For the subsequent analysis, the coupled beam-wave system is assumed to be a periodic structure so that considering a single unit cell is sufficient to derive characteristic quantities. In order to determine the impact of an electron beam coupled to the delay line, a beam model similar to the one presented in Section2.2.2 is used. The modulation of the beam is thus modeled by velocity and space-charge density amplitudes, u and ρ, that are assumed to be small with respect to their DC counterparts.
This description of the electron beam is coupled to the EC model of the FW line outlined in Section 3.3. All in all, a four-by-four transmission matrix can be set up to relate the relevant signals in one plane of the periodic unit cell to those in the other. This yields
where V2,3 and I2,3 denote the voltages and currents at the respective ports, and ua,b and ρa,b are the amplitudes of velocity and charge density modulation in the respective planes. The coupled unit cell is illustrated in Figure 4.1, where [Z] indicates the three-port impedance matrix of the delay line unit cell. Rotational symmetry of the beam tunnel region is assumed.
Figure 4.1: Schematic of coupled beam-wave unit cell.
The transfer matrix of the coupled system [Tc] is obtained numerically by evalu-ating the interaction of the delay line and the electron beam in one unit cell. The axial electric field shape, as described in Section 3.3.4, is crucial in this analysis, because it defines the coupling between the two systems. It determines the field the beam is subjected to (Equation (2.9)) and also the current induced in the delay line (Equation (2.19)). The field shape of KlysTOP discussed in Section 3.3.4 is used for the following analysis. Once the matrix [Tc] is set up, the eigenvalue problem
[Tc]·
is solved numerically. Again, as in Section3.3.1, the eigenvalues and -vectors can be interpreted physically. The phase (magnitude) of the complex eigenvalue ψ corresponds to the phase shift (gain) per unit cell. Forward and backward traveling modes can be distinguished and amplification can be identified by evaluating the eigenvalues and -vectors. This way a coupled, or hot, characteristic impedance can be defined.
Four modes exist in the coupled system, since each (n×n) matrix has exactly n eigenvalues including algebraic multiplicity. The same number of coupled modes is found from the solution of the determinantal equation as in Section2.2.3. Depending on the synchronism, these modes are more or less tightly bound to the delay line.
For example, when EM wave and electron beam are not synchronized, there are two modes on the delay line (forward and backward traveling mode) and two on the electron beam (slow and fast space-charge mode). For normal TWT interaction as discussed in Section 2.2.3 these modes couple and there are three forward traveling modes, one fast and two slow ones, in addition to a backward traveling mode. One of the slow modes grows exponentially, i.e., the operating mode of a TWT, while the other one decays.
0.0 0.5 1.0 1.5 2.0
Figure 4.2: Comparison of uncoupled (“Circuit” and “Beam”) and coupled disper-sion curves.
Sweeping over the operating frequency and solving the eigenvalue problem in Equation (4.2) results in the dispersion curves shown in Figure4.2for the uncoupled (red and blue lines) and coupled case (red dots). In this example, the EC parameters of the nose-cone loaded line (NC-FW in Table3.2) are used and coupled to a beam with V0 = 12.5 kV, I0 = 0.4 A, and rb = 0.25 mm. Figure 4.2(a) depicts the operating frequency f over the phase φ, while Figure 4.2(b) shows the operating frequency f over the magnitude |ψ| of the eigenvalues. Without synchronism the modes propagate independently as in the uncoupled case. Inside the passband the circuit mode propagates without attenuation (lossless case), and outside the modes are evanescently damped. The space-charge modes are undamped at any frequency.
When the slow beam mode and forward traveling mode have approximately the same phase velocity, two coupled modes with the same phase constant exist (Figure4.2(a)).
One of these modes has a growing, the other one a decaying amplitude as can be seen in Figure 4.2(b). This frequency range is denoted as the amplification bandwidth in the following. It ranges from approximately 39 to 44 GHz in this case.
Identifying the modes from the eigenvalue problem is not straightforward, es-pecially when done over a broad range of frequencies exceeding the amplification bandwidth or even the cold passband. A mere check of, e.g., the magnitude |ψ| is not sufficient, because an evanescent backward mode also exhibits|ψ| larger than unity, similarly to an amplified forward mode. The transported power has to be taken into account to identify the modes properly. Even with this consideration the coupled modes can only be identified by taking the electron beam parameters into account. For example, if the beam is synchronous with the backward traveling mode, a narrowband phenomenon known as backward-wave oscillation may occur [11].
36 38 40 42 44 46 48 50
Figure 4.3: Comparison of cold and hot characteristic impedance.
The interpretation of the eigenvalues and -vectors is fundamentally different for this kind of beam-wave interaction.
The characteristic impedance of the coupled system, or hot characteristic impedance, is calculated by relating the voltageVamp to the currentIamp of the amplified forward mode of the coupled system, i.e.,
Zchot = Vamp
Iamp. (4.3)
A comparison between the hot and cold characteristic impedance Zchot and Zccold is shown in Figure 4.3. Again, the delay line is assumed to be lossless. Figure 4.3 shows that for non-synchronous operation both impedances are essentially the same. Inside the cold passband they are purely real (no losses), and outside purely imaginary (evanescent mode). However, for frequencies inside the amplification bandwidth the hot characteristic impedance is obviously different. The real part is slightly modified and an imaginary part appears due to the beam. The imaginary part is one order of magnitude smaller than the real part. This fact and the small change of the real part illustrate the relatively weak coupling of circuit and beam modes despite the nose-cone loaded delay line. A similar analysis of a CC-TWT as in [40] shows a stronger change of the hot characteristic impedance due to the stronger beam-wave coupling for this delay line topology.