EXCITATION OF TRAVELING WAVES IN A CLOSED UNIFORMLY BENT WAVEGUIDE OF RECTANGULAR CROSS SECTION

E. V. PADUSOVA, V. I. SHIMANSKY, E. V. MILYUTINA

(Presented by the seminar of the Radio Engineering Faculty)

The production of traveling waves in a closed ring line is based on an idea expressed by Neiman [1], and can be accomplished by directed propagation of waves in the line with matching of the exciting elements to it.

In the case of directed propagation of waves in a closed homogeneous line, multiple superposition of waves traveling in one direction takes place; as a result, when the necessary phase relationships are satisfied, the energy of the traveling wave stored in the line increases.

This latter circumstance gives the right to call such systems traveling-wave resonators.

To create a traveling-wave resonator, long lines of various types may be used: two-wire lines, coaxial cables, waveguides of various cross-sectional shapes.

Waveguide traveling-wave resonators may be formed either by joining straight sections of waveguide by means of angles or bends¹, or by closing the ends of a uniformly bent waveguide having an angular length equal to \(2\pi\).

In a resonator of the first type, any wave type characteristic of waveguides of rectangular cross section may be excited; therefore, in designing the resonator, the theory of straight waveguides may be used. To analyze the field structure and the conditions of wave propagation in a resonator of the second type, the theory of wave propagation in bent waveguides may be applied; it makes it possible to construct the field structure in the waveguides, and also to determine the wavelength in the waveguide, the critical wavelength, and the phase velocity of wave propagation [2, 3].

We have investigated a resonator formed by bending a waveguide of rectangular cross section. Therefore, to construct the field structure in the waveguide and determine \(\lambda_g\) and \(v_\phi\), work [3] was used, which considers bent waveguides of rectangular cross section.

In considering waves in bent waveguides, a cylindrical coordinate system is used (Fig. 1).

Fig. 1. Arrangement of the coordinate system for a uniformly bent waveguide.

The types of waves in uniformly bent waveguides are classified according to the presence of field components \((E\) or \(H)\) along the \(z\)-axis.

The solution of Maxwell’s equations, carried out by means of the Hertz vector \(\overline{Z}\), oriented in the direction of the toroid axis \(z\): \(\overline{Z}_e=\Pi_e\overline{e}_z,\ \overline{Z}_m=\Pi_m\overline{e}_z\) \((\Pi_\varphi=\Pi_r=0)\), determines the field components for waves of type \(E\) and \(H\) as follows:

\[ \begin{aligned} E_r&=\frac{\partial^2\Pi_e}{\partial z\cdot\partial r},& E_\varphi&=\frac{1}{r}\frac{\partial^2\Pi_e}{\partial z\cdot\partial\varphi},& E_z&=k^2\Pi_e+\frac{\partial^2\Pi_e}{\partial z^2},\\[4pt] H_r&=-j\frac{k}{r}\frac{\partial\Pi_e}{\partial\varphi},& H_\varphi&=jk\frac{\partial\Pi_e}{\partial r},& H_z&=0 \end{aligned} \qquad \left\{ \begin{array}{c} \\[-10pt] \\ \end{array} \right. E \tag{1} \]

\[ \begin{aligned} E_r&=j\frac{k}{r}\frac{\partial\Pi_m}{\partial\varphi},& E_\varphi&=\frac{1}{r}\frac{\partial\Pi_m}{\partial r},& E_z&=0,\\[4pt] H_r&=\frac{\partial^2\Pi_m}{\partial z\partial r},& H_\varphi&=\frac{1}{r}\frac{\partial^2\Pi_m}{\partial z\partial\varphi},& H_z&=k^2\Pi_m+\frac{\partial^2\Pi_m}{\partial z^2} \end{aligned} \qquad \left\{ \begin{array}{c} \\[-10pt] \\ \end{array} \right. H \tag{2} \]

The expression for the potential function in the case of a wave of type \(E\) has the form:

\[ \Pi_e=D\cdot\cos\frac{m\pi}{a}z\,[I_n(\chi r)-P N_n(\chi r)]e^{\pm jn\varphi}, \tag{3} \]

and for waves of type \(H\):

\[ \Pi_m=D\cdot\sin\frac{m\pi}{a}z\,[I_n(\chi r)-P' N_n(\chi r)]e^{\pm jn\varphi}. \tag{4} \]

