ON THE CALCULATION OF SPACE HARMONICS IN DIAPHRAGM-LOADED WAVEGUIDES

E. S. Kovalenko, V. S. Kovalenko, B. G. Tsikin

With the advance of microwave engineering into the region of shorter waves, generators operating on higher harmonics of slow-wave structures are of considerable interest.

In this connection, it is necessary to carry out a comparative evaluation of the various methods used in calculating slow-wave systems, using as an example a diaphragm-loaded waveguide of rectangular cross section.

The most widely used method of calculation takes into account, in the space of the resonators of the structure, only one fundamental wave [1], [2]. This method gives good results in calculating the dispersion properties of the system, since the latter depend only weakly on the specification of the field on the boundary of the interaction space and the resonators. At the same time, as was shown in [3], for infinitely thin diaphragms this method entails appreciable errors in determining the amplitudes of the higher harmonics, and consequently also the efficiency of lamps operating on these harmonics.

The other existing methods [4], [5] are associated with attempts to refine the character of the field distribution at the edges of the diaphragms.

Fig. 1.

In the present work, the amplitudes of the harmonics of a diaphragm-loaded waveguide in the regime of $\pi$-oscillations will be calculated by the Fourier-series method, and the numerical results of the calculation will be compared with results obtained by other methods.

Consider a diaphragm-loaded waveguide of rectangular cross section, whose principal parameters, used below, are indicated in Fig. 1.

In [6] it is shown that the slowed in-phase waves in such a waveguide belong to the class of \(LE\)-waves and are described by the magnetic Hertz vector

\[ \Pi_{\mathrm{int}} = \sum_{s=-\infty}^{\infty} A_s \cos \frac{\pi}{a} x \,\sin k_y^s y \, e^{j\beta_s z} \]

in the interaction space, and

\[ \Pi_{\mathrm{res}} = \sum_{s=0}^{\infty} B_s \cos \frac{\pi}{a} x \cos k_0^s\left(\frac{b}{2}-y\right) \cos \frac{s\pi}{d}(z-t) \]

in the resonator space. Here \(A_s\) and \(B_s\) are as yet arbitrary constants,

\[ k=\frac{\omega}{c}; \qquad \left(k_y^s\right)^2=k^2-\beta_s^2-\frac{\pi^2}{a^2}; \]

\[ \beta_s=\beta_0+\frac{2\pi s}{D}; \qquad \left(k_0^s\right)^2=k^2-\frac{\pi^2}{a^2}-\frac{s^2\pi^2}{d^2}. \]

Matching the fields at the interface \(y=\pm g\) leads to the infinite system of equations

\[ X'_l=\sum_{n=-\infty}^{+\infty}\alpha_{ln}X'_n, \tag{1} \]

\[ l=0;\ \pm 1;\ \pm 2\ldots, \]

where

\[ X'_l=\beta_l A_l \sin k_y^l g\, e^{j\beta_l D/2} \]

and

\[ \alpha_{ln} = \frac{8}{Dd}\, \frac{\zeta_l^{\,2}}{k_y^l}\, \tan k_y^l g \left\{ \frac{ k_0 \tan k_0 h\, \sin \frac{\zeta_l d}{2}\, \sin \frac{\zeta_n d}{2} }{ 2\zeta_l^{\,2}\zeta_n^{\,2} } + \sum_{s=1}^{\infty} \frac{ k_0^s \tan k_0^s h\, \sin \frac{\zeta_l d+s\pi}{2}\, \sin \frac{\zeta_n d+s\pi}{2} }{ \left(\zeta_l^{\,2}-\frac{s^2\pi^2}{d^2}\right) \left(\zeta_n^{\,2}-\frac{s^2\pi^2}{d^2}\right) } \right\}. \tag{2} \]

Hence the dispersion equation of the \(LE\)-waves is written in the form

equal to zero of the determinant of system (1)

\[ \left| \delta_{ne}-z_{ne}\right|=0, \qquad \delta_{nn}=1;\quad \delta_{ne}=0,\quad n\ne e . \tag{3} \]

