Eddy Current Modeling and Measuring in Fast-Puld Resistive Magnets
Pasquale Arpaia1,2, Marco Buzio2, Giancarlo Golluccio1,2, Giuppe Montenero1,2
1) Department of Engineering, University of Sannio, Corso Garibaldi 107, 82100 Benevento, Italy. Ph : +39 0824305804-17, Fax: +39 0824 305840, E-mail:
arpaia@unisannio.it
2) CERN, European Organization for Nuclear Rearch 1211 Geneva 23, Switzerland. Ph: +41 22 767 6111, E-mail: lluccio@cern.ch; marco.buzio@cern.ch;
Abstract — A method for modeling and measuring electromagnetic transients due to eddy currents in fast-puld resistive magnets is propod. In particular, an equivalent-circuit model and a method for time-domain measurements of eddy currents are prented. The measurements are needed for an accurate control of the magnetic field quality to ensure adequate stability and performance of the particle beam in particle accelerators in dynamic conditions (field ramps up to about 700 T/s). In the cond part, the results of experiments for model definition, identification, and validation are discusd.
The tests were carried out on a quadrupole of Linac4, a new linear particle accelerator under construction at CERN (European Organization for Nuclear Rearch).
Keywords-Accelerator magnets, Eddy currents, Load modeling.
I.I NTRODUCTION
Linac2, the injector of the CERN Proton Synchrotron Booster (PSB), limits the performance of the proton accelerator complex owing to its low output energy (50 MeV). This bottleneck is going to be removed by means of a higher-energy linear accelerator under development (Linac4), which will double the brightness and the intensity of the beam delivered by the PSB and will ensure the “ultimate” beam needed for the Large Hadron Collider [1]. The beam is focud by means of veral narrow-aperture quadrupole electromagnets along the acceleration path. The magnet coils are air-cooled, thus only puld operation at 1 Hz is allowed. Conquently, the field is switched on and off in synchronization with the passage of particle bunches.
A difficult measurement problem aris from the dynamic requirements for the field quality inside such a magnet. The prent work focus on a so-called Type III quadrupole, installed in the Chopper line of Linac4 [2]. This 60 mm long, 29 mm aperture magnet is powered by the current cycl
e of Figure 1. The current ramp-up lasts 200 μs and is followed by a flat-top of 600 μs at a nominal current of 200 A, where the field is required to be stable within 10-3.The flat top stability is determined esntially by the decay of the eddy currents induced during the ramp, as well as by other parasitic effects [3]. In general, the main effects of eddy currents are: (i) to screen flux changes, (ii) to delay the achievement of nominal DC field, (iii) to introduce high harmonic content perturbing the field [4]. An equivalent circuit model is necessary to fully understand the eddy currents effects, as well as to plan possible corrective actions [5]-[6]. Eddy current effects are hard to predict accurately becau they depend on a number of uncertain parameters, such as the magnetic properties of the iron yoke, its temperature, the surface resistivity of the laminations [7], the mechanical tolerances leading to unwanted air gaps in the magnetic circuit, etc. As a conquence, an experimental approach is necessary to validate both design calculations and manufacturing quality [3].
In this paper, a method for modeling and measuring electromagnetic transients due to eddy currents in fast-puld resistive magnets is propod. In particular, in Sections 2 and 3, an equivalent-circuit model and a measurement method of eddy currents are respectively propod. In Section 4, the experimental results related to model definition, identification and validation carried out on a quadrupole of Linac4 are
discusd.
Figure 1: current excitation waveform for the laminated iron-core magnet “Type III quadrupole”.nwiz
978-1-4244-2833-5/10/$25.00 ©2010 IEEE
II. T HE PROPOSED MODEL The eddy currents are generated by electric fields arising from variable magnetic fluxes [7]. They build up during the current pul ramp-up and decay during the constant portion. In the propod model, their effect is considered as a bypass current, flowing through a chain of R/L circuits connected in parallel to the main coil (Figure 2).
