Wire Coaxial Resonators for Synchrotrons

B. A. Solntsev

(Presented by the scientific seminar of the Faculty of Physics and Technology)

In synchrotrons with accelerated-electron energies of 15–30 MeV, it is convenient to use quarter-wave coaxial resonators as the accelerating device; their conducting surfaces are formed by copper wires fastened to dielectric rings (Fig. 1). Resonators of this type were first used by Howard and Birnson in synchrotrons at 8 and 16 MeV [1], but they did not undergo further development. Meanwhile, the resonant properties of such resonators can be considerably improved.

Because of the bending of the resonator, the lengths of the individual wires that form its conducting surfaces differ from one another, and each pair of wires resonates at its own frequency. A resonator of this type may be regarded as a system of coupled two-wire lines, each of which is equivalent to a parallel oscillatory circuit. The properties of a system of coupled circuits are characterized by the pass band—a quantity reciprocal to the quality factor of the system. It is known that the pass band of such a system is the wider, the greater the difference in the natural frequencies of the circuits composing it. Therefore the difference in the lengths of the individual wires reduces the quality factor of a wire coaxial resonator. Equalization of the electrical lengths of the wires should lead to an increase in the quality factor of the resonator.

Fig. 1

Two methods of compensating the electrical lengths are possible. The first method consists in changing the width of the dielectric rings along the perimeter of the resonator.

The natural wavelength of each of the two-wire lines forming the resonator is

\[ \lambda = 4l\xi, \]

\[ \xi = \sqrt{1 + \frac{P}{S}(\varepsilon - 1)}, \tag{1} \]

where \(l\) is the length of the line,

\(\varepsilon\) is the dielectric constant of the material of the rings.

The quantities \(P\) and \(S\) are indicated in Fig. 2.

Fig. 2

The shape of the rings can be chosen so that the natural frequencies of all the lines prove to be equal. In this case the shape of the rings is described by the equation

\[ \frac{\left(\dfrac{r_x}{r_b}\right)^2 - 1}{\dfrac{r_x}{r_b}(\varepsilon - 1)} \cdot s_b + p_x \frac{r_x}{r_b} = p_b . \tag{2} \]

All dimensions are shown in Fig. 2. From Fig. 2 it is seen that the ratio \(\dfrac{S_b}{P_b}\) cannot be less than 1, and the quantity \(a_n\) must be greater than \(O\). Consequently, the condition for applicability of this method has the form:

\[ \varepsilon > \left(\frac{r_n}{r_b}\right)^2 . \tag{3} \]

The greater the difference between the outer and inner radii of the chamber, the higher the dielectric constant of the material of the rings must be.

The second method consists in making the geometrical length of all the wires forming the resonator identical. The condition for applicability of such a compensation method, following from the construction in Fig. 3, is the following:

\[ r_b > \frac{r}{2}. \tag{4} \]

The inner radius of the chamber \(r_b\) must be greater than half the outer radius.

The second method, although it leads to the appearance of a radial component of the accelerating electric field (Fig. 3), gives better results, since it does not require increasing the amount of dielectric introduced into the resonator.

In developing a high-frequency system for a synchrotron, it is very important to know the resonant resistance and the quality factor of the resonator that is proposed for use.

Despite the fact that the theory of hollow resonators has been developed fairly well, practical formulas for determining these quantities of a resona-

...of resonators of the type under consideration is not available. The formulas proposed below make it possible to avoid the cumbersome calculations that are inevitable in the usual computation.

Fig. 3

The resonance resistance of a resonator in the form of a quarter-wave section of a long line is determined by the expression [2]

\[ Z_p=\frac{8Z_0^2 f\varepsilon}{Rc}. \tag{5} \]

where
\(Z_0\) is the characteristic impedance of the line in ohms,
\(f\) is the frequency in cycles,
\(R\) is the resistance of the line per unit length in ohm/m,
\(c\) is the speed of light in m/sec.