In what follows the regime of \(\pi\)-oscillation \((\zeta_0 D=\pi)\) will be considered. This makes it possible to reduce equation (2) to a form convenient for calculation,

\[ \begin{aligned} z_{ne}={}& \frac{8}{Dd}\, \frac{(2e-1)^2\pi^2}{\kappa_\gamma^e D^2}\, \tan k_\gamma^e g \Biggl\{ \frac{ k_\gamma^e \tan k_n^0 h\, \sin \dfrac{\zeta_e d}{2}\, \sin \dfrac{\zeta_n d}{2} }{ 2\zeta_e^2 \zeta_n^2 } \\[4pt] &\qquad\qquad + \sum_{s=1}^{\infty} \frac{ \kappa_s^0 \tan k_s^0 h\, \sin \dfrac{\zeta_e d+s\pi}{2}\, \sin \dfrac{\zeta_n d+s\pi}{2} }{ \left(\zeta_e^2-\dfrac{s^2\pi^2}{d^2}\right) \left(\zeta_n^2-\dfrac{s^2\pi^2}{d^2}\right) } \Biggr\}, \end{aligned} \tag{4} \]

where

\[ \zeta_e=\frac{(2e-1)\pi}{D}; \qquad \zeta_n=\frac{(2n-1)\pi}{D}. \]

Introduce new variables:

\[ x_e=(-1)^e X'_e+(-1)^{e+1}X'_{-(e+1)}, \]

\[ y_e=(-1)^e X'_e-(-1)^{e+1}X'_{-(e+1)}. \]

Hence

\[ X'_e=\frac{1}{2}(-1)^e(x_e+y_e), \tag{5} \]

\[ X'_{-(e+1)}=\frac{1}{2}(-1)^{e+1}(x_e-y_e). \]

After substituting (5) into (1), we have

\[ 4(-1)^e(x_e-y_e) = \sum_{n=0}^{\infty} \left\{ z_{en}(-1)^n(x_n+y_n) + (-1)^{n+1}z_{e,-(n+1)}(x_n-y_n) \right\}, \]

\[ (-1)^{e+1}(x_e-y_e) \sum_{n=0}^{\infty} \left\{ z_{-(e+1),n}(-1)^n(x_n+y_n) + z_{-(e+1),-(n+1)}(-1)^{n+1}(x_n-y_n) \right\}. \]

Adding and subtracting the last two equalities and using the relations

\[ z_{en}=z_{-(n+1),-(e+1)}; \qquad z_{-(e+1),n}=z_{e,-(n+1)}, \]

\[ z_{e,-(n+1)}=z_{-(e+1),e}; \qquad z_{ee}=z_{-(e+1),-(e+1)}, \]

we obtain

\[ \begin{aligned} [[unclear: continuation of the derived system begins on the next page]] \end{aligned} \]

\[ (-1)^e y_e-\sum_{n=0}^{\infty}(-1)^n\{y_{en}+z_{-(e+1),\,n}\}y_n; \tag{6} \]

\[ (-1)^n x_e-\sum_{n=0}^{\infty}(-1)^n\{x_{en}-z_{-(e+1),\,n}\}x_n, \tag{7} \]

where

\[ z_{en}-z_{-(e+1),\,n} = \frac{8}{Dd}\,\frac{\beta_e^{2}}{\kappa_y^{e}}\,\tg \kappa_y^{e} g\, \sum_{s=1}^{\infty} \frac{ \kappa_s^{0}\tg \kappa_s^{0}h\, \sin \dfrac{\beta_n d+s\pi}{2} }{ \left(\beta_e^{2}-\dfrac{s^{2}\pi^{2}}{d^{2}}\right) \left(\beta_n^{2}-\dfrac{s^{2}\pi^{2}}{d^{2}}\right) } \cdot 2\sin\frac{s\pi}{2}\, \cos\frac{2e+1}{2}\,\frac{\pi d}{D}; \tag{8} \]

\[ \begin{aligned} x_{en}-z_{-(e+1),\,n} &= \frac{8}{Dd}\,\frac{\beta_e^{2}}{\kappa_y^{e}}\,\tg \kappa_y^{e}g \Bigg\{ \frac{ \kappa_0^{0}\tg \kappa_0^{0}h\, \sin \dfrac{\beta_e d}{2}\, \sin \dfrac{\beta_n d}{2} }{ \beta_e^{2}\beta_n^{2} } \\ &\qquad +\sum_{s=1}^{\infty} \frac{ \kappa_s^{0}\tg \kappa_s^{0}h\, \sin\left((2n+1)\dfrac{\pi d}{D}+s\pi\right) }{ \left(\beta_n^{2}-\dfrac{s^{2}\pi^{2}}{d^{2}}\right) \left(\beta_e^{2}-\dfrac{s^{2}\pi^{2}}{d^{2}}\right) } \,2\cos\frac{s\pi}{2}\, \sin(2e+1)\frac{\pi d}{2D} \Bigg\}. \end{aligned} \]

As is seen, systems (6) and (7) are independent, have different coefficients and, consequently, different eigenvalues \(K\). Let \(K\) be an eigenvalue of system (7), i.e., in this case \(K\) satisfies the relation

\[ \left|\delta_{ne}-\left(x_{en}-x_{-(e+1),\,n}\right)\right|=0. \tag{9} \]

Then \(x'_e\ne 0\), but the determinant (6) is different from zero and, consequently, \(y_e=0\). Using (5), we conclude that the solution of (7) is waves whose harmonic amplitudes are subject to the conditions

\[ X_e=-X_{-(e+1)}; \tag{10} \]

\[ e=0,1,2\ldots \]

By direct substitution of (10) into the expressions for the field components one can verify that system (7) determines waves with a maximum of the longitudinal electric field at the centers of the cells.

