ON THE CALCULATION OF ELECTRODYNAMIC CHARACTERISTICS OF PIN STRUCTURES

R. G. Kaminskaya, L. F. Chernogalova

(Presented by the seminar of the Microwave and Traveling-Wave sector of the Research Institute of Nuclear Physics)

As accelerating systems in high-energy accelerators, it is desirable to use waveguide structures possessing high efficiency over a wide passband and having relatively small dimensions.

From this point of view, the most suitable are single-row and double-row pin combs, since in such structures \(\lambda_{\mathrm{cr}}\) of the fundamental TEM wave is \(\sim 4h\).

The calculation of single-row and double-row combs (Fig. 1) can be carried out by the electrodynamic method [1]. In this case the waveguide space is divided into 2 regions (Fig. 1). The region for \(y<0\) is considered as a section of a multiconductor line, and the field in it is represented in the form of an expansion in the natural waves of the multiconductor line (TEM waves). The field in the region \(y>0\) is found in the form of an expansion in spatial harmonics (an \(E\)-wave). We assume that the wave propagates along the \(y\)-axis and that the excited types of TEM-wave oscillations are divided along \(y\) into symmetric and antisymmetric, and along the \(z\)-axis into in-phase and out-of-phase [1].

Fig. 1. Pin structures: \(a\)—single-row comb, \(b\)—double-row comb.

The dispersion equation of these systems has the form

\[ \frac{\operatorname{ctg} kh}{k}=K_c(\psi)\cdot C(\psi,k), \tag{1} \]

where \(k=\dfrac{\omega}{c}\) is the wave number in free space,

\(h\) is the pin height,

\(K_c(\psi)\) is the dimensionless wave impedance of the multiconductor line,

\(C(\psi,k)\) is the capacitance at the ends of the pins.

Since the acceleration process is influenced mainly by the in-phase symmetric wave, for this wave, taking into account the homogeneity of the field between the pins, for a two-row comb \(K_c(\psi)\) and \(C(\psi,k)\) are expressed as follows:

\[ K_c(\psi)= \frac{1}{ 4\xi\sin^2\frac{\psi}{2} + 2\sin\frac{\psi}{2} \sum_{s=-\infty}^{\infty} B_s \frac{\sin \xi\psi_s/2}{\xi\psi_s/2} \left(Cth\,\eta\psi_s+th\,\eta_1\psi_s\right) }, \tag{2} \]

instead of the capacitance \(C(\psi,k)\), the dimensionless capacitance \(C_0=C/D\) is introduced.

\[ C_0= \sum_{m=1}^{\infty} \frac{ Cth\left(\sqrt{\psi^2-\varphi^2+\dfrac{(2m-1)^2\pi^2}{4\tau_1^2}}\,\Lambda\right) }{ \sqrt{\psi^2-\varphi^2+\dfrac{(2m-1)^2\pi^2}{4\tau_1^2}} \left[ \psi^2+\dfrac{(2m-1)^2\pi^2}{4\tau_1^2} \right]\tau_1 } \cdot f_m^2(\psi), \tag{3} \]

\[ f_m= 4\sin\frac{\psi}{2}\cdot \sin\frac{\xi\psi}{2}\, \frac{ \cos\left[(2m-1)\cdot\frac{\pi}{2}\cdot\frac{2\tau-\eta}{\tau_1}\right] - \cos\left[(2m-1)\frac{\pi}{2}\frac{\eta}{\tau_1}\right] }{ (2m-1)\frac{\pi}{2}\cdot \frac{\xi}{\tau_1} } \]

\[ -2\sin\frac{\psi}{2}\, \frac{\sin\dfrac{\xi\psi}{2}}{\dfrac{\xi\psi}{2}} \left[ Cth\,\eta\psi\cdot \sin(2m-1)\frac{\pi}{2}\frac{\eta}{\tau_1} + th\,\eta_1\psi\cdot \sin\left[(2m-1)\frac{\pi}{2}\frac{2\tau-\eta}{\tau_1}\right] \right]. \]

