Copper coil magnetic system study in a 60 GHz gyrotron design

Abstract

This paper presents the study of a gyrotron magnetic system with a field intensity of 2.45 T and a triode magnetron injection gun system to achieve electron trajectories into the cavity operating zone to generate a 60 GHz radiofrequency used in electron cyclotron resonance heating applications. The operating mode of the gyrotron is TE_4,3. A copper coil system arrangement of four solenoids with a stationary current of 750 A and a magnetic compression of 16.53 with a magnetic field homogeneity of 0.10% over a length of 56 mm in the operating zone was designed. A very good approximation of the magnetic field profile of the copper coil system was obtained using a novel formulation derived from the geometry of the coils. The magnetron injection gun parameters were obtained applying the trade-off equations, resulting in a cathode radius of 13.5 mm. The helicoidal and laminar trajectories of the electrons with a transverse-velocity of 1.175 are obtained from the magnetic and gun systems. The resulting trajectories are generated using the 2-D EGUN code. The study represents an effort to create an Experimental Theoretical Fusion Platform of the High Magnetic Field Program at the Autonomous University of Nuevo León (UANL).

Keywords

Gyrotron magnetic field system magnetron injection gun system

1. Introduction

A gyrotron is a coherent electromagnetic radiation source capable of generating hundreds of kilowatts of power in millimetre and submillimetre wavelength regions from an annular helicoidal electron beam. This generated power, whether as a continuous wave or as a large pulse, oscillates between 0.1 and 1 MW, covering frequencies from 28 to 170 GHz [1]. The gyrotron system was initially conceived in September 1964, in the Institute of Applied Physics of the Russian Academy of Sciences, in the city of Gorki, Russian Federation [2]. The current gyrotron development system includes a great number of technologies and applications, mainly applied in the area of plasma physics: radiofrequency plasma production, plasma heating, non-inductive current drive, plasma stabilization, active plasma diagnostics. In thermonuclear fusion research, this system generates an attractive plasma behaviour and an interesting complex phenomenon for heating applications such as low hybrid current drive (LHCD), electron cyclotron resonance heating (ECRH) and electron cyclotron current drive (ECCD) [3].

In contemporary gyrotrons, the microwave radiofrequency (RF) generation system is the main part of the ECRH device, together with ion cyclotron resonance heating (ICRH) and neutral beam injection (NBI) external heating systems is one of the most attractive and developed area on physics and engineering projected to be used in nuclear fusion reactors. The generation of an RF field is widely used not only for pre-ionizing, heating and eliminating magnetic islands in fusion plasmas, it also has applications in spectroscopy and diagnostics. The most important application of this microwave heating system at present is on the International Thermonuclear Experimental Reactor (ITER) where 24 gyrotrons would enhance the primarily conditions toward a fusion nuclear reaction [4].

The development of the present gyrotron system is oriented to the global design of an electron cyclotron resonance heating (ECRH) acting on magnetically confined plasmas. The gyrotron system will be used to achieve a long pulse, generating an accelerated electron beam into a cavity where an external strong magnetic field would modify the electron trajectories resulting in high frequency waves that will be used to heat a fusion plasma. The main focus of the gyrotron system is to be used in the elimination of magnetic islands on the plasma as well as for heating of plasma flat-top in a given operating time of (5 ms) in the Mexican Low Aspect Ratio “T” Tokamak that is being developed as a part of the Experimental Theoretical Fusion Platform of the High Magnetic Field Program of the Universidad Autónoma de Nuevo León (UANL) [5] and [6].

2. Physics and elements of a gyrotron system

A gyrotron is an accelerator particle system able to generate a coherent cyclotron stimulated radiation from the transversal kinetic energy of the electrons that are relatively turning around in a constant magnetic field, interacting resonantly with an electromagnetic wave. To reach the radiation phenomenon the electrons need to achieve the orbital bunching mechanism in a perpendicular plane at the beam direction. With this mechanism, the electrons collide among themselves, entering a deceleration phase which extracts the orbital momentum of the electrons, inducing with this the required electromagnetic radiation. To achieve this bunching mechanism, it is required to accomplish the cyclotron resonance condition: $\begin{eqnarray}\displaystyle {\omega}_{RF}-k_{z}v_{z}\approx s{\omega}_{c},\quad s=1,2,\ldots & & \displaystyle\end{eqnarray}$ (1) where ω_RF is the electromagnetic wave angular frequency, k _z is the axial wave number, v _z is the axial velocity in the electron beam direction, ω_c is the gyrating (rotational) frequency or electron cyclotron frequency, and s is the harmonic number. Early works describing this phenomenon can be found in [7,8] and [9].

