Novel compact and lightweight coaxial C-band transit-time oscillator

where E_r and B_θ are the electric field and magnetic field generated by the electron beam in the radial direction and angular direction, respectively. The radial velocity and angular velocity of the electron beam are ignored, and only the axial velocity v_z is considered.

When the foil is loaded, equation (2) can be expressed as

$$ \begin{eqnarray}\displaystyle \frac{\partial }{\partial t}\left(\gamma m\displaystyle \frac{\partial r}{\partial t}\right)=-e({E}_{{\rm{r}}1}-{E}_{{\rm{r}}2}-{v}_{z}{B}_{\theta }),\end{eqnarray}$$ (3)

where E_r1 is the radial electric field generated by the electrons, and E_r2 is the radial electric field due to the foil.

We established the model shown in Fig. 3 to analyze the radial electric field E r2 generated by the foil. Supposing that the charge density of the foil is σ, the observation point P is in the Z - Y plane, and the foil with radius R 0 is in the X - Y plane.

Figure 3.Schematic diagram of foils in coordinate system.

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According to the point charge field strength formula, the radial electric field at point P can be expressed as

$$ \begin{eqnarray}\displaystyle \iint \displaystyle \frac{\sigma R{\rm{d}}R{\rm{d}}\varphi ({r}_{0}\sin {\theta }_{0}-R\sin \varphi )}{4\pi {\varepsilon }_{0}{r}^{3}}.\end{eqnarray}$$ (4)

From the axisymmetric property, it can be known that E y is the radial electric field intensity generated by the foil at each point in space. 1/ r 3 in formula (4) can be expanded using Taylor series as follows:

$$ \begin{eqnarray}\displaystyle \frac{1}{{r}^{3}}\approx \displaystyle \frac{1}{{{R}^{^{\prime} }}^{3}}\left(1+\displaystyle \frac{3t}{2}+\displaystyle \frac{15{t}^{2}}{{2}^{2}}+\displaystyle \frac{3\cdot 5\cdot 7{t}^{3}}{{2}^{3}}+\cdots \right),\end{eqnarray}$$ (5)

where

R^{'} = {(R^{2} + r_{0}^{2})}^{1 / 2}

and t = 2Rr₀ sin θ₀ sin φ/R’². By substituting Eq. (5) into Eq. (4) and utilizing the first two terms of the series, E_y can be expressed as

$$ \begin{eqnarray}{E}_{y}=\displaystyle \frac{\sigma \sin {\theta }_{0}}{2{\varepsilon }_{0}}\left[\displaystyle \frac{3{r}_{0}{R}_{0}^{2}+2{r}_{0}^{3}}{2{({r}_{0}^{2}+{R}_{0}^{2})}^{3/2}}-\displaystyle \frac{{r}_{0}}{{({r}_{0}^{2}+{R}_{0}^{2})}^{1/2}}\right].\end{eqnarray}$$ (6)

From Eq. (6), the radial electric field generated by the opening foil on the beam envelope at the distance d from the origin can thus be expressed as

$$ \begin{eqnarray}\begin{array}{lll}{E}_{{\rm{r}}2} & = & \displaystyle \frac{{\sigma }_{1}{r}_{{\rm{c}}}}{2{\varepsilon }_{0}d}\left[\displaystyle \frac{3d{r}_{{\rm{out}}}^{2}+2{d}^{3}}{2{({d}^{2}+{r}_{{\rm{out}}}^{2})}^{3/2}}-\displaystyle \frac{d}{{({d}^{2}+{r}_{{\rm{out}}}^{2})}^{1/2}}\right]\\ & & -\displaystyle \frac{{\sigma }_{1}{r}_{{\rm{c}}}}{2{\varepsilon }_{0}d}\left[\displaystyle \frac{3d{r}_{{\rm{a}}}^{2}+2{d}^{3}}{2{({d}^{2}+{r}_{{\rm{a}}}^{2})}^{3/2}}-\displaystyle \frac{d}{{({d}^{2}+{r}_{{\rm{a}}}^{2})}^{1/2}}\right]\\ & & +\displaystyle \frac{{\sigma }_{2}{r}_{{\rm{c}}}}{2{\varepsilon }_{0}d}\left[\displaystyle \frac{3d{r}_{{\rm{b}}}^{2}+2{d}^{3}}{2{({d}^{2}+{r}_{{\rm{b}}}^{2})}^{3/2}}-\displaystyle \frac{d}{{({d}^{2}+{r}_{{\rm{b}}}^{2})}^{1/2}}\right]\\ & & -\displaystyle \frac{{\sigma }_{2}{r}_{{\rm{c}}}}{2{\varepsilon }_{0}d}\left[\displaystyle \frac{3d{r}_{{\rm{in}}}^{2}+2{d}^{3}}{2{({d}^{2}+{r}_{{\rm{in}}}^{2})}^{3/2}}-\displaystyle \frac{d}{{({d}^{2}+{r}_{{\rm{in}}}^{2})}^{1/2}}\right].\end{array}\end{eqnarray}$$ (7)

From Eq. (7), we can qualitatively analyze the effect of distance d and the opening size on the radial electric field. A larger distance d and opening size result in a smaller radial electric field. The variation of E_r2 with opening size is shown in Fig. 4.

Figure 4.Plot of electric field E_r2versus opening size.

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