are both
located at a radius of R , as shown above. The flux through the coil at an
c
q
angular orientation is:
q +D /2
F (q ) = NLq +(cid:242)D B/2 (R ,q )R dq = NL (cid:243)(cid:244) (cid:229) ¥ C(n)(cid:230)(cid:231)(cid:231) Rc (cid:246)(cid:247)(cid:247) n- 1sin(nq - na )R dq
r c c (cid:244) n c
Ł R ł
q - D /2 ı n=1 ref
q - D /2
n
F (q ) = (cid:229) ¥ 2NLRref (cid:230)(cid:231)(cid:231) Rc (cid:246)(cid:247)(cid:247) sin(cid:230)(cid:231) nD (cid:246)(cid:247) C(n)sin(nq - na )
Ł ł n
n Ł R ł 2
n=1 ref
w q = d
If the coil rotates with an angular velocity and is the angular position
q w d.
at time t = 0, then = t + The flux as a function of time is then:
n
F (t) = (cid:229) ¥ 2NLRref (cid:230)(cid:231)(cid:231) Rc (cid:246)(cid:247)(cid:247) sin(cid:230)(cid:231) nD (cid:246)(cid:247) C(n)sin(nw t + nd - na )
Ł ł n
n Ł R ł 2
n=1 ref
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US Particle Accelerator School on Superconducting Accelerator Magnets, Jan. 22-26, 2001, Houston, TX.
A Tangential Coil (or B Coil)
r
Y
D
B
r
Number of Turns = N R
c
Length of the coil = L
q
D
Opening Angle =
X
Angular position at
d
time t=0: w
n
F (q ) = (cid:229) ¥ 2NLRref (cid:230)(cid:231)(cid:231) Rc (cid:246)(cid:247)(cid:247) sin(cid:230)(cid:231) nD (cid:246)(cid:247) C(n)sin(nq - na )
Ł ł n
n Ł R ł 2
n=1 ref
n
F (t) = (cid:229) ¥ 2NLRref (cid:230)(cid:231)(cid:231) Rc (cid:246)(cid:247)(cid:247) sin(cid:230)(cid:231) nD (cid:246)(cid:247) C(n)sin(nw t + nd - na )
Ł ł n
n Ł R ł 2
n=1 ref
The voltage signal induced in the tangential coil is:
n
V(t) = - (cid:230)(cid:231) ¶F (cid:246)(cid:247) = - (cid:229) ¥ 2NLR w (cid:230)(cid:231)(cid:231) Rc (cid:246)(cid:247)(cid:247) sin(cid:230)(cid:231) nD (cid:246)(cid:247) C(n)cos(nw t + nd - na )
Ł ¶ t ł ref Ł R ł Ł 2 ł n
n=1 ref
The amplitude of the voltage signal is proportional to the angular velocity.
For analysis based on voltage signals, it is essential to control the angular
velocity and make corrections for any speed fluctuations. The integrated
voltage signal gives the flux, which is independent of angular velocity.
The radius, R , of the coil should be maximized to get good signal strength for
c
D
higher harmonics. The opening angle, , should be large enough to give
D
enough signal and small enough so that sin(n /2) does not vanish for higher
D p D
harmonics of interest ( << 2 /n ). Typically, ~ 15 degrees.
max
Page 6
US Particle Accelerator School on Superconducting Accelerator Magnets, Jan. 22-26, 2001, Houston, TX.
Sensitivity of a Tangential Coil to Various Harmonics (R = R )
c ref
1.2
1.1
D
= 10 deg.
1.0
0.9
0.8
2) 0.7 D = 15 deg.
/
D
n 0.6
(
n
i 0.5
S
0.4
D
0.3 = 20 deg.
0.2
0.1
0.0
0 2 4 6 8 10 12 14 16 18 20 22 24
Harmonic Number (n )
Typically, one is interested in precise measurement of about 15 harmonics. It
is clear from the above plot that a tangential coil with an opening angle of 20
degrees rapidly loses sensitivity for higher harmonics, although it is more
sensitive to lower harmonics as compared to a coil with 15 degrees opening
angle. On the other hand, the sensitivity of a coil with an opening angle of
only 10 degrees peaks at 36-pole term, which is of little relevance for
accelerator physics. Such a narrow angle, therefore, sacrifices sensitivity for
lower harmonics of interest with no useful outcome. It is clear from this plot
D
that optimum value of the opening angle is ~ 15 degrees.
Page 7
US Particle Accelerator School on Superconducting Accelerator Magnets, Jan. 22-26, 2001, Houston, TX.
A “Dipole Coil” : Radial or Tangential ?
A “Dipole Coil” is a coil with a
Y
specific geometry which has the
“Dipole Symmetry”, namely an
R =R
2 c
p
antisymmetry under rotation by
radians. The flux through this coil
R
c
can be calculated by treating it as q '
a radial coil with R = –R and
1 c q
D = p
q
R = +R , oriented at an angle ,
2 c
as shown. The flux through the X
coil can also be calculated by
treating it as a tangential coil
p
with an opening angle of
radians, oriented at an angle of
R =–R
q ’= q +p /2. Both approaches 1 c
give the same result.
