METHOD FOR DETERMINING A QUALITY FACTOR OF AN ACCELERATING CAVITY OF A PARTICLE ACCELERATOR

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First Claim
1. A method for determining a quality factor of an accelerating superconducting cavity of a particle accelerator, the method comprising:
 determining a heat load to which a cryomodule having the accelerating cavity and a bath of cryogenic fluid is subjected; and
determining a quality factor based on the determination of the heat load during the operation of the particle accelerator.
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Abstract
The method for determining a quality factor of an accelerating superconducting cavity of a particle accelerator, in particular a linear particle accelerator, the method includes
determining a heat load to which a cryomodule having the accelerating cavity and a bath of cryogenic fluid is subjected, then
determining a quality factor based on the determination of the heat load during the operation of the particle accelerator.
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20 Claims
 1. A method for determining a quality factor of an accelerating superconducting cavity of a particle accelerator, the method comprising:
determining a heat load to which a cryomodule having the accelerating cavity and a bath of cryogenic fluid is subjected; and determining a quality factor based on the determination of the heat load during the operation of the particle accelerator.  View Dependent Claims (2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20)
1 Specification
The invention relates to a method for determining a quality factor of an accelerating cavity of a particle accelerator, in particular a linear particle accelerator. The invention also relates to a method for operating a particle accelerator. The invention furthermore relates to a device for determining a quality factor and to a particle accelerator comprising such a device.
Within particle accelerators, in particular linear particle accelerators, superconducting accelerating cavities (referred to as RF cavities) perform the acceleration of particles. These cavities are manufactured from a very low temperature superconductor such as niobium, and are submerged in a volume of cryogenic fluid, such as in particular helium. The performance of an accelerating cavity, and in particular the maximum power able to be accepted by the cavity, depends on its quality factor. This factor is directly related to the surface state and to the geometry of the cavity. Under certain conditions, the cavity may lose its superconducting state, this triggering an effect known as quench that leads to a complete stoppage of the beam in the particle accelerator. Degradation of the quality factor also degrades the accelerating capacity of the cavity in question.
Currently, quality factor is measured using a probe that measures an electromagnetic field in the cavity. This measurement is carried out when the accelerator is offline, generally before the startup of the installation or during maintenance periods. In particular, the document “Improvement of the Qfactor measurement in RF cavities” by Wencan Xu, S. Belomestnykh, and H. Hahn (BNL Technical Note, BNL98894(Ipac 13): 24892491, 2013) describes such a method. However, it is impossible to use this measuring method when the particle accelerator is in use.
The aim of the invention is to provide a method for determining a quality factor of an accelerating cavity of a particle accelerator that remedies the aforementioned drawbacks and that improves the devices and methods known from the prior art. In particular, the invention allows continuous measurement of quality factor. The obtained measurement has the advantage of being able to be carried out when the accelerator is in use. Moreover, the determination of the quality factor according to the invention may allow deterioration of the cavity to be detected.
The invention relates to a method for determining a quality factor of an accelerating superconducting cavity of a particle accelerator, in particular a linear particle accelerator, the method comprising the following steps:
 determining a heat load to which a cryomodule comprising the accelerating cavity and a bath of cryogenic fluid is subjected, then
 determining a quality factor based on the determination of the heat load during the operation of the particle accelerator.
The steps of determining the heat load and of determining the quality factor may be carried out simultaneously and in real time.
The step of determining the heat load may comprise the use of a state observer.
The state observer may comprise an estimation of a mass flow rate of cryogenic fluid passing through a valve of the cryomodule taking the form
{dot over (m)}=β_{T}·{dot over (m)}_{comp}+(1−β_{T})·{dot over (m)}_{incomp }
in which:
{dot over (m)}_{comp }is a mass flow rate of cryogenic fluid in compressible form through the valve,
{dot over (m)}_{incomp }is the mass flow rate of cryogenic fluid in incompressible form through the valve, and
β_{T }is a coefficient of isothermal compressibility of the cryogenic fluid.
The state observer may comprise an estimation of a density and of a specific internal energy of the bath of cryogenic fluid.
Said estimation may be carried out based on:
 a volume of cryogenic fluid in the liquid state, which is calculated based on a measurement of a height of the cryogenic fluid in the liquid state and/or calculated based on a measurement of the amount of cryogenic fluid entering and exiting the bath of cryogenic fluid; and
 a static heat load and a dynamic heat load received by the bath of cryogenic fluid; and
 an input specific enthalpy and an output specific enthalpy of the cryogenic bath, based on a measurement of the pressure of the bath of cryogenic fluid, or
 an output temperature of the bath of cryogenic fluid, based on a measurement of the pressure of the bath of cryogenic fluid and on the input mass concentration of the bath of cryogenic fluid.