Here \(\chi=\dfrac{x_{nq}}{r_2}\), \(m\) and \(q\) are the numbers of field variations along the axes \(z\) and \(r\)

\[ P=-\frac{I_n(X_{nq})}{N_n(X_{nq})} =-\frac{I_n(CX_{nq})}{N_n(CX_{nq})}, \tag{5} \]

\[ P'=-\frac{I'_n(X_{nq})}{N'_n(X_{nq})} =-\frac{I'_n(CX_{nq})}{N'_n(CX_{nq})} \tag{6} \]

\[ C=\frac{r_1}{r_2}. \]

The field variation along the bent axis of the waveguide occurs according to the law \(e^{\pm jn\varphi}\); here \(n\) is the angular propagation constant. Depending on whether \(n\) is real or imaginary, the waves in the wave-

in the waveguide will be propagating or nonpropagating. The critical condition is \(n=0\).

The propagation of electromagnetic oscillations in a uniformly bent waveguide can be characterized by two velocities:

by the angular phase velocity \(v_{\varphi} = \dfrac{\omega}{n}\) and by the linear phase velocity

\[ v_{\varphi}=\frac{\omega}{n}r. \]

The angular phase velocity is a constant quantity and determines the uniform rotation of the wave front in the waveguide relative to its central axis. The linear phase velocity varies along the radius of the waveguide, reaching its maximum value at the outer radius and its minimum at the inner radius.

It follows from this that, with an appropriate calculation of the waveguide, it is possible on some circumference of radius \(R\) to obtain a phase velocity equal to the velocity of light.

The angular and linear phase velocities correspond to the angular and linear wavelengths

\[ \lambda_{\varphi}=\frac{2\pi}{n} \quad \text{and} \quad \lambda_g=\frac{2\pi}{n}r. \]

The linear wavelength, like the phase velocity, varies over the cross section of the waveguide.

When calculating the geometrical dimensions of the waveguide, it is convenient to specify the wavelength in free space. The generator wavelength can be determined from the equality

\[ k^2=x^2+\gamma^2,\quad \text{where } \gamma^2=\left(\frac{m\pi}{a}\right)^2,\quad k^2=\omega^2\mu\varepsilon, \]

whence

\[ \lambda_a= \frac{2}{\sqrt{\left(\dfrac{X_{nq}}{\pi r}\right)^2+\left(\dfrac{m}{a}\right)^2}} = \frac{2(r_2-r_1)} {\sqrt{\dfrac{X_{nq}^2(1-C)^2}{\pi^2}+\left(\dfrac{m\beta}{2}\right)^2}} \tag{7} \]

\[ \beta=\frac{2(r_2-r_1)}{a}. \]

If \(n=0\), then the wave in the waveguide will not be able to propagate. In this case \(\lambda_a=\lambda_{\mathrm{cr}}\),

\[ \lambda_{\mathrm{cr}}= \frac{2(r_2-r_1)} {\sqrt{\dfrac{X_{0q}^2(1-C)^2}{\pi^2}+\left(\dfrac{m\beta}{2}\right)^2}}. \]

When the bent waveguide is closed into a ring and excited by a single exciter, a toroidal resonator is formed in which, when the resonance condition is satisfied, a standing wave exists.

To excite a traveling wave, the toroidal resonator must be excited by directed exciters, in the particular case by two capacitive pins arranged at a distance

\[ \frac{l\lambda_g}{4} = \frac{l\,2\pi}{4n}, \]

where \(l\) is an odd integer.

For a waveguide length equal to an integer number of wavelengths, the traveling waves will be resonant. The resonance will be expressed in an increase of the field strength of the wave by a factor of $\frac{1}{1-\alpha}$, where $\alpha<1$ is a quantity characterizing the attenuation of the wave in the waveguide over one revolution of the wave around the ring.

The main purpose of the experiment was to verify the practical possibility of exciting traveling waves in a uniformly bent closed waveguide.

When a traveling wave arose, it was necessary to measure the linear phase velocity at the small, middle, and large radii of the waveguide.

The experiment was carried out on two closed waveguide systems with the following dimensions:

The resonant frequencies for resonators with these dimensions were determined by formula (7). The resonators were made of sheet brass with $\delta=0,8$ mm and $\delta=0,6$ mm.

Table 1 summarizes the results of the calculation and experimental verification of the resonant frequencies of the resonators for waves of type $E$.

Table 1

In the experimental verification of the resonant frequencies, the resonators were excited by GSS-12, GS-10, and GSS-D generators. The frequencies of the generator and of the resonator were measured with a VST-D medium-accuracy wavemeter.

The field structure in the resonator was checked by specially developed capacitive field indicators.

