5.3 Vsim Setup
5.3.7 Distributed Cathode With Spatial and Time Varying Current
currents can also be used. Spatially-time varying currents have been simulated in magnetrons [13, 93–95] and have shown improvements to efficiency and demonstrated phase control, among other things. For simplicity, in the simulation, discrete emission locations are not used, and the current varies continuously along x. To implement this in VSim, the relMacroFluxFunc function is used.
The relMacroFluxFunc is a function whose inputs are spatial and time coordinates, and output is a value from 0-1. Each time there is a particle load attempt, a random number from 0-1 is generated, and if the random number is less than the value of relMacroFluxFunc, then the particle is loaded; otherwise it is not loaded. This can create non-uniform loading density and variations in time. Note that with a specified number of macroparticles to be emitted per time step and because not all particles are actually emitted, there will be less current actually emitted than what is specified.
The number of macroparticles per timestep needs to be increased so that the average number of particles emitted yields the desired current.
Four different profiles are studied in this work: two spatial and two time-varying. By varying the spatial profile, the beam injection shape can be tailored to see the effects on gain, efficiency, and noise. The effects on cathode length and the current density vs. length are studied. The time-varying periodic profiles can pre-bunch the beam synchronously with the RF wave to improve gain, efficiency, and noise on the output. These profiles are described in the following sections.
5.3.7.1 Spatial Current Profiles
The two spatial profiles are a uniform profile and a linear profile. The uniform current density profile is constant throughout the cathode, and the total current equals the desired current. Eq. (5.1) describes the uniform profile current density. The total current isItot and the sweep variable here is the length of the cathode, Le, to find the effect from the length of the cathode. The second profile is a linear profile described by Eq. (5.2) where the terms are added when the slope is positive and subtracted when negative. fDC is the fraction of the current to be emitted uniformly, and Jp is the peak current density in[A/m]and is chosen to obtain the desired total current. The current densities given here are in [A/m]because they vary only in the x−direction and the current is uniform in the z−direction. Fig. 5.24 shows three different spatial profiles. Je(x) = Itot Le A m (5.1)
Je(x) =Jp fDC ± 1−fDC Le (x−xstart) A m (5.2)
Figure 5.24: Three spatial profiles: linear profile with positive slope and no DC current in blue, linear profiles with negative slope and 50% DC current in magenta, and a uniform profile in green.
5.3.7.2 Sine Wave Emission Profile
This profile was the easiest time-varying profile to implement and allows for a smooth current density transition from current density peaks to nulls which may have an effect on noise. The sine wave profile is shifted to be between 0-Jp and travels through time and x. Eq. (5.3) shows the simplest form of the equation, with no DC current. ω is the angular frequency, β is the wave number associated with the retarded wave, and
φis the phase offset used to synchronize maximums in beam currents with minimums of the x-component of the electric field. φ = φt−φx +φ90, where φt accounts for a time offset, φx shifts the sinusoid starting point to under the input coax, and φ90 shifts the maximum to be under the input coax. φof f set ranges from 0 to π and is the controlled phase offset used determine the optimum synchronization between the beam profile and the RF wave. Fig. 5.25 shows a plot of the current distribution compared with the RF wave at ωt =φt with φof f set = 0 rad. The maximums in the emission profile follow the minimums in the RF wave with φof f set = 0 rad.
Je(x, t) =Jp 1 2+ 1 2sin (βx−ωt+φ+φof f set) A m (5.3)
Figure 5.25: The sine wave electron emission profile compared to the ERF x field with φof f set = 0 rad at ωt=φt. In this case the profile peaks are in the accelerating regions of the RF wave (out of phase).
5.3.7.3 Square Pulse Emission Profile
This profile can vary the current density peaks and can test the effect of synchronous current on the RF wave. It uses square pulses that travel synchronously with the RF wave. This profile is similar to the sine wave profile but has the ability to use varying pulse widths to provide tightly focused beams to better interact with the RF wave on the circuit. The approach is designed to optimize injection of synchronous current. Each pulse can be defined by two Heaviside functions which travel in time as shown in Eq. 5.4. Each pulse would have to be defined explicitly for the entire simulation time. To make it easier, the sine wave function is used in conjunction with the max and ceil functions, shown in eq. (5.5). The function max(a, b) takes the maximum value a and b, ceil(a) rounds up to the nearest integer, and yLp is the y value corresponding with the desired pulse width Lp. Using eq. (5.5), all the pulses are defined for all time. Fig. 5.26 shows the square pulse emission profile atωt=φt compared with the x component of the RF field on the circuit.
Jpulse =Jp H x+ Lp 2 −xof f set−vpt −H x−Lp 2 −xof f set−vpt A m (5.4)
Je(x, t) =Jp(ceil(max(yLp,sin (βx−ωt+φ+φof f set)−yLp)))
A m
(5.5)
Figure 5.26: The square pulse electron emission profile compared to the RF electric field in the x-direction (ERF x) at ωt=φt. In this case the pulse is “out of phase” with the RF field.
5.3.7.4 GFEA Modulated Current Characteristics
Modulation of the current requires control over when the cathode turns on and emits current and when it turns off. The calculation of GFEA power consumption uses data from the GFEAs studied at Massachusetts Institute of Technology (MIT) [25–27] discussed in Sec. 2.6.3.2. To turn the current mostly off, the gate-to-emitter voltage must be.15 V. To modulate the current from100 mA/cm2to∼0 mA/cm2, a voltage swing of 30 V is needed (15 V to 45 V). But if “off” was considered to be 1% of the maximum current(1 mA/cm2), a voltage swing of only15 Vis needed. Approximately
15 V is needed to change the current by 2 orders of magnitude at all currents. This reduces the power consumption of the cathode modulation significantly.