The invention also relates to a method for operating a particle accelerator, in particular a linear particle accelerator, comprising at least one accelerating cavity, the operating method comprising implementing the method for determining a quality factor of at least one accelerating cavity such as defined above and a step of modifying at least one operating parameter of said accelerating cavity depending on its quality factor.
Said operating parameter may be a power setpoint value for a radiofrequency wave emitted in the accelerating cavity, and the modifying step may comprise decreasing the value of the power setpoint if the quality factor of at least one accelerating cavity crosses a preset threshold, the other cavities of the particle accelerator, when they exist, being able to continue to operate.
The invention also relates to a device for determining a quality factor of at least one accelerating cavity of a particle accelerator, the determining device comprising hardware and/or software elements that implement the method such as defined above, in particular hardware and/or software elements designed to implement the method such as defined above.
The invention also relates to a particle accelerator, in particular a linear particle accelerator, comprising at least one determining device such as defined above.
The particle accelerator may comprise at least one cryomodule comprising an accelerating cavity or a plurality of accelerating cavities and a bath of a cryogenic fluid.
The invention also relates to a computer program product, comprising programcode instructions stored on a computerreadable medium, for implementing the steps of the method such as defined above, when said program runs on a computer, or a computer program product that is downloadable from a communication network and/or stored on a computerreadable and/or computerexecutable data medium, comprising instructions that, when the program is executed by a computer, lead the latter to implement the method such as defined above.
The invention also relates to a data storage medium that is readable by a computer, on which is stored a computer program comprising programcode instructions for implementing the method such as defined above, or computerreadable storage medium comprising instructions that, when they are executed by a computer, lead the latter to implement the method such as defined above.
With reference to
The helium in the liquid state is denser than the helium in the gaseous state. Under the effect of gravity, the helium in the liquid state therefore occupies a lower volume of the enclosure 6 of the cryomodule 3 whereas the helium in the gaseous state occupies a higher volume of the enclosure. The helium bath therefore behaves as a phase separator, i.e. a bath in which the equilibrium reached between the gaseous state and the liquid state of the same fluid is dependent on the pressure and temperature conditions. In the rest of the document, the expression “phase separator” will therefore also be used to refer to the helium bath contained in the cryomodule 3. The cryomodule is equipped with a level sensor LT, which is able to measure the height of helium in liquid form within the enclosure of the cryomodule.
The phase separator is subjected to a heat load that may be decomposed into two parts. On the one hand the phase separator is subjected to a measurable static heat load Q_{static}, due to heat exchange by conduction, convection and radiation between the external environment of the cryomodule at room temperature (i.e. about 300 K) and the cryogenic fluid at a temperature of 4 K. On the other hand, the phase separator is subjected to a dynamic heat load Q_{dynam }due to the power of the electromagnetic field in the cavity and/or to the passage of particles through the cavity. This dynamic load will be determined (in other words estimated, simulated or calculated) according to the description given below. From a thermodynamic point of view, the cavity 4 has no other effect than delivering additional heat to the helium bath.
The heat load may reflect the radiofrequency power injected into the cavity but not solely. It may reflect a degradation in the isolation vacuum, lowenergy electron emission from a radiofrequency coupler or in the cavity, field emission or online loss of the beam.
The thermodynamic model of a cryomodule 3′ equipped with two accelerating cavities 4 is equivalent to the thermodynamic model of a cryomodule 3 equipped with a single accelerating cavity 4. Only three parameters of these models differ the volume of the enclosure Vol containing the cryogenic fluid and the static heat load Q_{static }and dynamic heat load Q_{dynam }acting on the cryogenic fluid 5. The invention will be described in detail through the example of a cryomodule equipped with a single accelerating cavity. Those skilled in the art will be able to transpose these teachings to a cryomodule comprising two or more accelerating cavities.