Lacking exact data on the distribution of current over the surface of the conductors forming the walls of the resonator, we shall assume that the current is uniformly distributed over half of the surface of the conductors facing inward into the resonator, in a layer whose thickness is equal to the penetration depth of high-frequency currents into the metal \(\delta\) (Fig. 4). Then the resistance of the line per unit length will be equal to:

\[ R=\frac{2\rho}{\delta\pi rN}, \tag{6} \]

where
\(\rho\) is the resistivity of the conductor material in ohm/m,
\(r\) is the radius of the cross section of the conductors in m,
\(N\) is the total number of conductors on each surface of the resonator,
\(\delta\) is the penetration depth of high-frequency currents into the metal in m.

To determine the characteristic impedance of the line one may use the known relation:

\[ Z_0=\frac{30}{C_0\sqrt{\varepsilon}}. \]

Here \(C_0\) is the capacitance per unit length of the line in the absence of a dielectric. The calculation of the capacitance per unit length of a coaxial line whose conducting surfaces are formed by a large number of conductors is a very difficult problem. On the basis of experience we have obtained an empirical formula giving a good approximation within the limits considered:

\[ C_0=\frac{31N^{[[unclear: exponent]]}(a+b)^2}{(d/r)^{1/2}}. \tag{7} \]

In this formula \(a\) and \(b\) are the semiaxes of the outer cross section of the accelerating chamber in \(m\),

\(d\) is the distance between the centers of the wires forming the outer and inner surfaces of the resonator (Fig. 4).

Fig. 4

After substituting the corresponding values, we obtain:

\[ Z_p=3.9\cdot 10^{-8}\frac{f\cdot d\cdot N^{1/3}}{(a+b)^{4/3}}. \tag{8} \]

In the case where the conducting surfaces of the resonator are formed by copper wires, \(\rho=1.74\cdot 10^{-3}\ \frac{\Omega m}{m}\) and \(\delta=\frac{6.62\cdot 10^{-2}}{\sqrt f}\ m\), formula (8) takes the form:

\[ Z_p=1.5\cdot 10^{-3}\frac{d f^{1/2}N^{1/3}}{(a+b)^{1/3}\xi}. \tag{9} \]

The resonant resistance increases with increasing distance between the inner and outer surfaces of the coaxial line and does not depend on the radius of the wires forming the surfaces of the resonator. The number of wires has only a weak effect on the \(Q\) factor of the resonator.

On the basis of formula (9), the quality factor of the resonator \(Q\) can also be determined. It is known that

\[ Q=\frac{\pi Z_p}{4Z_0}. \tag{10} \]

Making the corresponding substitution, we obtain the expression for the quality factor of the resonator in the form

\[ Q=\frac{\pi f d^{3/2} r^{1/2} N^{1/2}}{\rho(a+b)^{2/3}}. \tag{11} \]

The quality factor of a resonator formed by copper wires is:

\[ Q=0.12\frac{f^{1/2}d^{3/2}r^{1/2}N^{1/2}}{(a+b)^{2/3}}. \tag{12} \]

For the 20-MeV synchrotron, three wire resonators were fabricated. For all resonators, \(d=3\cdot 10^{-3}\ m;\ r=2\cdot 10^{-4}\ m;\ N=70;\ a+b=7.6\cdot 10^{-2}\ m;\ f=365\ \text{MHz}\).

The value of the quality factor calculated by formula (12) is 168. The quality factor measured by the frequency-detuning method was found to be 128 for the resonator without length compensation, 134 for the resonator with dielectric compensation, and 141 for the resonator with electrical compensation.

As was to be expected, the resonator with geometrical compensation has the highest quality factor. The value of the quality factor obtained from formula (12) agrees well with the actual value.

LITERATURE

1. Coward F. K., Barnes D. E., Experimental 8 Mev Synchrotron for Electron Acceleration. Nature, No. 4012, 158, p. 413, 1946.

2. Handbook of Radio Engineering, edited by B. A. Smirenin. Gosenergoizdat, 1950.