Figure 1 is a general sketch of a gyrotron design with a simple cavity, it offers an overview of the main components conforming a basic gyrotron device. The magnetron injection gun (MIG) is mainly composed by a cathode (A) and an anode (B). Build into the cathode is tungsten ring (C) where the electrons are emitted in a thermionic emission process operating at high temperatures (∼1000 °C) [10] and potentials of hundreds of kilovolts. In modern gyrotrons such cathodes are complex matrix structures of tungsten with several types of impurities. The resonator or cavity (D) is a cylindrical waveguide structure with a slightly conical cross-section. This cavity is responsible to provide the suitable environment for the efficient beam–wave interaction. In the operation region (G) the electrons are circulating and interacting with an electromagnetic field, yielding its kinetic transversal energy into the electromagnetic wave. The decline collector (E) traps the remaining energy beam after the required radiofrequency is obtained. In the output window (F) the produced radiation emerges converted into a Gaussian mode [11]. (I), (H) and (J) form the intense external magnetic field conformed by the MIG, cavity and collector coils respectively. The gyrotron magnetic system is fundamental to generate a stimulated radiation due to the dependence of the electron frequency on the magnetic field intensity.

Fig. 1.

A gyrotron typical configuration is composed as follows: The MIG (A, B and C), the operating zone (D and G), the magnetic system (H, I and J), and the beam collector and the output window respectively (E and F).

3. Magnetic field system

In recent gyrotron external magnetic system designs, copper and high temperature superconductivity materials are used [12]. In the present design, copper coils materials operating with a DC current are considered. This is due to a short turn-on time, required continuous operation and economic reasons as mentioned in [13]. For this purpose, a short circuit time of 300 ms is considered in the power supply for the magnetic field copper coils. Based on the cyclotron frequency wavelength (∼5 mm), a magnetic field intensity of 2.45 T is obtained in the operating zone. The electron beam is guided into a helicoidal path using the external magnetic field lines. The magnetic system is designed as an arrangement of four copper wire solenoids with a steady current close to 750 A and cooled by water at 10 °C [14]. This particular arrangement requires four coils to achieve the characteristic profile of gyrotron devices. That is, a profile showing homogeneity in the operating zone. In addition, the magnetic compression ratio in Eq. (19) relating both cathode and operating zones must be fulfilled. These considerations in the design produce the required homogeneous magnetic field profile inside the resonant operation cavity.

3.1. Magnetic field intensity

In a gyrotron device, the external magnetic field maintains the electron gyrating frequency ω_c close to the radiofrequency field ω_RF. From Eq. (1) it follows that the required magnetic field intensity in the operating zone for a radiofrequency radiation with a wavelength 𝜆_RF [2] is: $\begin{eqnarray}\displaystyle B_{o}=\frac{2{\pi}m_{o}c}{e}\frac{{\gamma}_{o}}{s{\lambda}_{RF}}(1-n{\beta}_{z_{o}}), & & \displaystyle\end{eqnarray}$ (2) where m _o, e and c are the electron rest mass, electron charge and light speed respectively. Here 𝛾_o = 1 + eV _o∕m _o c ² is the energy of the normalized electron beam or relativistic mass factor, where V _o is the beam voltage. The refraction index of the electromagnetic waves is given by n = ck _z∕ω and the beam normalized axial velocity is given by 𝛽_{z
_o} = v _{z
_o}∕c. Characterizing the electron beam with a voltage of V _o = 67.5 kV (see Table 2), and considering that an axial wave number k _z ≈ 0 is a typical value in a gyrotron design [15], the magnetic field for the wave–beam interaction region treated in this work corresponds to a maximum value of B _o = 2.48 T. In addition, if an electron cyclotron frequency with no relativistic effect (𝛾_o = 1) is considered, then the minimum magnetic field value is B _o = 2.14 T. Therefore, the magnetic field intensity for the operating zone ranges from 2.14 ≤ B _o (T) ≤ 2.48 depending on the relativistic consideration mentioned before.

