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A.C. Magnets
Neil Marks,
CCLRC,
Daresbury Laboratory,
Warrington WA4 4AD.
[email protected]
Neil Marks; DLS/CCLRC
Cockcroft Institute 2005/6, © N.Marks, 2006
Philosophy
1.
Present practical details of how a.c. lattice magnets differ from d.c.
magnets.
2.
Present details of the typical qualities of steel used in lattice magnets.
3.
Present an overview of the design and operation of power supply
systems, both d.c. (for storage rings) and cycling (for cycling
accelerators).
4.
Give a qualitative overview of injection and extraction techniques as used
in circular machines.
5.
Present the standard designs for kicker and septum magnets and their
associated power supplies.
Neil Marks; DLS/CCLRC
Cockcroft Institute 2005/6, © N.Marks, 2006
Contents
Core Syllabus
Variations in design and construction for a.c.
magnets;
Effects of eddy currents;
‘Low frequency’ a.c. magnets
Coil transposition-eddy loss-hysteresis loss;
Properties and choice of steel;
Inductance in an a.c. magnet;
‘Extension’
Power supply systems – d.c. and a.c.;
Injection and extraction schemes;
‘Fast’ magnets;
Kicker magnets-lumped and distributed power
supplies;
Septum magnets-active and passive septa;
High frequency ferromagnetic material;
Neil Marks; DLS/CCLRC
Cockcroft Institute 2005/6, © N.Marks, 2006
Differences to d.c. magnets
A.c magnets differ in two main respects to d.c. magnets:
1.
In addition to d.c ohmic loss in the coils, there will be ‘ac’
losses (eddy and hysteresis);
design goals are to: correctly calculate a.c. losses;
minimise a.c. losses.
2.
Excitation voltage now includes an inductive (reactive)
component; this may be small, major or dominant
(depending on frequency);
this must be accurately assessed.
Neil Marks; DLS/CCLRC
Cockcroft Institute 2005/6, © N.Marks, 2006
Equivalent circuit of a.c. magnet
Rac
Rdc
Lm
Im
Cleakage
Neil Marks; DLS/CCLRC
Cockcroft Institute 2005/6, © N.Marks, 2006
A.C. Magnet Design
Additional Maxwell equation for magneto-dynamics:
curl E = -dB/dt.
Applying Stoke’s theorem around any closed path s enclosing area A:
 curl E.dA =  E.ds = V loop
where
Vloop is voltage around path s;
 - (dB /dt).dA
= - dF/dt;
Where F is total flux cutting A;
So:
Vloop = -dF/dt
Thus, eddy currents are induced in any conducting material in the alternating
field. This results in increased loss and modification to the field strength and
quality.
Neil Marks; DLS/CCLRC
Cockcroft Institute 2005/6, © N.Marks, 2006
Eddy Currents in a Conductor I
x
Rectangular cross section
resistivity  ,
B sin t
breadth 2 a ,
thickness  ,
length l ,
cut normally by field B sin t.
Consider a strip at +x, width  x , returning at –x ( l >>x).
Peak volts in circuit = 2 x l  B

Resistance of circuit = 2 l /(   x )
Peak current in circuit = x  B   x / 
-a -x
Integrate this to give total Amp-turns in block.
Peak instantaneous power in strip = 2 x2 l 2 B2   x / 
Integrate w.r.t. x between 0 and a to obtain peak instantaneous power in block
= (2/3) a3 l 2 B2   / 
Cross section area A = 2 a 
Average power is ½ of above.
l
0
x a
Cross
section A
Power loss/unit length = 2 B2 A a2/(6  ) W/m;
a 10 mm x 10 mm Cu conductor in a 1 T sin field, loss = 1.7 kW/m
Neil Marks; DLS/CCLRC
Cockcroft Institute 2005/6, © N.Marks, 2006
Eddy Currents in a Conductor II
x
Circular cross section:
resistivity  ,
radius a ,
length l ,
cut normally by field B sin t.
Consider a strip at +x, width  x , returning at –x ( l >>x).
Peak volts in circuit = 2 x l  B
Resistance of circuit = 2 l /{ 2 (a2-x2)1/2  x }
Peak current in circuit = 2x  B (a2-x2) 1/2  x / 
Integrate this to give total Amp-turns in block.
Peak instantaneous power in strip = 4 x2 l 2 B2 (a2-x2) 1/2  x / 
Integrate w.r.t. x between 0 and a to obtain peak instantaneous power in block
= (p/4) a4 l 2 B2  / 
Cross section area A = pa
Average power is ½ of above.
a
x
Power loss/unit length = 2 B2 A a2/(8  ) W/m;
Neil Marks; DLS/CCLRC
Cockcroft Institute 2005/6, © N.Marks, 2006
Eddy Currents in a cylindrical vacuum vessel
total flux cutting circuit at angle q:

