EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH CERN - BEAMS DEPARTMENT CERN-BE-Note-2010-014 (CO) The LHC AC Dipole system: an introduction Javier Serrano, Matthieu Cattin Abstract: The LHC AC Dipole is an instrument to study properties of the LHC lattice by inducing large transverse displacements in the beam. These displacements are generated by exciting the beam with an oscillating magnetic field at a frequency close to the tune. This paper presents the system requirements and the technical solution chosen to meet them, based of high-power audio amplifiers and a resonant parallel RLC circuit. Geneva, Switzerland May 2010 1
Transcript
EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH
CERN - BEAMS DEPARTMENT
CERN-BE-Note-2010-014 (CO)
The LHC AC Dipole system:an introductionJavier Serrano, Matthieu Cattin
Abstract:The LHC AC Dipole is an instrument to study properties of the LHC lattice by
inducing large transverse displacements in the beam. These displacements are generatedby exciting the beam with an oscillating magnetic field at a frequency close to the tune.This paper presents the system requirements and the technical solution chosen to meetthem, based of high-power audio amplifiers and a resonant parallel RLC circuit.
The Large Hadron Collider (LHC) at CERN will probe a new energy domain in ParticlePhysics using collisions of two 7TeV proton beams against one another. The 27 kmcircumference of this accelerator is mainly occupied by electromagnets of different sorts,whose main duty is to keep the beam inside the vacuum chamber and guarantee that itstransverse characteristics are optimal for collisions.Dipole electromagnets bend the beam horizontally towards the center of the circum-
ference to keep it from drifting away in a straight line. The sum of the angular kicks ofall dipole magnets in the circumference is therefore equal to 2π.A perfect accelerator only containing dipole magnets would constrain a perfect beam
inside the pipe, but there are many reasons to enhance this basic magnet lattice1 withmagnets of higher order. These reasons all have to do with the fact that neither thebeam nor the accelerator are perfect. The first needed enhancement to the dipole lat-tice consists of adding quadrupoles to focus the beam. One can prove [14, 4] that aquadrupole focuses a beam of particles in the same sense that a convergent lens focusesa beam of light. The major difference is that a quadrupole which focuses in one plane(e.g. the horizontal plane) will defocus in the orthogonal plane (e.g. the vertical one).There are two types of quadrupoles used in accelerators for focusing: horizontally focus-ing quadrupoles (QF) and vertically focusing (i.e. horizontally defocusing) quadrupoles(QD). Fortunately, it can be shown [14] that a suitable arrangement of interleaved QFand QD quadrupoles will provide focusing in both planes. Particles drifting too muchhorizontally will sooner or later hit a QF that will correct their trajectory. Since thestrength of a quadrupole kick is proportional to the transverse particle displacementfrom its center, we get overall horizontal focusing because particles cross QFs at largerhorizontal displacements than QDs. The same argument applies of course for verticalfocusing.Let’s stop here our lattice enhancements and have a look at what happens to a parti-
cle which, for some reason, finds itself transversely displaced from the ideal trajectory.At some point it will find a quadrupolar field and experience a transverse kick, thenanother one, and so on. Since the strength of the quadrupole kicks is proportional to thetransverse displacement of the particle, this looks a bit like a simple harmonic oscillatorwith a spring constant replaced by the strength of the quadrupoles. The only differenceis that the spring constant depends on the longitudinal coordinate of the trajectory, typ-ically denoted s. It is zero outside quadrupoles, a positive number in QFs and a negativenumber in QDs. To get a bit more concrete, let’s choose a coordinate system as the onedepicted in Figure 1.1.
1The arrangement of all the magnets that make up an accelerator is called the “lattice”.
3
1 Introduction
y
sx
Figure 1.1: Coordinate system used for describing beam dynamics in circularaccelerators.
The coordinate system moves with the particle being studied. At each instant, −→spoints tangentially to the trajectory in the sense of particle movement, while −→x pointsin the radial outwards direction and −→y points vertically upwards.Using this coordinate system, we can write for the movement of a particle in the
horizontal plane:d2x
ds2 + k (s) · x = 0 (1.1)
where k(s) represents the quadrupole “spring constant” varying along the longitudinalcoordinate s. This is Hill’s equation, and it can be shown that its solution can beexpressed as:
x(s) =√ε · β(s) cos(ϕ(s) + δ) (1.2)
where ε (the emittance)2 and δ depend on the initial conditions, β is the so called betafunction, which depends only on the lattice, and φ(s) is called the phase advance. Sothe particles execute oscillations around the nominal orbit as they travel around theaccelerator. The number of these oscillations per accelerator turn is called the tune,and is typically (in Europe) termed Q. The tune is determined by the strengths of allquadrupoles around the accelerator, and it is a very important parameter for a variety ofreasons which are beyond the scope of this note. For our purposes, it is enough to viewthe beam as a system which likes to oscillate in transverse space at a natural frequencywhich is the tune. If it is excited at that frequency, the transverse excursions will be verylarge, eventually leading to complete beam loss. This is exactly analogous to exciting aresonator with a signal at its resonant frequency.Some accelerator diagnostics operations require driving the beam at large displace-
ments with respect to the nominal orbit. The most obvious one is the measurement ofdynamic aperture, which is the available effective aperture for the beam after all mag-netic imperfections of the lattice are taken into account. This is of course smaller than
2This definition of emittance, as the area of the ellipse drawn by the particle in its trajectory intransverse phase space divided by π, is taken from the LHC Design Report [3]. Other sources,such as [4], define emittance as this area without dividing by π. With this alternative definition ofemittance, there would be a
√π in the denominator of Equation 1.2.
