SOME RESULTS OF COMPARING DISPERSION EQUATIONS OBTAINED BY VARIOUS METHODS WITH EXPERIMENT

L. F. CHERNOGALOVA

(Presented by the scientific seminar of the Sector of Superhigh Frequencies and Accelerator Theory, Research Institute of Nuclear Physics, Electronics, and Automation)

The dispersion properties of diaphragm-loaded waveguides and other systems are usually investigated by the method of partial regions, i.e., the electromagnetic fields are specified independently in the space of the resonators and in the interaction space in the form of a sum of spatial harmonics. By matching the tangential components at the boundary separating the regions, one obtains the dispersion equation in the form of infinite determinants. However, solving such a dispersion equation is very cumbersome, since one has to solve a complicated transcendental equation. The degree of accuracy of the solution of this equation will depend on the number of terms retained in the infinite system. Moreover, this method is valid for the case of partial diaphragms. At the same time, practice shows that the number of diaphragms in a waveguide electron accelerator per generator wavelength usually does not exceed 3–4.

Fig. 1. Section of a waveguide chamber

Taking into account the usual operating conditions of diaphragm-loaded waveguide accelerators, V. V. Vladimirskii [1] proposed studying the propagation of electromagnetic waves in a circular waveguide on the basis of the normal-wave method. The circular diaphragm-loaded waveguide was considered by him as a chain of coupled cylindrical endovibrators with infinitely thin membranes and small apertures for passage of the beam. The advantage of such a method is that the dispersion equation obtained on its basis is a simple algebraic equation with respect to the propagation constant \(\beta_n\). In calculating the dispersion of the system here it is not necessary to solve complicated transcendental equations. Moreover, already in the first approximation the method correctly takes into account the field singularity at the edges of the diaphragms and makes it possible to obtain good agreement with experiment if the number of diaphragms per wavelength in the waveguide is small.

In view of the fact that rectangular diaphragm-loaded ... in accelerator technology and microwave instrument technology are of great interest,

waveguides; in [2, 3], diaphragmed waveguides of rectangular cross section were studied by the normal-wave method.

In these works, the propagation of in-phase \(LE\)-waves (for \(\dot{L}E\)-waves \(E_x=0\)) along a waveguide of rectangular cross section was considered. In contrast to the mode-matching method, here the field is specified at once over the entire cell, and then, according to the Floquet condition, is matched at the cell boundaries:

\[ z=\pm \frac{1}{2}D;\quad \pm \frac{3}{2}D;\quad \pm \ldots \]

The components of the \(LE\)-waves are expressed through the \(x\)-component of the Hertz magnetic vector of the \(m\)-th cell, which for in-phase waves has the form [2, 3]:

\[ \Pi_x^m = e^{-jm\psi}\cdot \cos \frac{\pi}{a}x\cdot \sum_{n=0}^{\infty} \left[ A_n\sin \beta_n(z-mD)+B_n\cos \beta_n(z-mD) \right] \]

\[ \cdot \sin (2n+1)\frac{\pi}{b}y; \]

\[ m=0,\ \pm 1,\ \pm 2,\ldots \quad \text{— cell number;} \]

where

\[ (2m-1)\frac{1}{2}D<z<(2m+1)\frac{1}{2}D; \]

\[ \beta_n= \sqrt{ \kappa^2-\left(\frac{\pi}{a}\right)^2-(2n+1)^2\left(\frac{\pi}{b}\right)^2 } \quad \text{— propagation constant,} \]

\[ \kappa=\frac{\omega}{c} \quad \text{— wave number in free space;} \]

\(0\le \psi\le \pi\) characterizes the phase shift between oscillations in neighboring cells and is related to the phase velocity of the wave by the relation

\[ v_{\phi}=c\frac{\kappa D}{\psi}=\frac{\omega D}{\psi}. \tag{2} \]

