3.4 Fabrication and Characterization of Folded-Waveguide Delay Lines 39
3.4.3 Measurement of Complex Propagation Constant
Since the previous measurements could not be conducted across the whole passband, a different characterization approach is now used that can be evaluated more easily.
In general, it is not necessary to obtain the scattering parameters of the delay line under test, since the complex propagation constant is of interest. For this purpose a measurement procedure as in [34] is applied and briefly outlined here.
The measurement setup is depicted in Figure 3.25(a). It requires a calibration standard in addition to the actual delay line. The calibration standard is shown in Figure 3.25(b). For both measurements the vector network analyzer (VNA) is calibrated to the characteristic impedance of the connecting waveguide.
The measured scattering parameters [Sline] and [Sthru] can be transformed to the wave cascade parameters [Tline] and [Tthru], respectively [35]. They can be further separated into
[Tline] = [Tsteps,1]·[TFW]·[Tsteps,2] and (3.48) [Tthru] = [Tsteps,1]·[Tsteps,2], (3.49)
where [TFW] is the cascade matrix of the FW delay line, and [Tsteps,1] and [Tsteps,2] are the cascade matrices of the input and output stepped transformers, respectively.
Multiplying [Tline] by the inverse of [Tthru] results in
[Tline] [Tthru]−1 = [Tsteps,1]·[TFW]·[Tsteps,1]−1. (3.50) The matrices [Tline] [Tthru]−1 and [TFW] are mathematically similar and thus exhibit the same trace. A homogeneous piece of TL of length ` has a cascade matrix of the form
[T] = e−γ` 0 0 e+γ`
!
, (3.51)
where γ = α+jβ is the complex propagation constant with attenuation constant α and phase constant β. Therefore, the complex propagation constant can be computed from
γ = arcosh0.5·trace[Tline] [Tthru]−1
` . (3.52)
The two stepped impedance transformers in waveguide technology are connected to the VNA and the FW line via flanges. The transformers are optimized in Ansys EM to exhibit a low reflection across the passband. In the “thru” standard the same transformers are used. However, a short piece of waveguide with the same cross section as the input of the FW line (3.7×0.9 mm2) is placed between the transformers instead of the delay line. Therefore, the influence of the change in waveguide width is also included in the calibration. For the “line” and “thru”
measurements the same transformers are used, i.e., after one measurement they are disconnected from the setup and reused for the other. This immensely improves the accuracy of the approach. Otherwise, mechanical inaccuracies during the manufacturing of the waveguide components would deteriorate the measurement results, because the cascade matrices would be different and would therefore not cancel out as in Equation (3.50).
The measured S-parameters for the “line” and “thru” setup are shown in Fig-ures 3.26(a) and 3.26(b), respectively. The passband is clearly visible in the “line”
measurement. The stepped transformers yield low reflections across the band for both the “line” and the “thru” setup.
From these results the complex propagation constant γ =α+jβ can be derived by applying Equation (3.52) with ` = 62p = 68.2 mm. The phase constant β calculated from the imaginary part of the inverse cosine hyperbole is ambiguous by increments of 2π. This is corrected by minimizing
RMSβ` =h(β`)HFSS −(β`)Meas+ 2πni2 (3.53) inside the passband and with an integern. The real part ofγ is uniquely defined and needs no further processing. The results forβ and α are shown in Figures 3.27(a)
36 38 40 42 44 46 48 50
Figure 3.26: Measurement results for line “Mo2”.
and 3.27(b), respectively. In Figure 3.27(a) the obtained results are additionally compared with the eigenmode results. The agreement is excellent with deviations of less than 1 % across the passband. Only the lower cutoff frequency for “MoCu2”
and at the upper one for “Mo2” and “MoCu2” exhibit slightly larger discrepancies.
The attenuation over frequency in Figure 3.27(b) is also quite similar for all lines.
They exhibit a relatively low attenuation of approximately 0.5 dB/cm. There are some artificial peaks in Figure3.27(b) due to the evaluation algorithm, presumably when the transmission through the delay line is low. Small peaks around 42 GHz are proof of slight asymmetries of the electron beam tunnel that could not be identified with the resonant measurement setup. If the beam tunnel would be off-centered, a stopband would occur at this frequency. The placement of the beam tunnel seems to be acceptable for all manufactured lines, since the dispersion in Figure 3.27(a) shows no discernible stopband.
The obtained attenuation α has an order of magnitude comparable to that of typical helical delay lines in this frequency range which exhibit approximately 0.1 dB of attenuation per wavelength. Therefore, it should be relatively easy to extract the amount of loss-induced heat, since the FW delay line consists of bulk metal.
An accurate thermal prediction calls for involved multi-physics simulation and is outside the scope of this thesis. However, the distribution of the surface current in the lossless case can be used to estimate the impact of the nose cones on the power loss distribution. The edges of the nose cones might lead to strongly increased electric field and surface current density amplitudes. The distribution of the latter obtained by eigenmode simulation with and without nose cone is compared for this purpose. The maximum surface current density along the center of the FW line is slightly increased by 10 % due to the nose cones. This means that local power
1.0 1.2 1.4 1.6 1.8 2.0 βp/π
36 38 40 42 44 46 48 50
f
in (G Hz )
Mo1Mo2 MoCu2 HFSS
(a) Dispersion.
36 38 40 42 44 46 48 50
f
in (GHz)
0.0 0.5 1.0 1.5 2.0 2.5 3.0
α
in (d B/ cm )
Mo1Mo2 MoCu2
(b) Attenuation.
Figure 3.27: Comparison of derived propagation constants.
conversion is only marginally changed, not taking the slightly longer path length into account.
The obtained results verify the reproducibility and feasibility of nose-cone loaded FW lines. Additionally, the agreement with full-wave results confirms the field enhancement achieved by means of the nose-cone topology.
Operation
In Section3.3.2 models for delay lines with ideally matched couplers and severs have been introduced. The ideal matching condition was derived from the characteristic impedance of the delay line circuit. Coupling the delay line to an electron beam, however, changes the propagation constant and characteristic impedance of the line [36, 37]. In the following, all quantities related to the first case are denoted as cold, while those related to the latter case are referred to as hot. The approach outlined in [36,37] is applied here to an FW-TWT. Alternative modeling approaches of space-charge induced reflections and hot operation can be found in [38, 39].