A cryogenic system 10 comprises the cryomodule 3 and three valves CV001, CV002, CV005, allowing the cryomodule 3 to be connected to a circuit 11 for distributing helium. A first valve CV001 is a helium inlet valve and is connected to a lower portion of the helium bath, at a point where the helium is in liquid form (once the temperature of the helium has been decreased to its operating temperature). A second valve CV002 is also a helium inlet valve and is connected to an upper portion of the helium bath, at a point where the helium is in gaseous form. A third valve CV005 is a helium outlet valve and is connected to the upper portion of the helium bath, at a point where the helium is in gaseous form. The first valve CV001 may be used to fill the enclosure of the cryomodule with helium. The cryogenic system 10 need not comprise this first valve CV001 if the enclosure is fillable with helium in some other way. The first valve CV001 is not used for regulating purposes. The second valve CV002 may be used to regulate the helium level in the enclosure of the cryomodule. The third valve CV005 may be used to regulate the pressure in the enclosure of the cryomodule. The first and second valves CV001, CV002, which are referred to as supply valves, are connected to a line that supplies twophase helium. The third valve CV005, which is referred to as the exhaust valve, is connected to a return line. These lines are not shown in
A triplet of variables is associated with each input or output of the assembly of components: the internal pressure P (expressed in bars absolute), the specific enthalpy H (expressed in J/kg) and the mass flow rate {dot over (m)} (expressed in kg/s) represented by the letter “M” in
The cryogenic system also comprises a heliumpressure sensor PT (illustrated in
There are two ways of implementing the method for estimating quality factor: either the calculations are carried out by the programmable logic controller PLC and the result is communicated to the humanmachine interface HMI, or the logic controller transmits the data from the sensors PT, LT to the humanmachine interface HMI and the latter performs the calculations. The programmable logic controller PLC and the humanmachine interface HMI are computers and comprise means for implementing the method for estimating quality factor, and in particular a memory and a processing unit. The first option is recommended since it makes it possible to avoid potential bugs in the humanmachine interface (programmable logic controllers being designed to minimize the risk of bugs). In the rest of the description, it will therefore be assumed that the method is implemented by the programmable logic controller PLC.
The various valves and sensors of the cryogenic system are connected to one or more controllers CTRL that are able to regulate the helium pressure and the liquidhelium level inside the cryomodule by controlling the valves CV002 and CV005. With reference to
The control structures CTRL and Coord use the estimation of the heat load on the helium bath to improve the overall stability of the system, as will be described in detail below.
One way in which the method for determining a quality factor Q0 of a cavity may be executed will now be described through six steps E1 to E6, which are carried out in succession. As shown in
The method is implemented in real time, i.e. the estimations of the dynamic load Q_{dynam }and of the quality factor Q_{0 }are calculated instantaneously and constantly repeated. By “real time”, what is meant is that the determining steps are executed at a rate suited to the variation in the dynamic load Q_{dynam }and in the quality factor Q_{0}. For example, new values of the dynamic load Q_{dynam }and of the quality factor Q_{0 }may be calculated at a frequency higher than or equal to 1 Hz, or even higher than or equal to 10 Hz, or even higher than or equal to 1 kHz. In addition, the method may be carried out while the cavity is in the process of operating, i.e. while the particle accelerator is being used to accelerate particles, in particular for experimental purposes. The method is not necessarily implemented during an operation dedicated to the measurement of quality factor. The method may therefore be implemented parallel to an experiment during which all the systems of the accelerator are in operation. The estimation may also be repeated when a sensor of the cryogenic system records a significant variation.
When not specified in the description, the physical unit associated with a given physical quantity is an SI unit (SI being the wellknown abbreviation of International System of Unit).
In a first step E1, characteristics of the phase separator and of the regulating valves are determined. These characteristics depend directly on the design of the cryomodule and of the valves. They may be measured or calculated. These characteristics are:
 the volume of the enclosure containing the cryogenic fluid,
 the static heat load Q_{static }borne by the cryogenic fluid, i.e. the energy transmitted to the cryomodule by heat exchange with the exterior of the cryomodule,
 the function f1 defining the height h of liquid helium as a function of the volume Vliq of liquid helium in the enclosure: h=f1(Vliq). This function depends on the geometry of the enclosure containing the cryogenic fluid. It may be calculated by means of a numerical model of this enclosure or be defined empirically.
For the second valve CV002 and the third valve CV005, these characteristics are:
 The flow coefficient CV of each valve, i.e. the coefficient expressing the flow rate of a fluid passing through a valve, at a given temperature, that causes a given drop in pressure.
 The rangeability Rv of each valve, i.e. the ratio of the maximum and minimum flow rates between which the characteristic of a valve is maintained within certain precision limits.
In a second step E2, a thermodynamic model of the cryomodule is produced. Such a model is illustrated macroscopically in
In a first substep E21 a model of the valves is established. This first substep E21 allows the amount of helium entering into the enclosure {dot over (m)}_{in }and the amount of helium exiting from the enclosure {dot over (m)}_{out }to be defined depending on the boundary conditions BC_{in}, BC_{out}, and on the positions of the three valves POS_{CV001}, POS_{CV002}, and POS_{CV005}, and on the pressure in the cryomodule.