Similarly, it can be shown that if system (6) has a corresponding nonzero solution, then it leads to waves whose longitudinal electric field in the grooves is zero on the average. Therefore the solution below is carried out only for system (7).

The coefficients of system (7), by means of (8), are represented in the form of infinite series. Obviously, for this reason their calculation is rather difficult, and first of all we shall find approximate expressions, but ones sufficiently accurate for a number of practical cases, for the coefficients in closed analytic form.

Using the obvious equality

\[ \sin\left(\frac{\zeta_e d}{2}+\frac{s\pi}{2}\right) \sin\left(\frac{\zeta_n d}{2}+\frac{s\pi}{2}\right) = \frac{1}{2}\left\{ \cos\frac{(\zeta_e-\zeta_n)d}{2} - (-1)^s\cos\frac{(\zeta_e+\zeta_n)d}{2} \right\}, \]

and taking into account that

\[ \kappa_n^s \sim i\,\frac{s\pi}{d}; \qquad \operatorname{tg}\kappa_n^s h \sim i, \]

one may write the sum on the right-hand side of equation (8) in the form

\[ \begin{aligned} S_{ne} &= \frac{1}{2}\sum_{s=1}^{\infty} \frac{s\pi d} {\left(\zeta_e^2-\dfrac{s^2\pi^2}{d^2}\right) \left(\zeta_n^2-\dfrac{s^2\pi^2}{d^2}\right)} \cos\frac{(\zeta_e-\zeta_n)d}{2} \\ &\quad -\frac{1}{2}\sum_{s=1}^{\infty} \frac{(-1)^s s\pi d} {\left(\zeta_e^2-\dfrac{s^2\pi^2}{d^2}\right) \left(\zeta_n^2-\dfrac{s^2\pi^2}{d^2}\right)} \cos\frac{(\zeta_e+\zeta_n)d}{2}. \end{aligned} \tag{11} \]

Carrying out the summation in (11), we obtain for \(\gamma_{en}\), \(\gamma_{en}=\gamma_{[[unclear: index relation]]}\), and \(\gamma_{nn}\) expressions of the form

\[ \begin{aligned} \gamma_{ne} &= \frac{8}{\pi^3}\, \frac{1}{(2n+1)^2(2e-1)} \sin\frac{\zeta_e d}{2}\, \sin\frac{\zeta_n d}{2}\,A \\ &\quad -\frac{(2e+1)} {\pi^2 (n-e)(n+e+1)} \sin\frac{\zeta_e d}{2}\, \sin\frac{\zeta_n d}{2}, \end{aligned} \tag{12} \]

\[ \times \left\{ 2\psi\!\left(\frac{\zeta_e d}{2\pi}\right) -2\psi\!\left(\frac{\zeta_n d}{2\pi}\right) -\frac{2\pi}{\zeta_e d} +\frac{2\pi}{\zeta_n d} +\pi\operatorname{ctg}\frac{\zeta_e d}{2} -\pi\operatorname{ctg}\frac{\zeta_n d}{2} \right\} \]

and

\[ \begin{aligned} \gamma_{nn} &= \frac{8}{\pi^3}\, \frac{\sin^2\dfrac{\zeta_n d}{2}} {(2n+1)^3}\,A + \frac{2d}{D\pi^2} \sin^2\frac{\zeta_n d}{2} \left\{ \psi'\!\left(\frac{\zeta_n d}{2}\right) + \frac{2\pi^2}{\zeta_n^2 d^2} \right\} \frac{d}{D}, \end{aligned} \tag{13} \]

where

\[ A=\frac{D^2}{d}\,\kappa_0^0\operatorname{tg}\kappa_0^0 h, \]

\(\psi(x)\) is the logarithmic derivative of the gamma function \(\Gamma(x)\).