4
1 Introduction
the physical aperture which is determined by the shape of the beam pipe and other phys-ical obstacles such as collimators. The way of measuring dynamic aperture is drivingthe beam to larger and larger amplitudes until it is lost. The problem in the LHC isthat the beam is very “rigid”. It has a lot of momentum and it is very difficult to kickit out of its nominal trajectory.The AC Dipole principle uses the fact that the beam shows a large response to exci-
tations at frequencies close to the tune to generate large displacements using relativelylittle power. In the LHC the revolution frequency (the inverse of the time it takes aparticle to complete one turn) is 11245 Hz. The fractional part of the tune is expectedto be around 0.3 in both the horizontal and the vertical planes. This means that we cangenerate large transverse deviations in the beam if we power a dipole with an oscillatingcurrent at a frequency of around 11.245·0.3≈3.4 kHz. Other frequencies are also pos-sible [8], but we will concentrate on 3 kHz operation in the beginning because there areno compelling arguments at this point to try other options.Another significant advantage of the AC Dipole type of excitation is that it causes
virtually no emittance growth in the beam, so that it can be reused many times over fordifferent measurements if needed.
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2 AC Dipole Specifications
The emittance used in equation 1.2 scales as 1/p, where p is the momentum of theparticles in the beam. It is convenient to define a normalized emittance which, underideal conditions, stays constant through particle acceleration in the LHC, i.e. from450 GeV to 7 TeV in energy:
εn = εγrβr (2.1)
where γr and βr are the relativistic gamma and beta factors:
βr = v
c(2.2)
γr = 1√1− β2
r
(2.3)
(v is the particle velocity and c is the speed of light in vacuum). For LHC, the normalizedemittance is [3] εn = 3.75µm·rad. This emittance is the average of the individual particleemittances in the beam. Looking at equation 1.2 it seems natural to define the transversebeam size in a given point s in the accelerator as:
σx,y(s) =√εx,y · βx,y(s) (2.4)
where the x and y subscripts denote horizontal and vertical beam sizes respectively.For the LHC AC Dipole system, we will use the four MKQA magnets originally de-
signed to kick the beam with high amplitude current pulses. There is one magnet perbeam and per transverse plane. Table 2.1 summarizes the beam sizes at each of thefour locations for these magnets, both for injection and collision energy. Note that thenormalized emittance is expected to degrade a bit through acceleration. Also note thatthere is no separate mention of horizontal and vertical beam sizes. Beams in the LHCare assumed to be “round”.AC Dipole theory tells us that a good approximation to the displacement experienced
by a test particle under the influence of an oscillating dipole field is [2]:
x(s, n) = BmL
4πBρδ
√β(s)β0 · cos (2πνmn+ φ(s) + φ0) (2.5)
where:
• Bm (T ) is the amplitude of the oscillating magnetic field in the AC Dipole.
• L (m) is the length over which the AC Dipole field acts, i.e. the effective magnetlength of the AC Dipole.
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2 AC Dipole Specifications
Table 2.1: Beam sizes for injection and collision energy for each of the four AC Dipolemagnet locations.
Injection (450 GeV) Collisions (7 TeV)Parameter β εn ε Beam size (σ) εn ε Beam size (σ)Units m µm · rad µm · rad mm µm · rad µm · rad mm
• Bρ (T ·m) is to be taken as a single symbol and represents the magnetic rigidityof the beam. If the particle momentum p is expressed in GeV/c, the magneticrigidity can be computed in T·m as 3.336p.
• νm is the AC dipole tune, defined as the ratio between the AC Dipole frequencyand the beam revolution frequency.
• δ is the resonance proximity parameter, defined as the difference between the ACdipole tune and the beam betatron tune.
• βs and β0 are the beta functions (in meters) at the observation point and the ACDipole location respectively.
• n is the turn number.
• φ(s) is the phase advance of the free oscillation measured from the location of theAC Dipole.
• φ0 is the phase of the AC Dipole field when it first kicks the beam.
Since both the beam size and the response to the excitation scale as the square root ofthe beta function at the observation location, it is customary to express the requirementsfor beam displacement in units of beam size, therefore producing a specification whichis independent of the place of observation.
2.1 The loadThe magnet to be driven by the AC Dipole generator can be represented as an inductorin series with a resistance, as shown in figure 2.1. This magnet was originally foreseenfor another application where the current waveform is a pulse with a fast rising edge. Toavoid reflections, an impedance adapter network was placed in parallel with the magnet.In order to handle the large currents involved with little loss, ten coaxial cables areparalleled together to decrease series equivalent resistance. Taking into account thatthe real part of the characteristic impedance of the cables is 40Ω, this gives an effective
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2 AC Dipole Specifications
Coax1
Coax2
Coax10
Relay
Lbox
Lmagnet
200nH
1.315µH
Rmagnet
2mΩ
Csnubber
120nF
Rsnubber
4Ω
ReZ0=40Ω
Figure 2.1: Load to be driven by the AC Dipole generator
Ldz RdzCdz Gdz
Figure 2.2: Model of differential length dz of a lossy transmission line
ReZ0 of 4Ω, well matched to the impedance shown by the snubber network at highfrequencies. The selection between the pulse generator and the AC Dipole generator isperformed using a high power electromechanical relay.Since the magnet is shared by the pulse and AC Dipole generators, there is no way
to disable the termination network for AC Dipole operation. Fortunately its impedanceat 3 kHz is much higher than that of the magnet, so its influence on the operation canbe neglected to first order. However, it is important to keep in mind its existence toquantify the current it will draw during normal operation and make sure it is compatiblewith the ratings of the components.The cables have a specified attenuation of 0.14 dB/km at a frequency of 1 kHz. Using
the model depicted in figure 2.2, and applying the appropriate√f scaling to account
for skin effect, we can calculate the series resistance of the cable per unit length at3 kHz[12], which turns out to be R = 2.2 mΩ/m. Taking into account the length of thecables (20 meters) and the fact that ten of them are paralleled together to form a lesslossy transmission line, we therefore expect the cables to give a resistive contribution ofaround 4.4 mΩ to the overall load.For our purposes, we will neglect G (the shunt conductance) and use the values quoted
in the datasheet for the other two parameters: L = 224 nH/m and C = 140 pF/m. Thenif we define Z = R + jωL and Y = G + jωC = jωC, we can find the characteristicimpedance of the line, which for a lossy line is a complex number defined by
Z0 =√Z
Y(2.6)
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2 AC Dipole Specifications
1688 A
200 ms min. 200 ms min.~200 ms
Figure 2.3: AC Dipole signal envelope specifications.