In a second substep E22 an energy model of a phase separator is established. This second substep allows the density ρ of the helium (expressed in kg/m^{3}) and the specific internal energy u (expressed in J/kg) of the helium contained in the cryomodule to be defined depending on the amount of helium entering into the enclosure {dot over (m)}_{in }and the amount of helium exiting from the enclosure {dot over (m)}_{out}, on the static load Q_{static}, and on the dynamic load Q_{dynam}, and on the input and output specific enthalpy H_{in }and H_{out }of the cryomodule or on the output temperature of the cryomodule and on the input mass concentration of the cryomodule.
In a third substep E23 a model of the physical properties of the helium bath is established. This third substep allows the height h of liquid helium in the enclosure, and the internal pressure P in the enclosure to be defined depending on the density ρ of the helium and on the specific energy u of the helium.
We will now describe in detail each of the models established in these three substeps.
The first substep E21 allows a model of the valves to be established. The method will be described in detail through the example of any one particular valve among the three valves CV001, CV002, CV005. Firstly, the expansion that occurs in the valve is considered to be isenthalpic, i.e. without addition of energy from the exterior. Thus, the enthalpy of the helium upstream of the valve is identical to the enthalpy downstream of the valve, i.e.: H_{out}=H_{in}. The valve is also considered not to accumulate fluid. Thus, it is also possible to write the equation {dot over (m)}_{out}={dot over (m)}_{in}. According to standard ANSI/ISA75.01.01, the flow rate of a compressible fluid through a valve is written according to the following formula F2:
in which:
where:
 K is a coefficient of conversion between the imperial system of units and the international system of units (K=7.59×10^{−3}).
 ρ_{in }is the density of the helium (expressed in kg/m^{3}) upstream of the valve. This density may be interpolated from a table of a property of helium if the pressure and enthalpy upstream of the valve are known.
 P_{in }is the pressure upstream of the valve (expressed in Pa).
 P_{out }is the pressure downstream of the valve (expressed in Pa).
 γ is the ratio of the specific heats (unitless), defined as the ratio of the specific heat at constant pressure Cp (expressed in J/(kg.K)) to the specific heat at constant volume Cv (expressed in J/(kg.K)) of the helium, i.e.
This ratio may be interpolated from a table of a property of helium if the pressure and enthalpy upstream of the valve are known.
 CV is the flow coefficient of the valve (unitless), which may be calculated using the formula:
in which:
 CV_{max }is a dimensioning constant of the valve, chosen so that the degree of openness of the valve remains in a suitable operating range (i.e. so that the valve does not open or close 100% during its use).
 R_{v }is the rangeability of the valve (unitless).
 “open” is an openness of the valve that varies from 0 when the valve is completely closed to 100 when the valve is completely open. The openness “open” represents the respective open position of each valve POS_{CV001}, POS_{CV002}, and POS_{CV005 }in percent.
Moreover, the flow rate of an incompressible fluid through a valve is written according to the following formula F3:
{dot over (m)}_{incomp}=K·CV·√{square root over (ρ_{in}·(P_{in}−P_{out}))}
in which the variables have the same meaning as in formula F2.
The isothermal compressibility β_{T }is defined as a factor indicating the variation in the volume of a system when the pressure in the system varies while its temperature remains constant. This factor indicates to what point a fluid is compressible. Thus, β_{T}=0 when the fluid is incompressible and β_{T}=1 when the fluid is compressible. At a given temperature, the factor β_{T }may be calculated using the following formula:
This factor is used to weight the flow rate of the fluid flowing through the valve according to formula F2 or F3 presented above. Thus, the calculation of the flow rate through the valve may be written with the following formula:
{dot over (m)}=β_{T}·{dot over (m)}_{comp}+(1−β_{T})·{dot over (m)}_{incomp }
The second substep E22, in which the energy model of the phase separator is established, will now be described in detail. It is assumed that the helium bath is in liquidgas equilibrium. Therefore, the density ρ of the helium and the specific energy u of the helium (in other words its energy density per unit mass) are distributed uniformly in the enclosure.
Firstly, a physical relationship between the total mass m_{tot }of helium in the bath of the cryomodule, its density ρ and the volume Vol of the enclosure containing the helium is established with the following physical equation:
m_{tot}=ρ·Vol
This formula may be differentiated so as to be written:
{dot over (m)}_{tot}={dot over (ρ)}·Vol
Next, a mass balance of the cryomodule allows the total mass m_{tot }of helium inside the enclosure, the entering mass of helium m_{in }and the exiting mass of helium m_{out }to be related by the following equation:
{dot over (m)}
_{tot}
={dot over (m)}
_{in}
−{dot over (m)}
_{out }
The relationship relating the total energy U stored by the helium, the specific energy u of the helium and the total mass m_{tot }of helium is written:
U=m
_{tot}
·u
This formula may be differentiated so as to be written:
{dot over (U)}={dot over (m)}
_{tot}
·u+m
_{tot}
·{dot over (u)}
Lastly, an energy balance applied to the helium bath is written with the following formula:
{dot over (U)}={dot over (m)}_{tot}·u+H_{in}·{dot over (m)}_{in}−H_{out}·{dot over (m)}_{out}+Σ_{i}Q_{i }
in which equation:
 Σ_{i}Q_{i }is all of the heat loads acting on the helium bath, i.e.:
Σ_{i}Q_{i}=Q_{static}+Q_{dynam}.