To modulate the current, a square gate-to-emitter voltage pulse can be used to turn on and off the current. The GFEA consists of a ground plane with tips and a gate plane above and surrounding the tips separated by a dielectric. This is a capacitive load and can be modeled as a capacitor. To turn on and off the current using a square pulse, the cathode consumes the power as calculated by Eq. (5.6). Using Vpp = 15 V, estimating C = 3 nF/cm2, and using the operating frequency of the NU CFA, f = 150 MHz, the power consumption of the GFEA is ≈ 50 W/cm2. The square pulse also limits the upper frequency that can be used, limited by the charge-up time of the capacitor whose cutoff frequency,fc, is determined by Eq. (5.7), where R is the series resistance.
Pgf ea= 1 2CV 2 ppf (5.6) τrc =RC = 1 2πfc (5.7) Rather than using a square gate-to-emitter pulse to turn on and off the current on the GFEA with minimal power consumption, a sinusoid input signal and a resonant circuit can be used instead. This reduces the bandwidth of the device, but reduces the power consumption dramatically. By using the resonant RLC circuit shown in Fig. 5.27, the load impedance can be designed to be purely real and matched to the transmission line, which is 50 Ω. The GFEA is modeled by the capacitance, Cgf ea, and L and R are the added elements to adjust the load. By sizing L = 1/ω2
0C, the reactive impedance is near zero. R is sized such that the real impedance is equal to
50 Ω after accounting for the line resistances. The power dissipated into this load is given by Eq. (5.8) where Vamp =Vpp/2. Using Vpp = 15 V and R = 50 Ω, the power consumption of the GFEA is ≈ 0.5 W. This is much lower power consumption than using a square pulse and is not dependent on the area of the GFEA used since the inductor is sized accordingly.
ZLOAD=R+jωL+ 1 jωC =R+ jω0 ω2 0C − j ω0C =R Pgf ea= 1 2 Vamp R 2 (5.8) CFEA ZT ZLOAD VRF+VDC
Figure 5.27: Proposed resonant circuit to minimize consumed power by the GFEA.
This resonant circuit would decrease the bandwidth of the device due to the resonant circuit itself and may prove difficult to implement. Another proven design from Calame [97], explained in Sec. 2.8.5, can be used instead. The Calame circuit when used as a resonant circuit, would have drive powers similar to the resonant circuit proposed before (≈0.5 W). When used as a non-resonant circuit to achieve a higher bandwidth, the RF drive power would be comparable to square pulse estimate
Emitted electrons when using a sinusoid input signal on the gate-to-emitter input forms a pulse shape due to the exponential I-V relationship shown in Fig. 2.21. The pulse shape for two sinusoidal inputs are shown in Fig. 5.28. The input to the GFEA is sinusoidal with an offset. Fig. 5.28(a) shows a the current density output for a sinusoid which varies from 20−50 V. The pulse width is less than half the period. Changing the sinusoid to vary from 35−50 V results in the pulse shape shown in Fig. 5.28(b). The pulse width is slightly larger, but maintains the same shape. These current density pulses created by the sinusoidal input are ideal for use in the modulated cathode. The sinusoidal input can turn on and off the current efficiently.
ωt/π
0 0.5 1 1.5 2
Current Density [A/cm
2 ] 0 1 2 3 4 (a) ωt/π 0 0.5 1 1.5 2
Current Density [A/cm
2 ] 0 1 2 3 4 (b)
Figure 5.28: Calculated electron current density pulses from the MIT GFEAs [25] for a sinusoidal input from (a) Vg = 20−50 V and (b) Vg =
CHAPTER 6
EXPERIMENTAL AND SIMULATED RESULTS AND
DISCUSSION OF BSU CFA
6.1
CFA Experiments
The injected beam experiment at Boise State University using the SW1 and SW2 con- figuration was performed many times without success. No electron beam interaction with the RF wave was observed. At first, arcing on the cathode was common, and the cathode was constantly damaged. The setup eventually minimized this damaging arcing, and this cathode section functioned for many hours. Low amplitude arcing somewhere else, however, is a constant occurrence at higher voltages, but these arcs do not noticeably damage the cathode. The arcing rate was also observed to increase when the RF wave and electron beam were turned on simultaneously. The reason for the arcing is currently unknown. There are many locations where electrodes are close to each other and have a high potential difference, so pinpointing where the arcing occurs has been difficult.
To determine the reason there was no RF-wave-beam interaction is discussed in the following sections. The potential problems are listed below. The first two potential problems are studied in detail; however the last one is much more difficult to test with the cathodes available to the group and can only be studied by simulation in
Vsim.
1. SW circuit dispersion: extensive study on this is provided in Sec. 6.2 and is ruled out as a problem
2. Poor electron optics: extensive study on this is provided in Sec. 6.3 and remains a potential problem
3. Low injected current from the GFEA: simulations results are discussed in Sec. 6.4 and indicate this as the major problem
6.2
Meander Line Study
The main goal of these experiments was to find the RF phase velocity at the operating frequency. For brevity, the focus of this will be on SW2. The same analysis was performed on SW1 to determine the predicted circuit retardation.