LS Note 218: Fundamental Mode RF Power Dissipated in a Waveguide Attached to an Accelerating Cavity

LS Note 218

Fundamental Mode RF Power Dissipated in a Waveguide Attached to an Accelerating Cavity

Y. W. Kang
RF Group
Accelerator Systems Division
Argonne National Laboratory

February 9, 1993

Contents:

Introduction
Coupled Waveguide Modes
Power Dissipation in a Matched Load Terminated Waveguide
Power Dissipation in a Short-Circuited Waveguide
Power Dissipation in a Coaxial Transmission Line Section
Results

I. Introduction

An accelerating RF cavity usually requires accessory devices such as a tuner, a coupler, and a damper to perform properly. Since a device is attached to the wall of the cavity to have certain electrical coupling of the cavity field through the opening, RF power dissipation is involved. In a high power accelerating cavity, the RF power coupled and dissipated in the opening and in the device must be estimated to design a proper cooling system for the device. The single cell cavities of the APS storage ring will use the same accessories. These cavities are rotationally symmetric and the fields around the equator can be approximated with the fields of the cylindrical pillbox cavity. In the following, the coupled and dissipated fundamental mode RF power in a waveguide attached to a pillbox cavity is discussed. The waveguide configurations are 1) aperture- coupled cylindrical waveguide with matched load termination, 2) short- circuited cylindrical waveguide, and 3) E-probe or H-loop coupled coaxial waveguide. A short-circuited, one-wavelength coaxial structure is considered for the fundamental frequency rejection circuit of an H-loop damper.

II. Coupled Waveguide Modes

Figure 1 shows a pillbox cavity with a cylindrical waveguide attached to it. For TM₀₁₀ mode, fields in the cavity are

where J_n is the n-th order Bessel function of the first kind. The fundamental mode RF power dissipated in the cavity with the fields in equation (1) is

where R_s is the surface resistance of the cavity metal

For the cavity input power P_in, the fields are

where

The magnetic field in the aperture S_a is assumed to be uniform and -directed.

In a circularly cylindrical waveguide, the mode functions are

for TM modes, where

and

for TE modes, where

a is the guide radius, k is the free space wave number, and x_np and x'_np are the p-th zeros of J_n and , respectively.

The aperture field can be expressed in a two-dimensional Fourier-Bessel series as

where h_np are the normalized mode vectors with

From equation (7), the Fourier expansion coefficients are found as

For the uniform aperture field in equation (4),

III. Power Dissipation in a Matched Load Terminated Waveguide

For each mode, the dissipated power in the waveguide is the dissipated power in the waveguide wall plus the dissipated power in the matched load. The dissipated power in the load equals the time average power flow through the waveguide. The total power dissipation

where E and H are the fields inside the waveguide supported by the aperture field in equation (7).

The np-th modal electric field is

where the wave impedances are given as

for TM modes and

for TE modes. At a frequency below the cut-off for the dominant TE₁₁ mode of the waveguide, ignoring the wall loss, all the waveguide modes have purely reactive input impedance. The input impedance of the matched load terminated waveguide at the aperture may be found using the transmission line theory. If the length of the waveguide is sufficiently long, the resistive component of the input impedance is negligible and, therefore, the time average power flow in the waveguide is negligible.

IV. Power Dissipation in a Short-Circuited Waveguide

In a terminated cylindrical waveguide, the magnetic field wave of the np-th mode is

where

the propagation constant

and H⁺ and H^- are the incident and the reflected waves, respectively. In a short-circuited waveguide, H^-/H⁺ = 1 at the short z = . The total power lost in the waveguide section is the power dissipated on the waveguide wall plus the power dissipated on the short.

At a frequency below the cut-off of the waveguide mode, the modal field evanesces rapidly and the dissipated power on the short can be ignored if the length is sufficiently long.

V. Power Dissipation in a Coaxial Transmission Line Section

Figures 2(a) and 2(b) show the coaxial transmission line attached to a cavity through a probe and a loop, respectively. The current induced on the probe is

where A is the probe surface area, and the voltage induced on the loop is

where S is the area of the loop. The coaxial transmission line has a diameter much less than a wavelength and thus coupling of the TM and TE waveguide modes will be ignored. In the coaxial line, only a TEM mode will be excited with no frequency limit. For a short-circuited coaxial transmission line section, the standing wave fields are

where is the free space wave impedance. The voltage induced on the E-probe is

The total power dissipated in the inner and outer conductor walls and on the short is

Note that, if P_sc is ignored, the power dissipation is inversely proportional to the conductor radius of the coaxial structure for a fixed characteristic impedance Z₀.

VI. Results

In the following, the cases of the cavity to waveguide power coupling described above are discussed with computation results. A cylindrical pillbox cavity whose TM₀₁₀ resonance occurs at 352 MHz is used in the computation. The radius and the height of the pillbox are 0.325m and 0.365m, respectively. 100KW of cavity input power is used.

Transverse magnetic modes have magnetic fields closed in the transverse plane. For equation (9), results show that only the TE_1p modes are excited in the waveguide due to the aperture field H^a. The modal power distribution of the dissipated power in the waveguide is shown in Table 1. The cumulative power dissipation along the axis of the 8cm radius waveguide is shown in Figure 3. The power dissipation versus the diameter of the attached waveguide is shown in Figure 4. The waveguide is terminated with a resistive load which is assumed to be matched for all waveguide modes. For TE₁₁ mode, the attenuation constant of the 8cm radius waveguide is = 21.8 Nepers/m. This shows that the RF power at z = 30 cm is more than 50 dB below the input power. Therefore, the time average power flow through the waveguide can be ignored.

If the guide length L > 0.1m and the aperture radius r < 8cm, from Figure 3, the power dissipations on the short and on the wall due to the reflected field become negligible. Thus, the power dissipation in a short-circuited cylindrical waveguide attached to a pillbox cavity must be similar to the results shown in Figure 3.

The power dissipation in a short-circuited 50 coaxial transmission line section versus the radius of outer conductor of a coaxial line is shown in Figure 5. The length of the coaxial section is 1 at 352 MHz and the characteristic impedance of the line is 50. The result shows that a coaxial line with greater conductor radius dissipates less, as expected. Since TE₁₁ can still couple to the coaxial structure, combining the above result with the power dissipation shown in Figure 4, the optimum radius of a coaxial transmission line may be found.