 H_{in }is the enthalpy of the helium entering into the enclosure of the phase separator.
 H_{out }is the enthalpy of the helium exiting from the enclosure of the phase separator.
By combining the aforementioned equations, an equation of the thermodynamic model of the cryomodule is obtained:
Lastly, the third substep E23, which allows the height h of liquid helium in the enclosure, and the internal pressure P in the enclosure, to be defined depending on the density ρ of the helium and the specific energy u of the helium, will now be described in detail.
The internal pressure P in the enclosure of the cryomodule may be determined directly depending on the density ρ of the helium and its specific energy u by exploiting the physical properties of helium. To this end, an interpolation function integrated into a simulation software package such as Hepak© and/or a C++ library such as “CoolProp” will possibly advantageously be used. By way of example, a first table of a property of helium is illustrated in
By virtue of a second table of a property of helium, it is also possible to determine the mass concentration X of the helium depending on the density ρ of the helium and its specific energy u. This second table is shown by way of example in
m_{liq}=m_{tot}(1−X)
in which formula m_{liq }is the mass of the liquid helium in the enclosure and m_{tot }the total mass of helium.
The volume of helium in liquid form V_{liq }is defined by the equation:
in which ρ_{liq }is the density of the liquid helium.
Lastly, by virtue a third table of a property of helium, which is illustrated by way of example in
Finally, by virtue of the three substeps E21, E22, E23 described above, a thermodynamic model of the cryomodule is obtained, this model relating, via the equations, the boundary conditions BC_{in}, BC_{out}, the positions of the three valves POS_{CV001}, POS_{CV002}, POS_{CV005}, the static load Q_{static}, the dynamic load Q_{dynam}, the height h of liquid helium in the enclosure and the pressure P in the enclosure.
The precision of the thermodynamic model of the cryomodule may be verified by comparing the measured height of liquid helium h_{mes }with the height of liquid helium h_{calc }estimated using the model, and likewise by comparing the pressure measured in the enclosure P_{mes }with the pressure P_{calc }estimated using the model, when the degree of openness of the valves CV002 and CV005 is varied. For the sake of this verification, the dynamic heat load Q_{dynam }will possibly be kept at a zero value and the valve CV001 kept shut.
In a third step E3, various parameters of the cryogenic system are stored in the memory of the programmable logic controller PLC. In particular, the internal pressure P delivered by the pressure sensor PT is stored. The height h of liquid helium in the enclosure, which is delivered by the level sensor LT is also stored. The boundary conditions are also stored, i.e.:
 the helium pressure P_{in }upstream of the admission valves CV001, CV002,
 the helium pressure P_{out }downstream of the outlet valve CV005,
 the input specific enthalpy of the supply valve: H_{in}.
The boundary conditions are dependent on the helium distribution circuit 11 and may be measured and/or calculated by means of suitable sensors positioned in the distribution circuit 11. As a variant, and to simplify the calculations, the boundary conditions could be considered to be constant Such a simplification however leads to a less precise estimation of the dynamic load.
The thermodynamic model of the cryomodule obtained at the end of the second step E2 comprises equations that are complex to solve. In order to facilitate the solution of these equations, the invention makes provision, in a fourth step E4, for a linearization of the thermodynamic model, i.e. for an approximation of the thermodynamic model by a set of differential equations that are linear about a preset operating point.
In a first substep E41, an operating point about which the model will be linearized is defined. This operating point may be determined depending on constraints that the cryomodule must respect. For example, it is possible to define the operating point by an internal pressure P of the helium bath equal to 1200 mbar and a height of liquid helium equal to 90% of the total height of the enclosure.
In a second substep E42, the boundary conditions BC_{in }and BC_{out }of the system are defined and the opennesses POS_{CV001}, POS_{CV002}, POS_{CV005 }of the valves that allow the model to stabilise to the operating point defined beforehand are sought. The openness of the first valve POS_{CV001 }may be set to 0% (i.e. completely closed) because this valve is used only to fill the enclosure. Two PID regulators (PID being the wellknown acronym of proportionalintegralderivative) may be used to determine the opennesses POS_{CV002}, POS_{CV005 }of the two other valves. In this step, the radiofrequency antenna may or may not be activated, depending on the operating point about which it is desired to linearize the thermodynamic model.
