LS Note No. 200
April, 1992 Effect of Eddy Current in the Laminations on the Magnet Field
Y. Chung and J. Galayda
Advanced Photon Source
Argonne National Laboratory
Argonne, IL 60439
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In this note theory and measurements of the effect of the eddy current in the laminations on the magnet field will be prented. The theory assumes a simple solenoid–type magnet with laminated iron core and ignores the end field effect. The measurements were made on the input voltage and current, and the dipole component of the magnetic field in the middle of the magnet bore. The amplitude and pha relations between the quantities give the field attenuation fac-tor, the pha delay, and the resistance and inductance of the magnet as functions of frequency. Comparisons of the results with the theory will be discusd.
1.Introduction
The propod corrector magnets to be ud for global and local orbit corrections in the storage ring have six poles like xtupole magnets, and in respon to the beam motion, a current of up to approximately 50 Hz will be applied varying with time. The corresponding time–vary-ing magnet field in the iron core induces an eddy current in the magnet laminations, which not only decreas the field strength but also produces power loss due to ohmic heating.
The eddy current effect can be reduced significantly by using thin laminations for the iron core. The current design thickness of the lamination is 0.025”. From the viewpoint of curb-ing the eddy current effect, the thinner the laminations, the better. However, the larger number of thinner laminations required to asmble a magnet of given length will drive up the cost of magnet manufacturing.
This study investigates the effectiveness of the current design for the xtupole/corrector magnet in terms of the attenuation and pha shift of the magnet field and the power efficiency.
2.Impedance of Electromagnetic Fields
For time–varying electromagnetic fields with harmonic time dependence e–iωt in conduc-tors and fer
romagnetic materials, the complex field impedance Z can be obtained from consider-ation of Poynting’s theorem for harmonic time variation of the fields,1
1 2ŕ
V
J*@E d3x–ŕVǒE@ēD*ēt–H*@ēBētǓav d3x)ŏS S@n da+0,(2.1)
where n is the unit vector outward normal to the surface and S is the complex Poynting vector defined by
S+1
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E H*.(2.2)
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B B
B1
B1sin a1
B (t)+h ȍ
R n +*R B n e *i(n w t *αn ) .(3.1)
属狗最佳婚配属相h is a unit vector in the direction of the magnetic field H , and αn is the pha delay of the n–th harmonic with respect to H . We now assume that the conductivity σ is real and that the radiation loss is negligible. Then the resistance R can be written as, from Eqs. (2.8) and (3.1),
R +1ŤI i Ť
2ŕV 1s ŤJ Ť2d 3x )w ŤI i Ť2ŕV sin α1H 1B 1d 3x .(3.2)where we ud
1T ŕT
0e i w (1*n)t dt +d n,1 .(3.3)
教槽料
As shown in Eq. (3.2), all other harmonic terms other than n = 1 vanish, and when α1 is greater than 0, there is a net loss of power proportional to the area enclod by the ellip in Fig. 3.1(b).
Another effect of the hysteresis is the reduction of the magnet inductance, as can be en from Eq. (2.9). When the conductivity σ is real, and if the load is purely inductive (X = ωL), the magnet inductance L is, from Eq. (2.10)
L +1ŤI i ŤŕV cos a 1H 1B 1d 3x.(3.4)
Equations (3.2) and (3.4) show that the hysteresis effect and the eddy current effect can-not be parated. In general, the pha delay α1 is a function of the field amplitude H 1 as well as of the space, and the field amplitude H 1 (and B 1 also) is in turn affected by the eddy current in the conductor. In the following ction, we will derive the eddy current distribution in the mag-net lamination and the field attenuation as functions of the lamination thickness and the skin depth.
4.Eddy Current Effect
In this ction, we will consider the effect of the eddy current in the magnet lamination on the magnet field and the resulting change in the resistance R and the inductance L. With
finite conductivity of the magnet iron, time–varying current applied on the coil winding will pro-duce an eddy current in the core in the direction canceling the original magnetic field. This results in the decrea of the magnet field efficiency and the pha shift of the field with respect to the current in the coil winding.
斑马卡通In order to reduce this undesirable effect and also for the convenience of manufacturing,a magnet core is made of many thin laminations, which are electrically insulated from each
other. Since the current flux line must clo on itlf, the eddy current circulates confined within a lamination and does not cross over to adjacent ones. In the limit of infinitely thin laminations,the current flux lines cancel each other macroscopically and there is no eddy current effect.
Becau of the finite conductivity of the magnet iron, eddy currents produce Joule heat-ing and increa the resistance. Reduction of the magnetic field decreas the magnet induc-
tance. This can be shown as follows. Let us for the moment ignore the effect of the hysteresis (α1 → 0) and focus on the effect of the eddy current with real conductivity σ. We also assume that the magnet is purely inductive. In this ca, using the notation of Section 3, Eqs. (2.8) and (2.9) reduce to
R +1ŤI i Ť
2ŕV 1s ŤJ Ť2d 3x ,(4.1)and
L +1ŤI i Ť2ŕV H 1B 1d 3x .(4.2)
Equation (4.1) shows that the resistance of a magnet is the overall power loss due to
Joule heating divided by the square of the input current amplitude. The Joule heating occurs pri-marily in the coil winding and in the magnet lamination. If the resistance change of the coil winding over the frequency range of interest is negligibly small, the dominant source of resis-tance change with frequency is the eddy current in the magnet lamination. Another effect of the eddy current in the lamination is reduction of the magnet inductance due to the partial cancella-tion of the magnetic field as shown in Eq. (4.2).
In the following, we will derive the current distribution J and the field H inside the mag-net lamination of finite thickness. The distribution of the eddy current in the laminations and the resulting field in the magnet bore will be obtained as functions of the lamination thickness and the skin depth.
4.1.Distribution of the Eddy Current and the Field in the Lamination
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Consider a simple magnet with M laminations of thickness d, with coil wound around it.Let N be the number of windings and I = I i e –i ωt be the current in the coil. L x , L y , and L z are dimensions along x, y, and z, respectively. The schematic of the magnet is shown in Fig. 4.1.The magnet volume V m occupied by the laminations is then
V m +MdL y L z .
(4.3)
Ignoring the displacement current ∂D /∂t and assuming ε = ε0, we have
J +ʼn H and H +*i 1mws ʼn J ,(4.4)where we ud J = σE . Since the current flux lines cannot cut across the boundary between lam-inations, there will be a circulating current contained inside the narrow lamination. Now, we can imagine two opposing current fluxes of the same magnitude flowing through the gap between adjacent laminations, so that there is a circulating current around each lamination. If we assume the relative permeability K m (= µ/µ0) to be very large, the effect of the this imaginary circulating current on other laminations will be negligible, since all the field lines will be nearly perpendicu-lar to the magnet surface.
