60 Ghz-300 Kw Gyrotron General Design for the Mexican Tokamak “T”

1 FIP/P4-33

60 GHz-300 kW Gyrotron General Design for the Mexican Tokamak “T”

J.A. Gonzalez1, M.Salvador1, J.Mart´ınez1,2, A.Nieto1, O.A. Muñoz1, J.González1, J.R. Morones3, R.M.Chávez1, G.R. Cavazos1, V.M.Arredondo1, S. Mart´ınez1, M.A. Sanroman1, I.E. Morales1, A. Acosta1, J.V. Guzman1 and C.A. Briseño1

1Facultad de Ingenier´ıaMecánicay Eléctrica- Universidad Autónomade Nuevo León, San Nicolásde los Garza, México 2ComisiónFederal de Electricidad, Monterrey, México 3Facultad de Ciencias F´ısicoMatemáticas - Universidad Autónomade Nuevo León,San Nicolásde los Garza, México

Corresponding Author: [email protected]; [email protected]

Abstract: We present the preliminary design of our source for ECRH system applied in our Toka- mak device, a gyrotron device 60 GHz-300 kW which is currently developed by the Fusion Research Group (GIF, Spanish acronyms) at the Universidad Autónomade Nuevo León (UANL, Spanish acronyms) in Monterrey, Mexico. A gyrotron is a source of coherent electromagnetic radiation capable of providing hundreds of kilowatts of power in regions with millimeter and sub-millimeter wavelength. This power generated either continuous wave (CW) or long pulse, oscillates between 0.1 - 1 MW. In this device, a magnetron injection gun (MIG) is used to generate a gyrating electron beam, then, it interacts whit the eigen- mode of a cavity transforming part of the kinetic energy into the microwave energy. The design of the MIG requires an optimized electron beam analysis. For this purpose, a theory based on the conservation of the angular momentum and the adiabatic electron motion is regularly used. As a first step in our gyrotron design, we present an array of four copper coils achieving a maximum configuration of magnetic intensity of 2.56 T, required into the cavity to arise 60 GHz high-power millimeter frequency in the fundamental harmonic, in addition to the nominal beam parameters such as interaction region to cathode ratio of a magnetic compression of 13.68; a beam voltage of 100 kV, a beam current of 3 A, and transverse to axial velocity ratio of 1.5, were established. An initial design has been obtained, from analytical adiabatic trade-off equations by Baird and Lawson, of a diode type magnetron injection gun characterizing our trajectory parameters beam in the Tokamak “T” gyrotron. The mean radius of the emitter (7.3 mm), slant length of the emitting surface (8.8 mm), cathode modulating anode gap (11.5 mm), slope angle of emitter (40) are obtained. These results are supported by 2D computer simulations. FIP/P4-33 2

1 Introduction

In Monterrey, Mexico a medium size low aspect ratio Tokamak design is under development [1]. This facility has as primarly objective provide a good experimental facility with national mexican designs, at this fisrt stage we consider the use of conventional materials but our mechanical design could bring us the opportunity to use those advanced materials to obtain a long-term discharge in our Tokamak called “T”, the toroidal field requires to be larger than the poloidal field (BT BP ) we consider at this sense a toroidal field BT of 1.6 T. An important factor in the confinement regime, is the deter- 2 mination of the maximum plasma current (Ip), we use Ip < (2π a BT )/(µ0 q R) [2]. The determination of temperatures for our Tokamak has been established with Artsimovich 2 1/3 1/2 studies [3,4], where Ti = (1.29 ± 0.11)(IBR hni) /Ai eV, given us an ion temperature (Ti) of 280.53 eV. For the electron temperature, we have considered the ion energy balance could be analyzed independently of the electron energy balance, thus applying 3/2 1/2 f(Te) = Te −(3.7 Te )/(11 Ti )− Ti = 0 and solving numerically, we found a (Te) of 516 eV. For sustain and achieve attractive plasma conditions is fundamental the development of excellent heating systems working with ions and electrons; electrons are important due the plasma complexity behaviour present in a discharge in MCF, we present at this work a development of one ECRH system source.