Figure 2: equivalent model circuit of eddy current effect
For a short magnet such as a Type III, the contribution of the fringe field may be dominant [8]-[9], thus two R/L chains are considered in order to take into account: (i) the eddy currents parallel to the main coils (i.e. across the laminations), which have a short time constant, and (ii) the eddy current circuit, in the plane of the end laminations (i.e. perpendicular to the magnet axis) due to the fringe field. The model also takes into account the parasitic capacitance of the magnet coil that could influence the main field at high frequency (about 1 MHz or above). The bypass eddy currents (I 1 and I 2) are negligible with respect to the main current, thus can be considered as mutually independent and the analysis can be simplified [8]. The current I m flowing through the main coil of the magnet is given by:
p m m m m m m m I L L R L L L L L R L
L L I I +⎪⎪⎭⎫ ⎝
⎛⎪⎪⎭⎫ ⎝⎛++-⎪⎪⎭⎫ ⎝⎛++-=2221110exp exp 1(1) where I 0 is the current source, L m the main coil inductance, R 1, R 2, L 1, and L 2 are the equivalent resistance and inductance of the eddy current circuits, respectively. The current I p flowing through the parasitic capacitor will introduce small ripples in the flux. Assuming the static field B to be simply proportional to the magnet current, the main field component can be described by:
ac B t t B B +⎪
⎪⎭⎫
⎝
⎛⎪⎪⎭⎫ ⎝⎛--⎪⎪⎭⎫ ⎝⎛--=210exp exp 1τβτα (2)
where B 0 is the main field at the end of the double exponential
decay and B ac is the high-frequency ripple introduced by I p ; α and β are the amplitudes of the exponential decays, and τ1 and τ2 are the time constants delaying the field respon.
scholarshipIII. T HE PROPOSED MEASUREMENT METHOD
f. V C induced in the coil has to be integrated in order to obtain the linked flux φ. Assuming that the quadrupole is excited starting from a demagnetized state at t=t 0, the flux φ en by a arch coil with N t turns inrted in the magnet is calculated as:
虚拟语气倒装
⎰=t
t C t
dt t V N t 0)(1)(φ (3) In the ideal (linear) ca, this flux would be simply proportional to the excitation current via the mutual inductance L CM , depending on the geometry and on the magnetic permeability of the iron core [10]:
φ (t)= L CM I (t) (4)
In actual conditions, a current difference ∆I can be defined as:
)()
(t I L t I CM
-=
∆φ (5)
The current difference (5) includes contributions from all non-ideal effects, builds up during the ramp and decays over the flat-top. Assuming that the transient has completely died out at the end of the flat top, i.e. ∆I(t S )=0, one can derive:
⎰=s t C s t CM
郑州培训dt
t V t I N L 0
)()(1 (6)
By considering that the measured signals are affected by electrical noi, mains hum, power supply ripple and so on, the accuracy of the ratio (6) can be incread by averaging both the terms, the current excitation and the voltage coil integral over a short time interval t S -∆t ≤ t ≤ t S .
IV. E XPERIMENTAL RESULTS
In the following, the experimental results related to (i) the model definition, (ii) the model identification, and (iii) the model validation carried out on a Type III quadrupole are illustrated.
A. Model Definition
In the model definition, the main problem is to take into account both the eddy current circuit in the iron yoke lamination and the other circuit created by the fringe field in the plane perpendicular to the magnet axis.
To understand how the different eddy current circuits influence the time domain integral measurements a small coil of 4.5 mm diameter was ud to scan the field inside the aperture in different positions with a 5 mm longitudinal step. The obrved variation of the eddy current decay is function of the position inside the magnet aperture [11]. The decay includes two contributions: a slow one which tends to disappear when the coil is in the middle of the magnet
aperture, and a fast one which is prent along the whole magnet length.
B.Model Identification
The equivalent magnet impedance was measured in order to determine the values of the model para
meters from 500 Hz up to 1 MHz. A gain pha analyzer (Powertek® GP102) was ud to measure the pha and the amplitude of the impedance. The measurement was performed at low current (200 mA), thus the model does not take into account the effects due to the nonlinearity of the permeability (saturation effects of the iron yoke) that might affect the measurements. The obtained data were validated through a crosscheck with a CERN-built system, the PROMIT [12], capable of analyzing gain and pha up to 20 kHz by forcing a sinusoidal current up to 5 A.
