Optimization of couplers and severs is in practice commonly performed based on the characteristics of the cold delay line. As is well known, an ideal match for cold operation does not result in optimum operating conditions of the active device, i.e., with an electron beam. Therefore, it is worthwhile to study the impact of
matching for cold and hot operation using the abstract models for couplers and severs presented in Section 3.3.2. Figure 4.3 suggests that the amplified forward mode experiences a mismatch, if the tube is terminated by the cold characteristic impedance. Depending on the gain and the magnitude of the reflection this can lead to instabilities and a gain ripple over frequency. These effects are studied in the following.
4.2.1 Stability
Three main causes can be identified for instabilities in vacuum tubes: backward-wave, drive-induced, and regenerative oscillations [41]. The first type, as mentioned above, depends mainly on the operating point of the tube and can be observed without any input signal. For helices this type of oscillation occurs mostly for large beam radii and can be tuned by changing the accelerating voltageV0of the tube [11].
The second oscillation type is a large-signal phenomenon. A tube that is zero-drive stable, or even stable under small-signal conditions, may start to oscillate when the input power is increased. Figuratively, the electrons become slower and thus the electron beam line may intersect the dispersion curve close to a band edge at φ=π or φ = 2π. Because the coupling can be high at these frequencies and power cannot propagate, the tube may start to oscillate [41]. Instabilities of the third type occur due to mismatches along the TWT delay line with an amplifying path in between, essentially leading to a loop gain larger than unity. In the following the focus is on oscillations of this type.
These oscillations can be described by means of control theory as an unstable control loop with a loop gain (expressed in dB)
gloop = Γ1+ Γ2+G+B >0, (4.4) where Γi, i = 1,2, are reflection coefficients at delay line discontinuities, and G and B are amplification coefficients (including loss) in forward and backward direction, respectively.
In a linear beam device such as a TWT, the feedback path is created by dis-continuities along the line, e.g., by couplers and severs. The distributed nature of the TWT results in the formation of resonant frequencies at which the oscillations occur. For an inhomogeneous delay line this is more involved, because the dispersion changes along the line.
A TWT with homogeneous delay line is therefore studied in this section to demonstrate the impact of matching for hot operation only, i.e., without any additional effects. The EC and beam parameters from Section 4.1 are used here to illustrate the impact of input and output matching on stability. A generic, lossless TWT with one section and Nc = 25 cells is simulated using the small-signal version of KlysTOP with couplers exhibiting a specified reflection. A sketch of
Figure 4.4: Sketch of a single-section folded-waveguide traveling-wave tube with signal paths indicated.
the interaction space of the considered tube is shown in Figure4.4. A signal flow chart is included in the figure for convenience. The backward mode experiences a mismatch Γ1 at the input coupler and the amplified forward mode is reflected at the output coupler assuming a reflection of Γ2. For simplicity Γ1 = Γ2 = Γ, Γ∈ [0,1], is assumed in the following.
The calculated small-signal gain Gss of the tube is depicted in Figure 4.5(a) for several values of Γ. An ideal match for hot operation, i.e., Γ = 0, yields a smooth gain curve. Increasing the local reflection at the couplers leads to the formation of local maxima and minima, i.e., a gain ripple. Its period ∆f is determined by the length and dispersion of the delay line, while its amplitude ∆G depends on the magnitude of Γ. The respective input reflections for the different matching conditions are shown in Figure 4.5(b). It can be clearly observed that the amplifier gain as well as the hot input reflection grow indefinitely for Γ = 0.4 (≈ −8 dB) at discrete frequencies. The maximum stable gain can be calculated from Equation (4.4) for a given reflection Γ. Taking into account the loss of the delay line relaxes the stability limit, because loss effectively decreases the loop gain.
Further conclusions can be drawn from Figure 4.5(b). A perfect match for the amplified forward mode results in non-zero hot input reflections. The other two forward modes that are inevitably excited at the input coupler of the TWT [11]
experience a mismatch, while the amplified mode does not. The main contribution to the input reflection stems from the fast forward mode, because the other slow mode is exponentially damped and therefore has a negligible amplitude at the end of the tube. On the other hand, a small reflection for the operating mode, e.g., Γ = 0.1, already leads to a significantly larger input reflection, because this mode is amplified. The mismatch at the end is thus magnified by the amplification.
Matching the amplified forward mode is consequently the optimum load condition in terms of stability.
Furthermore, it is obvious that a perfect match for hot operation can only be obtained if the TWT is mismatched in the cold case with a specific load condition.
(a) Small-signal gain.
39.0 39.5 40.0 40.5 41.0 41.5 42.0 42.5 43.0
f
in (GHz)
39.0 39.5 40.0 40.5 41.0 41.5 42.0 42.5 43.0
f
in (GHz)
Figure 4.5: Comparison of input match and gain for different levels of input and output coupler reflection Γ.
This can be observed in Figure 4.5(c) which shows the cold input reflection for the different cases. The reflection at the end of the tube has to exhibit the proper amplitude and phase for minimum hot input reflection. Additionally, the ideal load condition depends on the electron beam, i.e., accelerating voltage V0 and beam current I0, as well as on the operating frequency f of the tube.
4.2.2 Large-Signal Behavior
The derivation of the coupled characteristic impedance is performed using a small-signal approach. The derived matching condition for hot operation is now applied to the single-section TWT of the previous section at an operating frequency f = 40.5 GHz for maximum gain and the input power is changed to analyze how the reflection is altered. The large-signal version ofKlysTOP is used for this analysis.
As before, the input coupler and output coupler are assumed to be perfectly matched to the backward and amplified forward mode, respectively.
20 10 0 10 20 30 40 50 60
Figure 4.6: Results at f = 40.5 GHz for different input power levelsPin. Figure 4.6 summarizes the obtained results. Figure 4.6(a) shows the gain over input power at 40.5 GHz. Relatively large input powers are necessary to drive the tube into the nonlinear regime, since the tube has only 25 cells. For input powers up to 30 dBm the amplifier is linear. For higher powers the gain drops drastically. The respective hot input reflection is depicted in Figure 4.6(b). For small input powers the match is very good with reflections smaller than −30 dB. In the nonlinear regime the input reflection rises up to −12 dB. The electrons are then strongly modulated and have lost a non-negligible portion of their kinetic energy. This shifts the synchronism to higher frequencies and thus changes the propagation and ideal matching condition for hot operation at the operating frequency. In a real tube this knowledge could be used to optimize the load of the amplifier for a specific output power.