A Gyrotron is a source of coherent electromagnetic radiation capable to provide hundreds of kilowatts of power in regions with millimeter and sub-millimeter wavelengths. This power generated, either on continuous wave (CW) or long pulse, oscillates between 0.1 and 1 MW at frequencies ranging from 28 → 170 GHz [5]. The use of RF through this devices was initially conceived in the mid-60s at the Institute of Radiation Physics Re- search in Gorki, Russia and actually its development satisfy a wide range of technologies and applications [6], on these applications we can mention those that are embedded in the Plasma Physics: production of plasma radiofrequency heating, generation of non-inductive current stabilization plasma, diagnostics, active plasma research of thermonuclear magnetic conﬁnement fusion such as generation of microwave frequency injection lower hybrid current drive (LHCD), electron cyclotron resonance heating (ECRH), generation nonin- ductive electron cyclotron current drive (ECCD) [7] among other waves.

In the Mechanical and Electrical Engineering Faculty (FIME) of the Autonomous University of Nuevo Leon (UANL), our Fusion Research Group (GIF) is currently working on one analytical and experimental design to develop a Gyrotron 60 GHz-300 kW for heating plasma purposes inside our Tokamak device. In this work we present in advance the primarly fast wave designs of our system. Inherent to a Gyrotron, one of its most important components is the intense external magnetic field. This field will guide the electron beam making on the electrons take a helical path through its field lines. Based on the wavelength (≈ 5 mm) a necessary field intensity is determined in the interaction region of 2.56 T. A set of four coils through which a stationary current flow within a range of 10-50 A, producing a continous field profile, desirable along the resonant cavity. 3 FIP/P4-33

2 General gyrotron source design

A Gyrotron is based on coherent cyclotron radiation mechanism comes from of the transverse kinetic energy of electrons through a gyromotion in a constant magnetic ﬁeld, inter- acting with an electromagnetic wave in resonance. The electrons will be able to achieve the radiation emission if a mechanism of “bunching” of electrons ﬂowing in a perpen- dicular direction to the plane of the beam, is established; through this mechanism the electrons collides, giving a deceleration phase which extracts the orbital momentum of the electrons inducing with this an electromagnetic radiation. The condition to achieve this bunching mechanism is to satisfy a resonance condition between the movement of the electrons and the electromagnetic wave in the interaction zone, as shown in (1):

∼ ω − kzvz = s Ω , s = 1, 2,... (1)

Where ω is the angular frequency of the EM wave, kz is the axial wave number into the direction of the electron beam, vz the axial velocity, Ω is the gyrofrequency (electron cyclotron frequency) and s is the armonic number. In general this radiofrequency (RF) technique have outstanding advantages as ad- ditional heat source in tokamaks: localized electron heating, pre-ionization and current start-up, MHD control, and the generation for non-inductive current drive in tokamaks, conventional RF systems has been developed depending of the frequency ranges applied: high, very high, and ultra-high. Our electron source design consider millimeter wave into the range of 25-250 GHz, which nominal values can be found on Table I.

TABLE I: NOMINAL BEAM PARAMETERS FOR MIG OF GIF- GYROTRON-60-300.

Parameters Values Beam voltage V0 100 kV Beam current Ib 3 A Magnetic ﬁeld in cavity B0 2.56 T

Average beam radius in cavity rg0 7.3 mm Velocity ratio α = v⊥/vz 1.5 Initial normalized energy γ0 1.195 Magnetic compresion ratio fm 13.68 5 Electric ﬁeld at the emitter Ec 100 × 10 V/m ◦ Slope angle of the emitter φc 40 Cathode modulating anode gap dac 11.5 mm Cathode radius rc 27 mm Emitter strip width ls 8.8 mm Larmor radius rL 0.36 mm Relative cathode loading Jc/JL 0.3 % FIP/P4-33 4

It is known as MIG to the region responsible to produce a focused annular electron beam with small helical orbits in its trajectories inside a gyrotron and also with a high relativistic kinetic energy orbital power. Into the MIG design is frequent the exploitation of the adiabatic theory with a first order approximation, see [8]. An adiabatic approximation is based on the assumption that the electrical and magnetical fields variations are small into the vicinity of the electron orbitals. When the magnetic field is slightly inho- 2 mogeneous, conservation of electronic angular momentum is established, p⊥/B = const. Under this consideration, a good design which achieves high power beam, seeks to have a high ratio of orbital-axial speed α = v⊥/vz (about 1.5 or more), the component of the orbital velocity must be as large as possible. Also, an acceptable propagation speed, either orbital or axial, is desirable, as is maintaining latter possibly small. Following the works of [9] and [10] a first approximation is achieved into the parameters that constitute the MIG GIF Gyrotron 60-300. Table I shows the parameters obtained with the use of the trade-off equations. Actually we still working on the optimization of these values with specialized software.