and we can use this characteristic impedance to calculate the input impedance seen bythe amplifier:
Zi = Z0
[ZL cosh (γl) + Z0 sinh (γl)Z0 cosh (γl) + ZL sinh (γl)
](2.7)
where ZL in our case is the load impedance, i.e. everything to the right of the coaxialcables in figure 2.1, γ is the so-called propagation constant defined by γ = α+jβ =
√ZY ,
and l is the length of the line. The propagation constant has a real part α called theattenuation constant which tells us the rate of exponential attenuation of a voltage wavetraveling through the coax, and an imaginary part β called the phase constant whichtells the amount of phase shift per unit length for the wave.At 3 kHz, we have, referring to the diagram in figure 2.1, ZL = (0.002 + 0.029j) Ω
and Zi = (0.006 + 0.037j) Ω, in agreement with our expectation of an increment of theresistive part by 4 mΩ due to the cables. This input impedance is that of a series Rs-Lscombination with Rs = 4 mΩ and Ls = 1.96 μH.
2.2 PowerIn LHC, to measure dynamic aperture at injection, we will need a 7σv displacement withδ=0.025. This high value of δ is mainly due to uncertainties in the tune of the beam andalso in the tune spread within the beam. Plugging these numbers in equation 2.5 andusing the location of the least favorable AC Dipole (ring 2 vertical), we get a necessaryintegrated field BmL = 17.5 mT·m, which for that magnet results in a current of 1688 Apeak, i.e. 1193 A rms. There is also a need to drive 4σv oscillations at collision energyfor δ=0.01, but the needed current in that case is less than for the dynamic aperturemeasurement at injection.The actual current waveform in the magnet will be a sine wave with a trapezoidal
envelope. The rising and falling segments need to have a minimum duration of 200 msto avoid emittance blow-up of the beam. The duration of the flat top at maximumcurrent is currently 200 ms. The excitation will not be sent to the magnet more oftenthan once a minute, in order to avoid potential heating problems in the high power relay.
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2 AC Dipole Specifications
There are two possible corrections which will need to be taken into account at alater stage: There will be some beta-beating at injection in the LHC, meaning that thenominal beta values will not be respected. Whether this will help our system or makeits life harder will only be known at LHC start-up. The estimated maximum magnitudeof this effect is 10%. Equation 2.5 is a simplification of the complete equation derived in[10], which works well when tunes are far away from half integers and δ is small. In thecase of LHC, and for the δ=0.025 required at injection, a correction of up to 8% couldbe needed [8].
2.3 BandwidthOnce we have found that we need to drive around 1700 A peak through a magnet for agiven amount of time, we still have to define the frequency range in which the systemshould be able to produce that drive. For LHC, the fractional part of the horizontaltune is expected to be comprised between 0.28 (450 GeV) and 0.31 (7 TeV) while thefractional part of the vertical tune should be between 0.31 (450 GeV) and 0.32 (7 TeV).In addition, these tunes should be allowed to vary within ±0.01 without compromisingthe operation of the AC Dipole. So in the worst case, and if we want to have a singledesign for both horizontal and vertical planes, we have to accommodate a tune spanbetween 0.27 and 0.33. If we add to this the requirement to move by 0.025 on either side(see 2.2), we end up with a bandwidth requirement of 11245*0.11 = 1237 Hz, centered at11245*0.3 = 3373 Hz. In other words, we need to be able to drive 1700 A peak throughthe magnet at any frequency between 2754 Hz and 3992 Hz.
2.4 Spectral purityThe spectral purity specifications for the system are very hard to quantify because thespectral contents of the excitation waveform at some offset frequency with respect tothe carrier are not enough to evaluate the effect on the beam. This effect depends veryheavily on the carrier frequency itself. It is assumed that the spectral quality of highfidelity audio amplifiers (see 3.2), enhanced by the filtering action of the RLC resonantcircuit (see 3.1), is enough for our purposes.