Formulas (12) and (13) give sufficiently accurate expressions for the coefficients of the system, provided only that the height of the diaphragms is not very small. Although the latter condition is always satisfied for slow waves in the \(\pi\)-oscillation regime, if desired one could make a further refinement by retaining terms proportional to \(\delta_s\),

\[ \left(\frac{\operatorname{tg}\kappa_n^s h}{\kappa_n^s}\,d/s\pi\right). \]

We shall not do this, since for those values of the parameters for which the system will be solved, the errors due to neglecting the corrections are negligible.

To solve system (8), let us introduce the relative amplitudes of the harmonics

\[ X_n=\frac{x_n}{x_0}. \tag{14} \]

Next, from the first equation (8) we express \(\lambda\) as a function of \(X_n\) and of the waveguide dimensions; the remaining equations give an inhomogeneous system with respect to the harmonics.

As was shown in [3], for \(D=d\) the system obtained belongs to the type of completely regular systems. If \(D/d\) is not equal to 1, but is close to 1, then from the analyticity of the coefficients it follows that regularity or complete regularity is preserved. It can be violated only for small \(d/D\). Thus, if \(d/D\) is not very small, system (8) will belong to the regular type, and the reduction method may be applied to solve it.

The solution of (8) was carried out in the eighth approximation, i.e., it was assumed that \(X_n=0\) for \(n>9\). This is equivalent to a calculation taking into account the first 18 harmonics, 9 forward and 9 backward. The results of calculating the harmonics for various \(d/D\) are given in Fig. 2. The accuracy of the calculation for

\[ \frac{d}{D}=1 \]

can be estimated rigorously and is \(2\text{–}3\%\) for the first harmonics, increasing to \(15\%\) for the eighth harmonic. For

\[ \frac{d}{D}<1 \]

the accuracy is somewhat worse; however, one may expect that the accuracy of calculation of the first five harmonics nevertheless does not go beyond \(10\text{–}15\%\). A rigorous estimate of the accuracy is difficult here, but an accuracy of \(10\%\) is quite sufficient for all practical cases. The values \(X_n\) obtained by solving the infinite system (8) will hereafter be called the true values of the harmonics.

Fig. 2.

Fig. 3.

Knowledge of the quantity \(\Lambda\) makes it possible to determine the wave number \(k\) in the \(\pi\)-oscillation mode. The results of its calculation depend very little on \(d/D\), as is illustrated in Fig. 3.

In accordance with the purpose of the present work, let us compute the amplitudes of the harmonics in the approximation of a prescribed field at the interface \(y=g\). A number of variants are used here; moreover, as has already been noted, the approximation that takes into account one harmonic in the resonators is used most often. In this case

\[ E_z = \begin{cases} 0, & 0<z<t,\\[4pt] \cos \dfrac{\pi}{a}x \cdot \mathrm{const}, & t<z<d+t,\\[6pt] 0, & d+t<z<D. \end{cases} \tag{15} \]

The approximation of the form (15) gives the relative amplitudes of the harmonics in the form

\[ X_c^{\mathrm{I}} = \frac{\sin \dfrac{\zeta_c d}{2}} {\sin \dfrac{\zeta_0 d}{2}} \, \frac{\zeta_0}{\zeta_c} (-1)^c . \tag{16} \]

This formula can be obtained directly from the infinite system (7), if in (7) all terms are neglected except those that are proportional to \(\Lambda\).

A more accurate approximation, proposed by Walkinshaw [4], and also by Chu and Hansen [5] for a circular waveguide, takes into account the singularities of the field \(E_z\) at the edges of the diaphragms:

\[ E_z = \begin{cases} 0, & 0<z<t,\\[4pt] \displaystyle \cos \dfrac{\pi}{a}x \, \frac{\mathrm{const}} {\sqrt{(z-t)(z-d-t)}} , & t<z<d+t,\\[10pt] 0, & d+t<z<D. \end{cases} \tag{17} \]

According to this approximation, as \(z\to t\) and \(z\to d+t\) the field \(E_z\) tends to infinity with exponent \(1/2\). The harmonics corresponding to (17) are given by the expression

\[ X_c^{\mathrm{II}} = \frac{ J_c\!\left(\dfrac{\zeta_c d}{2}\right) }{ J_0\!\left(\dfrac{\zeta_0 d}{2}\right) } (-1)^c . \tag{17a} \]