In a third substep E43, the thermodynamic is represented as a linear dynamic system. The linear system is defined by the following state representation:
with
 A, B, C and D state matrices of the system,
 v a vector of controllable inputs defined by:
{dot over (m)}_{in }being the mass flow rate entering into the cryomodule.
{dot over (m)}_{out }being the mass flow rate exiting from the cryomodule.
 w a vector of uncontrollable inputs defined by:
Q_{static }being the static load, Q_{dynam }being the dynamic load and H_{in }being the enthalpy within the cryomodule.
 x a state vector, the latter being equal to the output vector y and being defined by:
where ρ the density of the helium and u its specific internal energy.
This linear system describes the dynamics of the method about the operating point defined in substep E41 and defined by:
By virtue of the thermodynamic module established beforehand in the second step E2, it is possible to determine the values of the state matrices of the system, A, B, C and D. To this end, it is possible to use a linearization function of a computational tool such as the “linearize” function of Matlab®. In practice, as x=y, the state matrix C is equal to a unit matrix and the state matrix D is equal to a zero matrix. The linear system thus obtained describes the thermodynamic behaviour of the cryomodule about the operating point defined in substep E41.
In a fifth step E5, a heatload observer OBS such as illustrated in
In a first substep E51, the signals delivered by the sensors, are filtered so as to decrease noise. A first order filter of the following form may be used:
in which τ_{filter }is a time constant of the filter, chosen depending on a time constant τ_{method }of the method, such that:
τ_{filter}«5·τ_{method }
In a second substep E52, the density ρ and the internal energy u of the helium bath are calculated based on the level of liquid helium h and the internal pressure P in the enclosure. Initially, the volume of liquid helium V_{liq }is calculated based on the measurement of the level using the formula V_{liq}=f_{1}(h), in which f_{1 }is the function giving the volume of liquid helium V_{liq }depending on the height indicated by the level sensor LT. Based on the volume of liquid helium it is possible to deduce the mass concentration X of the fluid contained in the cryomodule using the formula:
in which m_{gas}, ρ_{gas}, V_{gas }are respectively the mass, density and volume of the helium in gaseous form, and m_{liq}, ρ_{liq}, V_{liq }are respectively the mass, density and volume of the helium in liquid form.
As explained above, since it is a question of a saturated liquid, the density τ_{liq }of liquid helium may be determined by virtue of the third table of a property of helium, depending on the internal pressure P.
Analogously, since it is a question of a saturated liquid, the density ρ_{gas }of the gaseous helium may be determined by virtue of the fourth table of a property of helium, depending on the internal pressure P.
It is thus possible to calculate the mass concentration X, then, based on fifth and sixth tables of properties of helium, to deduce therefrom the density ρ and the internal energy u of the helium bath, respectively.
In a third substep E53, the flow rate {dot over (m)} through each of the valves CV001, CV002 and CV005 is calculated in accordance with the logic diagram illustrated in
The substep E531 aims to compensate for deviations and drifts observed between the simulated mass flow rate and the observed mass flow rate through a valve CV00i (i being equal to 1, 2 or 5 depending on the valve in question). In order to compensate for a deviation between the simulated mass flow rate and the observed mass flow rate, a static offset CV00i_{offset}_{stat }is defined, which is applied to the measurement of the position of valve i via the following formula:
CV00i_{offset}_{stat}=CV00i_{pos}_{mes}_{nom}−CV00i_{pos}_{sim}_{nom }
where:
 CV00i_{pos}_{mes}_{nom }is the measured nominal value of the position of value i at the operating point defined in substep E41.
 CV00i_{pos}_{sim}_{nom }is the simulated nominal value of the position valve i at the operating point defined in substep E41.
In order to correct a drift of the mass flow rate through the valve, it is also possible to define a dynamic offset by the following formula:
CV00i_{offset}_{dyn}=(CV00i_{pos}_{mes}−CV00i_{pos}_{mes}_{nom})·gain
where:
 CV00i_{pos}_{mes }is the current measured value of the position of valve i.
 CV00i_{pos}_{mes}_{nom }is the measured nominal value of the position of valve i at the operating point defined in substep E41.
 gain is a coefficient of proportionality to be adjusted based on measurements carried out on the system.