Writing the boundary conditions

\[ E_z\left(\frac{b}{2},\,z\right)=0 \quad \text{for } -\infty<z<+\infty; \]

\[ E_y(y,z)=0 \quad \text{for } g<y<\frac{b}{2}, \]

\[ (2m-1)\frac{1}{2}D\le z\le (2m+1)\frac{1}{2}D \tag{3} \]

and supplementing them with periodicity conditions for the Hertz vector \(\Pi_x^m\) and its normal derivative \(\dfrac{\partial}{\partial z}\Pi_x^m\), after a number of transformations in [2] the following integral equation was obtained with respect to \(F(\eta)\):

\[ \cos^2\frac{\psi}{2}\cdot \operatorname{tg}\frac{\beta_0D}{2} - \sin^2\frac{\psi}{2}\cdot \operatorname{ctg}\frac{\beta_0D}{2} \cdot \sin\frac{\pi}{b}y \cdot \int_{-g}^{+g} F(\eta)\sin\frac{\pi}{b}\eta\cdot d\eta = \]

\[ = - \int_{-g}^{g} G(y,\eta)F(\eta)\,d\eta, \tag{4} \]

where \(F(\eta)\) is the form of the field specified at the diaphragm aperture. When the normal-wave method is used, the field is specified with allowance for the singularities at the diaphragm edges.

\(G(y,\eta)\) is the kernel:

\[ G(y,\eta)=\sum_{n=1}^{\infty} \frac{ \cos^{2}\frac{\psi}{2}\cdot \tan \frac{\beta_n D}{2} -\sin^{2}\frac{\psi}{2}\cdot \cot \frac{\beta_n D}{2} }{\beta_n} \cdot \sin (2n+1)\frac{\pi}{b}\,y\cdot \sin (2n+1)\frac{\pi}{b}\,\eta . \tag{5} \]

The summation in (5) is carried out over the wave types of an individual cell.

The approximation of this method consists in the fact that, beginning with some \(n\), depending on the proposed accuracy of the solution of the equation, the dependence of \(\beta_n\) on frequency is neglected and one writes

\[ (\beta_n)=(2n+1)\frac{\pi}{b}. \]

In addition, it is assumed that under the conditions usually present in electron waveguide accelerators, the dependence of the kernel \(G\) on \(\psi\) and \(D\) is very weak and decreases exponentially with increasing \(D\). This makes it possible, for large \(n\) and not very small \(D\), to set \(g_n(\beta_n)=1\), where

\[ g_n= \frac{ \cos^{2}\frac{\psi}{2}\cdot \tan \frac{\beta_n D}{2} -\sin^{2}\frac{\psi}{2}\cdot \cot \frac{\beta_n D}{2} }{\beta_n}. \]

By specifying the form of the field at the diaphragm aperture and solving the integral equation (4), one can obtain the dispersion equation with any desired degree of accuracy.

In [2], equation (4) was solved by expanding the integrand in a Fourier sine series. No conditions were imposed here on the magnitude of the ratio \(g/b\).

In the quasistatic approximation, where the dependence of \(\beta_n\) on frequency was neglected for \(n\ge 1\), this gave an equation of the form

\[ \cos \psi=\beta_0D-\frac{b\beta_0}{\pi}\cdot \cot^{2}\frac{\pi g}{b}\cdot \sin \beta_0D . \tag{6} \]

Similarly, the dispersion equation can be obtained in the second approximation, neglecting the dependence of \(\beta_n\) on frequency beginning with \(n\ge 2\), i.e., taking into account two wave types,

\[ \cos \psi=a_1-\sqrt{a_1^2-a_2}, \tag{7} \]

\[ a_1= \frac{\cos \beta_0D+\operatorname{ch}|\beta_1|D}{2} +\frac{|\beta_1|b}{6\pi}\cdot \frac{1-\sin^{6}\xi}{\sin^{6}\xi}\cdot \operatorname{sh}|\beta_1|D - \]