3.2. Off-axis magnetic field

In a gyrotron, the configuration of the external magnetic field system is a critical design stage. As mentioned earlier, this system guides the electron beam from the cathode region, passing through the operating zone (cavity), to finally arrive to the electron collector. The Ampere’s Law is fundamental for the solenoids design, it states that a magnetic field with axial symmetry will be formed when an electric current passes in a circular spiral. In this work, full elliptical integrals are used following the works of [16] and [17]. The analytic expressions were obtained for the axial and radial components of the off-axis magnetic field. The axial and radial magnetic field components are defined on the point (r, z) due to fact that the electric current flux I describes a circular path of radius r ₁ with centre in (0, 0) on the z axis: $\begin{eqnarray}\displaystyle B_{r}(r,z)\text{} & =\text{} & \displaystyle \frac{{\mu}_{0}I}{4{\pi}}\frac{z}{r}\left(\frac{m}{r_{1}r}\right)^{1/2}\left[\frac{2-m}{2-2m}E-K\right],\end{eqnarray}$ (3) $\begin{eqnarray}\displaystyle B_{z}(r,z)\text{} & =\text{} & \displaystyle \frac{{\mu}_{0}I}{4{\pi}}\frac{1}{r}\left(\frac{m}{r_{1}r}\right)^{1/2}\left[rK+\frac{r_{1}m-(2-m)r}{2-2m}\right],\end{eqnarray}$ (4) where, $\begin{eqnarray}\displaystyle m\text{} & =\text{} & \displaystyle \frac{4rr_{1}}{(r+r_{1})^{2}+(z+z_{1})^{2}},\nonumber\\ \displaystyle K(m)\text{} & =\text{} & \displaystyle \int _{0}^{{\pi}/2}\frac{d{\theta}}{(1-m\sin ^{2}{\theta})^{1/2}},\nonumber\\ \displaystyle E(m)\text{} & =\text{} & \displaystyle \int _{0}^{{\pi}/2}(1-m\sin ^{2}{\theta})^{1/2}∼d{\theta},\nonumber\end{eqnarray}$ and I is the electric current in A and μ₀ = 4π × 10⁻⁷ Tm/A is the magnetic permeability constant in the vacuum. K(m) is the full elliptical integral of first type and E(m) is the full elliptical integral of second type, both are dimensionless parameters depending on 0 < m < 1. To determine the magnetic field configuration generated by a coil of radius r ₁ and centred on the point (0, z ₁), a change of variable must be considered from z to z − z ₁ in Eq. (3) as well as in the definition of parameter m.

3.3. Axial magnetic field

The axial magnetic field should be as uniform (homogeneous) as possible at the wave–beam operating zone. It is to be noted that Eqs (3) and (4) are not defined for a value of r = 0, where the magnetic field intensity is required. To deal with this issue, a new analytic expression is needed to calculate the magnetic field intensity profile on the axial axis. To derive this expression, the toroidal coil of a rectangular transversal section in a cylindrical coordinate system must be considered first.

In Fig. 2 a stationary electric current density J $=J\hat{\boldsymbol{{\phi}}}=I/{\rm\Delta}z{\rm\Delta}r\hat{\boldsymbol{{\phi}}}$ is circulating in a single-turn coil of height Δr = r ₂ − r ₁ and width Δz = z ₂ − z ₁. From the Biot-Savart’s Law it is known that the volume differential dV = r dr d𝜙 dz generates a magnetic field intensity differential d B at point z ₀ which belongs to the z-axis. Having $\mathbf{R}=-r\hat{\mathbf{r}}+(z_{0}-z)\hat{\mathbf{z}}$ as the relative position between z ₀ and dV, the function B(z ₀) that represents the magnetic field profile on the z-axis due to a stationary current I passing through a single-turn coil is of the form: $\begin{eqnarray}\displaystyle \mathbf{B}(z_{0})\text{} & =\text{} & \displaystyle \frac{{\mu}_{0}}{4{\pi}}\iiint \frac{\mathbf{J}\times \mathbf{R}}{R^{3}}∼dV\nonumber\\ \displaystyle \text{} & =\text{} & \displaystyle \frac{{\mu}_{0}}{4{\pi}}\int _{r_{1}}^{r_{2}}\int _{0}^{2{\pi}}\int _{z_{1}}^{z_{2}}\frac{Ir∼dr∼d{\phi}∼dz∼\hat{\boldsymbol{{\phi}}}\times \{-r\hat{\mathbf{r}}+(z_{0}-z)\hat{\mathbf{z}}\}}{{\rm\Delta}z{\rm\Delta}r\{r^{2}+(z_{0}-z)^{2}\}^{3/2}}\nonumber\end{eqnarray}$ resulting in a general case of the form: $\begin{eqnarray}\displaystyle B(z)=\frac{I{\mu}_{0}}{2{\rm\Delta}z{\rm\Delta}r}\ln \frac{\left\{\frac{r_{2}+\sqrt{r_{2}^{2}+(z-z_{1})^{2}}}{r_{1}+\sqrt{r_{1}^{2}+(z-z_{1})^{2}}}\right\}^{z-z_{1}}}{\left\{\frac{r_{2}+\sqrt{r_{2}^{2}+(z-z_{2})^{2}}}{r_{1}+\sqrt{r_{1}^{2}+(z-z_{2})^{2}}}\right\}^{z-z_{2}}} & & \displaystyle\end{eqnarray}$ (5) where I and μ₀ are defined as in Eqs (3) and (4).

