1 Single-stage depressed collector analysis

A collector is one of most important component of gyrotron. As axial magnetic field reducing, the gyration radius of electron is increasing and transverse energy convert into axial kinetic energy dissipated in collector wall with heat. Because only transverse energy of electron beam can interact with RF field and the interaction efficiency is relatively low, plenty of electron energy have no contribution to RF output power and are wasted in vain. This behavior limits the whole efficiency of high power gyrotron as well as the power capacity of collector. The depressed collector is capable of recovering axial electron energy with retarding electrostatic field produced by potential difference between collector and body of tube. For the sake of easy fabrication and cost effectiveness, we adopt single-stage depressed collector here.

The operating parameters of electron beam are Vbeam=80 kV , Ibeam=70 A , pitch factor α=1.3 , an average guiding center radius of 10.1 mm in cavity. The 35% electronic efficiency has been used to calculate the beam power after interaction with RF for a gyrotron at a frequency of 170 GHz and the residual beam power is approximately 3.64 MW. The energy distribution of spent electron is obtained by beam-wave interaction code. The choice of reasonable decelerating potential need to avoid reflected electrons and recover residual energy as more as possible.

The collector is electrically isolated from body of tube with a high voltage ceramic insulator. The decelerating potential is 30 kV, the maximum electric field strength is supposed to satisfy vacuum constraint which the breakdown strength is 10 kV/mm , otherwise the unwished pulse shortening and damaged surface may appear. The depressed collector has been simulated and optimized by using CST Particle Studio software. The optimal collector radius and thickness are 400 mm and 3 mm, respectively. Figure 1 shows the equipotential lines and electron trajectory on collector. The maximum electric field strength is 1.12 kV/mm which is low enough to achieve the vacuum condition. The collector efficiency ηc and the total efficiency η are calculated by

where Pbeam is total beam power, ηe is electronic efficiency equally the ratio of RF output power to total beam power and Pc is recovered power. The calculated collector efficiency is 57% with ηe 35% . Simultaneously, the total efficiency η exceeding 55% has been achieved. The dissipated power density distribution is showed in Fig. 2. The Figure 2 indicates that the spent electrons impinge inner collector wall with a deposited area nearly 450 mm. The maximum power density is about 482 W/cm2 within the limit condition 500 W/cm2 for OFHC copper. But the power density distribution is inhomogeneous and the maximum power density is still in a high level remaining a small safety margin, resulting in a higher requirement for the design of collector cooling system. So we employ the magnetic field sweeping system for collector in order to smoothen power density distribution as well as to reduce power density further.

Fig. 1 The equipotential lines and electron trajectory

图1 等电势线和电子轨迹图

Fig. 2 The dissipated power density distribution along z axis

图2 沿z轴耗散功率密度分布图

2 Simulation and analysis of TFSS

The TFSS usually consists of two or three pairs of solenoid coils called TF_coils which are generating transverse sweeping magnetic field and one solenoid coil called VF_coil which is generating static longitudinal magnetic field. The former is powered by AC power supply while the latter is powered by direct current power supply. Here we employ three pairs of coils namely 6 coils which have 60 degrees phase shift between two adjacent coils. That is to say the phase constant equals 0° ,  60° , 120° , 180° ,  240° , 300° for each coil respectively ^[10]. Six TF_coils generate rotating magnetic field which would cause elliptic shape of electron trajectory rather than circle as usual. This behavior spreads out electron deposition area and increases trace length so that lowering the peak power density obviously. The Fig.3 show elliptic electron trajectory clearly from different views.

(a) left side view

（a） 左视图

（b) top view

（b） 俯视图

Fig.3 The elliptic electron trajectory

图3 椭圆电子轨迹

To find out the influence of number of TF_coils on dissipated power density, we keep it constant that the size and radial height of TF_coil to see variation of the peak power density and spread trace length by changing coil numbers. The simulated results are listed in table 1.

From table 1, it is seen that azimuthal distance between adjacent coils decrease with the number of TF_coil increasing, the trace length of spent electron increase and peak power density on collector surface decrease. When the number of TF_coil increases from 6 to 7, the peak power density reduces by approximately 7 W/cm2 , while the trace length of dissipated electrons increases by nearly 300mm , which means the influence on peak power density generated by number of TF_coil has become weaker and weaker and results in larger collector longitudinal size. Besides, thinking the condition of current power supply for solenoid coil, it’s feasible to choose 3 pairs of coils namely 6 coils as transverse field sweeping system.

Other parameters of TFSS geometry are listed in table 2. The whole profile of TFSS is showed in Fig.4.

(a) front view

（a） 前视图

(b) left side view

（b） 左视图

Fig. 4 The whole profile of TFSS.

