Theoretical Analysis of Fowler Nordheim Parameterization and RLC Characteristics for Ring Cathode Field Emitter Arrays f
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analytical model of a field emitter is used to estimate Fowler Nordheim A and B parameters, effective resistance and
capacitance of the array under several GHz modulation, signal propogation lengths, total current and current density, and effects of emitter non-uniformity on the basis of array geometry and materials. Estimates of inductance, resistance, and capacitance are made to estimate the drive power required to produce a bunched electron beam for Inductive Output Amplifier applications. An electronic efficiency of 32% with 15 dB gain may be possible from an array producing 260 mA peak, 71 mA average, current at 10 GHz using a TWT helix 1.51 cm long.
INTRODUCTION Due to their high current density capabilities and instant turn-on, field emitter arrays (FEAs) have been recognized as a promising alternative to thermionic emitters for use in Inductive Output Amplifiers (IOAs) [1]. The steady evolution in the performance of field emitter arrays, particularly in the form of ring cathodes designed for IOAs, motivates an optimism that FEAs may usher in a new class of rf amplifiers due to their ability to create density modulated beams at GHz frequencies. Here, we investigate the performance levels required of an emission-gated FEA-TWT by using an analytical model of a field emitter. Previous approaches to modeling an FEA cathode in an emission gated TWT (Twystrode) [2] treated the FEA and tube physics separately, and joined analyses using the Fowler-Nordheim AFN and BFN parameters, which were therefore independent quantities. However, by extending a recently developed analytical model of an FEA unit cell, AFN and BFN can be predicted on the basis of the geometry, materials, and uniformity of the array, allowing for a seamless relation of fabrication issues to tube performance. While a rigorous estimation of the actual performance of an emission-gated FEA-based TWT (a "Twystrode") requires particle-in-cell simulations, the simplified analysis here gives qualitatively valid results. ANALYTICAL MODEL OF A FIELD EMITTER The field at the apex of a hyperbolic emitter may be well approximated by [3,4] F[tip =lk ( as-tan 2fl ; where k = -L 86 + a'g cot 0/) "l+nkag / a S) ca. 54( a.)~ i
1
where a. and ag are the tip and gate radii, respectively, Vg is the gate voltage, and /Pe is the cone angle. The factor k can be extrapolated from Boundary Element simulations; however, we have found that k scales with geometry as given in Eq. (1), and typical examples of which are shown in Fig. (la) for geometries similar to field emitters produced by MIT-Lincoln Labs and SRI. The coefficient of Vg = Vgate is the field enhancement factor fig, and is shown in Figure (lb). The current density is approximated by JFN(F) = af, F2 exp(-bfnIF). Integrating the current density over the surface of the emitter can be shown to be Itip(Vg) = 2 7c a2 cos2(3c) (
2
sin (fc] JFN(F ) S ++ sn2(ic) bf tip(2) tip
The coefficient of JFN is called the area factor barea. Due to the presence of Ftip in the coefficient, it is a gate voltage dependent quantity
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