Fig. 2.

Single-turn copper coil carrying a stationary current density J.

3.4. Magnetic system arrangement

In the external magnetic system, four multi-turn copper coils (p ₁ to p ₄) are built in the cavity region as shown in Fig. 3. Each of the multi-turn coils consists of 11 × 9 single turn coils and each of these coils is defined as a 2 mm square cross-section carrying a 750 A maximum stationary electric current in a 300 ms short circuit time. The short circuit current permitted by the cable size is determined using the work done by the Insulated Cable Engineers Association (ICEA) [18].

Fig. 3.

Transversal view of the multi-turn coil p _n showing the single-turn copper coil N _{i, j} with centre in (z, r) and with an overall steady current of 750 A.

Table 1 shows these parameters in addition to the magnetic system coil positions. In this geometric configuration the magnetic field homogeneity has a value of ΔB∕B = 0.10% over a length of 56 mm throughout the operating zone located in the region 110 ≤ z (mm) ≤ 166. Figure 4 shows two axial magnetic field profiles from the magnetic field system in Table 1. The profile “axial” comes from the result obtained in this work, Equation ((5)), and the profile “off-axis” is derived from Eqs ((3)) and ((4)) using r = 0.001 mm as an approximation to the centre of each turn of the cable. A good agreement in both profiles is observed. It should also be noted that as r, Δz and Δr approach to zero, a better agreement between both profiles is gradually being obtained. From the profile “axial”, the values B _o = 2.4531 T (z = 138 mm) and B _c = 0.1484 T (z = 10 mm) are calculated for the cavity interaction region and the cathode region respectively. Therefore, the resulting magnetic compression F _m = B _o∕B _c has a value of 16.53, which is under the restriction value (<50) thus preventing electric arc creation problems [15].

Fig. 4.

Comparison between the two axial magnetic field profiles calculated from the “off-axis” (Eqs (3) and (4)) and “axial” (Eq. (5)) approaches respectively.

Table 1

Magnetic coil system

Multi-turn coil	z ₁ (mm)	r ₁ (mm)	Total turns	I (A)
p ₁	81	22	11 × 9	750
p ₂	111	31	11 × 9	750
p ₃	141	31	11 × 9	750
p ₄	171	22	11 × 9	750

4. The magnetron injection gun

The usual magnetron injection gun (MIG) is composed of a cathode, an anode and an electron emission ring. The optimization of its design depends heavily on numerical simulations, which have as its principal objective the determination of a high-performance electron beam. A reasonable first approach to an optimal MIG design could be obtained by using the trade-off algebraic equations developed in the work of [19]. These algebraic equations assume that such electron trajectories use the principles of angular momentum conservation, adiabatic processes and low spatial charge effects.