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3 Fixed resonance frequency solution
As explained in 2.1, the AC Dipole generator must drive a load composed of the relaycontacts followed by 20 meters of coaxial transmission line (made up of 10 coaxial cablesin parallel) and the magnet itself. At the frequency of interest, this can all be lumpedinto a network composed of an inductance and a resistor in series to account for all thelosses. The magnet inductance is very low, so the parasitic inductances in the signalpath are far from negligible. This series RL network can be converted to a parallel RLusing the formulae in Figure 3.1.Once we have the parallel arrangement, we can add a capacitor in parallel to form a
parallel resonant RLC circuit. Since the frequency of excitation is conveniently locatedin the audio range, we can use standard off the shelf audio amplifiers as exciters. At theRLC resonant frequency, these amplifiers will see only the R as a load, which has manychances of not being the optimal one for maximum power delivery. Therefore, it wasdecided that a transformer would be inserted between the RLC and the amplifiers tomatch this load. In fact, to satisfy the requirements from section 2.2 using commercialamplifiers, we must use two of them and combine their outputs using transformers.These transformers play therefore a triple role:
• they provide galvanic isolation so that we can use the amplifiers’ mono bridgeoutput configuration (see 3.2) even if the magnet return is grounded.
• they allow the combination of the power out of two amplifiers.
• they match the load (using the appropriate turns ratio) to allow optimal powertransfer.
The overall system can be seen in figure 3.2. This arrangement is similar to the one usefor the AC Dipole in FNAL [7].
3.1 RLC resonatorAs represented in figure 3.1, a series combination of a resistor Rs and an inductor Ls(reactance Xs) can be converted, for a given operating frequency, into an equivalentparallel network of a resistor Rp and an inductor Lp (reactance Xp). The quality factorQ is defined as Xs/Rs for the series network and as Rp/Xp for the parallel network. Itis easy to verify that the conversion formulae in figure 3.1 leave Q invariant.Now the question is: in which respect does the capacitor we add help the amplifier? In
principle, if we treat the amplifier as a voltage source V and we take the parallel RpLp
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3 Fixed resonance frequency solution
Rs
jXs
Rp jXp
Xs
RsXsXp
22
Rs
XsRsRp
22
22
2
XpRp
XpRpXs
22
2
XpRp
XpRpRs
Figure 3.1: Parallel-Series conversion formulas.
Amplifier 1
Amplifier 2
LoadCap bank
Figure 3.2: Overall system diagram.
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3 Fixed resonance frequency solution
model of the magnet, the active power needed will be V2/Rp irrespective of whether weinclude the capacitor in parallel or not, so it is not a matter of active power.To better investigate the issue, it is important to understand that all audio amplifiers
have a maximum voltage and a maximum current they can deliver. The amplifier candeliver simultaneously the maximum voltage and current for an optimal value of loadimpedance Vmax/Imax. Lower impedances will result in current limiting and higherimpedances will produce voltage limiting. In both cases, the power delivered to the loadwill be less than the rated power for the amplifier.Now let’s consider a non-tuned circuit and see how much current we can put through
it. Its impedance is Znon−tuned =√X2
s +R2s = Rs
√1 +Q2. This impedance is probably
lower than the optimal one, so we insert a transformer with primary to secondary turnsratio N1
N2=√
Zopt
Znon−tuned. This gives us a final current in the magnet Inon−tuned =
Imax
√Zopt
Rs
√1+Q2
.Let’s now turn to a tuned solution. At resonance, the amplifier only sees Rp, which
again has many chances of being lower than the optimal impedance value. Now weinsert a transformer with turns ratio N1
N2=√
Zopt
Rp, which gives us a resistor current
IR = Inon−tuned
(1 +Q2)−0.25 after some Math. But the total current in the magnet is
IR
(1 +Q2)0.5 so we have a gain in current with respect to the non-tuned solution:
Ituned
Inon−tuned=(1 +Q2
)0.25(3.1)
In our case, with an estimated Q of 6.35 (see figure 3.3) the gain in magnet currentexpected by using a capacitor bank is therefore 2.5. Note that a current factor of 2.5 isequivalent to a power factor of 6.25, i.e. we would need to couple the power of 6 or 7amplifiers to achieve a similar current gain. Another important difference between thetuned and non-tuned cases is that the former leaves the amplifier with a purely resistiveload, while the latter shows a very inductive one. Depending on the amplifier technologyone could have problems if there are big phase differences between the output voltageand current.The magnets used in the AC Dipole project are built around a metalized ceramic
vacuum chamber (see figure 3.4). The metallization consists of a 5 μm thick layer of Ti-tanium, which ensures that image currents in the beam pipe will not hit a high impedancewhen they arrive at the AC Dipole. On the other hand, the layer is thin enough to guar-antee that the magnetic field will reach the beam with virtually no degradation, sincethe skin depth of Ti at 3 kHz is much larger than the layer thickness. Indeed:
δ =√
2ρωµ
(3.2)
So the skin depth of Ti at 3 kHz (ρ=0.420μΩ·m, μ≈μ0=4π·10-7H/m) is 6 mm. Themagnet itself is made of a single turn of solid copper to achieve a very low inductance.This has the advantage of guaranteeing a very fast response, but the negative side is
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3 Fixed resonance frequency solution
0.000
0.050
0.100
0.150
0.200
0.250
0.300
0.350
0.400
0.450
0.500
0.00
1.00
2.00
3.00
4.00
5.00
6.00
7.00
1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
Vm
ag
ne
t/Im
ag
ne
t (O
hm
)
Ima
gn
et/
Ige
n
f (kHz)
CROWN I-T800 amplifier with 760uF capacitor Q=6.35 Rp=0.46 Ohm
Imagnet/Igen
Vmagnet/Imagnet
Figure 3.3: Measurements of Imagnet/Igenerator and Vmagnet/Imagnet for a frequencysweep between 1 and 5 kHz using 760μF in parallel with the magnet. Thevalues of the peaks of the two curves give us Q (roughly) and Rp respectively.
that a lot of current will be needed to achieve a given magnetic field inside the beampipe. The yoke of the magnet is made of stacked C cores, which are themselves madeby winding 50 μm Si-Fe tape to form toroids, and then cutting off a part of the toroidto get the C shape. The whole assembly can be split in half to allow easy mounting anddismounting without touching the beam pipe.