The corrected position CV00i_{pos}_{corr }of the valve CV00i may then be obtained by adding the measured position, the static offset and the dynamic offset, i.e. via the following equation:
CV00i_{pos}_{corr}=CV00i_{pos}_{mes}+CV00i_{offset}_{stat}+CV00i _{offset}_{dyn }
The block referenced E531 in
In the second substep E532, the model established in substep E21 is used. This model allows the mass flow rate through a valve CV00i to be calculated depending on the corrected position CV00i_{pos}_{corr }of the valve CV00i calculated beforehand, on the pressure P_{in }upstream of the valve, on the pressure P_{out }downstream of the valve and on the enthalpy H_{in }upstream of the valve (which is assumed to be identical to the enthalpy downstream of the valve). Substep E53 is then repeated for each of the valves CV001, CV002 and CV005 of the system so as to determine the mass flow rate through each of these valves depending on the current measured value CV00i_{pos}_{mes }of the position of valve i.
In a substep E54, a state observer, referred to as the Kalman observer, is implemented in accordance with the diagram defined in
The resulting system is an invariant linear system, for which there is a Kalman estimator obtained by solving a Riccati difference equation, for example using the “lqr” function of Matlab® with L==lqr (A, C, Q, R) in which Q and R are weighting matrices. In other words, it is a question of finding the gain L that minimizes the following criterion:
∫=Σ_{k=1}^{∞}x^{T}Qx+u^{T}Ru, with u=−Lx
For example, the matrices Q and R may be written in the following form:
The state observer thus implemented allows the dynamic load Q_{dynam }to be determined and observed in real time.
Advantageously, the calculating method thus developed may be validated by means of an experiment on a cryogenic system when the latter comprises a cavity equipped with a device for generating heat such as a resistive heater of variable supply, also referred to as a “Joule heater”. Such a heater allows a supply of heat identical to that which would be produced by the operation of the cavity in a particle accelerator to be simulated. The resistive heater delivers heat equivalent to a dynamic heat load Q_{dynam}.
In a sixth step E6, the quality factor Q_{0 }of the cavity 4 is calculated. The quality factor is a measure of the damping ratio of an oscillating system. The quality factor depends on the temperature T of the internal wall of the cavity, which is assumed to be uniform, on the material of the cavity and on its geometric shape. It is defined by the ratio of the energy U stored in the cavity to the energy P_{loss }dissipated in the walls of the cavity, per period of oscillation. The quality factor may therefore be expressed by the following formula:
in which ω is the resonant angular frequency of the cavity.
To calculate the energy U stored in the cavity and the energy P_{loss }dissipated in the walls of the cavity it is assumed that there is a perfect vacuum in the cavity and that the resistivity of the walls of the cavity is uniform over all of their surfaces. In light of the fact that the energy stored in the electric field is equal to the energy stored in the magnetic field, and of the fact that the internal energy of the cavity is calculated over the volume and that the losses are concentrated on the surface of the cavity, it is possible to express U and P_{loss }via integrals over the volume of the cavity and over the surface of the walls of the cavity, respectively. U and P_{loss }may therefore be expressed using the following formulae:
in which formulae:
 μ_{0 }is the magnetic permeability of free space,
 R_{s }is the surface resistance of the cavity, the value of which depends on the temperature of the cavity,
 H is the magnetic field inside the cavity.
Thus, the quality factor Q_{0 }may be expressed in the following form:
where G is called the geometry factor of the cavity, and is defined by:
The geometry factor G is a known and invariant datum that may be calculated directly from a radiofrequency model of the cryomodules. Considering the geometry factor G to be known, it is still necessary to find the expression of the surface resistance R_{s}(T) in order to be able to deduce the quality factor therefrom. The surface resistance R_{S}(T) respects the following equation:
R_{S}(T)=R_{BCS}(T)+R_{res }
in which equation:
 R_{res }is the residual resistance of the niobium,
 R_{BCS}(T) is a variable resistance defined by the following equation:
in which equation:
 T is the temperature of the internal wall of the cavity,
 A is a constant dependent on the properties of the material used to manufacture the walls of the cavity (niobium in particular),
 f is the resonant frequency of the cavity,
 Δ is the energy gap of the material used to manufacture the walls of the cavity,
 k_{B }is Boltzmann'"'"'s constant.
Among all these parameters, only the temperature T is unknown and subject to variations. All the other parameters A, f, Δ, k_{B }are set values that are known or measurable.
The temperature T of the internal wall of the cavity may be estimated from the power dissipated in the cavity and the temperature of the helium bath. Specifically, the heat dissipated on the interior surface of the cavity is transmitted to the helium bath by conduction through the niobium walls of the cavity. Assuming that the cavity dumps all its heat into the helium bath, the energy P_{loss }dissipated in the walls of the cavity may be deduced to be equal to the dynamic heat load Q_{dynam}.
