EXPERIMENTAL STUDY OF THE DISPERSION PROPERTIES OF A DOUBLE-ROW LADDER SYSTEM IN A CIRCULAR WAVEGUIDE

A. N. DIDENKO, R. G. KAMINSKAYA

A number of works [1–3] have been devoted to the choice of an accelerating system for a cybernetic accelerator. In these works, accelerating systems were proposed in the form of circular iris-loaded waveguides with slots and drift tubes, accelerating systems of the “cloverleaf” type, post systems, and systems of the double-row ladder structure type.

If the frequency of the accelerating field varies during the acceleration process, then, among all these systems, preference is given to an accelerating system of the double-row ladder structure type in a circular waveguide (the Schnell system) [2]. An experimental comparison of this system with an accelerating system of the circular iris-loaded waveguide type is given in [3]. However, theoretically the Schnell system has remained little studied up to the present.

In [4] an attempt was made to approximate a circular waveguide by a rectangular waveguide with ridges, as shown in Fig. 1. The parameters of the rectangular waveguide with ridges were chosen in such a way that: a) the height of the waveguide with ridges was equal to the diameter of the waveguide, and the perimeter was equal to the perimeter of the circular waveguide, and b) the height of the waveguide with ridges was equal to the diameter of the circular waveguide, and the sum of the areas of two squares was equal to the area of the cross section of the circular waveguide. In this case, the calculation can be carried out in the same way as the calculation of ladder-type slow-wave systems in a rectangular waveguide.

The calculation of such systems is most conveniently carried out by the method of multiconductor lines [5]. In this case it is assumed that only TEM waves exist in the system (with respect to the direction of the posts); expressions are written for the fields in each of the partial regions, and these are matched at the boundary of separation [6]. The dispersion equations of such a system for symmetric and antisymmetric waves have the form:

\[ \frac{Y_1(\Theta)}{Y_2(\Theta)}=\operatorname{tg}kh_1\cdot \operatorname{tg}kh_2, \tag{1} \]

\[ \frac{Y_1(\Theta)}{Y_2(\Theta)}=-\frac{\operatorname{tg}kh_1}{\operatorname{tg}kh_2}. \tag{2} \]

Each of these equations corresponds to a definite distribution of the potential along the conductor. Thus, for example, in equation (1) the expres-

...expressions for the potential and current in each region can be written in the following form:

for the in-phase wave

\[ V_{n}^{(1)}(y)=B_{0}^{(1)}\sin \kappa y\,e^{-jn\Theta}, \]

\[ I_{n}^{(1)}(y)=jB_{0}^{(1)}Y_{1}(\Theta)\cos \kappa y\,e^{-jn\Theta}, \]

\[ V_{n}^{(2)}(y)=B_{0}^{(1)}\frac{\sin \kappa h_{1}}{\cos \kappa h_{2}}\cos \kappa (y-h_{1}-h_{2})\,e^{-jn\Theta}, \]

\[ I_{n}^{(2)}(y)=-jB_{0}^{(1)}\frac{\sin \kappa h_{1}}{\cos \kappa h_{2}}Y_{2}(\Theta)\sin \kappa (y-h_{1}-h_{2})\cdot e^{-jn\Theta}, \]

\[ V_{n}^{(3)}(y)=-B_{0}^{(1)}\sin \kappa (y-2h_{1}-2h_{2})\,e^{-jn\Theta}, \]

\[ I_{n}^{(3)}(y)=-jB_{0}^{(1)}Y_{1}(\Theta)\cos \kappa (y-2h_{1}-2h_{2})\,e^{-jn\Theta}; \tag{3} \]

for the out-of-phase wave:

\[ V_{n}^{(1)}(y)=B_{0}^{(1)}\cos \kappa y\,e^{-jn(\Theta-\pi)}, \]

\[ I_{n}^{(1)}(y)=jB_{0}^{(1)}Y_{1}(\Theta)\sin \kappa y\,e^{-jn(\Theta-\pi)}, \]