In the case of a single-row comb, the wave impedance \(K_c'(\psi)\) can be obtained from (2); for this, \(th\,\eta_1\psi_s\) is replaced by \(Cth\,\eta\psi_s\), and the first term of the denominator is doubled. The expression for the capacitance in this case is written as

\[ C_0'= \sum_{m=1}^{\infty} \frac{ Cth\left(\Lambda\sqrt{\psi^2-\varphi^2+(2m-1)^2\dfrac{\pi^2}{4\tau^2}}\right) }{ \tau\sqrt{\psi^2-\varphi^2+(2m-1)^2\dfrac{\pi^2}{4\tau^2}} } \cdot \frac{ f_m^{\prime 2}(\psi) }{ \psi^2+(2m-1)^2\dfrac{\pi^2}{4\tau^2} }, \]

\[ f_m'(\psi)= \frac{ 4\sin\psi/2\cdot \sin\xi\psi/2 }{ \dfrac{\xi\psi}{2}\,th\,\eta\psi } \left\{ \cos\left[(2m-1)\frac{\pi}{2}\frac{\xi\zeta}{\tau}\right] + \tau\psi\,th\,\eta\psi\, \frac{ \sin(2m-1)\dfrac{\pi}{2}\dfrac{\xi\zeta}{\tau} }{ (2m-1)\cdot \pi/2 } \right\}, \tag{4} \]

where

\[ \xi=\frac{d_1}{D_1},\qquad \eta=\frac{d_2}{D_1},\qquad \varphi=kD_1,\qquad \Lambda=\frac{g}{D_1},\qquad \tau=\frac{D_2}{D_1}, \]

\[ \zeta=\frac{l}{d_1},\qquad \eta_1=\frac{a}{2D_1},\qquad \tau_1= \frac{\dfrac{a}{2}+2l+d_2}{D_1}. \]

The dispersion equation (1), with (2)—(4) taken into account, makes it possible to calculate the electrodynamic characteristics \(c/v_{\phi}\), \(v_{\mathrm{gr}}\). But the principal quantity,

characterizing the efficiency of the system is the ratio \(R_{\mathrm{sh}}/Q\), which is determined according to [2]

\[ R_{\mathrm{sh}}/Q=\frac{4\pi}{\lambda}\beta_{\mathrm{gr}}R_{\mathrm{c}}, \tag{5} \]

where \(\lambda\) is the generator wavelength,
\(\beta_{\mathrm{gr}}=v_{\mathrm{gr}}/c\) is the reduced group velocity,
\(R_{\mathrm{c}}\) is the coupling resistance.

From the definition of \(R_{\mathrm{c}}\) we have

\[ R_{\mathrm{c}}=\frac{\left|E_x\right|^2}{2\beta_s\cdot P}, \tag{6} \]

where \(E_x\) is the field strength at the location of the beam;
\(\beta_s=\psi_s/D_1\) is the wave propagation constant;
\(P\) is the power flux through the cross section, which can be determined as

\[ P=v_{\mathrm{gr}}\cdot W_{\mathrm{st}}, \tag{7} \]

\(W_{\mathrm{st}}\) is the energy stored per unit length of the waveguide.

Using the expressions for the fields in [1], one can determine \(R_{\mathrm{c}}\) of a two-row comb for the \(s\)-harmonic:

\[ \begin{aligned} R_{\mathrm{c}} &= \sum_{m=1}^{\infty} \frac{ \left( \dfrac{\sin \psi/2}{\psi/2}\cdot \dfrac{\sin \xi\,\psi_s/2}{\xi\psi_s/2} \right)^2 \sin^2 ky }{ \sin^2 kh\cdot \operatorname{Ch}^2\psi_s\eta_1\, \dfrac{\varepsilon_0 v_{\mathrm{gr}}}{4D_1} \left[ \dfrac{1}{\sin^2 kh} \left( 2h-\dfrac{1}{k}\sin 2kh \right)\sigma_1+ \dfrac{f_m^2\sigma_2}{D_1Lk_y^2(\beta_s^2+k^2)} \right] },\\[1.2em] \sigma_1 &= 8\xi\sin^2\frac{\psi}{2} + \psi_s \left( \frac{ \sin\dfrac{\psi}{2}\cdot \sin \xi\dfrac{\psi_s}{2} }{ \dfrac{\psi_s}{2}\cdot \xi\dfrac{\psi_s}{2} } \right)^2 \left[ \operatorname{th}\psi_s\eta_1+C\,\operatorname{th}\psi_s\eta \right],\\[1.2em] \sigma_2 &= k_y^2 \left( g-\frac{1}{2k_y}\sin 2k_yg \right) + (\beta_s^2+k^2) \left[ g+\frac{1}{2k_y}\sin 2k_yg \right],\\[1.2em] L&=\frac{a}{2}+2l+d_2,\qquad k_z=(2m-1)\frac{\pi}{2L},\qquad k^2=k_z^2+k_y^2+\beta_s^2. \tag{8} \end{aligned} \]

\(\varepsilon_0=8.85\cdot 10^{-12}\ \dfrac{\mathrm{F}}{\mathrm{m}}\) is the relative dielectric permittivity of the medium.

From (8), after simple transformations, one can obtain an expression for \(R_{\mathrm{c}}\) of a single-row comb, analogous to [3].

