  A simple and fast model for numerically calculating small-signal gain in
serpentine waveguide traveling-wave tubes (TWTs) is described. In the framework
of the Pierce model, we consider one-dimensional electron flow along a
dispersive single-mode slow-wave structure (SWS), accounting for the
space-charge effect. The analytical model accounts for the frequency-dependent
phase velocity and characteristic impedance obtained using various equivalent
circuit models from the literature, validated by comparison with full-wave
eigenmode simulation. The model includes a relation between the modal
characteristic impedance and the interaction (Pierce) impedance of the SWS,
including also an extra correction factor that accounts for the variation of
the electric field distribution and hence of the interaction impedance over the
beam cross section. By applying boundary conditions to our generalized Pierce
model, we compute both the theoretical gain of a TWT and all the complex-valued
wavenumbers of the hot modes versus frequency and compare our results with
numerically intensive particle-in-cell (PIC) simulations; the good agreement in
the comparison demonstrates the accuracy and simplicity of our generalized
model. For various examples where we vary the average electron beam (e-beam)
phase velocity, average e-beam current, number of unit cells, and input radio
frequency (RF) power, we demonstrate that our model is robust in the
small-signal regime.