Moreover, an equation of thermal conduction applied to the walls of the cavity is expressed in the following way:
in which equation:
 λ(T) is the thermal conductivity of niobium,
 S is the area of the exchange surface between the cavity and the helium bath
 e is the thickness of the wall of the cavity,
 T_{cavity }is the temperature of the internal wall of the cavity,
 T_{bath }is the temperature of the helium bath.
Since it is a question of a saturated liquid, the temperature T_{bath }of the helium bath may be interpolated from a table of a property of helium if the internal pressure P of the helium bath (regulated about a value of 1200 mbar) is known.
It will be noted that the internal pressure of the helium bath is assumed to be uniform in this model. The model could be refined by considering the pressure as a function of the height of the point in question in the helium bath. It would then be possible to define a temperature gradient in the helium bath rather than to consider the temperature to be uniform.
Finally, as the thermal conductivity of niobium λ(T), the area S of the exchange surface between the cavity and the helium bath, and the thickness e of the wall of the cavity are known quantities, and as an estimation of the dynamic heat load Q_{dynam }is available (delivered by the state observer) it is possible to calculate the temperature T_{cavity }of the internal wall of the cavity. Once this temperature has been determined, it is possible to calculate the value of the variable resistance R_{BCS}(T), then the surface resistance R_{S}(T), and lastly the quality factor Q_{0}.
Finally, the quality factor Q0 may be expressed by the following formula:
The invention also relates to a method for operating a particle accelerator comprising implementing the method for determining the quality factor Q0 such as described above, and in particular steps E4 to E6, and a step E7 of modifying at least one operating parameter of the accelerating cavity depending on the quality factor Q0. For example, the operating parameter may be a setpoint value of the power of a radiofrequency wave emitted in the cavity by the antenna 8. The modification may consist in decreasing the value of the power setpoint until the emission of the radiofrequency wave stops if the quality factor of at least one accelerating cavity crosses a preset threshold, the other cavities of the accelerator, when they exist, being able to continue to operate. The decrease in the value of the setpoint may optionally be continued until the emission of the radiofrequency wave stops. The invention may also be implemented while the particle accelerator is being powered up, the value of the setpoint for example being gradually increased depending on the determined quality factor. The modification may also consist in any other modification of the configuration and/or of the regulation of the particle accelerator.
Concretely, the regulating means 12, which is incorporated into the programmable logic controller PLC, may for example compare the estimation of the quality factor with a threshold value. If the quality factor Q0 is higher than a preset threshold, then the regulating means 12 may send a control signal to the radiofrequency system in order to decrease the power of the waves emitted by the radiofrequency antenna 8 integrated into the cavity. The regulating means 12 may optionally comprise a plurality of thresholds beyond which the power of the waves emitted by the radiofrequency antenna will be successively decreased until a power of zero is reached. Thus, it is possible to use each cavity at a power that is optimal given its quality factor without having any impact on the operation of the other cavities of the particle accelerator. Preferably, the operating method comprises a plurality of iterations of steps E4 to E7.
During operation, the control structures CTRL and Coord illustrated in
The measuring principle illustrated through this invention is fundamentally different from the conventional measurement because it is based on the thermal state of the bath of cryogenic fluid in which the cavity is submerged and not on a direct measurement of the radiofrequency field in the cavity.
The heat load is estimated using sensors that belong to the cryogenic system, and the determination of the quality factor Q_{0 }does not require a pickup probe, a network analyser or any other dedicated means for determining quality factor. In addition, the estimation of the quality factor is carried out during operation (and not offline). This determination may be carried out in real time or not in real time, but in any case during the operation of the particle accelerator, which was not the case in the prior art. In other words, the invention allows the accelerating potential of an accelerating cavity (the maximum power that it is able to accept) at any time to be estimated and the power emitted by the radiofrequency antenna to be adjusted accordingly. This estimation does not require any physical modification of the existing system, i.e. it does not require sensors or any other measuring devices to be added, and also does not require knowledge of a voltage applied to the cavity. Knowledge of the quality factor, in particular in real time, allows the stability of both the cryogenic behaviour and the behaviour of the radiofrequency system acting on the cavity to be improved with a view to achieving reliable operation of the accelerator. The device achieved is sufficiently economical in computational resources to be able to be implemented via a programmable logic controller.
The method according to the invention is executable from the moment that radiofrequency power is injected into the superconducting cavities and even before a beam has been formed. It is also executable with the beam and it allows certain possible anomalies that cause an abnormal heat load to be placed on the cavities to be diagnosed.