B sin  t
Wall
conductivity 
 2 ( R cos q )( B sin t );
/t  2 (R cos q )(B cos t);
voltage round unit length of loop (top only) :
V
q
  2 (R cos q )(B cos t);
R
resistance of unit length of loop :
 2  / (  R q)
q
eddy currents in top and bottom loops :
Ie
  2 R 2  (cos q)  B (cos  t) q /  ;
total eddy current in cylinder is integral of

above w.r.t. q between 0, p/2 :
Ie = - 2   R2 B (cos t) / 
Geometry of cylindrical vacuum vessel,
It can be seen that the eddy currents vary as the square of the cylindrical radius R and
directly with the wall thickness t.
Neil Marks; DLS/CCLRC
Cockcroft Institute 2005/6, © N.Marks, 2006
Perturbation field generated by eddy currents
m=
Field perturbations Be(X) generated by the eddy
currents will depend on the vessel’s magnetic
environment. If the imposed field is generated by
coils on a ferro-magnetic yoke, eddy currents will
couple to that yoke. To establish the amplitude and
distribution of the field perturbation, a yoke
geometry has to be considered.
g
R
0
x
Magnet geometry around vessel
radius R.
Using:
Be= m0 Ie/g;
Amplitude ratio between perturbing and imposed fields at X = 0 is:
Be(0)/B = - 2 m0   R2 /  g;
Phase of perturbing field w.r.t. imposed field is:
qe = arctan (- 2 m0   R2 /  g )
Neil Marks; DLS/CCLRC
Cockcroft Institute 2005/6, © N.Marks, 2006
Distributions of perturbing fields
Cylindrical vessel:
Be(X)


2R  Bcos t
R 2 - X2 .
g

2 R 2   Bcos t 
X2
X4
X6
5 X8
1 






..........
..
2
4
6
8


g
8R
16 R
128 R
 2R



Rectangular vessel (semi axies a, b):
Be(X)

2 m 0  (B cos t)  (a 2  X 2 )


 

ab


g
2


Elliptical vessel (semi axies a, b):
Be(X)
m0   B cos t 
2

g

Neil Marks; DLS/CCLRC
2 1/2
2
2 1/2  

 2
  tan -1  (a  b ) ( a - X )  


2
2
2
2 1/ 2  
(a 2  b 2 )  
b
X
(a
/
b
1)

a


a2 b


Cockcroft Institute 2005/6, © N.Marks, 2006
‘Low frequency’ a.c. magnets
1.5
We shall deal separately with ‘low frequency’ and ‘fast’ magnets:
‘low frequency’
– d.c. to c 100 Hz:
0
0
50
100
150
‘fast’ magnets
frequency (Hz)
– pulsed magnets with rise times from 10s ms to < 1 ms.
200
time (ms)
(But these are very slow0 compared to r.f.~10systems!)
Neil Marks; DLS/CCLRC
Cockcroft Institute 2005/6, © N.Marks, 2006
Coils for up to c 100 Hz.
Coil designed to avoid excessive eddy currents. Solutions:
a) Small cross section copper per turn; this give large number of turns - high
alternating voltage unless multiple conductors are connected in parallel; they
must then be ‘transposed’:
a b
c d
b a
d c
d c
b a
c d
a b
b) ‘Stranded’ conductor (standard solution in electrical engineering) with
strands separately insulated and transposed (but problems locating the
cooling tube!):
Flux density at the coil is predicted by f.e.a. codes, so eddy loss in coils can
be estimated during magnet design.
Neil Marks; DLS/CCLRC
Cockcroft Institute 2005/6, © N.Marks, 2006
Two examples:
Note that eddy loss varies as 2 ; B2 and (width)2 .
NINA :
ISIS:
E
= 5.6 GeV;