3.2 Power amplifiersThe availability of high power class-D (switching) audio amplifiers is very convenient forthe project. Three audio amplifiers have been studied:
• Crown I-T8000.
• Lab.Gruppen FP13000.
• PowerSoft DIGAM K20.
Items to bear in mind for the choice include:
• Nominal output power. This is of course crucial but is only half of the story. Theother half is for how long can this output power be sustained. For the AC Dipoleproject we take the figure for 1 second, since longer excitations are not foreseenfor the time being.
• Mono bridge mode. In this mode, the two output channels (A and B) in a stereoamplifier are combined into a single one with twice the power of each channel.This is done by generating an inverted version of input A and feeding it to channelB. The output is then taken between the hot terminals of A and B, thereforeachieving twice the voltage as each stereo channel. The down side is that the load
Figure 3.4: Details of the AC Dipole magnet design. Courtesy of L. Ducimetière.
return cannot be grounded in this case. This is one of the reasons why we usetransformers (see figure 3.2).
• Spectral quality. Initial commissioning with beam has shown that the spectralquality delivered by high fidelity audio amplifiers is enough for our purposes.
• Availability of a digital audio input. It can be very convenient if we are forced toplace our low level signal generator far away from the power electronics rack.
• Availability of an open standard (RS-485, Ethernet...) to communicate with theamplifiers from a Linux PC in order to control (ON/OFF) and to retrieve diag-nostics information.
• Power grid friendliness. Power Factor Correction (PFC) is certainly a plus. Also,soft starts and other techniques to minimize large surges on the 220V grid are veryimportant. This has an impact on the choice of electrical installation equipment,such as the type of breakers used to protect the installation. For example, type Kbreakers designed for continuous operation at a rated current In, will not trip onshort current spikes smaller than 8·In.
The final decision has been to select the K20, mainly due to its higher output power.
3.3 TransformersMany types of magnetic materials are available to form the core of the transformersneeded to couple the output power of the two audio amplifiers. One important trade-offconcerns the magnetic losses vs. saturation B field for a given material. Iron-basedmaterials typically have big losses and big saturation fields. On the other extreme,ferrites have extremely low losses but also lower saturation fields.
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3 Fixed resonance frequency solution
In general, the magnetic losses in a given material follow the following empirical law:
PF e = k · F a ·Bb (3.3)
where:
• PFe are the magnetic losses in mW/cm3.
• F is the frequency in kHz.
• B is the magnetic field in mT.
• k, a and b and empirical parameters for each material.
For normal applications without any forced cooling, 60mW/cm3 is considered appro-priate for a working cycle of 100% (system continuously on). In our case, we specifiedthat users can only drive a trapezoidal excitation into the AC Dipole every minute, andthis excitation will never exceed 1 second in duration. We can therefore go higher inmagnetic losses if needed.The material we chose belongs to the amorphous family, which is somehow half way
between the iron powders and the ferrites. If has lower losses than the former and higherBsat than the latter. We decided to operate at a Bmax of 1.1 T which gives us lossesof 411mW/cm3. The material chosen is Ener-GTM from AMS, and its parameters arek = 7.53 · 10−5, a = 1.5 and b = 1.98. Once we specify the currents and voltages in theprimary and secondary, the turns ratio is defined by their ratios. The absolute numberof turns is derived from these quantities, the core size, the frequency and the Bmax wewant in the core.The chosen core has a double-C geometry and the windings are made of copper foil.
This copper foil is thicker in the secondary, where current is higher, so that the powerdissipation is uniform across the transformer.The transformers used in the four operational LHC systems were manufactured by
CME, and tests so far have been very satisfactory. The primary leakage inductance isin the order of 100µH, and is seen by the amplifiers in series with the load. Far frombeing a problem, this allows the amplifiers – which are in principle voltage sources – todrive a load which contains parallel capacitors.
3.4 Configurable capacitor bankThe capacitive part of the RLC resonant circuit used in the AC Dipole is implementedwith a set of capacitors in parallel. The frequency range to cover in the excitation of thedipole is quite large, and a fixed resonant circuit of reduced bandwidth is not adequate,so it was decided to have a 5-stage capacitor bank, with the capacitance Cn being twicethe value of that for Cn−1, therefore resulting in 25 = 32 possible combinations. In thisbank, depicted in figure 3.6, there is a bulk capacitance component and then additionalcomponents that can be added to this bulk value by screwing copper plates that connecta given stage to the bulk area. In the future, these copper plates may include MOSFET
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3 Fixed resonance frequency solution
Figure 3.5: Transformer for the AC Dipole system.
transistor switches, so that the screwing/unscrewing action is replaced by configurationbits driven by a computer, therefore allowing remote reconfiguration. A physical viewof the bank can be seen in figure 3.7.In order to evaluate the losses induced by imperfect capacitors on the system, one
needs a model of these capacitors which includes these imperfections. There are threetypes of losses typically modeled:
1. The most important one is dielectric losses, i.e. the losses induced by the fact thatcharges in the dielectric move around (although they are not set free) and collidewith other charges, heating the material. These losses are typically modeled by aresistor in parallel with the perfect capacitor. A typical way of specifying theselosses is through a figure of merit call ‘tangent δ’ or ‘dissipation factor’, and defined
−
+
−
+
Input 2CC 4C 8C 16C
Rs
LsCbulk
Figure 3.6: Capacitor bank and its place in the overall system.
