# Fields and Q factor for a high-power, annular beam klystron

The reason there has been a long interval since my last post is that I have been working intensively on Aether, our new 3D electromagnetic code. The program is performing beautifully in tests and is well on schedule for the September 30 release.

In this post, I want to complete an example started in a previous one (http://www.fieldp.com/myblog/?p=47). The previous work described a resonant frequency calculation for the loaded klystron cavity shown in the first picture. Since that time, we have completed all functions of the Aether RF mode and added many capabilities to the Aerial post-processor. It is now possible to compute and display the fields of the resonant mode and to make an accurate calculation of the loaded Q factor.

The previously-determined resonant frequency f = 1.4209 GHz was used in the new calculation. The Aether script contained the following lines:

Freq = 1.4209E9
NPeriod = 40 2
The relatively large number of RF periods was necessary to ensure that the fields reached equilibrium before conversion of the time-domain solution to phasor form. Some care must be taken to represent drive currents accurately in a loaded RF solution. In this case, the drive was an annular beam with outer radius 0.5 cm and inner radius 0.3 that extended along the length of the transport tube. The second figure shows the beam placement. The harmonic component of current at frequency f in the bunched beam had amplitude I = 100 A. The discrete representation of the beam cross-section included 48 elements, each with area 1.0E-6  m2. The current density to generate 100 A was jz = 2.083E6 A/m2.

The third figure shows a plot of Hy(t) at the probe position, confirming that the solution had reached a steady state. The run time was 1 hour and 20 minutes. One method to calculate cavity loading is to use the energy and power integrals recorded in the listing file. The quality factor is given by

Q = 2πfU/P,
where U and P are time-averaged values of the electromagnetic energy in the cavity and power dissipated in the ideal absorbing layer at the end of the transmission line. The values determined in the Aether solution were U = 0.0485 J and P = 14.53 MW. The equation implies that Q = 29.8. The Q value can also be determined from the signal envelope in the third figure. The theoretical variation is

Hy(t) = Hy0 sin(2πft) [ 1 – exp (-πft/Q)].

Measurements of the signal using the Probe utility program  imply that Q ≈ 31.5.

The final activity was to inspect the mode fields. The last figure shows the variation of |E| over the xy plane at z = 0.0. Here, the amplitude symbol refers to the peak value in time of the sum of the spatial components of the electric field. Note that the uniform value in the transmission line indicates a wave traveling to the right in the transmission line with no reflection at the absorbing layer (VSWR = 0.0). Using the Line integral command in Aerial, the voltage in the transmission line was determined to be V = 45.2 kV at a phase of 240°. The corresponding power flux is

P = V^2/2Z = 13.62 MW.

The peak electric field value in the solution of 2.26 MV/m occurred on the tip of the smaller nose. A line integral of electric field across the axis of the cavity gave a cavity voltage V = 295.4 kV at 180°. The predicted beam power was P = IV/2 = 14.8 MW. Within the accuracy of the line integrals, the three methods for estimating the RF power were consistent.