3 Magnetic System

In a Gyrotron, an external magnetic field sustains the cyclotron frequency (Ω) near of the frequency of the EM field (ω). From equation (1) it follows that the strength field required in the interaction zone for a wavelength corresponding to a frequency of 60 GHz cyclotron is:

2πm c γ B = 0 0 (1 − nβ ) (2) 0 e s λ z0

Where m0, e and c are the electron rest mass, its charge and the speed of light respectively. γ0 is the initial energy of the normalized beam or the relativistic mass factor. n = ckz/ω is the refractive index of the EM wave and βz0 = vz0 /c is the normalized axial velocity of the beam. Characterizing our electron beam with a voltage of Vb = 100 kV, and considering that the axial number wave kz ≈ 0 is a typical value in a Gyrotron [7], our field value on the interaction zone wave-beam corresponds to B0 = 2.56 T. The configuration and the respective parameters of the external magnetic field presents in the gyrotron are of vital importance, because these will be the responsibles for guiding the electron beam from the cathode side through the beam tunnel area (cavity region) to finally reach the area of the collector. As it is well known, Ampere’s law lays the foundation for the design of the solenoids, which says when passing an electric current in a circular loop an axisymmetric magnetic field will occur. There, in literature concerning we found three analytical methods for a good approximation of these fields: (1) Legendre polynomials, (2) complete elliptic integrals and (3) expansion series of their present values on one axis. The method least used for poor convergence is the Legendre [7], however the other two methods gives an approximation very accurate of the fields present both inside and outside of the axis, see e.g. [11,12]. 5 FIP/P4-33

Previous researchers has used [11] to arrive to one approximation in a magnetic ﬁeld with the application of the full elliptic integrals K(m) and E(m), namely,

" 2 2 2 # r1 − r − (z1 − z) 2 2 −1/2 Bz (r, z) = M1 2 2 E(m) + K(m) (r1 + r) + (z1 − z) (3) (r1 − r) + (z1 − z)

2 2 2 z1 − z h r1 + r + (z1 − z) i 2 2 −1/2 Br(r, z) = M1 E(m) − K(m) (r1 + r) + (z1 − z) 2 2 r (r1 − r) + (z1 − z) (4) Where Bz and Br are the axial and radial components respectively, measured in Teslas localized in the point (r, z). Here M1 = µ0 I/2π, r1 is the coil radius, whose center is (0, z1), I is the amperes of electric current, and µ0 is the magnetic permeability constant in vacuum. K(m) is the ﬁrst type complete elliptic integral, similarly, E(m) is the complete elliptic integral of the second type, both are dimensionless parameter depending on m, where,

4rr1 m = 2 2 (5) (r1 + r) + (z1 + z) Nevertheless we consider the research of [12] to calculate the amount of Teslas in a magnetic ﬁeld, at each point (r, z) due to the ﬂow of an electrical current (I) through a circular loop of radius r1 and centered at the point (0, 0) in the z axis, then we follow the next equations,