17
3 Fixed resonance frequency solution
as tan (δ) = CωRd
, where Rd represents the parallel resistance which accounts fordielectric losses. If all losses were dielectric, we see that the quality factor of acapacitor would just be the inverse of tan(δ). As an example, the dissipationfactor of polypropylene at 1 kHz is tan (δ) = 2 · 10−4, corresponding to a qualityfactor of Q=5000.
2. There are also losses due to the fact that the dielectric does not isolate perfectly.It has a finite resistivity, and there is therefore a minuscule conduction currentthrough it. This effect is modeled by another parallel resistor, which is muchbigger in value than Rd, representing the fact that this effect is negligible withrespect to dielectric losses.
3. Finally, there is series resistance Rs in the capacitor contacts and electrodes, sincethey are made of metal with a finite conductivity. This resistance is placed inseries with the parallel C//Rd combination.
The combined effect of these three mechanisms for losses can be represented by a singleEquivalent Series Resistance (ESR) in series with the perfect capacitor. As an example,at 3 kHz, the ESR for Electronicon model E53.R60-333T20 is 0.52 mΩ, which combinedwith its nominal capacitance of 33 μF gives a quality factor Q=1196, lower than the5000 we obtain if only the losses in the polypropylene are taken into account. It is easyto see that putting these capacitors in parallel with the magnet will not alter the Q ofthe system, since their equivalent parallel resistance will be much greater than the onefor the magnet (in fact their ratios will equal the Q ratios at resonance), so the parallelcombination of these resistances will be almost equal to the magnet’s parallel resistance.The losses in the capacitors can therefore be neglected in our system if our focus is onthe accounting of overall losses. Of course they have to be taken into account to properlydesign the capacitor bank itself.The family we chose for the capacitor bank is the E53 series from Electronicon, which is
based on rolls of low loss polypropylene dielectric film with metal coating. The genericname of this technology is MKP (Metalized Polypropylene). The coating is a self-healing mixture of Zinc and Aluminum, and the whole roll is impregnated with solidifyingpolyurethane resin in order to protect the electrodes from oxygen, humidity and otherperturbations. The dielectric is therefore the mixture of polypropylene and resin. MKPtechnology features low losses and a low temperature coefficient (−200 · 10−6K−1).One of the options to tune the resonant frequency of the RLC tank automatically
is to replace the mechanical switches by MOSFET ones. Preliminary choices for thecomponents in the switches needed to activate each capacitor bank stage are
• XFN180N20 MOSFET transistor.
• V-INFINITY CONVERTER DC/DC 12V out, 1W, in order to generate the iso-lated supply needed by the drivers. This component generates 12V from a 5Vsupply such as those found in PCI busses. If stronger decoupling from the controlPC is needed, an external power supply powered off the mains would be used.
18
3 Fixed resonance frequency solution
Bulk
Bit 0
Bit 1
Bit 2
Bit 3
Bit 4
Figure 3.7: Mechanical drawing of the capacitor bank layout and packaging of the Elec-tronicon E53 series of power electronics capacitors.
• FOD3180TV MOSFET drivers.
These switches can be configured statically before an excitation. If the bank is arrangedusing capacitor values scaling in powers of two, the system behaves as a binary wordselecting an effective capacitance value. The more bits we have, the finer the control.For even finer control, we can consider making the switches change state during theexcitation. Their duty cycle would then select the effective capacitance, but one hasto be very careful in selecting an appropriate algorithm for switch control: trying toimpose a voltage step on a capacitor will result in enormous currents due to the i =C dV
dt law. We have a preliminary choice for the solid state switches we could use inorder to implement a remotely configurable capacitor bank. The XFN180N20 MOSFETtransistor can handle DC currents of up to 180A and has an ON resistance of 10 mΩ.One can make a bidirectional switch with two MOSFETs in series . A single MOSFETwould allow current flow in the off state if negative VGS were applied, due to forwardbias of its internal parasitic diode. By connecting two MOSFETs in series, source tosource, the internal diodes are opposed, blocking unwanted current of either polarity inthe OFF state. Through simulations, we can calculate how many amperes each of thestages in the capacitor bank will have to support in different configurations, and usethis data to decide how many MOSFET switches we should use in parallel to fulfill thatrequirement. MOSFET’s ON resistance has a positive temperature coefficient, thereforeallowing easy paralleling of many devices without risks of thermal runaway.
19
3 Fixed resonance frequency solution
3.5 Local controlThe waveform sent to the amplifiers comes from a PCI card sitting in an industrial PCrunning the Linux operating system. The selected card for this task is the X3-Servofrom Innovative Integration. It features six 1 MS/s DAC channels and six 250 kS/sADC channels, all connected to a Xilinx Spartan 3 FPGA. In addition, it features 48digital I/O lines which can be used to control the switches in the capacitor bank in alatter phase of the project. One of these lines is used as an input for an external triggerpulse which starts the generation of the waveform. This trigger pulse comes from aCERN-standard CTRI timing receiver card, which is programmed to provide a pulseupon reception of a special AC Dipole trigger message on the timing network. TheADC channels in the card are used to monitor several current and voltage waveformsin the system. Some of these signals can be used to perform current feedback for safetypurposes and also to make the audio amplifier (a voltage source by definition) behave asa current source. For the purpose of monitoring current waveforms, the different currentsin the system are translated into voltages using current transformers manufactured byPearson Electronics. Another important aspect of the controller card concerns safety.Thanks to its programmable nature, some limitations can be hard-coded in hardwarein the future so that it is impossible to operate the AC Dipole in dangerous ways. Anexample is the minimum ramping time for the excitation. It has been proven that aminimum ramping time of 200 ms guarantees that the beam will take at least 100 turnsto be lost, even if the excitation frequency is exactly equal to the tune. This is enoughtime for independent protection systems to react.