In the present paper, results are presented for calculating the dependences of \(R_{\mathrm{sh}}/Q\) on \(\beta_{\mathrm{gr}}\) (Fig. 2) for a single-row comb with dimensions \(D_1=2.5\ \mathrm{cm}\), \(d_1=1.5\ \mathrm{cm}\), \(2l=1\ \mathrm{cm}\), \(h=1.5\div 2\ \mathrm{cm}\), \(d_2=0.5\div 2\ \mathrm{cm}\), and for a two-row comb with dimensions \(D_1=2.5\ \mathrm{cm}\), \(d_1=1.5\ \mathrm{cm}\), \(2l=1\ \mathrm{cm}\), \(g=1\ \mathrm{cm}\), \(d_2=2\ \mathrm{cm}\), \(h=2\ \mathrm{cm}\), \(a=1\div 3\ \mathrm{cm}\).

The dimensions were chosen on the assumption that for the oscillation mode \(\dfrac{\pi}{2}\)

the phase velocity of the wave \(v_{\phi}\) must be close to or equal to the speed of light \(c\).

Figure 2 also gives curves of \(R_{\mathrm{sh}}/Q\) for the Snell system and for a circular iris-loaded waveguide, taken according to the data of [2]. The dependence of \(\beta_{\mathrm{gr}}\) on the dimensions of the comb is shown in Fig. 3.

Fig. 2. Dependence of \(\dfrac{R_{\mathrm{sh}}}{Q}\) and \(R_{\mathrm{sv}}\) on \(\beta_{\mathrm{gr}}\): 1 — single-row comb, variable \(d_2\); 2 — single-row comb, variable \(h\); 3 — two-row comb, variable \(a\); 4 — Snell system; 5 — circular iris-loaded waveguide.

Fig. 3. Dependence of \(\beta_{\mathrm{gr}}\) on the geometrical dimensions of pin structures.

As can be seen from the figures, pin structures are characterized by a higher value of \(\beta_{\mathrm{gr}}\), which indicates the broadband nature of these systems. For \(\beta_{\mathrm{gr}} \geq 0.31\) the two-row comb is somewhat more effective, while for \(\beta_{\mathrm{gr}} < 0.31\) the single-row comb is more effective.

Table 1

Calculations showed that \(R_{\mathrm{c}}\) and \(R_{\mathrm{sh}}/Q\) for a two-row comb strongly depend on the distance \(a\) between pins and much less on the other dimensions.

For a single-row comb, \(R_{\mathrm{c}}\) and \(R_{\mathrm{sh}}/Q\) depend strongly on the distance between the pin and the side wall \(d_2\); in addition, decreasing the pin height \(h\) leads to an increase in \(R_{\mathrm{sh}}/Q\) as a result of an increase in \(\beta_{\mathrm{gr}}\), while the coupling resistance \(R_{\mathrm{c}}\) remains almost unchanged.

For comparison, the table presents the electrodynamic characteristics of several structures.

Here,

\[ \Delta f = \frac{2\left|f_{\pi}-f_0\right|}{f_{\pi}+f_0} \]

is the width of the operating passband,

\[ \frac{2b}{\lambda} \]

is the ratio of the inner diameter of the circular waveguide to the wavelength,

\[ \frac{g+h}{\lambda} \]

is the ratio of the height of the comb chamber to the wavelength,

\[ \frac{2D_2}{\lambda} \]

is the ratio of the chamber width of the single-row comb to the wavelength,

\[ \frac{2L}{\lambda} \]

is the ratio of the chamber width of the two-row comb to the wavelength.

All parameters are reduced to \(\lambda = 1.25\ \mathrm{m}\). The experimental investigation showed that the group velocity \(\beta_{\mathrm{gr}}\) is significantly smaller than the calculated value because of the decrease in the passband width. This result was obtained because, in the calculation, higher wave types were not taken into account and the field between the pins was assumed to be homogeneous. This led to a shift of the \(\pi\)-mode of oscillations into the region of short wavelengths.

Thus, on the basis of the results obtained, it may be concluded that pin combs can be more effective than the Schnell system and a circular diaphragm-loaded waveguide, and can be used as accelerating systems.

References

1. A. A. Vorob'ev, A. N. Didenko, L. N. Bezmaternykh [et al.]. Waveguide Synchrotrons. Moscow, Atomizdat, 1966.

2. Yu. A. Khlestkov, A. V. Shal'nov. In: Accelerator collection, issue XI. Moscow, Atomizdat, 1969, p. 155.

3. E. S. Kovalenko, V. S. Kovalenko. Proceedings of the Fourth Interuniversity Conference on Electron Accelerators. Vysshaya Shkola, 1964.