\[ -\frac{\beta_0b}{2\pi}\cdot \cot^{2}\xi\, \frac{1-\cos^{6}\xi}{\sin^{6}\xi}\cdot \sin \beta_0D, \qquad \text{where } \xi=\frac{\pi g}{b}, \]

\[ a_2= \cos \beta_0D\cdot \operatorname{ch}|\beta_1|D +\frac{|\beta_1|b}{3\pi}\cdot \frac{1-\sin^{6}\xi}{\sin^{6}\xi}\cdot \operatorname{sh}|\beta_1|D\cos \beta_0D - \]

\[ -\frac{\beta_0 b}{\pi}\cdot \operatorname{ctg}^2 \xi \cdot \frac{1-\cos^6 \xi}{\sin^6 \xi}\cdot \sin \beta_0 D \cdot \operatorname{ch} |\beta_1|D -\frac{\beta_0|\beta_1|b^2}{3\pi^2}\cdot \operatorname{ctg}^8 \xi \cdot \operatorname{ch}|\beta_1|D \cdot \sin \beta_0 D . \]

In [3], waveguides of rectangular cross section with small slot openings for the passage of the beam were investigated, i.e., \(g/b \ll 1\). Here the field at the diaphragm openings, in contrast to [2], was specified in the form of a polynomial. As a result, an equation similar to (6) was obtained if \(\operatorname{ctg} x\) is expanded for small argument, which is possible for a small value of the ratio \(g/b\), and one keeps only the first term,

\[ \cos \psi = \cos \beta_0 D - \frac{b\beta_0}{\pi}\cdot \left(\frac{\beta}{\pi g}\right)^2 \cdot \sin \beta_0 D . \tag{8} \]

Using equations (6), (7), and (8), the dispersion curves were calculated and compared with the experiment from [4]. The chamber dimensions were as follows: \(b=7.74\ \text{cm}\), \(a=7\ \text{cm}\); \(2g=2\ \text{cm}\), \(D=2.5\ \text{cm}\), \(t=0.6\ \text{cm}\). The wave type excited in the chamber was \(LE_{11}\). The results of the calculation and experiment are plotted in Fig. 2. It is evident from the figure that the experimental curve is shifted somewhat into the short-wavelength region relative to the theoretical curves. This shift evidently occurs due to the influence of the diaphragm thickness, since in the theoretical calculations the diaphragm thickness was not taken into account.

Fig. 2. Comparison of theoretical dispersion curves with experiment:
1 — curve calculated by formula (6); 2 — second-approximation curve calculated by formula (7); 3 — curve calculated by formula (8); 4 — experimental curve; 5 — curve calculated by the matching method in the zeroth approximation.

From the behavior of the curves it is evident that for

\[ \frac{\pi}{2}<\psi<\pi \]

the closest to the experimental curve (4) are curve (2), calculated in the second approximation by the method of integral equations, and curve (3), calculated by formula (8) of the normal-wave method. The course of these curves also shows that curves (2) and (3) agree fairly well with each other. Consequently, for a small value of the ratio \(g/b\) (in the present case \(g/b=0.13\)), calculation by formula (7) can be replaced by the simpler calculation by formula (8). Thus, from what has been considered it is clear that the normal-wave method in its various modifications can be used to investigate the dispersion properties of waveguides of rectangular cross section with a small number of diaphragms per wavelength in the waveguide, i.e., in operation in a mode close to the \(\pi\)-mode of oscillation.

REFERENCES

1. V. V. Vladimirskii, ZhTF, XVII, 1269, 1947.
2. E. S. Kovalenko, Theory of Accelerating Devices of Electron Synchrotrons. Dissertation, TSU, Tomsk, 1961.
3. A. N. Didenko, ZhTF, XXXV, 5, 1965.
4. A. P. Olshanskii, Problems of Excitation of the Resonator System of a Waveguide Cyclic Accelerator. Dissertation, TPI, Tomsk, 1963.