The fundamental parameters defined in the cavity zone (V _o, I _o, v _⊥o, v _zo, B _o, r _go, r _Lo) are later used to calculate the cathode region parameters (r _c, 𝜙_c, D _f, J _c∕J _L) in order to obtain a proper MIG design, as shown in Fig. 5. Here, the subscript “o” refers to the operating region (cavity zone) and the subscript “c” refers to the cathode region. The cathode electric field E _c, and the cathode-Langmuir current density ratio J _c∕J _L, are relevant in this design since they have well known restriction values of 100 kV/cm and 20 to 30% respectively, such as stated in [2] and [20]. In a gyrotron design it is common to consider two different types of MIGs: diode and triode. The incorporation of a control anode in the MIG allows for the modulation of the power into the electron beam by forcing the current to maintain a constant velocity. The transversal velocity is vital to produce a hollow electron beam in gyrotron design. A major advantage of a triode design lies in the fact that the transversal velocity can be varied by changing the voltage level without affecting the overall energy consumption. The triode MIG design as presented in this work, is a 60 GHz gyrotron currently under development at the Autonomous University of Nuevo León.

Fig. 5.

MIG system main parameters necessary to achieve an electron beam with helical trajectories, the triode type MIG is presented.

4.1. Trade-off equations

The trade-off equations as published by [19] represent an initial point in the design and simulation of the MIG. These equations are based in the physical principles that constitute and govern the electron beams in the cathode and operating zone, by using an adiabatic approximation based on the assumption that the variations of the electrical and magnetic fields are minimal in near electrons orbital regions. The adiabatic approximation allows the representation of the electrons trajectories as superposition circles distributed around the thickness of the annular electron beam. Such circles are called beamlets and have a Larmor radius of r _Lo = v _⊥o∕ω_c. When the interacting fields are quasi-homogeneous, the value of the orbital momentum of the electrons obeys the equation $p_{\bot }^{2}/B=$ constant. At the moment of increasing the intensity of the magnetic field, the generated beam compression produces the corresponding increment into the orbital electron momentum. As previously mentioned, in an adiabatic compression over the surrounding z-axis, the following parameters can be considered to accomplish the relation B _z(r, z) = B _z(z).

Inside a gyrotron, the generated radiation takes place in a circular resonator with constant radius r _c. The main purpose of the resonator geometry is to extract as much energy as possible from the electron beam. It is important to note that the resonator radius determines the cutoff frequency for any TE (transverse electric) mode. In addition, it is desired that in the gyrotron resonator only one electric transverse mode TE_{m, p} must exist with a frequency greater than its cutoff frequency, which is given by: $\begin{eqnarray}\displaystyle f_{\text{cut}}=\frac{c∼x_{m,p}}{2{\pi}r_{c}}, & & \displaystyle\end{eqnarray}$ (6) where x _{m, p} is the p-th root of the Bessel derivative $J_{m}^{\prime }(x)$ , m is the azimuthal index, p is the radial index, and c is the speed of light. To achieve a deceleration phase (where the electron loses its kinetic energy which is transferred into the electromagnetic field) the cyclotron frequency f _co must be adjusted to a value slightly lower than the operating frequency f _RF, which in turn must be slightly higher than the resonator cutoff frequency f _cut. In this design it is considered a resonator with constant radius r _c = 10 mm with a corresponding value of x _4,3 = 12.6819 in the TE_4,3 operating mode and with a cutoff frequency of f _cut = 60.35 GHz, which is verified to be less than the cyclotron frequency f _co = 61.28 GHz.

The average beam radius r _go at the operating zone can be determined by the equation: $\begin{eqnarray}\displaystyle r_{go}=\frac{c∼x_{m-1,1}}{2{\pi}f_{\text{cut}}}. & & \displaystyle\end{eqnarray}$ (7)

When the value of x _3,1 = 4.2012 obtained from [21] is used, an initial beam radius of r _go = 3.3127 mm can be obtained.

The gyrotron is a device which commonly utilizes weakly relativistic electron beams with a voltage lower than 100 kV [3]. From this voltage limitation, a range of 50 ≤ V _o (kV) ≤ 100 was considered in order to calculate the normalized energy of the electron in a range of 1.0984 ≤ 𝛾_o ≤ 1.1967. The normalized energy of the electron is then given by the equation: $\begin{eqnarray}\displaystyle {\gamma}_{o}=1+\frac{eV_{o}}{m_{o}c^{2}}. & & \displaystyle\end{eqnarray}$ (8)