= 53 Hz;
Bpeak = 0.9 T.
E
= 800 MeV;

= 50 Hz;
Bpeak ≈ 0.2 T.
Neil Marks; DLS/CCLRC
Cooling
tube.
Transposed,
stranded conductor.
}
c 10mm x 10 mm solid
conductor with cooling hole.
Cockcroft Institute 2005/6, © N.Marks, 2006
Steel Yoke Eddy Losses.
To limit eddy losses, steel core are laminated, with a thin layer
(~2 µm) of insulating material coated to one side of each
lamination.
At 10 Hz lamination thickness of 0.5mm to 1
mm can be used.
At 50Hz, lamination thickness of 0.35mm to
0.65mm are standard.
Laminations also allow steel to be ‘shuffled’
during magnet assembly, so each magnet
contains a fraction of the total steel production;
- used also for d.c. magnets.
Neil Marks; DLS/CCLRC
Cockcroft Institute 2005/6, © N.Marks, 2006
Steel hysteresis loss
Steel also has hysteresis loss caused by the finite area inside the
B/H loop:
for a.c. excitation frequency f,
power loss Phys =
f
{
B
B.dH +
A
-

A
W/m3.

C
B.dH
B
DB.dH
-

C
CB.dH
}
B
D
D
A
Neil Marks; DLS/CCLRC
Cockcroft Institute 2005/6, © N.Marks, 2006
Steel loss data
Manufacturers give figures for total loss (in W/kg) in their steels
catalogues:
•for a sinusoidal waveform at a fixed peak field (European standard is at 1.5
T);
•and at fixed frequency (50 Hz in Europe, 60 Hz in USA);
•at different lamination thicknesses (0.35, 0.5, 0.65 & 1.0 mm typically)
• they do not give separate values for eddy and hysteresis loss.
3
Accelerator magnets will have:
•different waveforms (unidirectional!);
•different d.c. bias values;
•different frequencies (0.2 Hz up to 50 Hz).
0
0
How does the designer calculate steel loss?
Neil Marks; DLS/CCLRC
Cockcroft Institute 2005/6, © N.Marks, 2006
Comparison between eddy and hysteresis loss in
steel:
Variation with:
Eddy loss
Hysteresis loss
A.c. frequency:
Square law
Linear;
A.c. amplitude:
Square law
Non-linear-depends on level;
D.c. bias:
No effect
Increases non-linearly;
Total volume of steel:
Linear
Linear;
Lamination thickness:
Square law
No effect.
Neil Marks; DLS/CCLRC
Cockcroft Institute 2005/6, © N.Marks, 2006
Choice of steel
'Electrical steel' is either 'grain oriented' or 'non-oriented‘:
Grain oriented:
• strongly anisotropic,
• very high quality magnetic properties and very low a.c losses
in the rolling direction;
• normal to rolling direction is much worse than non-oriented
steel;
• stamping and machining causes loss of quality and the
stamped laminations must be annealed before final assembly.
Neil Marks; DLS/CCLRC
Cockcroft Institute 2005/6, © N.Marks, 2006
Choice of steel (cont).
Non-oriented steel:
• some anisotropy (~5%);
• manufactured in many different grades, with different
magnetic and loss figures;
• losses controlled by the percentage of silicon included in the
mix;
• high silicon gives low losses (low coercivity), higher
permeability at low flux density but poorer magnetic
performance at high field;
• low (but not zero) silicon gives good performance at high B;
• silicon mechanically ‘stabilises’ the steel, prevents aging.
