Fundamental Mode RF Power Dissipated in a Waveguide Attached to an Accelerating Cavity. Y. W. Kang

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1 ANL/ASD/RP DE 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 DISCLAIMER This report was prepared as an account of work sponsored by an agency of the United States Government. Neither the United States Government nor any agency thereof, nor any of their employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States Government or any agency thereof. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States Government or any agency thereof. DISTRIBUTION OF THIS DOCUMENT IS UNLIMITHrT

2 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) Eprobe or Hloop coupled coaxial waveguide. A shortcircuited, onewavelength coaxial structure is considered for the fundamental frequency rejection circuit of an Hloop damper.

3 II. Coupled Waveguide Modes Figure 1 shows a pillbox cavity with a cylindrical waveguide attached to it. For mode, fields in the cavity are E z == SJ o (k p.p'), H# = *,Ji (* />') (1) JUJC 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 P=ff R s \Hv{p',<l>',z)\ 2 ds (2) J Jcavity wall where R s is the surface resistance of the cavity metal For the cavity input power Pi n, the fields are k 2 E z = b n J 0 (k p,p'), JUJ6 H# = b n k p,j x {k pl p') where h p The magnetic field in the aperture S a is assumed to be uniform and <j>' directed. 8L x H a ~vify (4) In a circularly cylindrical waveguide, the mode functions are H B. 2HLT (I n\f>i n< t'f>~i kxz np a P <Jn\*pP)e e for TM modes, where y k

4 and r. jk s n T for TE modes, where j O /TO 10 x' a is the guide radius, k is the free space wave number, and x np and x' are the p th zeros of J n and J^, respectively. The aperture field can be expressed in a twodimensional FourierBessel series as n=0 p=l where h np are the normalized mode vectors with cross section \h np \ 2 ds = 1. (8) From equation (7), the Fourier expansion coefficients are found as a 2rr c np = J j,i a ( P, #) h np ( P, 4>) Pdpd4>. (9) 0 0 For the uniform aperture field in equation (4), H a = \H a \(k p cos<}> SL+siruff). (10)

5 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 Pd = Pwall + Pload = / f R s \H(a,<f>,z)\ 2 d<j>dz+rel f f E x H* a z pdpda (11) 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 Z = _E± = t H p k z for TE modes. At a frequency below the cutoff for the dominant TEu mode of the waveguide, ignoring the wall loss, all the waveguide modes have purely reactive input, impedance. The input impedance of the matched load teiminated 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.

6 IV. Power Dissipation in a ShortCircuited Waveguide mode is In a terminated cylindrical waveguide, the magnetic field wave of the np th H np (z) = Hi,e>»>'[1 T np (z)} (12) where the propagation constant? 2^"^, (13) 7 = a + j/3, and H + and H~ are the incident and the reflected waves, respectively. In a shortcircuited waveguide, H~/H + = 1 at the short z = t. The total power lost in the waveguide section is the power dissipated on the waveguide wall plus the power dissipated on the short. + Pghort 2K = J JR s \H(a,<j>,z)\ 2 d<j>dz + J JR s \H(p,$,e)\ 2 pdpd<f>. (14) At a frequency below the cutoff 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.

7 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 1 = [[[ toedv ~ jue 0 AE (15) "^ J J J probe where A is the probe surface area, and the voltage induced on the loop is v = ^~ II ^oh ds ~ ju:ij o SH (16) C* J J loop 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 shortcircuited coaxial transmission line section, the standing wave fields are a. V E = a. p E 0 sin(kz) = sin(kz) (l? a ) p ln(a/o) H = kj, cos(kz) (176) where77 is the free space wave impedance. The voltage induced on the E probe is V = Z in I. (18) The total power dissipated in the inner and outer conductor walls and on the short is Pd = Pic + Poc + 2n, <!>, z) 2 + tf*(a, t, z)\ 2 ) d4>dz 0 0 a 27r j JR^H^CP^)] + 2 Pd P d4>. (19) b 0 Xote 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 Z o.

8 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 TMoio resonance occurs at 3o2MHz 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^p 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 TEu mode, the attenuation constant of the 8cm radius waveguide is a = 21.8 Nepersfm. 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 shortcircuited cylindrical waveguide attached to a pillbox cavity must be similar to the results shown in Figure 3. The power dissipation in a shortcircuited 50 Q 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 1A at 3o2MHz and the characteristic impedance of the line is 50ft. The result shows that a coaxial line with greater conductor radius dissipates less, as expected. Since TE\\ 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. 8

9 IV V 'Ah Figure 1. A cylindrical waveguide attached to a pillbox cavity through an aperture b I' a ±±f hort circuit (a) Coaxial transmission line with Eprobe loop area S hort circuit I (b) Coaxial transmission line with Hloop Figure 2. Shortcircuited coaxial transmission line sections

10 Table 1 Spectral Power Density of Dissipated TElp Modes in the Wall of Cylindrical Waveguide vs. Guide Radius Cavity Input Power=iOOKW r= r0.024 r= *0.04 t0.048 p=0.056 r=0.064 =0.072 r=0.08 o p=2 E>=3 r4 P=5 p=6 p= O J J J p=8 r9 MO P=II p O ) * O76 0.O O ) D

11 Waveguide Radius = 8 cm, Cavity Input Power = 100KW O Pd(W) "T" O.O Depth(m) Figure 3. Cumulative Power Dissipation in a Cylindrical Copper Waveguide Wall with Respect to Depth

12 2,000 1,800 1,600 1,400 1,200 Input Power=10GKW, WaveguideLength=lm / Pd(W) 1, : ;.^y. 200 F 0 ' i i i ) i i i,,,,,,,, Aperture Radius(m) Figure 4. Power Dissipation in a Cylindrical Copper Waveguide Attached to Pillbox Cavity vs. Waveguide Radius

13 2,200 2,000 1,800 1,600 L : \ t \ "\ \ Coupling Loop Area = 1" x 1", Input Power = 100KW r 1 Pd(W) 1,400 1,200 [jj 1,000 ^^ h !...,.....,, Radius(m) Figure 5. Power Dissipation in a 50Q IX ShortCircuited Coaxial Line Section vs. Outer Conductor Radius

COAXIAL HIGHER-ORDER MODE DAMPER EMPLOYING A HIGH-PASS FILTER

COAXIAL HIGHER-ORDER MODE DAMPER EMPLOYING A HIGH-PASS FILTER .\ COAXAL HGHER-ORDER MODE DAMPER EMPLOYNG A HGH-PASS FLTER e Y. W. Kang and X. Jiang Advanced Photon Source, Argonne National Laboratory 9700 South Cass Avenue, Argonne, llinois 60439 USA A bstracr Two