Due to the helical motion of the electrons, which are subjected to quasi-homogeneous magnetic field lines in the operating zone, the cyclotron frequency f _co can be written in radian form as: $\begin{eqnarray}\displaystyle {\omega}_{c}=2{\pi}f_{co} & & \displaystyle\end{eqnarray}$ (9) where the electrons will have the following velocities in the operating zone: $\begin{eqnarray}\displaystyle v_{o} & = & \displaystyle c∼\sqrt{1-\frac{1}{{\gamma}_{o}^{2}}}\end{eqnarray}$ (10) $\begin{eqnarray}\displaystyle v_{\bot o} & = & \displaystyle v_{o}\frac{{\alpha}_{o}}{\sqrt{1+{\alpha}_{o}^{2}}}\end{eqnarray}$ (11) $\begin{eqnarray}\displaystyle v_{zo} & = & \displaystyle v_{o}\frac{1}{\sqrt{1+{\alpha}_{o}^{2}}}.\end{eqnarray}$ (12)

These velocities are important to determine the parameter 𝛼_o = v _⊥o∕v _zo that relates the azimuthal velocity v _⊥o with the axial velocity of the electron v _zo in operating zone. The parameter 𝛼_o defines the so-called pitch factor that characterises the angle of the helical curve in the trajectory of the electron.

The Larmor radius belongs to the circular trajectory that an electron describes when forming a beamlet. In operating zone, the Larmor radius is defined as: $\begin{eqnarray}\displaystyle r_{Lo}=\frac{v_{\bot o}}{{\omega}_{c}} & & \displaystyle\end{eqnarray}$ (13) and in the cathode zone, the Larmor radius is defined as: $\begin{eqnarray}\displaystyle r_{Lc}=\frac{v_{\bot c}}{{\omega}_{c}}. & & \displaystyle\end{eqnarray}$ (14)

As the resulting beamlets enter the homogeneous magnetic field in the operating zone, they start to move at a constant velocity and form a helicoidal trajectory with a constant radius r _Lo. The range in which 𝛼_o can operate is defined as 1.0 ≤ 𝛼_o ≤ 1.5 and is a common range value for gyrotrons [3]. By relating Eqs ((8))–((12)), it is possible to calculate a value for r _Lo ranging between 0.2271 ≤ r _Lo (mm) ≤ 0.3549. The cylindricity parameter is defined by [22] as: $\begin{eqnarray}\displaystyle {\mu}=\frac{1}{\sqrt{\left(\frac{r_{go}}{r_{Lo}}\right)^{2}-1}}. & & \displaystyle\end{eqnarray}$ (15)

In the present study, a value for the beam radius of r _go = 3.31 mm is calculated from Eq. (7) and the value range for the Larmor radius r _Lo. A new value range of 0.0687 ≤ μ ≤ 0.1078 can be obtained for the cylindricity by using Eq. (15), this is an acceptable range for a stable design with a cylindricity value between 0.07 ≤ μ ≤ 0.2 as recommended by [20].

In the electron source, the voltage of the control anode V _a is defined by: $\begin{eqnarray}\displaystyle \frac{eV_{a}}{m_{o}c^{2}}=\frac{\ln (1+D_{f}{\mu})}{\ln (1+2{\mu})}\left\{\left[1+\frac{4}{{\mu}^{2}}\left(\frac{1+{\mu}}{1+2{\mu}}\right)^{2}\left(\frac{{\gamma}_{o}^{2}-1}{R_{c}^{2}\cos ^{2}{\phi}_{c}}\right)\left(\frac{{\alpha}_{o}^{2}}{{\alpha}_{o}^{2}+1}\right)\right]^{1/2}-1\right\} & & \displaystyle\end{eqnarray}$ (16) here the normalized mean cathode radius is defined as R _c = r _c∕r _Lo and the normalized anode-cathode spacing factor as D _f = d _ac∕r _Lc. Where R _c is normalized according to the Larmor radius in the operating zone (r _Lo) and D _f is normalized according to the Larmor radius in the cathode zone (r _Lc), both parameters are design spacing factors that provide a radius change of one full Larmor parameter in its respective zones [19]. For the spacing factor a value of 3 ≤ D _f ≤ 7 was taken [20]. The emitter tilt angle 𝜙_c, required to generate a laminar beam and minimize the transverse spread velocity, has a value range of 20 ≤ 𝜙_c (deg) ≤ 50 as recommended by [23 ]. From the range values mentioned above, a control anode voltage range of 20.42 ≤ V _a (kV) ≤ 42.20 is obtained from Eq. (16).