3.6 Remote controlRemote control of the system from the CERN Control Center (CCC) makes use of a JavaGraphical User Interface (GUI) which communicates with the front-end PC throughthe Ethernet network using CERN-standard middleware. This application sends theappropriate configuration parameters to the front end, namely:
• The desired frequency of the excitation in tune units.
• The desired amplifier input voltage amplitude during the flat top in % of thefull-scale amplitude.
Additional features to be considered in the future include:• The possibility of forcing a tune measurement before the system is allowed to
operate.
• On-line display of beam intensity, energy and safe beam flag as read by the appro-priate decoder in the AC Dipole front end computer.
In any case, all security-related features of the system are dealt with at the hardwarelevel to avoid accidents due to misuse of the software. The overall control system canbe seen in figure 3.8.
20
3 Fixed resonance frequency solution
Front end PC
Waveform
generator
Timing receiver
Technical Network
CERN Control Centre (CCC)
LHC Point 4
Capacitor bank
Magnet
Figure 3.8: Overall AC Dipole control system
21
4 Modeling
This chapter sets the theoretical framework one can use to study ways of tuning theresonance frequency of the AC Dipole system in an automated way. Chapter 5 thendetails possible practical schemes. The electrical system can be modeled using the statespace formalism. In this formalism, inductor current and capacitor voltage are statevariables and can’t be arbitrarily modified (this would require infinite energy). To acton the system, one modifies either inductor voltage or capacitor current or a combinationof both. We need to develop these schemes using block diagrams. Voltage and currenttransformers are the actuators needed for such systems.Our first ideas for modeling the system include the use of functional representations
of all physical components. For example, a capacitor is modeled as a box with a currentinput and a voltage output, as depicted in figure 4.1. The transfer function is 1/sC.Conversely, an inductor has a voltage input and a current output, with a transfer function1/sL. Notice that both transfer functions contain an integrator. This is a consequence ofcausality: a physical system can only know its input at the present time and also in thepast, but it can never know what its input will be in the future. The choice for inputsand outputs of these two components reflects this fact.The outputs of these blocks will be state variables in our representation of the global
system. As mentioned previously, any external action on the system can only involvechanging the capacitor currents or the inductor voltages, i.e. the inputs to these blocksand never their outputs. A hybrid solution using a linear combination of these twoactions can also be considered. The final decision should be a function of how easy andefficient it is to change any of these parameters. Figure 4.2 depicts the two possible
1sC
Capacitor
Current Voltage
1sL
Inductor
CurrentVoltage
Figure 4.1: Capacitor and inductor functional models.
22
4 Modeling
1sC
Capacitor
1sL
Inductor
R
Resistance
Icadd Inductor current
1sC
Capacitor
1sL
Inductor
R
Resistance
Inductor current
Ucomp
-1
Figure 4.2: Influencing schemes for the AC Dipole resonant system.
influencing schemes. The first one consists of injecting an additional current Icadd tothe inductor current which is fed to the capacitor block. Physically, this would takethe shape of a current source wired in parallel with C. The second approach consistsof adding an external voltage Ucomp to the capacitor voltage before feeding it to theinductor block. Practically, this would involve connecting the secondary winding of avoltage transformer in series with the inductor. Then we could drive the primary of thattransformer with a voltage source.
23
5 Resonance frequency tuning schemes
The main problem of the fixed resonance frequency solution is that the Q of the systemis such that the resulting bandwidth – i.e. the frequency span over which we can drive1700 A peak into the magnet – is too narrow. In order to increase the current throughthe magnet at a given frequency there are two options: either increasing the total drivepower or taking the system’s resonance frequency closer to the frequency of interest. Thischapter describes the different solutions employed in other branches of power electronicsfor solving similar problems.One important remark is that tuning a resonance frequency is essentially the same
thing as compensating the power factor of a load or compensating its reactive power.These are all ways of speaking about the same phenomenon. Below is a list of thedifferent domains where this problem arises:
• Variable speed drives [9]. The existing solutions rely mainly on standard inverters,and no attempt at controlling reactive power. This corresponds to the “bruteforce” approach described above: whenever there is a need to increase speed, morepower is injected by the drive into the load.
• Power factor correction in electrical transmission and distribution [1, 13]. In thiscase, the real resonance is affected, i.e. there is a control of the reactive power re-quested by the load and this control is independent of the main source of power. Ofparticular importance are a new generation of schemes collectively known as Flex-ible AC Transmission Systems (FACTS), which include the switched capacitors,controlled reactors – both part of Static Var Compensators [5] – and STATCOMsolutions described below.
• Induction cooking and heating [6]. In this case, the requirement for tunable reso-nance frequency stems from the fact that either the inductive load varies in valueor the frequency of excitation needs to be controlled in order to achieve a givenpenetration profile in the heating of metals.
5.1 Switched capacitorsSwitched parallel capacitors are the easiest way to introduce tunability in a resonantsystem. Figure 5.1 shows the principle, without the driving source. Capacitors can beincluded or excluded from the circuit permanently by closing or opening a solid stateswitch in series with them. This implies discrete jumps in resonance frequency, since ingeneral emulating a variable capacitance by pulse-width modulating the switch controlis more complex and error-prone than doing it using a reactor (see section 5.2):
24
5 Resonance frequency tuning schemes
Figure 5.1: Switched capacitor principle.