F (q ) = (cid:229) ¥ NLRref ØŒ (cid:230)(cid:231)(cid:231) Rc (cid:246)(cid:247)(cid:247) n - (cid:230)(cid:231)(cid:231) - Rc(cid:246)(cid:247)(cid:247) nøœ C(n)cos(nq - na )
radial Œ œ n
n Ł R ł Ł R ł
n=1 º ref ref ß
= (cid:229) ¥ NLRref (cid:230)(cid:231)(cid:231) Rc (cid:246)(cid:247)(cid:247) n[1- (- 1)n] C(n)cos(nq - na )
n
n Ł R ł
n=1 ref
Similarly,
F (q ¢ )=F (q + p ) = (cid:229) ¥ 2NLRref (cid:230)(cid:231)(cid:231) Rc (cid:246)(cid:247)(cid:247) nsin(cid:230)(cid:231) np (cid:246)(cid:247) C(n)sin(nq + np - na )
tang. tang. Ł ł n
2 n Ł R ł 2 2
n=1 ref
= (cid:229) ¥ 2NLRref (cid:230)(cid:231)(cid:231) Rc (cid:246)(cid:247)(cid:247) nsin2(cid:230)(cid:231) np (cid:246)(cid:247) C(n)cos(nq - na )
Ł ł n
n Ł R ł 2
n=1 ref
The terms with n = even vanish in both the expressions. The flux through a
dipole coil is therefore given by:
¥ (cid:230) (cid:246) n
F (q )= (cid:229) 2NLRref (cid:231)(cid:231) Rc (cid:247)(cid:247) C(n)cos(nq - na )
Dipole n
n Ł R ł
n=1 ref
n=odd
A “Dipole Coil” is therefore sensitive to only the odd harmonics, i.e., dipole,
sextupole, decapole, etc. Such a coil is almost universally used in both radial
and tangential coil systems for “bucking” the main dipole field.
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US Particle Accelerator School on Superconducting Accelerator Magnets, Jan. 22-26, 2001, Houston, TX.
A Multipole Coil of Order m (A 2m-Pole Coil)
Y
q =+q /p (2 )
' m
R
c
p
/m
q
X
A multipole coil of order m, or a 2m-pole coil is a coil with special geometry
that has m loops connected in series, as shown in the figure. For any angular
q q
position characterized by the angle , the loops span the angular region of
q p q p q p q p q p
to +( /m), ( +2 /m) to ( +3 /m), ( +4 /m) to ( +5 /m), and so on. The
q
flux through such a coil as a function of can be easily calculated by treating
it as an array of m identical tangential coils with opening angle of D = p /m
and having angular positions of q ’ = q +(p /2m), q ’+2p /m , q ’+4p /m, and so
on.
The flux through the first segment of the coil is:
n
F (q )= F (q ¢ ) = (cid:229) ¥ 2NLRref (cid:230)(cid:231)(cid:231) Rc (cid:246)(cid:247)(cid:247) sin(cid:230)(cid:231) np (cid:246)(cid:247) C(n)sin(nq ¢ - na )
1 tang. Ł ł n
n Ł R ł 2m
n=1 ref
n
= Im(cid:229) ¥ 2NLRref (cid:230)(cid:231)(cid:231) Rc (cid:246)(cid:247)(cid:247) sin(cid:230)(cid:231) np (cid:246)(cid:247) C(n)ei(nq ¢ - na n)
Ł ł
n Ł R ł 2m
n=1 ref
Page 9
US Particle Accelerator School on Superconducting Accelerator Magnets, Jan. 22-26, 2001, Houston, TX.
A Multipole Coil of Order m (A 2m-Pole Coil)
Y
q =+q /p (2 )
' m
R
c
p
/m
q
X
The flux through the first segment of the coil is:
F (q )= F (q ¢ )=Im(cid:229) ¥ 2NLRref (cid:230)(cid:231)(cid:231) Rc (cid:246)(cid:247)(cid:247) nsin(cid:230)(cid:231) np (cid:246)(cid:247) C(n)ei(nq ¢ - na n)
1 tang. Ł ł
n Ł R ł 2m
n=1 ref
The contribution due to harmonics which are EVEN MULTIPLES OF m VANISHES
p
due to the sin(n /2m) factor. Let us now consider the total flux through the
array of loops:
¥ [ ]
F (q )=Im(cid:229) X einq ¢ 1+ei2p n/m +ei4p n/m +K(m terms)
n
n=1
=Im(cid:229) ¥ X einq ¢ ØŒ 1- exp(i2np ) øœ =0; unless (cid:230)(cid:231) n(cid:246)(cid:247) = integer
n º 1- exp(i2np / m)ß Ł mł
n=1
where, X = 2NLRref (cid:230)(cid:231)(cid:231) Rc (cid:246)(cid:247)(cid:247) nsin(cid:230)(cid:231) np (cid:246)(cid:247) C(n)e- ina n = 0 for (n/m)=even integer.
n Ł ł
n Ł R ł 2m
ref
Therefore, all terms in the summation vanish, except for those values of n
which are ODD MULTIPLES OF m.
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US Particle Accelerator School on Superconducting Accelerator Magnets, Jan. 22-26, 2001, Houston, TX.