\[ V_{n}^{(2)}(y)=B_{0}^{(1)}\frac{\sin \kappa h_{1}}{\cos \kappa h_{2}}\sin \kappa (y-h_{1}-h_{2})\,e^{-jn(\Theta-\pi)}, \]

\[ I_{n}^{(2)}(y)=-jB_{0}^{(1)}Y_{2}(\Theta)\frac{\sin \kappa h_{1}}{\cos \kappa h_{2}}\cos \kappa (y-h_{1}-h_{2})\,e^{-jn(\Theta-\pi)}, \]

\[ V_{n}^{(3)}(y)=-B_{0}^{(1)}\cos \kappa (y-2h_{1}-2h_{2})\,e^{-jn(\Theta-\pi)}, \]

\[ I_{n}^{(3)}(y)=-jB_{0}^{(1)}Y_{1}(\Theta)\sin \kappa (y-2h_{1}-2h_{2})\,e^{-jn(\Theta-\pi)}. \tag{4} \]

A similar result is also obtained for equation (2). Since one conductor enters the period of the slowing system, it is assumed that \(N=1\), \(\nu=0\), \(\Theta=\psi\). For equation (1) the distribution of the potential with respect to the center of the conductor is symmetric; for equation (2), antisymmetric.

Substituting into formulas (1) and (2) the expressions for \(Y(\Theta)\), calculated in [6] or more accurately in [7], one obtains dispersion equations for different types of oscillation. According to [6], for the expression for the waveguide conductances,

\[ Y_{i}^{\text{in-ph}}(\psi) = 4\frac{b}{q}\sin^{2}\frac{\psi}{2} + 2\sin\frac{\psi}{2}\frac{p-q}{p} \sum_{S=-\infty}^{\infty} (-1)^{s} \frac{ \sin \dfrac{(\psi+2\pi S)}{2}\cdot\dfrac{q}{p} }{ \dfrac{(\psi+2\pi S)}{2}\cdot\dfrac{q}{p} } \times \]

\[ {}\times \frac{ \sin \dfrac{(\psi+2\pi S)(p-q)}{2p} }{ \dfrac{(\psi+2\pi S)(p-q)}{2p} } \left[ \operatorname{cth}(\psi+2\pi S)\frac{Wi}{p} + \operatorname{th}\frac{(\psi+2\pi S)Q}{2p} \right], \]

\[ Y_{i}^{\text{out-ph}}(\psi) = 4\frac{b}{q}\sin^{2}\frac{\psi}{2} + 2\frac{p-q}{p}\sin\frac{\psi}{2} \sum_{S=-\infty}^{\infty} (-1)^{s} \frac{ \sin \dfrac{(\psi+2\pi S)}{2}\dfrac{q}{p} }{ \dfrac{\psi+2\pi S}{2}\cdot\dfrac{q}{p} } \times \]

\[ \times \frac{\sin(\psi+2\pi S)\dfrac{(p-q)}{p}} {\dfrac{(\psi+2\pi S)}{2}\cdot\dfrac{(p-q)}{p}} \left[ \operatorname{cth}(\psi+2\pi S)\frac{W_i}{p} + \operatorname{cth}\frac{(\psi+2\pi S)}{2}\frac{a}{p} \right]. \tag{5} \]

Using these expressions, one can determine the dependence of \(f\) on \(\Psi\) for various types of oscillations.

The calculations were carried out for a circular waveguide with parameters \(D=80\ \text{mm}\), \(P=27.5\ \text{mm}\), \(a=8\ \text{mm}\), \(b=6\ \text{mm}\). When approximating the circular waveguide by a waveguide with projections, the following were chosen:

\[ \text{1) } h_1/h_2=0.417 \quad \text{(Fig. 1),} \qquad \text{2) } h_1/h_2=1. \]

In the first case, the area of the circular waveguide differed from that of the rectangular one by \(2.59\%\). If the circular waveguide is approximated by rectangular waveguides with \(h_1/h_2=1\), then the difference in areas is \(4.5\%\).

Fig. 1.