图4 TFSS 结构图

The dissipated power density distribution is vital to next thermal analysis and its peak value determines maximum hot temperature of collector surface. Since the supplied current wave varies cyclically in time, the time-averaged power density has been employed to measure dissipated power density distribution.

To calculate time-averaged power density, the instantaneous power density and their corresponding weights in time must be gotten first, then multiplying them and getting a sum at different time. As for sweeping signal which is saw tooth or triangular wave, the weights are equal in any time. But sinusoidal signal has a various rates of descent or climbing corresponding to different weights. The variation between rates and weights are in opposite direction which means that high rates represent low weights and low rates represent high weights. Besides the value of weights are related to samples that we take in current cycle.

The profile of peak power density with different samples is showed in Fig.5. When samples are close to 36×29=18432 , the peak power density reaches to a nearly constant 128.7 W/cm2 with an increasing percentage of 0.12% only. For samples we take above, the single integral interval is small enough namely 0.02° for one cycle 360° as showed in Fig.6. And the whole dissipated power density distribution on collector wall with peak power density 128.7 W/cm2 is showed in Fig.7.

Fig. 5 The profile of peak power density with

Fig. 6 The divided intervals for one current cycle

Fig. 7 The whole dissipated power density distribution on collector wall with peak power density 128.7 W/cm2

图7 最大功率密度128.7 W/cm2对应的收集极壁耗散功率密度分布图

It’s inevitable to give rise to some unexpected errors in practical fabrication process due to misalignments and machining errors. The sensitivity analysis have been performed about the effect of the location of TFSS on peak power density. The simulated results are illustrated in Fig.8. The Fig.8 (a) shows the variation of peak power density as a function of axial deviation. The peak power density are fluctuated in a range from 127  W/cm2 to 134 W/cm2 with a maximum deviation of 4.1%. As showed in Fig.8 (b), the peak power density increases relatively quickly when radial deviation becomes larger than 3mm. But the peak power density is no more than 135  W/cm2 with a maximum deviation less than 5%.

(a)

(b)

Fig. 8 The peak power density as a function of axial deviation (a). and radial deviation (b)

图8 轴向偏移引起的最大功率密度变化（a）和径向偏移引起的最大功率密度变化（b）

In order to reduce peak power density further, a way of modulated transversal sweeping has been adopted. The lower peak power density alleviates the limited conditions of the design about water cooling system of collector, supports the stable and long time operation of gyrotron.

For modulated transversal sweeping, it’s a varied value in time that not only magnitude but also phase of coil current. In that case, the electron trajectory has a sweeping rotating motion on collector inner wall and moves back and forth along the symmetric z axis at the same time. The local hot spots will be shifted up and down scattering to ambient area so that the peak power density is going be reduced obviously.

We utilize the modulated signal waveform of sinusoid. The optimal modulated frequency 10 Hz has been employed to simulate the influence of modulation depth between the average and the max range of current magnitude on peak power density. The simulation result is showed in Fig. 9. The larger modulation depth may result in higher peak power density due to reducing electrons strike areas. Finally the optimal modulation depth of 38.9% has been found with a peak power density only 117.1 W/cm2 and the power density distribution is showed in Fig.10. This behavior enhances the collector power density capacity by a factor of 1.09 and 1.76 compared to unmodulated transversal sweeping and SDC without TFSS leading to a higher safety margin for collector of gyrotron.

Fig. 9 The peak power density of different

图9 不同调制深度下最大功率密度变化

Fig. 10 The power density distribution with

图10 调制深度38.9%对应的功率密度分布

The table 3 gives a comparison between this schemes and other existing schemes from several aspects including operating frequency, collector dissipated power, adopted schemes and peak power density ^[13,14]. The schemes 1 operates in lower values both in operating frequency and collector dissipated power. The peak power density is 130  W/cm2 with TFSS only. The schemes 2 employs VFSS with peak power density above 400  W/cm2 . Furthermore, the power supply for VFSS is much more complex and expensive than TFSS’s. This schemes adopt the same techniques with the schemes 3. Nevertheless, the collector dissipated power and peak power density are nearly twice and quarter respectively compared to schemes 3. The lower power density guarantees higher operating safety margin as well as enhances power capacity of collector to a large extent. At the same time, it also offers the possibility of higher RF power output for single gyrotron at 170 GHz.

3 Thermal and fluid analysis

The thermal analysis of collector is necessary to ensure the safety and feasibility in actual situation. The enormous heat generated by dissipated power transfer from inner surface to outer surface of collector wall by conduction and then are absorbed by coolant due to convection motion. The influence of heat radiation is neglected in this case. The oxygen free high conductivity (OFHC) copper is used as collector material and deionized water is used as coolant. The non-uniform power density distribution are loaded on inner collector wall as heat flux in commercial available software ANSYS for more accurate simulations.