1/2 µ0I z m h 2 − m i Br(r, z) = E − K (6) 4π r r1r 2 − 2m

1/2 µ0I 1 m h r1m − (2 − m) r i Bz(r, z) = r K + E (7) 4π r r1 r 2 − 2m 2 2 Here K and E are as previously mentioned above, and m = 4rr1/[(r + r1) + z ]. In our research we develop entirely the mathematic method of [12] to determine all the points embedded in our magnetic configuration, with this aim we obtain the field due to a coil of radius r1 and centered at the point (0, z1) changing z by z − z1 into the equation (6), as well as their respective value of m. It should mention the following important points: A) Br = 0 at z = 0; B) Bz = µ0I/2r1 into the center (r = 0, z = 0); and C) When r → ∞ or z → ∞, then m → 0 and Br,Bz → 0. The magnetic field must be as uniform as possible in the area of wave-field interaction respect to the axial axis. It is easy to see that the expression (6) of magnetic component of the axial field, is not defined for r = 0. Hence an expression desired analytically is necessary to determine the strength field in the magnetic axial direction. In this work, to generate the external magnetic field, also we consider a rectangu- lar cross-section toroidal coil over a cylindrical coordinate system, obtaining our magnetic field profile equation (8), being a differential volume dV = rdr dφ dz localized at FIP/P4-33 6

−r rˆ + (a − z)ˆz with respect to the point z = a on the axis z. The height of the coil is ∆r = r2 − r1 and its width ∆z = z2 − z1. Let’s spend a stationary electric current density through this coil from Jφˆ. From the Biot-Savart law is known to have a diﬀerential of the magnetic ﬁeld strength on the B to point z = a, being the contribution of all the coil at this point in the form:

√ z−z1 r + r2+(z−z )2 2 √ 2 1 ZZZ 2 2 µ0 JdV × eR Nµ0I r1+ r1+(z−z1) Bz = = ln (8) 2 √ z−z2 4π R 2∆z∆r r + r2+(z−z )2 2 √ 2 2 2 2 r1+ r1+(z−z2)

Where I is the stationary electric current, µ0 is as mentioned above and N is the number of turns into the coil.

FIG. 1: Comparison of the strength of magnetic ﬁeld on the direction of the axial axis (analytical) and the strength of the axial magnetic ﬁeld component near the axis (data)

TABLE II: MAGNETIC DATA INPUT ARRANGEMENT.

Coil Parameters Coil ∆z ∆r N I (mm)(mm) (1) (A) Cathode 50 40 1618 10 Principal 400 40 17945 50 Secondary-A 100 28 341 50 Secondary-B 100 28 341 50 7 FIP/P4-33

From equation (8) the magnitude of the magnetic field profile generated is determined by a set of N toroidal coil turns solving the linear equations system. For our arrangement, we use, as first approximation, four coils (copper material), one primary, two secondary and one more for the cathode region. The set of the secondary coils support the principal coil to keep a constant profile as done in [13]. The coil of the cathode operates to control the field below the cut magnetic field BH . Here, in the emitter surface, the field should be kept as parallel as possible to it. If the cathode field Bc exceeded the Hull cutoff magnetic field, emitted electrons would not be able to achieve the anode [14]. In Table II the design parameters needed for equation (8) are deployed. In Figure 1 it shown a typical magnetic profile on the axial direction, the magnitude of the fiel along the longitudinal axis is 2.5 T, near to the analytical value of the GIF- Gyrotron-60. For the obtention of this Figure 1 we compare the equations (6) and (7) vs equation (8). Hence we obtained the magnetic range (0.18 − 0.4) in the cathode zone in Teslas, and the magnetic compression in a range of (6.4 - 14.2) dimensionless, these values are below of the restriction value (50) reported in [15], which avoids problems of the creation of electric arcs.

4 Conclusions

The Fusion Research Group presents its development design of its gyrotron ECRH source, with the main aim to establish at first one specialized device to heating plasma through the electron cyclotron resonance into our Tokamak experimental “T” facility, also, we generated an important physics area involving a general Gyrotron from its base in Mon- terrey, Mexico. This effort will serve to work in several heating derivations through our GIF-Gyrotron-60. Our general design parameters are arranged for four coils acting as a magnetic system, the electromagnetic theory was used to determine the field value in the region of interaction wave-beam producing a cyclotron frequency of electrons of 60 GHz. We developed a general equation to handle the magnetic external field based on previous scientific works, bring us the determination of the main characteristics of our ECRH system source.

5 Acknowledgements

The authors desires express their acknowledge to the Fusion Research Group, specially our gratitude to each and everyone that, in the time, has participated on this eﬀort in our Theoretical-Experimental Fusion Platform, present in our two Faculties and into our University.

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