Neil Marks; DLS/CCLRC
Cockcroft Institute 2005/6, © N.Marks, 2006
Solid steel
Low carbon/high purity steels:
• usually used for solid d.c. magnets;
• good magnetic properties at high fields
• but hysteresis loss not as low as high silicon steel;
• accelerator magnets are seldom made from solid steel;
(laminations preferred to allow shuffling and reduce eddy
currents)
Neil Marks; DLS/CCLRC
Cockcroft Institute 2005/6, © N.Marks, 2006
DK 70 (low silicon non-oriented steel)
Losses at 50 Hz: W/kg vs peak induction
(T) for 0.65 and 0.5 mm laminations.
Neil Marks; DLS/CCLRC
Magnetisation curves: B(T) vs H (A/m)
Cockcroft Institute 2005/6, © N.Marks, 2006
CK 27 (high silicon non-oriented steel)
Losses at 50 Hz: W/kg vs peak
induction (T) for 0.5 and 0.35 mm
laminations.
Neil Marks; DLS/CCLRC
Magnetisation curves: B(T) vs H (A/m)
Cockcroft Institute 2005/6, © N.Marks, 2006
M4, M5, M6 grain oriented steel.
Losses at 50 Hz: W/kg vs peak
induction (T) for 0.27, 0.30 and
0.35 mm laminations.
Neil Marks; DLS/CCLRC
Magnetisation curves: B(T) vs H (A/m)
Cockcroft Institute 2005/6, © N.Marks, 2006
Solid low carbon steel.
Neil Marks; DLS/CCLRC
Cockcroft Institute 2005/6, © N.Marks, 2006
Comparisons
Property:
DK-70:
CK-27:
27 M 3:
XC06 :
Type
Nonoriented
Low
0.65 mm
Nonoriented
High
0.35 mm
Grainoriented
0.27 mm
Nonoriented
Very low
Solid
Silicon content
Lam thickness
a.c. loss (50 Hz):
at 1.5 T peak
Permeability:
at B=1.5 T
at B=1.8 T
6.9 W/kg
2.25 W/kg
1680
184
990
122
Neil Marks; DLS/CCLRC
0.79 W/kg Not suitable
> 10,000
3,100
>1,000
>160
Cockcroft Institute 2005/6, © N.Marks, 2006
The ‘problem’ with grain oriented steel
In spite of the
obvious advantage,
grain oriented is
seldom used in
accelerator magnets
because of the mechanical
problem of keeping B
in the direction of the grain.
Rolling
direction.
B
Difficult (impossible?) to make
each limb out of separate strips
of steel.
Neil Marks; DLS/CCLRC
Cockcroft Institute 2005/6, © N.Marks, 2006
Magnet Inductance
F
Definition:
n turns,
L = n F /I
Inductance:
Dipole Inductance.
current I
For an iron cored dipole:
F = B A = µ0 n I A/(g +l/µ);
Where:
A is total area of flux (including gap fringe flux);
l is path length in steel;
g is total gap height
So:
Lm = µ0 n2 A/(g +l/µ);
Note that the f.e.a. codes give values of vector potential to
provide total flux/unit length.
Neil Marks; DLS/CCLRC
Cockcroft Institute 2005/6, © N.Marks, 2006
Inductances in series and parallel.
Two coils, inductance L, with no mutual coupling:
Inductance in series = 2 L:
Inductance in parallel = L/2:
ie, just like
resistors.
Neil Marks; DLS/CCLRC
Cockcroft Institute 2005/6, © N.Marks, 2006
But
Two coils, inductance L, on the same core (fully
mutually coupled):
Inductance of coils in series = 4 L
n is doubled, n2 is quadrupled.
Inductance of coils in parallel = L
same number of turns, cross
section of conductor is doubled.
Neil Marks; DLS/CCLRC
Cockcroft Institute 2005/6, © N.Marks, 2006