However, the degree of divergence of the fields to infinity is actually equal to \(1/2\) only for infinitely thin diaphragms. For diaphragms with \(t\ne 0\), the degree of divergence to infinity is smaller and equal to \(1/3\). This follows directly from the solution of the static problem for the field near an angle of \(\pi/2\). Therefore one should expect that a more accurate approximation will be

\[ E_z = \begin{cases} 0, & 0<z<t,\\[4pt] \displaystyle \cos \frac{\pi}{a}x \cdot \frac{\mathrm{const}}{\left|(z-t)(z-d-t)\right|^{1/3}}, & t<z<d+t,\\[8pt] 0, & d+t<z<D. \end{cases} \tag{18} \]

with harmonic amplitudes

\[ X_e^{(n)}=(-1)^e\, \frac{J_{1/3}\!\left(\dfrac{\beta_e d}{2}\right)} {J_{1/3}\!\left(\dfrac{\beta_0 d}{2}\right)} \left(\frac{\beta_0}{\beta_e}\right)^{1/3}. \tag{19} \]

The functions \(J_{1/3}(x)\) have been tabulated [7], so calculation by the last formula presents no difficulty.

Fig. 4.

Fig. 5.

The results of the calculation are presented in Figs. 3, 4, and 5. Curve 1 always represents the true values of the harmonics. Curve 2 is Walkinshaw’s approximation, curve 3 is approximation (18), and curve 4 corresponds to (15).

As follows from these graphs, even for \(\dfrac{d}{D}<1\), the errors to which the use of the simplest approximation (15) leads are rather large, although they are smoothed out as \(\dfrac{d}{D}\) decreases. For \(\dfrac{d}{D}=0.9\), the discrepancy for the first harmonic is by a factor of 1.6; for the second harmonic the results already differ by more than a factor of 2, and thereafter the difference is still greater. For \(\dfrac{d}{D}=0.7\), the difference for the first harmonic is very large, since (16) gives a result close to zero. The second harmonics for this case differ by a factor of 1.5. The approximation given by Walkinshaw’s approximation is considerably better. Even better

the approximation (18) is obtained. This is seen especially well from Fig. 4 for \(\frac{d}{D}=0.9\). The discrepancy between the true values and curve 3 for the first four harmonics is only a few percent.

Qualitatively, the course of all the curves is the same. For \(t=0\) the curves decrease monotonically; for \(t\ne 0\) they acquire an oscillatory character.

The largest value of the first harmonics is attained at \(\frac{d}{D}=1\); therefore, in devices operating on higher harmonics, the thickness of the diaphragms should, where possible, be reduced. The value \(\frac{d}{D}=0.9\) is permissible when operating on harmonics up to the third; however, \(\frac{d}{D}=0.7\) reduces the amplitude of the first harmonic by a factor of two and, consequently, the coupling resistance by a factor of 4. Such a value of \(\frac{d}{D}\) is already impermissibly small.

Thus, as a result of the direct solution of the infinite system of equations for a diaphragm-loaded waveguide, an estimate has been made of the accuracy of the approximate methods used for calculating the dispersion and amplitudes of the harmonics. The main conclusion at which we have arrived is that the approximation usually used, which consists in taking into account only one fundamental wave in the resonators, is unsuitable for calculating the operation of devices on higher harmonics. The coupling resistance in such an approximation may differ by several times. On the other hand, this approximation gives rather accurate results for calculating the dispersion properties of the waveguide. The approximation obtained with the aid of the Volkinshaw trial function (17) gives good results only for infinitely thin diaphragms. For calculating real waveguides with \(t\ne 0\), the trial function (18), of the Volkinshaw type but more accurately taking into account the singularities of the fields at the edges of the diaphragms, has been proposed. This function ensures considerable accuracy, at least for the first harmonic numbers.

Fig. 6.

LITERATURE

1. V. M. Lopukhin. Excitation of Electromagnetic Oscillations and Waves by Electron Beams. GITTL, Moscow, p. 192, 1953.

2. M. F. Stelmakh. On the interaction of an electron beam with a field of spatial harmonics. Radiotekhnika i elektronika, vol. 2, 4, 470, 1957.

3. E. S. Kovalenko and V. S. Kovalenko. On the theory of a diaphragm-loaded waveguide of rectangular cross section. Izv. vuzov MVO SSSR, Radiotekhnika, No. 1, 1961.

1. W. Walkinshaw. The Proc. of the Phys. Soc., V. 61, 3, No. 345, p. 246, 1948.

2. E. L. Chu, W. W. Hansen. Journal of Applied Physics, V. 18, No. 11, p. 996, 1947.

3. E. S. Kovalenko, V. I. Shimansky. In-phase waves in a diaphragm-loaded waveguide of rectangular cross section. Izv. vuzov MVO SSSR, Radiotekhnika, No. 2, 1960.

4. Watson. A Treatise on the Theory of Bessel Functions, Vol. II. Foreign Literature Publishing House, 1949.