Based on the previous value ranges for the parameters μ, D _f, V _a, 𝜙_c and r _Lo, a new value range for the electric field in the cathode E _c and the cathode-Langmuir ratio J _c∕J _L can be established, where J _c is the temperature limited cathode current density and J _L is the Langmuir space charge limited current density. The parameters E _c and J _c∕J _L are of great importance in the design of the MIG, as for a short pulse system (∼1 μs), the electric field E _c should be less than 100 kV/cm to avoid damages in the emitter ring [24] and for the ratio J _c∕J _L a value between 20 to 30% is typically used in order to avoid space charge problems in the control anode and cathode region [2] and [20].

Using the parameters presented before, the electric field around the cathode is given by: $\begin{eqnarray}\displaystyle E_{c}=\left(\frac{m_{o}c^{2}/e}{r_{Lo}}\right)\left(\frac{{\Phi}_{a}\cos {\phi}_{c}}{\ln (1+D_{f}∼{\mu})}\right)\left(\frac{1}{R_{c}}\right) & & \displaystyle\end{eqnarray}$ (17) and being the ratio of densities: $\begin{eqnarray}\displaystyle \frac{J_{c}}{J_{L}}=\left(\frac{2{\pi}∼r_{Lo}^{2}∼J_{c}(1+D_{f}∼{\mu})∼{\beta}^{2}}{14.66\times 10^{-6}(m_{o}c^{2}/e)^{3/2}\cos ^{2}{\phi}_{c}}\right)\left(\frac{R_{c}^{2}}{{\Phi}_{a}^{3/2}}\right), & & \displaystyle\end{eqnarray}$ (18) where $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \displaystyle {\beta}=\left({\zeta}+\frac{1}{10}{\zeta}^{2}+\frac{5}{300}{\zeta}^{3}+\frac{24}{9900}{\zeta}^{4}+\cdots ∼\right)e^{-{\zeta}/2}\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle {\zeta}=\ln (1+D_{f}∼{\mu})\nonumber\end{eqnarray}$ and Φ_a = eV _a∕m _o c ² represents the normalized control anode voltage.

In the present design, a value range for the temperature limited cathode current density of 4 ≤ J _c (A∕cm²) ≤ 6 was selected based on [3]. Considering the resulting value ranges of Eqs (17) and (18), the graph of Fig. 6 is obtained. Two limits are set at E _c = 65 kV/cm and J _c∕J _L = 25%, considering that both values are inside the acceptable ranges previously defined. By implementing these limits, the value range of the cathode radius corresponds to a value range of 12 ≤ r _c (mm) ≤ 15, with an average value of 13.5 mm.

Fig. 6.

Selected cathode radius in accordance with the limits E _c = 65 kV/cm and J _c∕J _L = 25%.

The remaining parameters of the trade-off equations are the magnetic compression, the cathode–anode gap and the cathode slang length. The magnetic compression F _m is defined as the ratio between the magnetic fields of the cavity and the cathode, as given by: $\begin{eqnarray}\displaystyle F_{m}=\frac{B_{o}}{B_{c}}=({\mu}R_{c})^{2}. & & \displaystyle\end{eqnarray}$ (19)

The cathode–anode gap d _ac is the normal distance between the cathode–anode surface, as defined by: $\begin{eqnarray}\displaystyle d_{ac}=\left(\frac{D_{f}∼{\mu}}{\cos {\phi}_{c}}\right)r_{c}. & & \displaystyle\end{eqnarray}$ (20)

Finally, the cathode slang length l _s is the necessary thickness of the electron emitting ring to produce an electron beam that meets the specified beam current I _o and cathode current density J _c, as shown below: $\begin{eqnarray}\displaystyle l_{s}=\left(\frac{I_{o}}{2{\pi}∼J_{c}}\right)\left(\frac{1}{r_{c}}\right). & & \displaystyle\end{eqnarray}$ (21)

By choosing the averaged value of the cathode radius, it is possible to set all the other parameters in a single point by recalculating the trade-off equations for μ, D _f, V _a, and so on (see Table 2).

Fig. 7.

Output of the electron trajectories simulation from the EGUN program using the magnetic coil arrangement shown in Table 1 and the beam design parameters shown in Table 2.

