# Electromagnetic Particle Algorithm for Beam-Wave Interaction in Traveling Wave Tube of Symmetry
## 5 Transmission Line Algorithms  
The circuit in Figure 1 can be written with loop and nodal equations, resulting in the telegraph operator's equation plus the driving term, under the constraints of small \(\Delta x\) and \(\Delta t\):  
\[\begin{array}{c}\frac{\partial V}{dt}=-\frac{1}{C_{0}}\frac{\partial t}{ \partial x}+\frac{1}{C_{0}}\frac{\partial Q_{0}}{\partial t}\\ \frac{\partial t}{\partial t}=-\frac{1}{L_{0}}\frac{\partial V}{\partial x} \end{array} \tag{27}\]  
At the source location and collector location:  
\[I_{N_{C}+1/2}^{t+1/2}=\left(V_{\rm o}-V_{N_{\rm C}}-(R_{\rm o}-L)I_{N_{\rm C}+ 1/2}^{t-1/2}\right)(R_{\rm o}+L)^{-1} \tag{28}\]  
where \(R_{\rm i}=e_{0}v_{\rm p}R_{\rm i}\), \(R_{\rm i}=e_{0}v_{\rm p}R_{\rm i}\). However, these terminations are matched (reflection free) only if there is no beam circuit coupling. The greater the coupling between the beam and the circuit, the greater the reflected signal, which pushes the loop gain beyond unity and the tube becomes an oscillator. To avoid self-oscillation, a segment of the transmission line is lossily compensated using the circuit segment shown in Figure 7.  
There are two options included in the algorithm. The first scheme has a smooth section located near the center of the line, starting from grid point \(N_{0}\), the spatial variation of the resistance is as follows:  
\[R_{i}=R_{0}(1-\cos\theta) \tag{29}\]  
In Formula (29), \(\theta=2\pi\frac{i-N_{0}}{N_{\rm C}-2N_{0}}\) and \(N_{0}<i<N_{\rm C}-N\). This loss is to attenuate the reflected signal before it reaches the input of the tube, while reducing the gain by a small amount. As the reflected wave approaches the lossy cross-section, the wave reflected from the collector end of the tube (which may be up to 10% of the incident signal) becomes larger relative to the forward (growing) wave and these two waves produce interesting effects in the lossy section the interferogram.  
To overcome these limitations, a nearly completely absorbing or non-reflective termination section was designed. The termination is a lossy line at the pipe end of length \(N_{1}\) units, where  
\[R_{i}=R_{t}\left((1+\gamma)^{i-N_{\rm C}+N_{1}}-1\right) \tag{30}\]  
where \(R_{t}\leq 10^{-5}\), \(\gamma\leq 0.2\) and \(i>N_{\rm C}-N_{10}\). In this section, the coupling between the beam and the circuit is also reduced in a tapered manner. It is not possible to abruptly terminate the coupling between the beam and the circuit, as this would also introduce reflections. The current decreasing loss form is as follows:  
\[\cos(2\pi(i-N_{\rm C}+N_{1})/4N_{1}) \tag{31}\]  
where \(x=\Delta x\) and \(t=\Delta t\), and Equations (18) and (19) become:  
\[\Omega V_{c}=\frac{K}{C}l_{c} \tag{32}\]  
\[\Omega l_{c}=\frac{K}{C}V_{c} \tag{33}\]  
Figure 7: Adding Resistive Elements to Transmission Lines.  
In the above formula, \(\Omega=\omega\sin c(\omega\Delta t/2)\) and \(K=k\sin c(k\Delta x/2)(\sin cy\equiv\sin y/y)\). The elimination of \(V_{\rm C}\) and \(I_{\rm C}\) produces transmission line dispersion relations:  
\[K^{2}=\Omega^{2}LC \tag{34}\]  
This equation shows two results. First, it should be written as:  
\[\sin^{2}(\omega\Delta t/2)=\frac{\Delta t^{2}}{LC\Delta x^{2}}\sin^{2}(k \Delta x/2) \tag{35}\]  
The right-hand side of the equation must be less than 1 for \(\omega\) to be correct; therefore, for a stable solution, we must make \(\Delta t\leq(LC)^{1/2}\Delta x\) as a condition. Formula (5) is then:  
\[v_{p}^{2}\equiv\frac{\omega^{2}}{k^{2}}=\frac{1}{LC}\frac{sinc^{2}(\omega \Delta t/2)}{sinc^{2}(k\Delta x/2)} \tag{36}\]  
This is the case if and only if \(v_{p}=\Delta x/\Delta t\), \(v_{p}=(LC)^{-1/2}\).