• Ideally the switch should be turned off at zero current, which means the capacitoris charged at peak voltage and will remain charged at that level after the switchis open. This can impose a doubled voltage stress in the non-conducting switchesand an increase in the number of switches in series might be needed in comparisonwith the controlled reactor case.
• Switch turn on is also a problematic situation since the capacitor has in the meantime lost a bit of its charge, and a connection with a mismatch in voltages can createlarge current transients. An external control circuit is responsible for evaluatingthe time at which a zero-voltage switch turn-on can be done.
5.2 Controlled reactorsThe principle of a controlled reactor – or inductor – for modifying resonance frequencycan be seen in figure 5.2. The switch is placed in series with the inductor and turn-on time is determined by sensing the line voltage and choosing an offset phase withrespect to its zero crossing. Assuming the turn-on happens while the line voltage ispositive, the current through the inductor then starts rising and it begins falling whenthe voltage crosses zero. The zero crossing of inductor current is a natural place to turnthe switch off, and the same cycle can be started on the other half of the sinusoidalperiod. The result is a periodic series of bumps in current with a period equal to that ofthe line voltage. The fundamental component of the Fourier expansion of this currenthas an amplitude which can be varied by playing with the duty cycle of the switchcontrol, therefore enabling the emulation of a variable inductance. The harmonics needto be filtered out, but in our case their effect would be less important because we areinterested in magnet current, and most of the high frequency current would flow throughthe capacitors rather than through the inductors.
25
5 Resonance frequency tuning schemes
Figure 5.2: Controlled reactor principle.
5.3 Audio STATCOMThe STATic COMpensator is a more evolved scheme allowing the injection of active andreactive currents with very little harmonic content. Although typically used at 50 Hz,there is no reason to believe it would not work at audio frequencies. The emulation of avariable reactance is achieved by means of a Voltage Source Converter (VSC) fed by acapacitor and whose output drives a large inductor. The series combination of the VSCand large inductor is placed in parallel with the resonant circuit (figure 5.3). Let’s firstsee what happens when the VSC drives into one end of the inductor a voltage signalequal in magnitude and phase to the line voltage. The inductor then sees no voltagedrop at its ends, so no current flows. The VSC-inductor series combination behaves inthat case as an open circuit. Now imagine we stay in phase with the line voltage butdrive the output of the VSC at a lower amplitude. As we decrease this amplitude, moreand more current will flow into the inductor, emulating smaller and smaller inductancevalues, until we hit the limit of zero amplitude at the VSC output. In this case, theeffective inductance is just Lstat. This is the smallest possible inductance value thesystem can emulate. If, on the other hand, we drive a voltage waveform on one end ofthe inductor whose magnitude is bigger than that of the line voltage – and still assumingperfect phase matching – the current will have a 180 phase shift with respect to theone which flowed through the inductor at lower excitation levels, i.e. the system nowemulates a capacitor. The lower bound of capacitance is 0 (when line voltage and VSCvoltage are equal in magnitude) and the upper bound is dictated by the upper outputvoltage limit of the VSC.One interesting feature of the STATCOM is that a slight de-phasing of its voltage can
allow the generation of active current in addition to the reactive current discussed so far.This effect can be used to charge the storage capacitor to a value set by a control system,compensating for switch losses inside the VSC. Since this value will also determine theoutput voltage of the VSC for a given modulation scheme, phase control alone is enoughto emulate all values of capacitance and inductance.
26
5 Resonance frequency tuning schemes
VlineVstat
StorageVSC
Lstat
Figure 5.3: STATCOM principle.
Another point worth mentioning concerns the spectral purity of the reactive currentsinjected and absorbed by the STATCOM. With a suitable PWM algorithm, the spectralspikes in the current spectrum can be pushed very high in frequency, resulting in easyfiltering and therefore a better magnet current in our application.It is very instructive to see that the STATCOM can be seen as a possible implemen-
tation of both schemes for influencing the resonator we described in chapter 4. Thecombination of a large inductor in series with a voltage source can indeed be seen as acurrent source modifying the current in the capacitor. At the same time, if we focus onthe large inductor of the STATCOM, we see that the VSC is effectively changing thevoltage across it. So changing current through a capacitor or changing voltage across aninductor – the two schemes we described to influence the system and change its resonancefrequency – are embodied by the same circuit diagram.
27
Acknowledgements
The LHC AC Dipole is a complex system. It could not have been successfully commis-sioned without the help of many people:
• The BE-CO-HT team developed the full system. The authors would like to singleout help from Olivier Barriere, Benjamin Ninet, Ioan Kozsar, Julian Lewis andErik van der Bij.
• The TE-ABT team graciously allowed the hosting of our generator in their rackand helped with integration.
• The BE-OP group developed the GUI which allows remote control of the AC Dipolefrom the CCC.
• The US-LARP collaboration helped with the capacitor bank and interpretation ofthe first current waveforms during hardware commissioning.
• The BE-ABP group helped with specifications and initial interpretation of acquireddata.
• The TE-EPC group helped modify the solenoid saver circuit to eliminate the 100Hz hum in the power relay.
Thanks are also due to the two successive BE-CO group leaders who supported thisproject so it could be brought to a successful end: Hermann Schmickler (the initiator ofthe project) and Eugenia Hatziangeli.
28
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