Table 2

The nominal design parameters for the 60 GHz MIG

Parameters		Values	References
Cavity beam voltage	V _o	67.5 kV	<100 kV
Control anode voltage	V _a	24.7 kV	<V _o
Cavity beam current	I _b	10.05 A
Cavity magnetic field	B _o	2.44 T
Cavity beam radius	r _go	3.31 mm
Cavity Larmor radius	r _Lo	0.282 mm	≪r _go
Cavity velocity ratio	𝛼_o	1.175	>1
Cavity normalized energy	𝛾_o	1.1328	<1.2
Magnetic compression	F _m	16.72	<50
Cathode electric field	E _c	49.4 kV/cm	<100 kV/cm
Cathode tilt angle	𝜙_c	30.5°	20 to 45°
Cathode–anode gap	d _ac	5.9 mm
Cathode radius	r _c	13.5 mm
Cathode slant length	l _s	2.52 mm
Cathode-Langmuir loading	J _c∕J _L	13.86%	20 to 30%

5. Electron beam trajectories

To validate the parameters presented in Table 2 and generate the electron trajectories from the gun emitter to the beam–wave interaction zone, the code 2-D EGUN was used. This code is a well know program used for calculating the trajectories of charged particles driven by static electromagnetic fields [25]. This code is validated in the following works [24,26–28]. Figure 7 shows the electron trajectories generated from a triode type MIG with the emitter set to z _c = 10 mm and the operating zone set to z _o = 138 mm, with the trajectories moving around this region. Additionally, voltage equipotential lines ranging from 6.178 kV to 23.476 kV can be seen in the region from z = 17.25 mm to z = 23.61 mm. The axial magnetic field profile in Fig. 7 is modelled using Eq. (5) and the coil array data from Table 1, with the inputs for the potentials given as 0 kV in the cathode, 25.71 kV in the control anode and 67.5 kV in main anode respectively. The annular electron beam formed at z = 138 mm, in the region of uniform magnetic field, has an average energy of ϵ = 67.509 keV with an average velocity ratio of 𝛼_o = 1.25 over an average radius at r _go = 3.33 mm and an average Larmor radius at r _Lo = 0.288 mm. Figure 8 presents a close-up of the electron trajectories up to z = 25 mm, here eleven electron beamlets can be seen forming a clear laminar flow. The electron beamlets are emitted perpendicular to the emitter and have an initial kinetic energy close to zero.

Fig. 8.

Close up to the eleven beamlets forming a laminar electron trajectory in the MIG region.

6. Conclusions

From the simulation results presented in this work, a design of the gyrotron magnetic system for the transportation of the electron beam is obtained. This gyrotron design includes one arrangement of four copper coils with a stationary current of 750 A that compose a magnetic system with a compression ratio of F _m = 16.72 and a magnetic field homogeneity of 0.10% over a length of 56 mm in the operating zone. A quasi-homogeneous axial magnetic field profile of B _z = 2.44 T is achieved in the beam–wave interaction zone. A very good approximation of this profile was constructed using a novel formulation of the magnetic field profile equation which is derived from the geometry of the coils and allows an optimal manipulation of the iterative process necessary to fix the magnetic field profile, and whose results are confirmed by specialized literature. With this magnetic field intensity, an electron beam inside the interaction zone will have an operating frequency near to 60 GHz and in consequence, the required radiation for plasma heating will be generated. As another system of the gyrotron, a magnetron injection gun of type triode with a cathode radius of r _c = 13.5 mm capable of producing an electron beam with helicoidal and laminar trajectories is presented. The trade-off equations based on the angular momentum conservation and the low space charge effects of the adiabatic electron trajectories are applied to obtain the MIG design parameters. Those parameters in conjunction with the external axial magnetic field are used to outline the beam electron trajectories. The particle trajectory simulation reports an electron beam with a low transverse-velocity spread of 𝛼_o = 1.175 and an energy beam of ϵ = 67.5 keV. The electron beam trajectories are plotted using the software EGUN. The work presented here is a national development, since this is the first time that an effort has been made in Mexico to research technology for both the accelerator gyrotron system applied to plasma heating and the high magnetic field confinement facilities. This gyrotron design and its future experimental phase is considered to be an essential part of the main electron cyclotron resonance heating system for the Experimental Theoretical Fusion Platform of the High Magnetic Field Program of the Universidad Autónoma de Nuevo León (UANL).

Footnotes

Acknowledgements

The present work was supported by the Fusion Research Group (GIF) constituted by the Faculty of Mechanical and Electrical Engineering and the Faculty of Physical and Mathematical Sciences, both faculties are part of the Autonomous University of Nuevo León.

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