The design of collector cooling system with axial grooves has been taken into account which has a capability of reducing the temperature of collector wall at some extent for more interface with coolant compared to the surface without grooves. The single groove geometry is showed in Fig.11. The parameters of groove geometry are groove width 5° , groove height 5mm, groove number 45, the corresponding hydraulic diameter is 8.8mm . Furthermore, the water flow rate is 900 L/min , the initial water temperature is 20℃ . The contour plot of temperature distribution on collector with/without TFSS are showed in Fig.12(a-b). A comparison is summarized in table 4. The maximum temperature of collector inner or outer surface without TFSS exceed two times higher than those with TFSS, as well as the constrained temperature 300 ℃ for collector material.

Fig. 11 The structure of single groove

图11 单槽结构

(a)

(b)

Fig. 12 The contour plot of temperature distribution on collector (a) with TFSS, (b) without TFSS

图12 收集极温度分布云图 （a）采用TFSS,（b）不采用TFSS

This high temperature value gives a serious result which plenty of copper molecule have been evapotranspired to high voltage ceramic insulator. So the insulating property has been deteriorated between collector and body of tube leading to a bad performance for gyrotron. This comparison implies that the implement of TFSS is indispensable to make sure high effective and stable operation of tube. It is also showed in Fig.13 that the profile of temperature distribution of collector inner wall along z axis corresponding to Fig.12 (a). The profile has a great agreement with dissipated power density distribution in Fig.10. There are two peak values positioned in approximately 200 mm and 500 mm from original point.

Fig. 13 The profile of temperature distribution of collector inner wall along z axis

图13 收集极内壁沿z轴温度分布

Besides the effect of water flow rate is also studied with identical groove geometry above. The result is obtained in Fig.14 and corresponding simulated calculating results for heat transfer coefficient and Reynolds number are showed in Fig.15.

Fig. 14 The effect of water flow rate

图14 水流速度的影响

Fig.15 The variation of heat transfer coefficient

图15 水流速度引起的换热系数和雷诺系数变化

With the increase of water flow rate, the maximum temperature of inner or outer collector wall reduce gradually whereas the heat transfer coefficient and Reynolds number increase. And when water flow rate becomes higher than 600 L/min , the declining rate of maximum temperature becomes lower and lower. The heat transfer coefficient alpha α and Reynolds number Re are determined by following two equations:

where B is a constant determined by the physical characteristics of coolant., υ is the velocity of water and the De is the hydraulic diameter ^[15]. Since the geometry of groove remains constant, the hydraulic diameter has no change and alpha will increase with the increase of water flow rate. The heat transfer coefficient α characterizes the capability of transferring heat from solid to fluid. Thus the higher alpha means more heat in collector wall will be absorbed by the coolant deionized water and of course the maximum temperature of inner or outer collector surface become lower.

where ν is the kinematic viscosity of the fluid ^[16]. As the geometry of groove and the characteristics of coolant remain constant, the De and ν have no change. So the Reynolds number Re is in direct proportion to the water flow rate. Finally, we take water flow rate 1100 L/min namely the water velocity 2.3 m/s for each groove as the optimal value within the experimental tolerance limit ^[15,17]. In this case, the alpha is 6000 W/m2∙℃ and the Re is 20062 which means the coolant DI water is turbulent flow so the capability of transferring heat is in the highest level among three kinds of liquid conditions, laminar, transient and turbulent. The temperature distribution of collector cooling system is listed in table 5. The maximum temperature of collector surface and outer water are lower than safety limit 300 ℃ and 70 ℃ , respectively.

4 Conclusion

The single-stage depressed collector enhances the total efficiency of tube exceeding 55%. The transverse field sweeping system spreads out spent electron trajectory length from 450 mm to 1100 mm. Thus, the peak power density reduces to 128.7 W/cm2 from 482 W/cm2 . As the supplied current of TFSS was modulated by 10 Hz sinusoid wave, the peak power density reduces to 117.1 W/cm2 with the modulation depth of 38.9% further. The number of TF_coils has been chosen to 6 according to peak power density and the feasibility of power supply. The thermal and fluid analysis are carried out by ANSYS code. The simulated results show that the maximum temperature of inner and outer collector wall are 156.09 ℃ and 140.35 ℃ respectively with groove number 45, groove width 3° , water flow rate 1100 L/min . The implementation of SDC and TFSS confirm the increase of total efficiency and the reduction of power density. These results bring specific guidance for the development of a MW-class gyrotron.

References