In Figure 3, the results of two typical measurements in the frequency domain at 0.2 and 3.0 A are reported as an amplitude bode diagram. In the range from 500 Hz to 20 kHz, the two curves overlap. This confirms that in this range the magnet has a linear behavior not related to the excitation current (from independent measurements, iron yoke saturation is expected at current values well above 100 A).
C.Model Validation
The model was validated according to the following procedure:
∙measure the magnet impedance vs. frequency;
∙evaluate the unknown model parameters R1, R2, L1, L2 and C p with a non linear complex fitting method [13];
∙compare the measurement data in the time domain
[11] with the DC model simulation results;
∙calculate the residual between the model and the measurements.
The results of the frequency-domain measurement were ud to determine the five unknown model circuit parameters. The main coil inductance L m was obtained from design specifications, while the resistance R m was calculated by considering the winding wire cross ction and length and cross-checked by four-wire measurement. A non linear least square method was ud to estimate R1, R2, L1, L2 and C p (e Table 1).
In Figures 4-5, the results of the fitting are shown. The model fits the impedance measurement with a residual less than 10 % for the absolute value of the impedance up to 750 kHz, and less than 10 % of error between the measured value and the simulated impedance pha. The maximum difference between the model and the measurement curves occurs for frequency higher than 600 kHz. In any c
a, in the range of interest (from 500 Hz to 450 kHz) the model is able to replicate the magnet behavior.
Figure 3: amplitude Bode diagram of the magnet.
Table 1: eddy currents parameter calculation
Parameter value
R1 (Ω) 110
L1 (µH) 957
R2 (kΩ) 1.42
L2 (µH) 583
C p (pF) 280
Figure 4: modeled and measured amplitude Bode diagram
of the impedance (up) and residual (down).
In Figure 6, the complete model for the transient analysis
is shown. The circuit reproduces the time domain measurement scheme propod in [11]. It simulates the flux
linked to the pick up coil ud to measure the field inside the
magnet aperture. The excitation current waveform (Figure 1)
is applied as input to the model in order to understand the
behavior of the time integral of the voltage generated at the
ends of the transformer condary winding (Figure 6) reprenting the pick-up coil. The resistance of 10 GΩ is the acquisition system resistance.
In Figure 7, the time respon of the equivalent circuit to
the excitation current is shown. The flux simulated at the end
of the acquisition chain, and the acquired actual flux expresd
滨才as described in (5), are shown.
The circuit is able to replicate the decay effect due to the
eddy currents but is still missing the contribution of the mutual
coupling of the parasitic circuits with the pick-up coil in the flux measurements. The difference between measurement and simulation in the time domain still evidences a very small-amplitude slow decay during the flat-top. In the future work all the mutual couplings between the coil and the circuits will
be taken into account.
Figure 5: modeled and actual impedance: pha diagram (up) and percentage residual error (down).
Figure 6: complete circuital model of the eddy current and
of the measurement system.
Figure 7: measured and modeled flux scaled by the mutual
inductance L m: time domain transients (up), and residual
(down).
V.C ONCLUSIONS
A model-bad method is propod for measuring
electromagnetic transients due to eddy currents in a short
quadrupole magnet in time and frequency domain. The effects
of eddy currents in the pole lamination and in the plane
perpendicular to the magnet axis are modeled by two R/L
chains.
The residual between equivalent impedance measurements
and model predictions below 750 kHz is less than 10% for
both the amplitude and pha.
The model describes the decay effect in the time domain
measurements by using an actual excitation current curve. In
this ca, the residual prents a slow decay effect during the
current flat-top probably due to direct coupling between the
coil and the two parasitic circuits.
地位英文
A CKNOWLEDGMENTS
Authors thank Ricardo Beltron Mercadillo, Lucette Gaborit
and Olaf Dunkel for their work and Alessandra Lombardi and
Felice Cennamo for their uful suggestions.
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