Fig. 2. Block diagram of the measuring setup

The experimental study of the circular waveguide with a double-row ladder structure was carried out on the setup whose block diagram is shown in Fig. 2. The type of oscillation was determined from the distribution of the field components along all axes by the perturbing-body method. As a result of the experiment, the existence was established of in-phase symmetric, in-phase antisymmetric, out-of-phase symmetric, and out-of-phase asymmetric types of oscillations. The most long-wave type is the in-phase symmetric type, which is the operating type. The experimental dependences of \(f\) on \(\psi\) for various types of oscillations for several operating zones are shown in Fig. 3.

The calculated and experimental dependences of the slowing coefficient obtained with the first approximation method for all types of oscillations are shown in Fig. 4. As can be seen from the figure, good agreement between the calculated and experimental results was obtained only for the in-phase symmetric type of oscillation. For the other types, this calculation gives a large error, which is a consequence both of the inaccuracy in calculating the wave conductances for the system with thin pins and of the change in boundary conditions, which most strongly affect the higher types of oscillations.

The experimental and calculated dependences of \(\lambda\) on \(\psi\) for both approximations of the in-phase symmetric type of oscillation are shown in Fig. 5. As can be seen from the figure, the calculated results agree well with the experimental ones, and the second approximation gives more satisfactory agreement.

The efficiency of the accelerating-system model for the in-phase symmetric type at the \(\pi/2\) mode of oscillation is \(R_{\mathrm{sh}}/Q=4.58\ \text{k}\Omega/\text{m}\), at \(2\pi/3\)

Fig. 3. Dependences of \(f\) on \(\Psi\) for various types of waves

Legend in the figure:

— △ — in-phase symmetric
— ○ — counter-phase symmetric
— × — in-phase antisymmetric
— □ — counter-phase antisymmetric

Fig. 4. Calculated and experimental dependences of the slowing-down coefficient

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

on \(\lambda\) for various types of waves

Legend in the figure:

× — in-phase symmetric, experiment
△ — in-phase antisymmetric, experiment
□ — counter-phase symmetric, experiment
○ — counter-phase antisymmetric, experiment

Figure annotations:

• in-phase antisymmetric
• counter-phase antisymmetric
• counter-phase
• in-phase symmetric

Fig. 5. Calculated and experimental dependences of \(\lambda\) on \(\Psi\) for the in-phase symmetric wave

Legend in the figure:

— calculated
— × — experiment

Figure annotations:

\[ h_1/h_2 = 1 \]

\[ h_1/h_2 = 0.414 \]

for the mode of oscillation, \(\dfrac{R_{\text{sh}}}{Q} = 9.6\ \text{k}\Omega/\text{m}\).

The effectiveness of a circular iris-loaded waveguide for this same wavelength is, for the \(\pi/2\) mode of oscillation, \(\dfrac{R_{\text{sh}}}{Q} = 4.2\ \text{k}\Omega/\text{m}\), and for the \(2\pi/3\) mode of oscillation, \(\dfrac{R_{\text{sh}}}{Q} = 4.6\ \text{k}\Omega/\text{m}\) [8].

Thus, the experimental study of the dispersion properties of a double-row ladder system in a circular waveguide has shown that the approximate calculation proposed in [4] agrees well with the experimental data and can be used for calculating the in-phase symmetric type of oscillation.

References

1. Scientific Proceedings of RAIAN, vol. IX, no. 2, Moscow, 1967.

2. W. Schnell. Proceedings of the Fifth Interuniversity Conference on Accelerators. Frascati, 1965.

3. Yu. A. Khrestkov, A. V. Shalnov. In: Accelerators, issue XI, MEPhI, Atomizdat, 1969.

4. Yu. G. Altshuler, A. S. Taranenko, R. I. Skorodumov. Izvestiya VUZov, Radiophysics, IV, 1, 126, 1961.

5. A. N. Didenko. Doctoral dissertation, Tomsk, 1966.

6. B. V. Zverev, N. I. Nechaev. Accelerators, issue XI, MEPhI, Atomizdat, 1969.