Resonators I and II were excited on a wave of type \(E_{111}\).

To avoid the occurrence of parasitic resonances, particular attention was paid to the precision of fabrication of the capacitive exciters (skews of the exciters relative to the end and side walls of the resonator were excluded).

Excitation of the resonator through the side wall excluded the possibility of the appearance of the wave \(H_{011}\).

The block diagram of the experimental setup for exciting a traveling wave is shown in Fig. 2.

To obtain a fixed phase shift, an ILD measuring line was used.

The coupling of the field indicator to the waveguide was very weak. In order to monitor the detunings introduced by the field indicator, one of the field indicators was fixed immovably.

Fig. 2. Block diagram of the experimental setup—1. GSS-12—standard-signal generator; 2—APP—variable absorbing attenuator; 3.—ILD—decimeter measuring line; 4.—UÉA—angular equivalent of the antenna; 5.—VST-D—decimeter medium-accuracy wavemeter; 6.—APF—fixed absorbing attenuator; 7—IPD—decimeter field indicator.

For the selection of the necessary phase relationships, provision was made for the possibility of moving the exciting elements along the resonator and along the measuring line within

\[ \frac{\lambda_g}{4} \pm \frac{\lambda_g}{8}. \]

Equality of the amplitudes of the exciters was established by changing the attenuation of a variable attenuator connected into one of the exciter cables.

As a result of careful matching of the exciting elements with the waveguide, implementation of a phase shift by \(\frac{\pi}{2}\), and selection of equal amplitudes, a traveling wave with TWR \(=0.75\) was obtained in the resonator.

Obtaining larger TWR values was made difficult by the following circumstances:

1. Nonideal fabrication of the resonator and exciters.
2. Instability of the frequency of the generator exciting the resonator.
3. Insufficient power of the generators, which made it difficult to regulate the amplitudes of the exciting fields.

Measurement of the relative phase propagation velocities of the traveling wave in the resonator,

\[ \frac{v_{\phi}}{c}, \]

gave the following results:

1. For the resonator with dimensions \(r_1 = 7.5\ \text{mm}\), \(r_2 = 270\ \text{mm}\), \(a = 100\ \text{mm}\):

\[ \frac{v_{\phi}}{c}=1 \quad \text{at the radius } R=29\ \text{mm}; \]

at \(r=7.7\ \text{mm}\), \(\frac{v_{\phi}}{c}=0.2\); at \(r=265\ \text{mm}\), \(\frac{v_{\phi}}{c}=9\).

1. For a resonator with dimensions \(r_1=7.5\ \text{mm}\), \(r_2=270\ \text{mm}\), \(a=275\ \text{mm}\)

\[ \frac{v_{\phi}}{c}=1 \quad \text{at the radius } R=55\ \text{mm}; \]

at \(r=7.7\ \text{mm}\), \(\frac{v_{\phi}}{c}=0.19\); at \(r=265\ \text{mm}\), \(\frac{v_{\phi}}{c}=4.75\).

Thus, on the basis of the experiments carried out, the following conclusions may be drawn.

1. In a toroidal resonator a traveling wave can exist stably when the resonator is excited by directional exciters.

2. The field structure of traveling waves fully corresponds to the field structure of coaxial resonators for which \(n\) is not equal to zero. Traveling waves are characterized by the absence of field variations in the coordinate \(\varphi\) and by the absence of a phase shift between the transverse components of the electric and magnetic fields.

3. In designing toroidal-type traveling-wave resonators, the theory of coaxial resonators may be used to determine the field structure, the geometrical dimensions of the resonator, and the choice of the resonant wavelength.

4. Excitation of a traveling wave by two capacitive exciters spaced by

\[ \frac{\lambda_g}{4}, \]

is possible, since with such an arrangement of the exciters there is compensation of the inhomogeneities introduced by the exciters, owing to a 180° phase shift of the reflected waves.

The experimental part of the work was carried out at the Department of Radio Transmitting Devices of the Moscow Aviation Institute.

REFERENCES

1. Neiman M. S., Production of traveling electromagnetic waves without power losses. Izvestiya elektropromyshlennosti slabogo toka, No. 1, 1939.

2. Krasnushkin P. E., On waves in curved tubes, Uchenye zapiski MGU, Physics, issue 75, book 2, part II, 1945.

3. Voskresenskii D. I., Uniformly curved waveguide of rectangular cross section, Trudy MAI, issue 73, 1957.

4. Sferazza P. I., Wave-travelling resonator, Tele-Tech, XI, 1955.

1. A similar case is considered in work [4]. ↩